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3 A radical rethinking of quantum gravity:

rejecting Einstein’s relativity and unifying Bohmian quantum mechanics with a Bell-neo-Lorentzian absolute time, space and gravity

Quentin Smith

Introduction

Replacing Einsteinian relativity with Lorentzian absolutivity in a Bell-Bohm-Lorentz quantum gravity theory

The major problem with contemporary physics is commonly acknowledged to be the incompatibility of Einstein’s general theory of relativity (GTR) and quantum mechanics (QM). The prospects of unifying these two theories into a quantum gravity (QG) theory seem insurmountable. Witten’s M-theory (and string theory in any form), perturbative QG (e.g. the Hartle-Hawking and Vilenkin proposals), topological QG, loop QG and other ideas about QG are not unifications but are speculations about what might belong to an approximation of a ‘‘complete quantum gravity theory’’ that is not yet known to be possible or even conceivable. No resolution of the incompatibility of Einstein’s GTR and QM has been achieved and there is none in sight.

I believe that if we ponder the implications of Bell’s theorems we should regard GTR the same way that Popper suggests we regard a Special Theory of Reference (STR). Writing about Aspect’s confirmation of Bell’s theorems, he writes:

we have to give up Einstein’s interpretation of special relativity and return to Lorentz’s interpretation and with it to . . . absolute space and time. . . . The reason for this assertion is that the mere existence of an infinite velocity entails [the existence] of an absolute simultaneity and thereby of an absolute space. Whether or not an infinite velocity can be attained in the transmission of signals is irrelevant for this argument: the one inertial system for which Einsteinian simultaneity coincides with absolute simultaneity . . . would be the system at absolute rest – whether or not this system of absolute rest can be experimentally identified.

(Popper 1982: xviii, 20)

If Einstein’s STR is endangered by Aspect’s confirmation of Bell’s theorems, it certainly seems to follow, indeed, ‘‘follow’’ in the sense of logical implication, that Einstein’s GTR is equally endangered. Does anybody really think that instantaneous, non-local, space-like, universe-wide relations of absolute simultaneity (and EPR causal correlations) are logically, mathematically and ontologically consistent with Einstein’s GTR? Of course not. Why is GTR not subjected to the same criticism as is STR? Could it be that people are thinking that we must ‘‘make do’’ with GTR ‘‘as approximately predicatively accurate in many cases’’ until a complete QG theory is developed, where this means a unification of GTR with QM? But what would be the point of unifying it with QM if we know it is disconfirmed even on the large scales where it is supposed to be most successful? The relation of instantaneous, non-local, absolute simultaneity is universe-wide; this is a large scale feature of the universe. It seems that to be consistent, we must treat GTR in the same way that we treat Newton’s theory, namely, that it is false, i.e. we know on the basis of observational evidence that it does not describe the nature of physical reality. Physical reality is not a Newtonian reality. Nor is it a general relativistic reality. Both Newton’s theory and GTR are useful for making predications within the approximate limited circumstances, but they do not give us a physical ontology. They must be interpreted instrumentally, as instrumentally useful in certain circumstances, but they cannot be given a realist interpretation. What is a scientific realistic to believe about the nature of physical reality?

A major task is to develop what may be called a Lorentzian GTR. Popper suggests we should adopt a Lorentzian theory of inertial motion or reference frames, what might be called a Lorentzian STR, where ‘‘STR’’ now means a Special Theory of Reference frames. It is a special theory since (like Einstein’s) it is only about inertial frames. A Lorentzian GTR is a General Theory of Reference frames; it is general since it is about both inertial and non-inertial reference frames. And just as Einstein’s GTR included a theory of gravity and a cosmology, so must our Lorentzian GTR.

The surprise is that this paper does not present some magnificent mathematical structure that is a new and closer approximation to ‘‘the Master Equation’’ of QG, such as Witten’s M-theory or the Hartle-Hawking theory, but rather that the results we reach show how easy it is to unify a classical (non-quantum) theory of space, time, gravity and cosmology with QM. What’s the catch? The key move is to (a) reject Einstein’s classical GTR and substitute for it a classical neo-Lorentz GTR; and to (b) select only Bohm’s 1952 interpretation of QM, which interprets QM as a supplement to or form of Newtonian mechanics. As physicists use these terms, a semi-classical theory quantizes the matter field in the space and time. QED and QCD are semi-classical theories. Einstein’s GTR, Lorentz’s theory and Newton’s theory are classical, in that they do not quantize space and time or matter fields. QG theory quantitates space and time as well as the matter fields. I believe quantitizing space, time and gravity can be achieved if we choose a neo-Lorentzian theory of space, time and gravity and unify it with Bohm’s 1952 interpretation of QM, which is classical in its approach and is ‘‘a form of classical mechanics.’’ Specifically, Bohm accepted Halpern’s (1952: 389) characterization of Bohm’s ‘‘quantum mechanics as a form of classical mechanics involving special quantum forces.’’ By classical mechanics Bohm here means Newtonian mechanics, including Euclidean space, absolute Newtonian time, and two of Newton’s three laws of motion and either including or being consistent with gravitational non-inertial motion. If we give these ideas a neo-Lorentzian formulation, we have a QG theory, and the main difficulty no longer is the QM-GTR unification, but developing a Lorentz GTR that can reproduce the accurate predications of Einstein’s GTR.

The reasons QG has seemed so difficult to Bohmians is the same sort of reason other physicists have. The problem with Bohmians is that they have been trying to unify de Broglie-Bohm theory with Einstein’s GTR, which is impossible, since the de Broglie-Bohm theory has a non-local, space-like, instantaneous, universe-wide, EPR causal correlation among events and this is logically incompatible with Friedman GTR’s basic laws that causal correlations are time-like, local, propagated at a finite velocity (not exceeding that of light) and are non-instantaneous. However, once we recognize that Aspect’s confirmation of Bell’s theorems disconfirms Einstein’s GTR no less than they do his STR, then we should look to the theory that has traditionally been regarded as observationally equivalent to Einstein’s STR, namely, Lorentz’s STR, and see if we can generalize this theory. Once we do, we will find that Bohm’s 1952 interpretation includes part of a Lorentzian GTR.

A Bohm-Lorentz QG consists of ideas and equations scattered throughout the physics literature and which have not been conjoined and organized and presented as a theory. An introductory outline of how these ideas can be unified is the project of this paper. A central claim is that a neo-Lorentzian GTR can incorporate the results of the neo-Newtonian cosmology and gravity theory presented by McCrea and Milne (1934) and developed by Bondi (1960), North (1965), Sciama (1971) Peebles (1993, esp. ), Harrison (2000: esp. pp. 323–338) and others. They show that a neo-Newtonian cosmology is observationally equivalent to Friedman’s cosmology on a cosmological scale. Harrison further develops this neo-Newtonian theory into a more comprehensive theory of gravity and shows it is observationally equivalent to Einstein’s GTR at smaller scales (Harrison, 2000: 334). But these physicists discuss the neo-Newtonian theory merely as a heuristic device for understanding Einstein’s GTR and do not discuss the new relevance it has given that this Neo-Newtonian theory, but not Einstein’s gravity theory, is consistent with the non-local EPR correlations. Furthermore, some of the problems with a more comprehensive neo-Newtonian gravity theory, such as Harrison’s theory, can be resolved if the neo-Newtonian equations are modified to become neo-Lorentzian equations. Bohm’s 1952 interpretation of QM contains laws of motion that are part of the neo-Lorentzian equations (the part that needs to be transformed by the Lorentz transformations). The mentioned parts of Bohm’s equations become complete Lorentzian equations through undergoing a Lorentzian transformation. Bohm’s 1952 ‘‘formally classical’’ was of interpreting of QM (where the quantum force Q is an additional force to the classical forces) either implies or is already unified or consistent with Newton’s first two laws of motion and his universal law of gravitation.

The phrase ‘‘quantum gravity’’ has been used to refer to a unification of Einstein’s GTR with QM. But that is not the meaning of this phrase, or at least it should not have such a limited meaning. I take it to mean a unification of a classical (non-quantum) theory of space, time, gravity and the universe with QM. QM has (supposedly) been unified with a classical theory of space and time (Minkowski space-time) in quantum electrodynamics and quantum chromodynamics. The classical (non-quantum) theory is Minkowski’s theory of space and time. Minkowski space-time is a flat space-time. Recall that Lorentz’s theory has traditionally been regarded as observationally equivalent to Einstein’s special theory of relativity (1905) or its space-time formulation in Minkowski (1908). If we substitute a flat Euclidean space and an absolute time (required by Aspect’s confirmation of Bell’s theorem) for Minkowski space-time, we have a theory that is observationally equivalent to Minkowski QED and QCD and also consistent with the absolute, instantaneous simultaneity that was observationally confirmed by Aspect’s confirmation of Bell’s theorems. This substitution makes all coordinate systems but one effective (merely apparent).

The further unification of the ‘‘relativist effects’’ (described by Lorentz transformation equations) with Newtonian mechanics and gravity shows us that a unification of the relativistic, Newtonian and ‘‘quantum effects’’ are already built into Bohm’s equations of motion. The classical potential V and quantum potential Q belong to the same equation of motion for particles, mdx/dt= –(V)–(Q), which provides a ‘‘built in’’ criterion for determining the strength of the quantum potential Q. The Newtonian laws in Bohm’s equations of motion are given a neo-Lorentzian formulation and these can provide classical predictions of observational data. The measurement of the degree of inaccuracy of the classical equations and the observational data is a measurement of the strength of the quantum potential’s Q contribution to the total causal force. However, some classical laws of motion are not implicit in Bohm’s theory, such as the law for gravitational acceleration, and these need to be built from the simpler laws that are implisit in Bohm’s equations of motion. An example is a non-gravitational equation for acceleration which I first formulate as a kinematic equation and then as a dynamical equation, which includes the mass and force. Bohm’s equations of motion, in their limit, include the kinematics of Newton’s second law of motion, i.e. d(v)/dt. A neo-Lorentzian formulation that modifies this Newtonian law of acceleration A is:

Neo-Lorentz kinematic equation of acceleration

v is velocity, v = dx/dt, the rate of change of position x with time. With Newton, we have d(v)/dt, but now we introduce into the kinematic equation the acceleration dv/dt divided by a Lorentzian factor. A is either the acceleration force or the acceleration field. But accurate predictions require that this be formulated in an equation with Bohm’s quantum potential Q which causally affects acceleration, as in the double slit experiment. A more comprehensive, dynamic acceleration equation will be given in Part Two.

To differentiate between the classical and quantum potentials, a sufficiently broad neo-Lorentzian theory of motion, gravity and cosmology needs to be developed.

Part one: Relativity is the omission of the Galilean transformation

The ‘‘advant garde’’ discussion in the last third of the nineteenth century and the early twentieth century centered around Maxwell’s laws, even though many universities still did not teach Maxwell’s (1865) theory because it was ‘‘too radical and new’’. What were these laws? In Maxwell’s equations, D is the electric displacement, H the magnetic field, p the charge density, l the charge velocity and c the speed of light. is the vector operator and the partial differential sign in (/x /y /z), where x, y and z are the three spatial variables.

What we shall focus upon is the implied role of space and time in these laws, i.e. the temporal variable t and that vector operator is a partial differential of the three spatial coordinates x, y, z

[Eq: 1]

(1)    D = p

(2)    H= 0

(3)    ×=D -1/4πc²H/ t

(4)    ×H= 4π(D/t)+p

The electric field due to a point charge can be determined at an absolute location x, y, z, and time t by Coulomb’s force law.

Where does this law obtain? According to Lorentz, it obtains in the absolute rest frame, where the relations of absolute simultaneity obtain. At the absolute time t, anything simultaneous with t is absolutely simultaneous. In the rest frame, whatever is at rest in the absolute rest frame denoted by x, y, z in Maxwell’s equations, is at rest relative to absolute space. This implies they are absolutely at rest. (a) Space is ‘‘absolute’’ in the sense of being substantival rather than relational (‘‘relational’’ meaning reducible to relations among events or sets of events or possible events). (b) It is also absolute in the sense that all physical laws are instantiated only by bodies that are at rest relative to absolute space. The absolute rest space (as distinct from the many ‘‘relative rest’’ positions) is where spatial and temporal coordinates measure the real (not merely apparent) spatial and temporal values. Maxwell, Lorentz, Fitzgerald and others we discuss posit a flat, non-dynamical space with an independent time variable. We shall see that this understanding of time and space corresponds to Bell’s and Bohm’s.

By ‘‘absolute simultaneity’’ I mean (a) a two-termed relation, such that even e can be simultaneous with event e´ without being also related by a relativity relation of ‘‘relative to the reference frame F’’. (b) If e and e´ sustain a two-term relation of simultaneity, then anything x simultaneous with events e and e´ is simultaneous with everything else y in the universe that is simultaneous with e and e´. Each relation of simultaneity can be decomposed into a two-term simultaneity. If event x is simultaneous with e and e´ this is equivalent to x being simultaneous with e, and with x being simultaneous with e´, and with e being simultaneous with e´. In addition (c) a maximal relation; there is nothing at all that is neither simultaneous nor not simultaneous with e and e´. It is a universe-wide simultaneity relation. Also (d) a relation that connects some space-like separated events and is non-local. If the line of absolute simultaneous events e, e´, etc. are space-like related to some light comes, the reason is that the distinction between ‘‘time-like’’ and ‘‘space-like’’ in relativity theory is about certain types of causal activities and their relation to light; it has nothing to do with time unless one stipulates that some mechanism on a world-line is going to be pragmatically used as a conventional ‘‘clock’’. It also means (e) an instantaneous relation; such that if e and e´ are absolutely simultaneous, they both are temporally located at the instant t, which is an instant and thus has zero duration. Anything x or y absolutely simultaneous with e and e´ is instantaneously occurring with everything else, across the universe, that is occurring at the same instant t as e and e´. Bohm, Lorentz, Aspect, Bell, etc. regard absolute simultaneously as possessing all these features. (f) Time is substantival, composed of instants and intervals, and not of physical or mental events. It can exist without events or matter.

Distinct from absolute simultaneously is a non-instantaneous ‘‘preferred simultaneity plane’’ or the time of a ‘‘privileged frame’’. For example, a Friedman universe is alleged to have a ‘‘preferred time’’ in this sense, but it needs some argument to show that two events simultaneous in this preferred time frame are absolutely simultaneous in my sense, or even that the imperfect homogeneity and isotropy of our universe makes a ‘‘privileged frame’’ an averaged out frame, and thus a conventional construct rather than an ontological reality. It is interesting that many defenders of a preferred time assume that this preferred time is not an ‘‘averaging’’ among imperfectly homogenous frames and imperfectly isotropic frames and is instead an ontological reality; but it would be real only if the universe had a perfectly homogeneous and isotropic frame.

Bell’s EPR correlations occur at the absolute time t. It is difficult in any theory, where non-local instantaneous causal relations or correlations obtain, to try and admit an absolute time but adopt a reductivist and relational theory of space. The correlation is causal, given suitable definitions of causality, such as David Lewis’ type counterfactual definitions (for example, see Maudlin 1994).

For two distant events to be EPR correlated, the correlating relation must extend from one event to the other and this requires a pathway in sub-stantival space. There are no candidate ‘‘events’’ or ‘‘bodies’’ to try and construct a reductivist theory of space. One would need continuum-many points extending from one place to another – the pathway of the EPR correlation. Positing such points is another way of saying one is positing sub-stantival space. The idea that a place could be a ‘‘possibility’’ for something to be located is not obviously intelligible. For example, what could it mean that an EPR correlation instantaneously and physically transmits information through continuum-many possibilities? Possibilities are abstract objects and cannot constitute something concrete such as space. With non-local instantaneous relations in the absolute frame, we require substantival space. Bohm’s 1952 space is also Euclidean, flat, non-dynamical and infinite. It has an absolute frame of simultaneity, detectable by observing EPR correlations.

According to Lorentz and Maxwell, absolute space is flat and non-dynamical. Considering some concepts pertinent to later twentieth century observations and theories, we may say that the Lorentz-Maxwell absolute space is not expanding or contracting and is not identical (even partially) with curves, dips, waves of space, holes and the like. The alleged ‘‘observational evidence’’ for such gravitational effects are the behaviors of large bodies, or, more precisely, large aggregations of fermions and bosons (such as the earth and its moon); the Einstein GTR postulate of curved space is observationally equivalent to a theory that what curves, dips, expands, etc. is not space but certain types of movements of aggregations of bosons (photons, gluons, weak vector bosons, etc.) or fermions (electrons, quarks, etc.). These trajectories are describable (perhaps) by a Bohm-Lorentz QG theory, where gravity is a ‘‘force’’ on bodies analogous to the ‘‘force’’ of the quantum potential Q (Bohm 1952).

According to Lorentz, any reference body moving in absolute space has its real spatial and temporal variables revealed through either being at rest in absolute space or through being related by a Galilean transformation equation to the rest system. The absolute frame and the Galilean transformation contains the real, absolute (Galilean) coordinates, which absolutely relate the coordinates of an inertial frame at rest in absolute space to another inertial frame that is uniformly moving in absolute space. As physicists now commonly state, the reintroduction of instantaneous, non-local absolute simultaneity and absolute space brings back the Galilean transformations, which, given recent evidence (mainly Aspect’s), makes these transformations ‘‘new’’ in the sense that they are needed in theories that accommodate Bell-Aspect-Bohm absolute simultaneity.

The real, absolute coordinates, i.e. the Galilean coordinates, are stated below in bold face. They state the spatial coordinates, x, y, z, of a body at rest in absolute space and they state the absolute time t, which is the same for resting or moving bodies. The regular type-face states the coordinates of a system moving in absolute space. The Galilean transformation equation that relates the moving body’s coordinates to the rest frame’s coordinates is:

x´ = x - vt, y´ = y, z´ = z, t´ = t

[Eq: 2]

The Galilean transformation, Lorentz, Fitzgerald, Lamor, Poincaré and others noted, did not allow Maxwell’s equations to be expressed similarly in all frames, but only in the frame absolutely at rest. The magnetic field H, the electric displacement D, the charge density p and the charge velocity u and the speed of light c obey Maxwell’s equations in the absolute rest frame, but different and more complicated equations are needed if we are to express something analogous to them in moving frames. For example, c+v or c-v would need to be substituted for c in Maxwell’s equations, since a moving body’s velocity would need to be included in the number for the velocity of light. The law of the addition of velocities really obtains only in the absolute rest frame, but the causal effects of moving through absolute space have as one of their consequences, that it could appear to a hypothetical observer on the moving body that the law of the addition of velocities, in particular regarding light, does not obtain.

If we want to retain the form of expression of Maxwell’s less complicated equations, we need to use, not the Galilean transformation, but the Fitzgerald- Lorentz transformation equations. The need for these new equations arose consequently upon the realization that the absolute rest frame could not be observationally distinguished from absolutely moving frames. What could be the cause of the observational indistinguishability of resting and moving frames?

The evidence offered for a causal explanation comes from the various and unsuccessful attempts by such physicists as Bradley (1728), Young (1804), Arago (1810), Fresnell (1810, 1811, 1815, 1818), Fizeu (1851), Hoek (1868), Michelson (1881), Michelson-Morley (1887) and others to discover the motion of the earth relative to absolute space or an aether at rest in absolute space. There is also ‘‘evidence of a kind’’ from some thought experiments by Einstein (see the letters interchanged between Alfred Einstein and Marvaria before 1905, where Einstein, assuming there is an aether absolutely at rest, thought about new ways to try and detect the earth’s motion relative to it). Fitzgerald (1889) was the first to develop the generally accepted and natural causal explanation, that the spatial length of an object is shortened in the direction of its motion by the factor (1-v²/c²)^-1/2 (due to the fact (then assumed by many) that since matter is electrical, or is bound together by electric forces, it will contract in the direction of motion as it moves through the electromagnetic field or aether or field. The usual histories of physics mistakenly say Fitzgerald (1889) presented an equation but no causal explanation. This is false. Furthermore, standard histories are not entirely correct when they say that Lorentz (1892) or Lamor (1900) first discovered the clock retardation equation. Since a light clock tipped in the direction of motion reveals the same contraction, the factor for clock retardation is easily obtained from Fitzgerald’s equation: (1-v²/c²)^-1/2

The ‘‘one mathematical step’’ that led twentieth century physicists along a relativistic path rather than a Fitzgerald-Lorentzian path, is that Einstein simply omitted the Galilean transformation equation in his 1905 essay, stated above, from Lorentz’s full transformation equation. (Poincaré had done the same.) Let us see how this one omission could lead to a relativism about space and time.

The Galilean variables for space and time in the absolute rest frame are in bold face, x, y, z, t.

The variables for a reference frame R^* that is moving in absolute space are starred: x^*, y^*, z^*, t^*.

The variables for a second reference frame R, which is moving in absolute space in such a way that it is also moving relative to R^*, are primed variables x´, y´, z´, t´. The full Lorentz transformation is:

[Eq: 3]

Lorentz shows by these equations several things. First, that since the electric displacement D of the electric field, and the magnetic field H, at rest in absolute space, are certain functions of the spatial and temporal variables, then to obtain a similar form to Maxwell’s laws in a system moving in absolute space, the electrical displacement and magnetic field of these moving systems should be a function of their spatial and temporal variables.

We recall that the vector operator D is a partial differential of the three spatial coordinates x, y, z. If our equations that hold in the absolute rest frame are a function of the coordinates, the same should be true for a moving system if Maxwell’s laws are to have the same form, which would allow observers on the two frames to each view themselves as at absolute rest and the other frame as moving.

But this sounds more like Poincaré (1902, 1905) and not like Lorentz, at least until Lorentz (1909, 1915). Before this, Lorentz emphasized that the only spatial and temporal coordinates are the Galilean ones, and that the ones for moving bodies are not coordinates of space or time, since space and time exist in their real nature only in the absolute rest frame. In moving frames, the coordinates record merely the contraction effects caused by motion through space (or, as Lorentz would prefer to say, the aether). How could t^*, for example, be considered a temporal coordinate if it at best measured merely a contracted and thereby distorted clock reading? How could t^* be a time? More basically, Lorentz viewed the non-Galilean coordinates as merely ‘‘effective variables,’’ as physically uninterpreted mathematical devices that were aids to computation.

Further, the Galilean transformations were not assigned values, since they were undetectable, but were left in to indicate that it is real time and space that we are talking about, even though only its phenomenal effects are perceivable.

We see here the basis for the absolutist theory that can be developed, added to and modified to produce the absolutist theory that can replace GTR. The Lorentzian approach fits in well with later observational evidence, e.g. Aspect’s experiments and with the theories of absolute time and space (of course, Lorentz’s aether, his electron and Maxwell’s equations need to be considerably updated, but here I am talking about Lorentz’s basic ideas). Lorentz developed transformation equations relating phenomenal frames to the absolute inertial frame, where physical systems have their real, absolute, temporal and spatial coordinates. Ignoring for the moment the adjustments required by non-inertial movement, this absolute rest frame is the only frame where absolute time and undeformed spatial shapes and sizes exist. Again, ignoring the adjustments required by non-inertial movement, this absolute frame is the absolute frame disclosed in the Aspect experiments, i.e. the frame wherein there exists the instantaneous, non-local EPR causal correlations. This absolute EPR frame, as shown by Bohm and Hiley, Valentini and other Bohmians, is the absolute frame along which the quantum potential Q (Bohm and Hiley 1993; Callender and Weingard 1994) or the phase S of the universal wave function (Valentini 2002a; Dürrrr et al. 2003) instantaneously acts at arbitrarily far, non-local, space-like distances. In fact, in de Broglie-Bohm theory, there is only one wave function, the universal wave function, and effective wave functions for particles or ‘‘beables’’ (to use Bell’s term) are obtained by abstracting from the universal wave function. The Bohmian causal Q-potential is universe-wide and any local application requires ‘‘mentally abstracting’’ an inseparable part of this universal, instantaneous causal relation. We also could follow Valentini and others and omit the Q-potential, identifying the causal element directly with the Bohmian causal phase S of the universal wave equation, but Bohm’s 1952 is more near the neo-Lorentzian approach of our other theories.

It is usually said that Einstein’s (1905) theory is at least equal in predictive power to Lorentz’s theory, since they are observationally equivalent. Many writers in the 1950s, 1960s and 1970s were fond of saying that Einstein’s (1905) theory was the most well-confirmed scientific theory in history (the same has also been said of QM and QFT). Given the observational equivalence and predictive power equivalence is acknowledged, it is at best misleading to say that Einstein’s (1905) is the most experimentally well-confirmed theory. One should say that Einstein’s (1905) theory, Lorentz’s, Poincaré’s, Fitzgerald’s, Lamor’s and others are tied for the most well-confirmed, since they are all observationally equivalent; whatever experiment result confirms any one of these theories, it confirms the other.

An important issue in Lorentz’s full equation lies in the type of causal explanation it gives to the various values of the spatial and temporal variables of the phenomenal frames. By regarding the absolute rest frame as ‘‘superfluous’’ in developing phenomenal physics, Einstein deprived his theory of virtually any explanatory causal power; it is a theory filled with ‘‘brute facts’’. In Fitzgerald’s and Lorentz’s theory, the rest frame with Galilean coordinates is interpreted as containing a force field that causally acts on moving frames and shortens their length x´ and x^*, in the direction in which they are moving and, as a part of this contraction effect, slowing the clock-motion processes t´ and t^*. The causal activity explains why lengths contract and physical processes are retarded. It also explains why the velocity of light c appears constant in every frame, enabling Maxwell’s laws to appear to be the same in any frame, absolutely at rest or absolutely moving. By characterizing the variables for moving frames as effective variables, Lorentz maintained that real spatial lengths, heights and widths, and real time, existed only in the absolute rest frame. But omitting the Galilean equation, Einstein’s theory lost this causal explanatory power. The coordinates (and all properties) of his inertial frames become ‘‘brute facts.’’ If one adopts the positivist interpretation of 1905 as a theory that the theory of phenomena is a theory of reality, and as not including the absolute frame as part of reality, the result is that there are different Maxwell equations held relative to each different co-moving inertial frame or reference frame. Lorentz’s parsimony is lost, for Lorentz posited only one real Maxwellian law and equations, those obtaining in the one, mind-independent absolute frame, with the phenomenal frames being mere appearances. Appearances, constructed by observers misinterpreting the distorted shapes, sizes and the slowed rate of physical activities of their moving system as being measurements of real sizes and real times.

If Lorentz’s theory has greater causal explanatory power then perhaps ‘‘Einstein’s’’ theory wins on grounds of ‘‘symmetry,’’ which has been a traditional view. But Lorentz’s theory in fact had greater symmetry. This is obscured by the metaphorical charge (the ‘‘derision’’ fallacy, as I call it) that Lorentz’s theory implied that nature was intrinsically paranoid that there was a ‘‘conspiracy of silence’’. This metaphorical derision is still widely believed, but what does it amount to beyond a ‘‘derision’’ attempt to try and make Lorentz’s theory sound ridiculous?

Examined precisely, it is seen that Lorentz’s theory meets none of the conditions of this ‘‘derision’’ fallacy of the ‘‘conspiracy of nature’’. The literal facts that lie behind this metaphor are in fact opposite to those one would expect if one assumed this to be an accurate way of metaphorically stating a charge against Lorentz’s theory. In Lorentz’s theory in the 1890s and 1900s, there is no conspiracy but a clear, well-defined, causal explanation of the observable phenomena and a greater symmetry than Einstein’s 1905 theory.

A typical statement of this ‘‘derision’’ informal logical fallacy, which tries to avoid mere metaphors more than most statements of this criticism, reads as follows: note that this charge is stated in a way that Lorentz’s theory superior symmetry is ‘‘twisted and ridiculed’’ so to speak, into some sort of strange type of asymmetry (even Einstein did not say anything like this; the asymmetry he and others were criticizing was Faraday’s, not Lorentz’s).

It is unlikely that Nature contains both deep asymmetries and compensatory factors which exactly nullify these asymmetries. Such a state of affairs is not logically impossible and to envisage it is not meaningless; but it is unlikely, or improbable, in the same intuitive sense in which a series of coincidences and accidents having a single global effect are improbable.

(Zahar 1973: 39)

What does this mean? What is a deep asymmetry that is exactly nullified by a compensatory factor? Does the ‘‘problem’’ consist merely in the type of emotive or rhetorical words used in stating the issue? I think so. What if the relevant clause in the first sentence is rephrased as:

Nature’s fundamental symmetry is manifest in the fact that phenomena are caused to possess the same sort of symmetrical structure that the one uncaused frame, the one at absolute rest, possesses. This symmetry makes Lorentz’s theory such an elegant and natural theory that Einstein’s elimination of the main symmetrical element results in an unacceptably less symmetrical theory?

But we get a clue about why a emotively negative ‘‘derision’’ is used to describe this exceptional symmetry. We can see where the basic misunderstanding of Lorentz’s theory lies, in a clause in the next sentence: ‘‘it is unlikely, or improbable, in the same intuitive sense in which a series of coincidences and accidents having a single global effect are improbable’’ (Zahar 1973: 39). If it can be seen that Lorentz’s theory is the opposite of something like a series of coincidences and accidents, this may cause critics to think exactly how the symmetry in Lorentz’s is somehow problematic, and not rest content with burying his symmetry beneath the officially accepted ‘‘derision’’ argument.

A series of coincidences or accidents having a single global effect are compounds of nomologically independent events that have a single global effect. A coincidence is inexplicable and the compound of events constituting the coincidence has necessary and/or sufficient conditions (or probabilistic conditions) that are independent of one another. (See David Owens, Causes and Coincidences.) However, in a Lorentzian theory, the so-called ‘‘compensatory factors which exactly nullify these asymmetries’’ are nomologically explained and have the same sufficient and necessary conditions (or have conditions that are not independent or nomically independent of one another). There is an aether in absolute space that transmits both the electromagnetic forces and molecular forces. This transmission causally affects (is a causally sufficient condition of) contracting the moving body in the direction of its motion. Lorentz deduces the Lorentz-Fitzgerald contraction hypothesis and the phenomenal constancy of the velocity of light from this causal hypothesis. The ‘‘ad hoc’’ derision fallacy is quickly dismissed. Compare Fitzgerald and Einstein. Fitzgerald looked at the Michelson-Morley experiment and developed a mathematically accurate causal explanation of it. Einstein looked at it and adopted it as a brute fact. Whether Zahar is right or not about the Molecular Forces Hypothesis, Lorentz’s theory is a good one in the respect in which it is similar to Fitzgerald’s. A new piece of evidence is discovered. One person looks at it, goes home and develops a theory that causally explains its existence. A second person looks at it and say ‘‘its just another brute fact. I will adopt it as an unexplained assumption’’. Whose theory will have greater theoretical virtue?

The Galilean transformation gives real temporal and spatial coordinates which, due to bodies departing from their natural equilibrium state (being absolutely at rest) are caused to be in the non-equilibrium state of being contracted. But the remarkable symmetry of Lorentz’s theory, borne out by the phenomena, is that the non-equilibrium state the moving body is caused to possess is caused to be symmetrical to the body’s rest state, the state the body would have if it were absolutely at rest and in its equilibrium state. Nature is symmetrical at such a basic level that the causal effect produced by motion is such that the cause produces an effect in the moving bodies that transfers to them a structure that makes them symmetrical to their rest states. If what occurs in the absolute frame is considered as the causal field, and the phenomenal arena as the arena where the resultant effects appear, then we can say this theory satisfies the old philosophical maximum, ‘‘causes produce effects that are alike the causes’’; ‘‘like causes, like effects’’. In fact, Pierre Currie (1894) can speak for us: ‘‘when certain causes produce certain effects the symmetry elements of the causes must be present in the effects which they reproduce’’. There is no clause better than this to capture a theoretical virtue of Lorentz’s theory. This is a marvelously symmetric theory. Einstein did not see it since he was interested in a phenomenal relation between two phenomena, the magnetic and conductor, which he wanted to explain in purely phenomenal terms, without recourse to the absolute frame. One need merely consider the phenomenal motion that the magnetic and conductor have relative to each other. Unfortunately, he did not realize that he could have retained his phenomenal symmetry, attained by explaining the experiment in terms of their relative motion, and at the same time retained the remarkable deeper symmetry of nature that Fitzgerald, Lorentz, Poincaré and others discovered.

We have explored some of the respects in which Lorentz’s theory had greater theoretical virtues than Einstein’s 1905 theory. And we know we need to develop an outline of neo-Lorentz-Bohm quantum field theory, which will not directly (by ‘‘third quantization’’) lead to a discussion of quantum gravity; rather, QG requires returning to Bohmian mechanics and its implicit Newtonian mechanics and some gravitational acceleration equations. This will lead to an outline of a neo-Newtonian gravitational field equation that can be unified with Bohmian QFT.

Part two: Lorentzian-Bohm quantum field theory and QG

I hope the burden of proof for justifying the unification of an absolutist theory with STR has by now shifted to some defenders of absolute simultaneity, such as some Bohmians. Pitowsky writes that ‘‘Bohm’s theory, as it stands, has no satisfactory special relativistic formulation’’ (1991: 343). I hope not! If it did, it would be falsified by Aspect’s observational data, assuming STR could even survive the criticisms made in the last section.

The unification of QM with STR is a desirable goal for most Bohmians and one wonders where the problem lies. STR and Bohmian QM are manifestly inconsistent and one must be eliminated if a ‘‘unification’’ of any sort is to take place. The underlying space-time of an adequate quantum field theory (QFT) cannot be Minkowskian, for that prohibits absolute simultaneity. The spatial-temporal structure of a 3+1 Lorentz space and time needs to be the flat space and absolute time on which quantum electrodynamics, quantum chromodynamics and quantum electroweak theory are formulated. This will be observationally equivalent to existent QED and QCD, except for the respects in which Minkowskian relativity renders the formulations inconsistent with the observational data, includes Aspect’s observations, and any changes required will be changes motivated by a desire to accommodate Bohm’s QM, not STR.

I wonder why Bohmians’ approach to quantum electrodynamics and quantum chromodynamics reveals a fundamental ambivalence. Maudlin expresses the typical ambivalence of the Bohmian to QED and QCD: ‘‘ . . . the most straightforward route to putting quantum theory into a relativistic space-time is to add something to the space-time, and, hence, at least implicitly, to reject Relativity as the complete story.’’ (Maudlin 1996: 295). The ‘‘added something’’ is ‘‘something like the classical notion of absolute simultaneity; a fundamental physical relation between events at spacelike separation’’ (Maudlin 1996: 295). It seems to me there should be no ambivalence and that another route (I believe hinted at by Maudlin) is the preferable one. This is best done through examining some ideas of Callender-Weingard (1997) about QFT, who adopt their own equations and make the attendant philosophical issues especially clear.

Note that Callender and Weingard (1997) write the following (the separation into numbered sentences is mine):

(1) According to contemporary physics, the quantum mechanics of non-relativistic particles is not fundamental.

(2) Instead, what we call particles are manifestations of relativistic quantum fields.

(3) Relativistic quantum field theory, then, is a more fundamental theory than elementary quantum mechanics, and it follows that believers in Bohm’s theory should apply it to relativist quantum fields.

(1996: 28)

The inference of (3) from (1) and (2) is invalid, since all that follows from (1) and (2) is

(4) According to contemporary physicists quantum mechanics is not fundamental unless it is unified with STR.

Perhaps ‘‘according to contemporary physicists’’ should read ‘‘according to the great majority of contemporary physicists’’. What Callender and Weingard could be discussing at this point is why (1) is considered by most physics to be true, even though it is false. Further, there are two distinct issues involved. One concerns the impossible task of unifying Bohmian QM with STR. The other concerns whether field theory is more fundamental than particle theory. Shall bosons be given a field theory ontology and fermions a particle ontology? Bohm and Hiley (1989, 1993) believe so. Or should both be given a field ontology (Cushing 1994; Valentini 1996)?

I suggest that the reasons are that the absolutists’ theories have been regulated to specialist branches. For example, one’s field of specialization could be the Aspect experiments. Or it could be Bohmian mechanics. But the great majority of physicists do not specialize in this area. This fact, along with the ‘‘officially sanctioned status’’ that STR and especially GTR has in the physics community, a specialist may say that the Aspect experiments do not lie in his own of specialization and with that remark proceed blithely with the assumptions of STR and GTR.

There are some good features to the compartmentalized, specialized, slow, cautious, hesitant, resistant way of dealing with fundamental changes in physics, specifically, rejecting GTR and replacing it with a neo-Lorentz gravity-Bohmian QM theory, i.e. a QG theory. I emphasized at the beginning of this essay that one part of the problem is that there is no picture of what physics will be left if GTR is jettisoned. They can’t change their beliefs if they don’t know what to change their beliefs to. I suggest we already have a replacement for Einstein’s GTR on hand, if we combine several difference sources, theories, ideas into one unitary framework.

One thing could change now regarding QFT. Consider Callender’s and Weingard’s guidance equation for a scalar field. The field values are indexed by the spatial position x. S is the phase of the wave function. φ(x^>) represents the differences or ‘‘displacement’’ of the field at successive times in the time evolution of the field φ(x^>). In the equation of motion for the time evolution of the field, one reads u(x^>) as a function that assigns the value of the field to each point x of 3-dimensional space.

This is the equation of motion for the time evolution of the scalar field u(x). Recalling that S is the phase of the Schrodinger wave function on a Bohmian interpretation of QM, it is immediately apparent that there is absolute simultaneity in an absolute frame along with non-local, space-like, instantaneous causal relations are propagated and that the temporal evolving displacements in the field du(x)/dt from one time to another are instantaneous and due to non-local, space-like causal correlations. In brief, there is a violation of Lorentz invariance of the field dynamics. But it will not be observationally detected. Callender and Weingard acknowledge this and say it is not ‘‘that bad’’ since ‘‘we will not detect a violation of Lorentz invariance, even though it is violated by the field dynamics’’ (1996: 30). In other words, relativist quantum field theory is ontologically inconsistent with Bohmian field theory, but at the observational level ‘‘we will not detect this’’.

Callender and Weingard add that in QFD we do not have

to give up the view that there is an underlying flat spacetime structure. What we have to accept is that there is additional, physically significant spacetime structure, namely, that there is a preferred frame. We do not. . . . postulate that Lorentz [Minkowski] metric breaks down at very small length scales.

(1996: 30)

I beg to differ. They adopt the standard convention in QFT that the ‘‘Lorentz metric’’ refers to Poincaré-Minkowski space-time. But ‘‘Minkowski spacetime’’ does not permit a privileged frame, let alone non-local, instantaneous, space-like causal connections and absolute simultaneity. ‘‘Adding spacetime structure’’ in the present instance is like adding volume to a point. The result is what should have been announced as their goal in the first place: to reject Minkowski space-time, replace it by a Lorentzian flat space, (not ‘‘Lorentzian’’ in the sense in which this word is used in Einsteinian relativity theory, but in the sense of a Heinrich Lorentz’s (updated) ontology), a flat space with an independent absolute time and an absolute frame along which there are transmitted non-local causal correlations. In particular, the absolute frame alone is the one in which the law φ(x)/t = S/φ(x) obtains and (the neo-Lorentzian idea here should be added that the movement relative to this absolute frame distorts the other frames and that the ‘‘relativistic effects’’ observed are causal effects of this movement). The ‘‘relativistic effects’’ are symmetrical to their absolute cause, have a sensory observable nature that enables the physicist to imagine (without conflict from observations except for Aspect’s) that he is on a Minkowski frame in a closed phenomenal system (i.e. with no absolute causal frame), or, more nearly on the right track, that his frame is the absolute rest frame and the others are the moving and distorted one or (exactly on the right tract) that he is on a frame and he cannot distinguish between the absolute rest frame and the distorted moving frame and that for pragmatic purposes he could adopt the phenomenal part of a Lorentzian absolute/phenomenal ontology and develop a theory, useful for pragmatic purposes, that the phenomenal part is a closed system (i.e. ‘‘all of reality, there being no absolute frame’’) whose spatial-temporal laws are Minkowskian. Exactly what justification do many Bohmians (I discussed Callender-Weingard merely as a representative example; their position is more or less the standard Bohmian approach to QFT) have for not adopting this neo-Lorentzian ontology that seems to me the fairly obvious approach that a Bohmian should take to QED and QCD? The ‘‘success’’ of these Bohmians in developing a theory that is ‘‘observationally Lorentz invariant’’ or that the absolute frame is ‘‘observationally undectactable’’ is not a success, for it requires physicists to misinterpret the sensible phenomena as a closed system that obeys Minkowski or Poincaré-Minkowski laws. This is not a ‘‘success,’’ for the theoretical requirement that the observations be misinterpreted is a requirement for a ‘‘failure’’ and for there to be a ‘‘success’’ there must be a theoretical requirement that the observations be correctly interpreted as mere phenomenal effects, symmetrical distortions caused by the absolute frame, with one of the frames being absolute by being observationally indistinguishable from the rest. But even this is wrong. Electrodynamics and chromodynamics are at the atomic and subatomic level at which Aspect’s EPR correlations were observed, which detected the absolute frame, and thus QED and QCD should be interpreted as capable of being observationally distinguished into the absolute frame and the ‘‘merely relativistically appearing’’ frames. In each case, the absolute frame is (if one performs an Aspect experiment) detectable, so we have to disagree with Callender-Weingard that Minkowski-QED and Bohm-Bell-Aspect QED are ‘‘observationally equivalent’’ in that the ‘‘lack of Lorentz invariance will not be observationally detectable’’.

This way of formulating QFT should be natural to the Bohmian. Bohmian QM implies the apparent ‘‘quantum effects’’ are mere appearances due to our ignorance of the unobserved causes and this ontology (they should acknowledge) carries over into QFT where the apparent ‘‘relativistic effects’’ are mere appearances due to our ignorance of the unobserved causes. One could say that QFT is about unifying one set of misinterpretations or ‘‘mere appearances’’ (Minkowski ‘‘relativistic’’ ones) with another set of misinterpretations or ‘‘mere appearances’’ (quantum ones). There is something real: the unobserved, absolute causation of the contractions/ retardations in the one case and the unobserved trajectories guided by a Qpotential or phase S of the wave function in the other. The question I am ultimately aiming to answer is how to integrate the absolute classical frame with the absolute quantum frame (QG, as it appears below).

Changes such as this to QED and QCD is how progress should be made towards a new basic physics. An objection, that QED is the most well-confirmed theory physicists have ever constructed, ought to be met with the same attitude as we now meet the frequent claims from 30 to 60 years ago that STR is the most well-confirmed theory physicists had ever constructed. In fact, the so-called confirmations of (Poincaré-Minkowski) STR were confirmations of a neo-Lorentz theory, which physicists have acknowledged is observationally equivalent to STR (as long as Aspect’s experiments are ignored). Aspect’s experiments pose no problem for a Lorentzian theory, but we know the situation is different with regard to STR. Bohmian QED on an absolute, neo-Lorentz space and absolute time is observationally equivalent to non-Bohmian QED on a flat Minkowski space-time, apart from the fact that the retention of a relativistic ontology renders QED falsified on the basis of the Aspect experiments.

What really is the problem with the Bohmians (virtually all Bohmians) who still cling to STR when doing QED and QCD? Surely, it is not an intellectual confusion, since they themselves note the contradictions. Is it due to the fact that Lorentzian physics has disappeared from the horizon of mainstream physics, rhetorically dismissed with the emotive rhetoric of ‘‘ad hoc’’ and ‘‘conspiracy of silence’’ informal fallacies of ‘‘derision’’ phrases? Insofar as ‘‘ad hoc’’ means something substantive, it is reducible to the theoretical virtue or ‘‘vices’’ I mentioned: lack of sufficient causal explanatory power, parsimony, symmetry, predictive power, conservativeness, mathematically simplicity (in the sense of the fewest independent mathematical equations used as axioms), etc. Neo-Lorentzian theory as it has been developed throughout the twentieth and early twenty-first century is both untaught, unknown to the point where virtually no one has even heard of the physicists’ names, and, even if one desires to learn it, one finds it is virtually inaccessible. The undergraduate and graduate training in physics does not include neo-Lorentzians such as Builder, McCrea, Ives, Stillwell, Procoviknic, Bastin, Jannossy, Sherwin, Keswani, and others. However, these thinkers developed only a ‘‘neoLorentz special theory of relativity,’’ adopting the positivist, and operationalist approach of Einstein in his 1905 STR. Furthermore, by ‘‘absolute simultaneity’’ they mean a privileged frame of reference, whose plane of simultaneity is the fundamental one. But they reject instantaneous, non-local simultaneity of the sort Lorentz, Bell, Bohm and others adopt. Further, their main focus is on defending the specific physical ideas of an aether that causes the contraction of moving bodies. However, I believe that absolute simultaneity is not a privileged plane of simultaneity, e.g. in Friedman’s or Prokhovnik’s sense, but a non-local instantaneous universe-wide relation. Further, I postulate no aether. It is the movement of a body in a flat Euclidean rest space that causes the contraction. No additional cause is needed, especially when the theory of its nature is varied constantly, depending on the Test advances in electrodynamics, elementary particular theory, and their attempts to accommodate Einstein’s GTR. Thus I am not a ‘‘neo-Lorentzian’’ in this ‘‘old sense’’. I am neo-Lorentzian in the ‘‘new’’ sense in which Bell’s chapter can be considered as presenting a neo-Lorentzian theory (1987). However, I do not agree with the specifics of Bell’s ideas, e.g. the manner in which he incorporates QM into neo-Lorentzian non-gravitational acceleration equations or his unwillingness to reject an ‘‘aether’’ that causes the contraction. But I agree with the basics; absolute simultaneity is instantaneous and non-local, and that bodies that move in absolute rest space have their lengths contracted and clock movements retarded. Unfortunately, Bell never connected his neo-Lorentzian theory to Bell’s theorems, to de Broglie-Bohm QM, to Einstein’s General Theory of Relativity, or to QG. In a certain sense, this essay may be seen as ‘‘going beyond Bell, but in the same direction’’ to develop the various new theories that he could have, but did not, develop or even hint at. Naturally, I could say the same of de Broglie-Bohm quantum physicists who integrated Bell’s theorems into their results. Both Valentini (1996) and Bohm and Hiley (1993) appear to have gone far in this direction, but Valentini still regards QG as a unification of Einstein’s GTR with Bohm’s QM and the Bell-Aspect EPR correlations. Bohm and Hiley (1993) become mystical and posit a ‘‘subquantum realm’’ at which an entirely new and unknown physics reigns, without the slightest justification for their assertion. Valentini perhaps has gone the furthest, especially with QED and QCD, but does not address the issue that Friedman’s universe is a solution of Einstein’s GTR and has a metric and Robertson-Walker line element that is logically, mathematically, and physically inconsistent with non-local and space-like, instantaneous, EPR causal correlations. York slicing does not help since it is also inconsistent with absolute simultaneity of this sort. And Valentini does not seem to address that the Friedman privileged plan of simultaneity is an ‘‘averaged out’’ concept of various non-connecting planes of simultaneity, thereby preventing it from being something unique and real, a real feature of the universe, rather than a concept constructed by averaging out the small divergences from perfect homogeneity and isotropy. I recommend to Valentini that he pursue a QG where space has a Euclidean metric, not a GTR metric (not even the ‘‘flat’’ metric of a Minkowski space-time), which automatically makes Bohm’s 1952 interpretation consistent with a non-Einsteinian and neo-Lorentzian general theory of inertial and non-inertial motion. This unifies classical space and time with quantum space and time, the main goal of ‘‘quantum gravity’’ as it is usually understood. It also shows how QED and QCD are consistent with gravitational acceleration, which is based on neo-Lorentzian equations. This achieves the goal usually associated with ‘‘quantum gravity’’ and make it a Bohm-neo-Lorentz-Valentini research program. It is indeed welcome news that Valentini, even if only in a brief paragraph about Bell (1987), indicates that Bell showed him how Lorentzian physics could be used instead of Einstein’s relativist physics. Valentini states that in QED

what emerges in equilibrium is not Einstein’s special relativity [Minkowski] but Lorentz’s earlier interpretation of the Lorentz transformations. The view of Lorentz – that one particular frame is at absolute rest – is perfectly consistently classically. A single frame with coordinates (x,t) is sufficient to describe all of physics – including the physical response of the moving equipment (See for example Bell 1987). . . . Relativist spacetime is a construct of equilibrium observers who are unable to see nonlocality directly; it is, like ‘quantum ontology, a misguided projection, a misguided projection of the contingent human limitations onto Nature herself.

(1996: 56)

Valentini, indicating his influence by Bell, takes Bell’s ideas one step further in QED and QCD. Valentini seems to be instantiating (at least in outline form) the particular ontological approach to QFT that belongs to a Bohm-neo-Lorentz QFT and the rarity of its instantiation is explained by how Valentini words his passage, namely, that he learned about a Lorentzian theory from reading Bell (1987).

Bell’s own approach to QFT involves the same problem of ‘‘adding structure’’ to a Minkowski space-time that we previously discussed:

The aspects of quantum mechanics demanding non-locality remain in relativistic quantum mechanics. It may well be that a relativistic version of the theory, while Lorentz invariant [i.e. having Minkowski metric] and local at the observational level, may be necessarily be non-local and with a preferred frame (or aether) at the fundamental level. Could we not then just omit this fundamental level and restrict the classical variables to some ‘‘observable’’ macroscopic level? The problem then would be to do this with clean mathematics, and not just talk.

(Bell 1987: 132–133)

The fundamental level that Bell asks if we could ‘‘omit’’ pertains to the level that Maudlin and Callender and Weingard discuss in terms of the ‘‘added structure.’’ In all three cases we have a Minkowski space-time and an absolute, non-local spatial-temporal structure added to it as the preferred frame on the non-observational, fundamental level

We are aiming to provide a space-time metric to QFT that does not involve adding a preferred frame to Minkowski space-time. This can be most simply achieved by using some of the many Riemannian geometric and generally covariant formulations of both Minkowski and neo-Newtonian space-time. On the geometric theory, it is possible to obtain a Euclidean space and absolute time by subtracting most of the structure of Minkowski space-time.

In a Minkowski space-time, there is a light cone at each point p. Each world-line through p is an inertial frame and for each of these world-lines there is a distinct plane of simultaneity. Each of these planes of simultaneity (considered in abstraction from the other frames of Minkowski space-time) has a Euclidean spatial metric, the same as a plane in an absolute frame in Lorentz’s space and time. Each world-line through p has as a set of successive planes of simultaneity and each of these world-lines has its own temporal metric, measuring the temporal distance or duration between its successive planes of simultaneity (e.g. see M. Friedman 1983: 126–127).

If we eliminate all but one of the world-lines through point p, the one remaining world-line will have a Euclidean spatial metric and an absolute time t, which is the one remaining frame. We eliminate its light cone structure (the conformal structure, a coefficient of the Minkowski metric g_ij)) and extend the space-like region of the (former) light cone to a maximal Euclidean space. But now it is not space-like related to the point. There is no structure that precludes a non-local, instantaneous causal correlation from connecting any point on the plane of simultaneity to any other point on the plane of simultaneity. There is a 3 + 1 Euclidean space and absolute time whose Lie group is O³ plus T (for an explanation, see Friedman 1983), which is the absolute frame of neo-Lorentzian space and time. The Minkowski metric g is dependent on the existence of the other inertial frames and the Minkowski metric cannot exist if there is only one inertial frame, the absolute Euclidean 3D space and the independent Newtonian time T.

But this not yet a complete description of a neo-Lorentzian absolute space and time. A neo-Lorentzian space and time contains moving bodies whose nature is described in terms of the absolute rest frame coordinate system. The laws describe in part how moving bodies have lengths, retardations, etc in the absolute frame; it describes the kinds of changes that are associated with kinds of movements.

But how could we describe the real laws if we have no local experimental evidence about whether we are on the absolute rest frame? It is true that we may be on the absolute, neo-Lorentz frame. Accordingly, if the observer assumes she is on the absolute frame and records moving systems as absolutely distorted and as bodies moving in the absolute frame. This neo-Lorentzian formulation is based on the symmetry of cause to effect described in Part 1 in Lorentz’s theory, that what the observer is assuming to be the absolute frame is similar in respect of its laws to the absolute frame, so she can grasp the nature of the real absolute laws, the nature of the laws stating the kinds of proportionate dependence of the length, rate of processes and amount of mass of a moving body to the absolute velocity of the body, its velocity relative to the absolute rest frame.

Bohm and Hiley (1993) mention a neo-Lorentz QFT in general terms but it is hard to avoid the impression that their theory is indefensible; they claim the microwave background radiation is the ‘‘ether’’ that causes contraction and retardation. Their theory may seem hard to evaluate since no details or equations are offered, but one does not need these since the presented statements are sufficient for an evaluation. The astronomical estimates tend to vary, but the figure of 340,000 years after the big bang is often identified as the time when the conditions of expansion of the material universe were suitable for the formation, ‘‘release’’ or ‘‘beginning to exist’’ of the background microwave radiation. The problems with this theory are many; for example, many elementary particles moved prior to this time and yet the (effective) light cone structure existed prior to this, which, even if ‘‘effective,’’ requires the contraction and retardation effects (relative to the absolute frame). More importantly, the background microwave radiation is not perfectly homogeneous and isotropic and this leads to the testable prediction that similar, moving bodies have degrees of contraction and retardation that vary proportionally to the variation in homogeneity of the radiation, a prediction falsified by the numerous tests that have been said to ‘‘confirm STR,’’ which we take to mean ‘‘confirm Lorentzian STR.’’ Third, the background radiation is more intense the in the more distant past, and if Bohm and Hiley’s theory is correct, what we should observe is that this radiation puts up a correspondingly stronger resistance to movement and consequently causes greater contraction and retardation, a prediction falsified by our astronomical observations of distant stellar and galactic systems.

Furthermore, the ether is the medium of electromagnetic propogation, which includes microwave propogation, and the medium of microwave propogation cannot be microwave propogation, since the medium through which some x travels is different than x. If microwave radiation is the medium of microwave propogation, this implies both x = x and the negation of x = x. The microwave radiation, further, is said to be at rest with respect to the expansion of space. This seems to suggest something analogous to Lorentz’s thesis that the ether is at rest with respect to space. But Bohm and Hiley cannot both accept and reject GTR; their classical forces V are Newtonian, not general relativist. On neo-Newtonian cosmology, microwave radiation moves through space; space does not expand. Rather, basic large-scale clusters of galaxies recede from each other within this stationary Euclidean space. One of the neo-Lorentzians, Builder (1958), seems to put forward a more plausible thesis when he says the neo-Lorentzian ether should be identified with space. But this is more ambiguous than might appear; there is not merely the question of whether space causes the contraction or whether it is a body’s state of movement that causes its other state of contraction (and the other effects, such as retardation) Rather, we have multiple possible positions available corresponding to the many definitions of causality in the philosophical literature.

Bohmian QFT should not accept Bohm’s view of the Heisenberg uncertainty principles. We must reject Bohm’s realist interpretation of the Heisenberg uncertainty principle and his groundless claim that there is an entirely new and unknown physics governing the behavior of processes that occur at the Planck dimensions. However, I think we should go beyond merely reporting our disagreement; these two theses are logically invalid in the context of the Bohm-de Broglie interpretation. A de Broglie-Bohm type of interpretation of QM and QGT must give the uncertainty principles a merely epistemic interpretation; a realist interpretation results in an inconsistent theory. For example, one of the inconsistencies that would arise concerns the fact that the Q-potential acts instantaneously, which imply the laws governing the Q-potential apply at times shorter than the Planck time 10^-43 second, and, second, this implies that there is a time shorter than the Planck time. Our quantum laws (e.g. the Shrödinger equation) and the Q-potential govern quantum processes at each instant, where an instant has zero duration, ruling out as logically inconsistent the Bohm-Hiley interpretation that QM does not apply but that, instead, ‘‘new unknown physics’’ apply, to processes shorter than 10^-43 second.

We need a neo-Lorentz acceleration equation, since our aim is to explain both non-gravitational non-inertial motions and gravitational acceleration. A significant early example is Lorentz’s acceleration equations (1904), whose equations continue to be born out by experimental evidence (Bell 2000). Lorentz developed an equation for smoothly accelerating moving particles in an electric field, p in particular for any electron e. Lorentz offers the equation dr = dte = p = + p²c². Here re is the position of the electron, m is matter and p is momentum. Lorentz definitely did not consider this as gravitational acceleration (Lorentz 1910), but as electromotor acceleration, but his notion of acceleration is verified (Bell 1987). Other work on the acceleration of electrons was done by Lienard (1898), ‘‘Champ Electrique et Magnetique produit par une change concentree en un point et animee d’un movement qualconque,’’ Eclairage Electrique, Vol. 16, pp. 5, 53, 106 and Wichert (1900) ‘‘Elektrodynamische Elementargesetz,’’ Arch. Neerll. 5 (2), 1900, p. 549. A. Lienard’s and Wichert’s works are discussed at length in W.K.H. Pnanofsky and M. Phillips, Classical Electricity and Magnetism, Addison-Wesley, 1964, eqs. 20–13, 2–15. J.S. Bell (1987) summarized some of Lienard’s, Wichert’s and Lorentz’s acceleration equations and developed his own neo-Lorentzian laws for acceleration. Bell briefly stated these laws or their more pertinent aspects (1987, 2004) and indicated they are highly confirmed by the observational evidence of contraction-retardation effects of smooth accelerations (Bell 2004: 78–79).

Bell presents an equation, using r and r as different symbols. The distance r of an accelerated particle from its rest source r can be calculated if we assume the source r has been at rest for some time and that r is initially just the instantaneous distance to the source r. Here r is the distance of the electron from the rest position r. N is the neutron. The electron e is gently accelerating away from the rest position r. Initially r is the instantaneous distance to the position r but the equations tell us how to keep track of it as it changes. The distance r subsequently changes in a law-like way, which can be seen by integrating the differential equation

[Eq:1]

which follows from

[Eq: 2]

on differentiating with respect to time,

[Eq: 3]

For much larger accelerations, there is different mathematics. Bell verifies that the acceleration transformations work with an example of a high-velocity acceleration. Bell incorporates some quantum mechanical ideas (which in my case will naturally follow from a version of Bohmian QM, which I explain in a later section, so I avoid QM in this section apart from explaining Bell’s equation). Bell uses as an example a hydrogen atom (Z = 1) with a Bohr radius:

[9]

Here a is the fine structure constant, 1/137, of photons, which (being the bosons constitutive of light) travel at the velocity c. The fraction 1/137 is the strength of the electromagnetic force relative to the strong force, set at 1. Bell notes that the hydrogen atom Z can have much higher values, e.g. Z = 70, and the Lorentzian contraction and retardation affects can be obtained.

These equations for the smooth acceleration of the orbits of an electron (Lorentz’s equation) and Bell’s more complete and general equation for smooth orbital acceleration, and Bell’s equation for larger orbital accelerations where Z = 70, are verifiable for some accelerations, as Bell indicates. But we need something more general for a general theory of acceleration and for a neo-Lorentz theory of gravitational acceleration. It will appear in the next section that a classical theory of gravity is most easily quantized by a Bohmian QM. Avoiding for now QM notions, let us see what we can do to obtain a general neo-Lorentz theory of gravity.

I do not think we can just say ‘‘we will add some sort of gravity in the quantum potential Q’’ since the question at issue is ‘‘what gravity’’? I do not follow the path of Pitowsky (1991) in his discussion of quantizing GTR in ‘‘Bohm’s Quantum Potentials and Quantum Gravity’’. He describes a situation where the only classical potential V present is that of gravity, where G is the gravitational constant:

Without the quantum potential Q present, the two particles will move towards each other. But with the addition of Q,

the sum of the effective gravitational potential [is] the sum of the classical potential in the equation and the quantum potential Q. We see that effective gravity is no longer attractive. Unlike classical gravity, the effective force depends not just on masses and distances, but also on the quantum state. . . . This means it might be possible to absorb the effective gravitational force in the general relativist framework’’

(Itamar Pitowsky (1991) ‘‘Bohm’s Quantum Potentials and Quantum

Gravity,’’ Foundations of Physics, 21 (3), 345)

But this is a suggestion of what is epistemologically possible in a QG theory; it is not itself such a theory. We do not even learn the nature of the ‘‘effective gravitational potential’’. Nor do we need the sort of effective gravitational potential that Pitowsky has in mind, since we are jettisoning GTR.

Both Pitowsky (1991) and Callender and Weingard (1994, 1997), among many others, believe a path integral over GTR spaces and matter fields is desirable for a unified GTR and Bohmian QM. This cannot possibly work, since the GTR spaces and matter fields do not permit non-local, space-like, instantaneous, causal connections. Regarding Callender’s and Weingard’s (1994, 1997), I would note that Einstein’s GTR 3-spaces or histories of ordered 3-spaces are defined in terms of their light cones, their relativistic metric, time-like curves and causal lines. The non-local, space-like causal correlations and the continuum-many, temporally successive Bohmian instantaneous simultaneity relations that relate instantaneously non-local, space-like, distant spatial regions completely shatter the Einstein GTR structure of each and every 3-space. Everything (or almost everything) in Einstein’s GTR is destroyed (or, less metaphorically) is mathematically, logically and physically inconsistent with these Bell-Bohm relations. Einstein says each GTR 3-space (we are dealing with those that are like or approximately like Friedman’s 3-spaces) has a light cone structure that sets the conditions upon causality, space and time. The limit upon causality is that it has to be time-like or null curve (if ‘‘time-like’’ is no longer the right word, say the inside of the light cone that is bound by the light cone). There cannot be a causal relation where the cause is space-like related to the effect. The limit upon space is that no spatial region can have a causal relation to another spatial region that is non-local. The limit upon ‘‘time’’ is that a hypothetical observer’s light clock determines the measured simultaneity relations. With Bell-Bohm, causality is frequently not time-like or null-like; it is space-like. With Bell-Bohm, each spatial region is instantaneously and non-locally causally connected to every other spatial region on that instantaneous 3-space. With Bell and Bohm, time is not measured by light signals, but is an absolute, instantaneous ‘‘equatable flow’’ on the absolute plane that is undetectable by methods used in GTR. Furthermore, the GTR metric is a metric for space-time, not for a 3-space or an order of 3-spaces. Time is the radius of the successive 3-spaces, or the density of matter in these 3-spaces, or a component of the curvature of space-time. Callender and Weingard talk about curved spaces. But there is no such thing in GTR. What is curved is space-time, not space.

I conclude that it is clear that Bell-Bohm theory cannot be made consistent or unified with Einstein’s GTR. Bohmians are looking in the wrong direction. They should be looking at a non-dynamical empty Euclidean space into which a material universe expands (see the last section on cosmology), and they should have an absolute time and space that coincides with a Bell-Bohm-neo-Lorentz theory. The major task is to fill out the multiple details and refinements of the Bohm-neo-Lorentz QG theory I am outlining in this essay. I believe the essential ideas are here. There is here a QG theory that has overcome any major obstacles, that is, ‘‘major obstacles’’ in the sense of those confronting Einstein GTR’s unification with QM, which no one yet has any idea of how to overcome. There are no such obstacles in a Bohm-neo-Lorentz QG theory. In a manner of speaking, its accomplishment is just as easy as the various attempts at Einstein-QM unification are hard (or, rather, impossible). What remains are the numerous minor obstacles that consist in developing this theory further, refining it, and working out ever more precise, physically interpreted mathematical equations.

We need some more general ideas about neo-Lorentzian acceleration than the ones obtained from Bell. Instead of using some GTR and some Newtonian gravity theory as source material for mathematics, could we somehow combine GTR and Newtonian gravity theory? Bondi thinks this can be done. There is a theory of gravity that is both non-Newtonian and non-Einsteinian if a gravitational field is defined in terms of the relative acceleration of neighboring particles. Newton’s theory is that a gravitational field is the gradient of the gravitational potential V. Suppose we adopt some suggestions made by Bondi (1979) for a theory of gravity that differs from both Newton’s and Einstein’s, although involving some mathematics used in both theories. There is a basic idea of Bondi’s that we will be able to use, namely that the gravitational field is the relative acceleration of two particles. But it remains to be seen if Bondi’s equations allow for an acceptable neo-Lorentz theory of gravity. Bondi uses Newton’s and Einstein’s mathematics, but not anything Lorentzian in nature. Using a tensor formulation, Bondi identifies the tensor that represents the gravitational field as the tensor linking the relative acceleration vector of neighboring particles to their separation vector. He introduces a relation between the relative acceleration vector f ⁱ and the relative displacement vector x^j

[12]

Here f is a function of force, accelerating particles. The tensor of rank 2 wholly characterizes the gravitational field. This enables the gravitational field to be characterized differently than in Newton’s equations, where represents the gravitational field. Bondi notes that two conditions need to be applied to S. S is the second derivative of a scalar -V), establishing the conservative character of the field,

[13]

The source of the field is the density p of matter, given by the familiar Poisson’s equation, where G is the constant of gravitation.

[13]

Now Bondi faces the problem that restrictions must be added to provide the Lorentzian transformational equations (but without having Lorentzian theory or ontology in mind). He tries to solve this problem in terms of the curvature tensor R, regarded as symmetric in (i, j) and (k, l) and anti-symmetric in (i, k) to ensure that the force function f cannot accelerate particles through the velocity of light and also to manifest the Lorentzian contraction and retardation effects. But the problem with introducing the curvature tensor R in this manner is that it eliminates absolute (instantaneous, space-like) simultaneity; the space-time becomes relativistic. What we want is an absolute space and time where the ‘‘relativistic effects’’ are due to the movement of bodies in the absolute rest frame. Bondi’s gravity theory is really relativistic, as distinct from being really absolute and only apparently relativistic due to the symmetrical contraction effects caused by movement in absolute space. A neo-Lorentzian gravity theory should give us merely apparent relativist effects. Bondi’s space-time is inconsistent with the absolute frame revealed in the Aspect experiments.

Further, Bondi’s neo-Lorentzian theory does not distinguish between gravitational acceleration and non-gravitational acceleration; in fact, he explicitly identifies gravity with acceleration. This is false since a body passing through a magnetic field accelerates towards the magnet and this is not gravitational acceleration. The acceleration is due to the interchange of photons between the magnet and body; it is not due to the gravitons emitted from the magnetic. We recall that in the standard model photons are said to ‘‘carry the electromagnetic force’’; weak vector bosons ‘‘carry the weak force,’’ gluons ‘‘carry the strong force’’ (and in an extension of the standard model, even if only based on an argument from analogy, gravitons ‘‘carry the gravitational force’’). It is already known that a ‘‘gravitons’’ vs. ‘‘curvature’’ interpretation of GTR are mathematically and observationally equivalent (but ontologically different), but gravitons can be used in a neo-Lorentzian gravity theory, but not ‘‘spacetime curves,’’ which presupposes the falsity of a neo-Lorentzian theory of space and time.

Nonetheless, we can develop a neo-Lorentzian gravitational theory where the gravitational field is the relative gravitational acceleration of two particles and not the gradient of Newton’s gravitational potential V. In fact, one of Bondi’s problems was to introduce GTR formulae into a reformulation of Newton’s theory. It seems to me the clearest path is to do the minimum that is necessary to generalize Lorentz’s transformation equation for inertial motion into an equation for acceleration and then proceed to add the conditions necessary to make it a neo-Lorentzian gravitational acceleration.

Let us begin with the familiar x, y, z, t as the absolute spatial and temporal coordinates of an event at the position vector P, where P = x, y, z, which is an event’s spatial coordinates at some time t. The coordinates x, y and z are the three components of the position vector P. We are describing the coordinates composing P on a body that is moving uniformly in the absolute rest frame. The velocity of an body is a velocity vector V, a function whose components are x, y, z, t, [1–(dx=dt; dy=dt; dz=dt)²=c²].

Consider the variable x, the variable for spatial coordinates on the x-axis. The rate of change of a body’s x coordinate, its change from one value on the x-axis to another, is mathematically represented by a fraction whose numerator is the differentiation of x with respect to time, dx/dt. The denominator of the fraction gives the object’s velocity along the x-axis,

[Eq: 15]

The velocity vector is a function which orders its components in the following way:

[Eq: 16]

This equation can be simplified as

[Eq: 15]

The rate of change of an object’s velocity V is the object’s acceleration. An accelerating (or decelerating) object is moving at a velocity that is increasing or decreasing. These basics enable us to formulate an acceleration function for the numerator of the fraction, the function being A (P, V, t), where A is a function of the position vector P, the velocity vector V and the scalar t. The acceleration function A(X, V, t) has the components

These components are the number of the fraction representing the acceleration of the body’s velocity along the x axis, namely, the numerator (which is itself a fraction) du/dt, which differentiates velocity component u with respect to time. Since u / dx=dt, the numerator of the fraction representing the acceleration along the x axis is A = d²x/dt²d, that is, the position variable x differentiated twice with respect to time. This information about position, velocity and time, enables us to formulate a numerator of a kinematic acceleration equation, i.e. an equation that does not take into account the dynamical aspects of acceleration, such as its cause. The numerator of the kinematic acceleration equation is:

Neo-Lorentzian physics requires that the acceleration of a body in the absolute frame includes a transformation equation relating the absolute accelerating body to this body considered as at rest in the absolute frame, where the body’s length is not contracted and its clock’s motions and other physical activities are not slowed down. This compares the absolute rest coordinates of the body to its contracted and retarded coordinates in acceleration motion. Since a body considered at rest has no absolute velocity, V = 0, and thus no absolute acceleration, dV/dt = 0. Bringing in the neo-Lorentzian transformation equation, we have the complete fraction,

But is not a fraction but is an undefined construction, since 0 divided by any number is 0.

The only components left are its position coordinates x, y, z, at an absolute time t. The real, absolute rest spatial and temporal coordinates of the resting body S´ can be obtained from its spatial and temporal coordinates in its acceleration function A(P, V, t, L), where L is one of our neo-Lorentzian transformation equations. (Recall that by a ‘‘neo-Lorentzian transformation equation’’ I do not mean one of ‘‘Lorentz’s transformation equations’’. I am not here doing a ‘‘history of physics’’ that recounts what Lorentz wrote a hundred years ago or so. Rather, I am developing, modifying, adding to his theory, as well as retaining some parts of it. Accordingly ‘‘one of our neo-Lorentz equations’’ does not mean ‘‘one of Lorentz’s equations’’.) The acceleration function is related to the body’s ‘‘absolute rest’’ position function P’ (x´, y´, z´) at time t´, when the body is at, or at least is hypothetically considered to be. The ‘‘approximately equal’’ symbol ≈ will be used to relate the absolute rest coordinates x´, y´, z´, t´ to the coordinates as they are contracted or retarded by the body’s acceleration. Since there is only one real coordinate system, the one that obtains in absolute rest space, the description of the resting and accelerating bodies will use the same coordinate system variables.

Since we need to construct some sort of neo-Lorentz transformation equation, we put

The approximate equality ≈ measures the degree to which the real rest coordinates x´, y´, z´, t´ are distorted by the acceleration. An alternate distortion equation is

In this equation as well, the right hand side of the approximation equation measures the magnitude of the distortion of the real, rest coordinates of the body. The acceleration equation can be abbreviated as

and if this is substituted in the right hand side of (RD) or (RD^*), it will be the right hand side of the acceleration distortion equation. Note that these equations, the various (RD) equations, are not transformation equations; nor are they equations that Lorentz or neo-Lorentzians used. Lorentz used, as I explain in the first part of the paper, the ‘‘Fitzgerald-Lorentz-Lamor equations,’’ with the Galilean equation as an intermediary.

We can derive phenomenal distortion equations if we imagine there to be a hypothetical observer on each system or body, e.g. one on the body S that is absolutely at rest and one on the accelerating body S´, and further imagine that the observer O´ is misinterpreting her contracted and retarded system as the system that is absolutely at rest. Suppose also that we have an observer O on S who cannot distinguish that his absolute rest frame, which is appearing to him veridically from a moving frame that is imagined to be at absolute rest. In this case we have phenomenal acceleration equations. The observer S on the absolutely accelerating frame imagines himself as being on the rest frame described by the left hand side of the distortion equation and imagines the absolutely resting observer to be really accelerating, thereby satisfying the right hand of the equation. Since movement in absolute rest space causes contraction and ‘‘clock’’ retardation, the velocity of light will appear to be c on absolutely moving bodies, insofar as there is a real or hypothetical observer who measures the velocity of light. Observer S will interpret S´ as accelerating relative to himself, and S imagines himself to be at rest, so these equations are imagined to be satisfied:

O imagines his own system S as absolute rest: (x, y, z, t),

And O imagines that the system S´ will have the coordinates of an absolutely accelerating system:

Bohm’s equation of motion implies Newton’s first and second laws of motion. To unify our classical (non-quantum) neo-Lorentzian theory with Bohm’s QM, we can formulate a neo-Lorentzian law, a dynamical law of acceleration that is modeled on Newton’s. This dynamical law includes both mass and the force producing the acceleration. By formulating a neo-Lorentzian equation, we can make the Bohm-Lorentz QG theory more adequate, although they will not be adequate until we add the quantum mechanical laws for Bohm’s quantum potential Q. For now, we will formulate a classical dynamical law of acceleration and a law of gravity, both being neo-Lorentzian versions of Newton’s laws, in conformity with the requirements of Bohm’s 1952 interpretation of the de Broglie-Bohm theory. The ‘‘m’’ stands for mass and ‘‘ – (V^A´)’’ stands for the negative gradient of the acceleration potential (Va). The ‘‘V’’ is used to make it correspond with Bohm’s equation, where ‘‘V’’ is the classical potential variable and ‘‘Q’’ the quantum potential variable.

Eq: 16

QG theory requires a specific form of acceleration, gravitational acceleration, and the addition of new gravitational variables into the acceleration equation will provide us with a classical gravity equation that appears in the classical part of Bohm’s equations of motion, namely, gravitational motion. I am including some neo-Lorentzian equations to the classical part of Bohm’s law of motion.

Is gravity a force or a field or a boson or something else? Whether gravity is a force or field or values of a field, and whether or not field values are identical with a system of gravitons (the fourth type of boson, in addition to photons, weak vector bosons, and gluons) moving at the speed of light c, and whether or not a gravitational force is identical with the force carried by gravitons, is an issue parallel to a similar issue about the electromagnetic field. Is electromagnetism an ‘‘electromagnetic field,’’ high values of an electromagnetic field, a configuration of photons, or a configuration of photons and electrons, or is it an ‘‘electromagnetic force,’’ or ‘‘photons that carry the electromagnetic force, for example, the force carried by the photons that are emitted from an electron and bind the electron to the protons in the nucleus of an atom, such that the interchange of photons between the electron and protons carry the electron in an orbit around the nucleus of the atom. These are issues that need not be decided here. The physical ontology is underdetermined by the conjunction of the observational evidence and the equations. Some physicists, such as Valentini, and philosophers of physics, such as Callender-Weingard, believe the ontology is not underdetermined and they argue for varying sorts of ontologies, but I will not enter that debate here and I will assume an ontological underdetermination. I use ‘‘gravitational field,’’ ‘‘values of a gravitational field,’’ ‘‘graviton configuration’’ and ‘‘gravitons carrying a gravitational force,’’ and ‘‘gravity force F’’ as different ways of referring to gravity and I do not commit myself to an ontology that might seem to be associated with some of these expressions.

A gravitational acceleration is distinguished by the source of the gravitational force or field values and by the nature of the force or values that cause the acceleration.

Let ‘‘gravity’’ mean the force exerted on the mass m by the distribution of other masses (we use only masses and their distribution for the sake of simplification) m_i at their various positions X_i, such that in our gravitational acceleration equation we are considering m as subject only to the gravitational force. Let F be a gravity force acting on a mass m that is moving in absolute space. Since the gravity force is acting on mass m, and since its source is the distributions or positions of the other masses, we need to indicate this in our gravitational acceleration equation. The acceleration of any mass is gravitational if the force F or the gravitational potential of the field is what is stated on the right hand side of the equation; we will assume that m₁ is the mass of a body at rest relative to absolute space and that m´₂ is the mass of a body that is moving towards m₁. The unprimed m₂ indicates that it is the rest mass of the body and the primed m₂ is the mass distorted by its movement. The neo-Lorentz law of gravity then takes this form, where V_g denotes the classical gravitational potential, and L² is the distance or ‘‘Length’’ between the two masses.

Eq: 19

If two bodies with mass m´₁ and m´₂ are absolutely gravitational accelerating in the same direction and at the same rate dV´ and time t´ our neo-Lorentz law of gravity takes the form, where distance between the two masses is written in a different form,

Eq: 20

It is clear that these are gravitational laws based on a neo-Lorentzian transformation of a Newtonian rather than an Einsteinian theory of gravity. One finds no indication here of Einstein’s equation, which in a simplified form reads G_uv = –(8πG/c²)T_uv. One reason is that Newton’s first and second laws of motion are contained in Bohm’s equation of motion (Bohm 1952) and a second reason is that Einstein’s theory is disconfirmed by the Bell/Aspect experiments but a neo-Lorentzian-Newtonian theory is not.

In Bohm’s 1952 theory, the wave function is governed by the equation of motion, Ψ = Aexp[iS(x)]. The Schrödinger equation is rewritten as a modified Hamilton-Jacobi equation, namely,

Eq: 21

Here usually pertains to the gradient of something but in the present case it means, mv, i.e., mass times velocity, i.e. momentum, p. This is the equation of motion for the wave function. The quantum potential Q is a ‘‘quantum force’’ additional to the usual classical forces. There is an equation for Q and an equation for the particles (called beables by Bell since they exist even when not observed – Bell’s substitute for observables).

The equation governing the particles is

(2)

The quantum potential Q is

(3)

Bohm’s 1952 QM is unified with neo-Lorentzian classical gravity in the following way.

The part of (2), mdx/dt = – (V) is a modification of the Newtonian second law of motion (ma = – (V)) or, written otherwise, (dp/dt = – (V)). Here m is mass, a is acceleration (in Newton’s sense), p is momentum and = – (V) is a classical force. Newton’s second law of motion is second order and the Bohmian guidance equation is first order, and adds a quantum potential Q to the classical force V. The Bohmian equation of motion contains Newton’s second law, since if Q = 0, then Newton’s law governs the particles, dp/dt = – (V), or, if you prefer, ma = – (V).

We know that Newton’s second law is disconfirmed by observations of bodies moving at velocities close to that of light. The so-called ‘‘relativistic effects’’ are relative to absolute rest space and pertain to the contraction of the length, retardation of clock, increase in mass, etc., which are causal consequences of moving in absolute rest space. Since Bohmian theory has a distinction between the absolute frame, where non-local, space-like, instantaneous causal EPR correlations are propagated, and relative phenomenal frames, where we cannot normally observe EPR correlations (unless we are performing an Aspect experiment) we normally do not know which frame is absolute. This parallels neo-Lorentzian ontology, with Bohm adding the Qpotential to the absolute frame and Lorentz adding the contraction of bodies that move relative to the absolute rest frame.

Bohm’s equations of motion imply Newton’s law of acceleration and we can deal with its failure to make correct predictions of ‘‘relativistic effects’’ either by rewriting it as a relativist equation or by substituting for it our neo-Lorentzian acceleration equation. Weinberg (1972: 31) formulates a relativist version of Newton’s second law but he does so in terms of the mathematics of a Minkowski solution to Einstein’s equation. But this leads us back into Einstein’s GTR and an inconsistency with QM. We can write a classical acceleration law that predicts the relativistic effects and yet is not a relativity law but a neo-Lorentzian law, which implies an absolute frame and the sort of absolute simultaneity that is consistent with Bohm’s theory. The neo-Lorentzian law of acceleration would have greater influence on particles’ behavior when the value of Q decreases and less effect when the value of Q is high.

Bohm’s laws of motion include both a classical law of inertial motion and a classical law of acceleration motion. Since our classical laws of motion are neo-Lorentzian, they already put Bohm’s theory in a form where it can be unified with a classical theory that predicts relativistic effects. The Bohmian equation of motion (in its Newtonian rather than Lorentzian form) would be reformulated in a Lorentzian manner and this would put Bohmian theory in a form where its fundamental level would be the form it has in Bohmian quantum field theory (with Euclidean space and an absolute time). In Bohmian quantum electrodynamics, electrons and photons are unified with neo-Lorentzian theory, in the way I discussed regarding Callender-Weingard’s and Valentini’s approach to Bohmian quantum electrodynamics. As I have indicated, I believe talk of ‘‘fields’’ can be translated into talk of fermions and bosons, e.g. electrons and photons. We can say that the laws of motion in QED are laws for changes in field values over time, or we can say they are laws for changes of electrons and photons, suggesting electrons and photons are individual particles, fermions and bosons. Quantum field theory is about a subset of the particles, the subset (electrons, photons) in the case of QED and (quarks, gluons) in the case of QCD. In Bohmian QG, the subject is values of the gravitational field or gravitons. Gravitons are bosons, just as are photons (in QED) and gluons (in QCD). The fermions in QED are electrons and the fermions in QCD are quarks and the protons and neutrons that quarks compose. QG is about gravitons and not merely the fermions of QED and QCD; gravitons affect photons and gluons as well. The observational effects that gravitons cause are most clearly noticeable in large aggregations of fermions and bosons, such as planets, galaxies, clusters of galaxies and superclusters of clusters of galaxies. Each of these large aggregations is composed of atoms and each atom (except for the lightest, hydrogen) is composed of electrons, protons and neutrons; the aggregations are also composed of the photons that travel between the electrons and protons and bind the electrons in their orbits around the nucleus of the atom. The nucleus consists of protons and neutrons and these consist of quarks; quarks are bound together by gluons and (derivatively) protons and neutrons are bound together by gluons. It would be accurate to say that QG is about the gravitons that are interchanged by large organized composites of electrons, photons, quarks and gluons. The gravitational acceleration law is about the manner in which the gravitons interact with the aggregations of electrons, photons, quarks and gluons. This is oversimplifying of course and not merely for the reason that it would take several books to characterize Bohmian QED, let alone the other matters. There are also other bosons and fermions, such as weak vector bosons, mesons, and there are photons that do not remain inside an atom but travel across large spaces, such as from the sun to the earth.

Many details are left out of this outline, for example, how the neo-Lorentz law of gravitational acceleration accurately predicts the orbit of Mercury (but see Jannossy 1971 for a neo-Lorentzian equation that makes this prediction) and predicts that the paths of photons will curve towards stars as they pass them, due to gravitational effects (also see Jannossy 1971 for a neo-Lorentzian equation that makes this prediction). But we understand the basic idea about how Bohmian QM can be unified with a neo-Lorentzian theory of gravity and a neo-Lorentz GTR (General Theory of Reference frames, inertial and non-inertial). There are no mathematical, logical or ontological inconsistencies within this Bohmian-neo-Lorentzian QG theory. For example, we do not need to worry about reconciling the causal determinism and the space-time continuum of Einstein’s GTR with the probabilistic and discrete nature of reality implied by a realist (rather than an epistemic) interpretation of Heisenberg’s uncertainty principles or the non- Bohmian interpretations of QM that have no causally determinist laws of motion in addition to the Schrodinger equation. In Bohm’s theory, the probabilities obtained from the Schrodinger equation are epistemic and do not belong to physical reality; there are no ‘‘probabilistic causes,’’ but only the deterministic causes implied by both Bohm’s equations of motion and the new Lorentzian equations of motion. Probabilities are merely a measure of our ignorance of the causally deterministic behavior that satisfies the Bohmian-Lorentzian equations of motion. The classical theory that is contained in Bohm’s equations of motion imply an absolute time and space and the classical equations, besides being modified by the Q-potential, are reformulated as neo-Lorentzian classical equations, which both imply there is an absolute time and space and the relativistic effects. Nor do we have to worry about the perturbative non-renormalizability of solutions to the Wheeler-deWitt equation, e.g., the Hartle-Hawking solution and the Vilenkin solution (or, rather, approximations to a solution). The Hartle-Hawking and Vilenkin theories are about Einsteinian GTR space-times and the Wheeler-DeWitt equation describes GTR 3-spaces. There is also no need for string theory or M-theory, since there is no inconsistency that needs to be resolved, namely, the inconsistency between QM and Einstein’s GTR. Einstein’s GTR has been replaced by a neo-Lorentzian GTR and, if we unify this with Bohm’s 1952 formulation of QM, the fundamental problems are not ‘‘resolved’’ but ‘‘dissolved’’.

The issue of whether or not a QG theory can make predictions is answered by the nature of the classical theory and quantum theory that are being unified. This QG theory makes approximately the same predictions as the classical theory does on large scales and the same predictions that quantum field theory does on subatomic scales. If there are new predictions, they are that the Planck dimensions, the Planck era, the Planck length, etc., are merely epistemic nature and that there is Bohmian-neo-Lorentzian physical reality smaller than these dimensions, all the way down to the spatial points and temporal instants in a spatial continuum and temporal continuum. The Planck dimensions are the limitations of our measuring equipment and prevent us from measuring these dimensions (at least until significant advancements are made in the area of measuring instruments utilizing EPR correlations).

Part three: Bohm-neo-Lorentz cosmology, present time, and modal necessetarionism

It is very often argued and noted that on the broadest cosmic level, Newton’s laws are observationally equivalent to Friedman’s solution to the Einstein equation in GTR. I will discuss this view later, but argue that neo-Newtonian cosmology (originated in 1931 by Lemaitre (1931) and developed by many others up to the present day) needs to be fundamentally changed to become a neo-Lorentz-Bohm cosmology, and that these changes render the cosmology observationally equivalent (in the mentioned respects) to Friedman cosmology. In fact, these changes are required to have a mathematically consistent, realistic and observationally confirmed cosmology. But the substance of this section is on Bohm-neo-Lorentz cosmology, which is where I begin.

On the cosmological scale, a static homogeneous universe has been found to be either impossible or extremely problematic, given either Newton’s or Einstein’s theory. On Newton’s static, infinite homogeneous universe, any motion brings matter collapsing towards the moving point. On Einstein’s static homogeneous universe model (1917), motion also brinks about a collapse towards the moving point. A static infinitely old universe would also result in light from stars filling everyplace, e.g. ‘‘night’’ on earth would be impossible, given the amount of starlight that would have arrived (Olber’s paradox). Prior to learning about the observational evidence (which began to be noted by Slipher in 1916 and was made more widely known by Hubble’s publications in the 1930s), several thinkers had realized that the nature of gravity, light and the distribution of matter possess intrinsic characteristics that put meta-scientific constraints on any cosmology, be it Newtonian or Einsteinian. An inhomogeneous and anisotropic universe would, as a theoretical virtue be relevant to prior probabilities, without (yet) considering any observational evidence, would make any such cosmological they has much lower prior probability that a cosmological theory that postulated large scale isotropy and homogeneity. Anisotropy and inhomogeneity is highly non-symmetric and violates would the equivocally used word ‘‘simplicity’’ is sometimes used to express, namely, lack of arbitrariness. Determining prior probabilities, a cosmology that has the theoretical virtue of postulating a non-arbitrary (‘‘simple’’ in this sense) structure of the universe has a greater prior probability than a theory that (for no reason or justification) postulates a universe that is highly arbitrary in its structure, e.g. inhomogeneous and anisotropic matter distributions. Given these several facts, there a cosmologist theory that postulates a universe that is homogenous, but non-static (due to the above-mentioned problems about static Einstein or Newtonian universes) would have much greater prior probability than a theory that postulated either an inhomogeneous universe (static or non-static) or a static and homogenous universe. If the matter in the universe is expanding or contracting on a large scale, it could be homogeneous and resolve the problems of a static universe, be the universe finite or infinite.

For various reasons, some thinkers began developing models of a universe in this sort prior to the observational evidence discovered by Slipher and Hubble. Some physicists realized that the assumption that the universe is static, an assumption adopted or ‘‘taken for granted’’ both by Newton in his cosmological thinking and in Einstein’s model universe of (1917), should be rejected. A dynamic universe would solve the host of problems that bedeviled the static universe theories. If we call ‘‘the universe’’ the largest aggregate of matter, e.g. the whole composed of either billions, trillions or of an infinite number of superclusters of galaxies, then the universe must be either expanding or contracting. It appears the first person to recognize this is Friedman (1922, 1924), the second Lemaintre (1931), the third Milne (1933), the fourth McCrea (1934) and after this time the results of Hubble’s observations were so well-known that the ‘‘Hubble Law,’’ as it came to be regarded, as an observational law of our universe. But this is mistaken; it is a fundamental theoretical hypothesis, a hypothesized Cosmological Law in cosmology, developed by Friedman, Lemaitre, Milne and McCrea; Slipher and Hubble merely confirmed this hypothesized cosmological law. The basic Cosmological Law is that the universe is homogeneous and isotropic on large scales and is either expanding or contracting. The Law states that if the universe is expanding, then it obeys a certain distance-velocity law, that the apparent recession velocity of a large-scale system (e.g. a supercluster of cluster of galaxies) is proportionate to its distance.

Hubble’s Law is

Here L is the present distance of between the current large scale plane of homogeneity, or some large system, e.g. a supercluster of clusters of galaxies, on this plane, to some spatially distant (of distance L) and long-past large system on the plane of homogeneity that used to exist, say, 5 billion years ago. This distant system would have the spatial distance, L = 5 billion light years, from the present plane of homogeneity or a supercluster on this plane. In Hubble’s Law, L/d(t) / L(t)=T, the symbol L/d(t) is the velocity of recession of the spatially distant supercluster, which is presently at distance L from the present plane of homogeneity, i.e. the plane at time T. Hubble’s constant is 1/T, where T is the present time.

Hubble’s constant 1/T specifies the present rate of mutual recession of the large scale homogenous and isotropic mass content, i.e., the how fast the superclusters are (present tense) receding from each other. The present value of the expansion rate (or of the Hubble constant) is approximately 75 km/ sec/Mps; that is, 75,000 kilometers per second per megaparsec (where a megaparsec may be translated as ‘‘20 quintillion miles’’ (where quintillion, comes after quadrillion, and quadrillion comes after trillion, and trillion comes after billion, etc.)). In everyday language, the present rate of mutual recession of superclusters is approximately 50,000 miles per second per 20 quintillion miles.

Since we ‘‘observe the past,’’ the plane at distance L is (we shall say) the plane of homogeneity that was present one billion years ago (with only the photons, etc., emitted from the superclusters still remaining in existence (existing in the present tense sense)). A photograph of this plane, which is 6 trillion miles away, and one billion years in the past, can be found in Powers of Ten, 1982, P. and P. Morrison, Freeman and Co. San Francisco. Such photographs provide an accurate image of a plane of isotropy and homogeneity as one reads about them in a text with no photographs.

The small d in Hubble’s Law, L=d(t) = L(t)=T, is the familiar differentiation symbol. L is not a ‘‘tenseless distance,’’ but the present distance of a large-scale particle, a past supercluster, from a large-scale particle located on the present plane of homogeneity, the plane of homogeneity at the present time T. The semantic content of ‘‘T’’ in Hubble’s law is the present time. Recall that this velocity-distance law is a basic law of neo-Lorentz cosmology and is not an observational law or statement. The earlier argument that the velocity-distance law is a basic cosmological law was an argument for this thesis. The semantic content of T is ‘‘whatever time is present’’ or ‘‘whatever time is the present time’’ and so no particular present time, such as 15 billion years after the big bang, is picked out as the present time. T ranges over all possible present times and picks out the one that is actually present.

T can be defined in past-tensed terms which also specifies the present time in relation to which some time is past (e.g. one billion years ago, i.e. one billion years earlier than whatever time is the present time). T can be defined in terms of a dating system as how long ago (tensed) the large scale particles (superclusters of clusters of galaxies) have been mutually receding from the big bang to the present.

In every physics article or book I have read, T is defined in presented tensed terms and is the only law in theoretical physics that is explicitly tensed. I will not quote one or two dozen passages from numerous sources as evidence of this, since it is obvious to physicists and philosophers can leaf through any book on cosmology or astronomy, or any article, and find ‘‘Hubble’s Law’’ and see that it is defined in present tensed terms.

The fact that Hubble’s Law is everywhere formulated in tensed terms is very strong primae facie evidence that there is an accepted (by consensus or virtual unanimity) a physically interpreted equation that is about whatever is the present state of the universe and that contains terms that refer to an objective, mind-nondependent present time. Stated in other’s words, Hubble’s law is accepted by all or virtually all physicists and in every case it is treated as the only law whose terms refer to the A-time in which the universe exists, or (stated in still other terms) that refers (by its sense; or, if you prefer, by the definition offered by physicists) to the objective, mind-nondependent present, past and future of the universe. Hubble’s law entails the tensed theory of time (the A-theory of time, presentism, Newton’s ‘‘flow’’ of time.) But more than prima facie evidence is needed to state that the tensed theory of time is posited by theoretical physics. I present some arguments for the irreducible tensed nature of Hubble’s Law.

The tensed formulation of Hubble’s Law is no ‘‘accident’’ that can easily be changed into a tenseless law by rewriting it. ‘‘T’’ denotes whatever time is present and this cannot be translated into tenseless language. The Hubble time T is a primitive or irreducibly tensed variable, since it cannot be stated in a tenseless language without rendering the Hubble law falsified. For example, on a date-version of the tenseless theory of time, ‘‘the present value of the Hubble time T is 15 billion years’’ would have for its truth conditions, truth-maker, (or be translatable by) a statement such as The value of T at 15 billion years later the big bang is 15 billion years later then the big bang, which renders Hubble’s Law useless because its output is merely tautologies.

But if a tenseless time theorist tried to used a different version of the tenseless theory of time, the so-called ‘‘token-reflexive version,’’ then the ontological significance of ‘‘the present value of the Hubble time, time T, is 15 billion years’’ would be the value of T simultaneous with the sentence-token S is 15 billion years. But then ‘‘the present value of the Hubble time T’’ becomes a temporal constant fixed at the time the sentence-token S occurs, and cannot be used to measure earlier or later values of T. More importantly, it provides no information about when the sentence-token S occurs, so astronomers at any time reading old astronomy texts would read, ‘‘the sentence-token S is simultaneous with the time that is 15 billion years after the big bang’’ and have no idea when that statement was made and would have no idea of whether or not the present time T is 15 billion years after the big bang. A series of such arguments showing that this tensed statement and similar ones cannot have their ontological significance stated or explained in tenseless terms can be found in Smith (1993), Craig (2000a, 2000b) and other works and I refer the reader to these texts.

Given the primitive, irreducible present tense in ‘‘the present value of T,’’ the result is that the basic law of cosmology is an irreducibly present tensed law. Since this law is instantiated by the universe, the present time that is denoted by the present tense is intrinsic to the universe. This argument can reach the conclusion about tensed Bohmian QM that is my view and that was formulated (but not endorsed or argued for) by Maudlin:

It is perhaps interesting to note that Bohm’s theory [in addition to Lorentz’s and Hubble’s] is deeply congenial to an ontology which maintains that all which exists is that which exists now, i.e., at a point in time classically conceived. Instantaneous velocities can of course be defined, but only as the limit of average velocities over finite periods of time. Those puzzled about the status of velocities in an ontology in which only an instant of time exists can happily adopt a Bohmian ontology of particles (with positions) and the wave function.

(Maudlin (1994) Quantum Non-Locality and Relativity, Oxford:

Blackwell Publishers, p. 124)

The view that time is tenseless was adopted (as Minkowski explicitly stated) for mathematical convenience, not ontological purposes, in Minkowski (1908). Minkowski’s assertion that he was assuming a tenseless time model for pragmatic purposes, not alethic purposes, later was ignored and his pragmatic model came to be misunderstood as ‘‘ontological model’’ that had alethic intention. The pragmatic and non-alethic model of a ‘‘tenseless spacetime view of the universe’’ in Minkowski’s mathematical model (and its use in Einstein’s GTR) led philosophers such as Russell, Smart, Quine and others to think that ‘‘the scientific view’’ is that time is tenseless. As this misunderstanding gradually became habitual, even physicists thought that ‘‘the scientific view of the universe is that it has tenseless time’’ was part of the definition of science, (forgetting that on a Newtonian or Lorentzian physics, time ‘‘flows’’ and time is not tenseless). Tenseless time is a peculiar artifact of the pragmatic models used in Minkowski-Einstein relativity theories, theories which have now been disconfirmed, by the Bell-Aspect experiments in favor of a Lorentzian science and its view of a ‘‘flow’’ of absolute time.

The basic cosmological law of distance/velocity, with its present value of Hubble’s constant as an explicitly present-tense and present-time defined part of the law, shows the burden of proof is on the tenseless theorist to show why physical cosmology is mistaken and to engage in a (I believe) futile task of trying to show that the scientific used tensed expressions either can be ‘‘translated’’ into tenseless ones, or have tenseless truth conditions, or that the deep structure of all possible natural and artificial languages is tenseless and is isomorphic to the ontological structure of reality.

The velocity/distance basic cosmological law is part of neo-Lorentzian cosmology and neo-Lorentzian GTR (general theory of reference frames, both inertial and non-inertial). It describes the gravitational behavior of the universe at its largest scales. Lorentz posited an infinite, flat, non-dynamical, substantival, absolute, Euclidean rest space. We have retained this since it provides the best explanation of contraction and dilation (namely movement in this space causes contraction and dilation and other Lorentzian effects, e.g. increase in mass as velocity increases). We do not need an ether. The material universe is the whole system of the expanding superclusters of clusters of galaxies, and this system is growing into an arbitrarily large system of matter and energy, a system that is accelerating (or decelerating) relative to absolute rest space. The inertial frame is at rest relative to absolute space or, more precisely, is absolute space itself. Infinite Euclidean space is homogeneous and isotropic. This implies it is symmetrical to the large-scale structures of the expanding universe, in respect of homogeneity and isotropy. But Euclidean space is not expanding. Nor is there a non-Euclidean space that is expanding. There is no observational evidence for a ‘‘space expansion’’ hypothesis. What is observed are superclusters of clusters of galaxies receding from each other with a velocity that is proportional to its distance.

In order to ‘‘expand’’ in a Euclidean non-dynamical space, the material universe must expand into an empty part of Euclidean space. Euclidean space is what set theorists or philosophers of mathematics call an ‘‘actual infinite’’ (an idea derived, of course, from Cantor’s transfinite mathematics) and the material universe is what Cantor calls a ‘‘variable finite’’ whose values can take on any arbitrarily high finite number. To take just one spatial dimension, e.g. the x axis, no matter how high one reaches on the real number line, a spatial size of the material universe is either below, at, or above that number. If the universe expands for ever, as present observational evidence (since the 1998 observations of two supernovae) suggests, then it is true that for any time t in the future, and any ‘‘size’’ or number on the real number line, the size of the universe will become larger than that size. This is symbolized by physicists by ‘‘+∞’’, or ‘‘–∞’’ which they call ‘‘infinite’’, which can be misleading by philosophers, who standardly use ‘‘infinite’’ in the sense of set-theorists, who use ‘‘infinite’’ to mean the number of all rational numbers, aleph-zero, ₀, and continuum-many to mean the number of all real numbers. When physicists say the future is open and thus infinite, they do not mean the universe is infinite in the sense of having omega, x, number of years later than the present year. Omega is the ordinal number of a set with ₀ or aleph-zero members, ordered with the natural order of all positive whole numbers {1, 2, 3, 4, . . . n, . . .}; the transfinite ordinal x is the order type of this set. Physicists mean the future is infinite in the senses that it has (in Cantor’s terminology) a variable finite number of years, +∞, meaning that for any finite number 10²⁰ on the positive real line, the future is finitely longer than 10²⁰ years. William Craig’s use of the phrase ‘‘potentially infinite’’ is much closer in meaning to ‘‘infinite’’ than what his philosophical critics mean by ‘‘infinite’’, which in set-theory means a set with at least an aleph-zero number of years..

This is not the Milne-Mcrea (1934) neo-Newtonian cosmology, which has been further developed and modified by Bondi (1960), North (1965), Sciama (1971), Peebles (1993), Harrison (2000), Hartle (2003) and a number of others. Milne and Mcrea (1934) derived from Newtonian physics a cosmology quite different than Newton’s own cosmology. Milne and Mcrea’s neo-Newtonian cosmology implied there is an infinite Euclidean space that was filled homogeneously with matter, with the density of the matter changing with time. Parts of this universe of the appropriate size can be considered as spheres, which are expanding in the Euclidean space. The Euclidean space is considered to be non-dynamical (not expanding or curved) and at rest, with all matter in motion relative to it. A hypothetical observer in this sphere on a large ‘‘particle’’ that is moving with radial motion will observe that large systems of matter (e.g. superclusters) are receding from himself and their velocity of recession increases proportionately with the distance of the superclusters from himself. The sphere obeys Hubble’s law of velocity-distance proportionality. If M(r) is the mass in the sphere of radius r and v the escape velocity from that mass, then the velocity needed for the ‘‘partice’’ to escape the sphere of mass M(r) is

where G is the gravitational constant. The behavior of the sphere is described by an equation,

where p is density and k the constant, where k = 0 corresponds to a parabolic path of the particle, +1 to an elliptical path and -1 to a hyperbolic path. This is Friedman’s equation, derived from Newtonian principles. Although this reflects the observational equivalence, the meaning of the terms is different in the Newtonian and Einsteinian interpretations. The constant k in the Newtonian interpretation refers to the total energy of the particles and in the Einsteinian interpretation it refers to curvature of space. The temporal variable refers to absolute time in the neo-Newtonian theory, but to Friedman’s ‘‘cosmic time’’ in the Einsteinian interpretation. Harrison (1981: 238), a proponent of Friedman-Einstein cosmology, remarks about this observational equivalence: ‘‘In all its applications, Newtonian theory is only approximately true, and yet in this most unlikely of all cases it yields the correct answer.’’ However, there are at least two insurmountable obstacles to considering this to be a realistic and consistent model. The neo-Newtonians, following Newtonian principles, estimate the gravitational potential in the sphere by ignoring the infinite amount of matter, the infinite number of superclusters, outside this sphere. Only in this way can the velocities of escape be calculated and can the sphere be considered to obey the neo-Newtonian interpretation of Friedman’s equation. Secondly, each sphere consists of particles that do not obey the Newtonian laws of motion; they obey neo-Lorentzian laws or (as a proponent of general relativity would say) relativistic laws.

These and other problems are overcome on our neo-Lorentzian cosmology. Our infinite Euclidean space is not completely filled with matter. Rather, part of it is filled with matter, and this part, the material universe, is expanding into an empty Euclidean space. The material universe may be open or closed, as in Friedman’s case. If it is closed, the expanding super-clusters will eventually begin contracting and contract at an ever increasing rate until they ‘‘crash into each other’’ in a ‘‘big crunch’’. The metaphorical talk is deliberate since there are no observations or known laws that explain what occurs at either the instant of the big crunch or the instant t = 0 at which the expansion began. There are also no observations or confirmed laws that enable us to say anything about the material universe before the beginning of its expansion. It is conceivable that some justified hypotheses can be made, especially once we take into account Bohmian QM, but an attempt to formulate any such hypotheses is beyond the scope of this paper.

If the material universe is open, as observational evidence suggests, it will expand forever and will expand to arbitrarily large distances; that is, for every distance, say, 15 billion light years, there is a greater distance to which the universe will eventually expand. The material universe will become less and less dense the further it expands, the superclusters becoming ever further apart, until they disintegrate into elementary particles or radiation, which themselves will keep receding from each other with no end to their mutual recession and with no end to future time.

The second problem is that the material universe is not composed of particles that obey Newton’s laws of motion; rather, they obey neo-Lorentzian laws. More exactly, they obey Bohmian-neo-Lorentz cosmological laws, which in effect reduces to neo-Lorentz cosmology, since there is no observational evidence that the quantum potential Q is greater than zero at cosmological scales. Some might think the presence of Planck’s constant in the equations imply that it cannot be the case that at cosmic scales Q = 0, but this is assuming a realist rather than an epistemic interpretation of Planck’s constant. Furthermore, the equations can be formulated without Planck’s constant, as Dürrrr, Goldstein and Zhanghi have shown (1996: 26). There is a phenomenal ‘‘cosmic time,’’ which is the perspective from which we say the material universe began to expand 15 billion years ago. But this does not mean ‘‘cosmic time’’ in the sense of Friedman’s universe, where there is no absolute time in the sense of non-local, space-like, instantaneous, universe-wide relations of simultaneity.

The so-called ‘‘clocks’’ introduced by Poincaré (1905) and Einstein (1905) are moving masses or light. Since movement in absolute space contracts the moving body in the direction of length, it ‘‘slows down time’’ in the unrealistic sense that, if a light clock is placed so that its top is in the direction of the motion, the light clock contracts. This has little if anything to do with time, especially since in the absolute, real frame, light has the velocity c and moving systems have a velocity of light c + v or c - v. The law of the addition of velocities is not in reality violated, even though it appears to be in the phenomenal realm. The fact that absolute moving systems have a light velocity of c + v or c - v is part of Lorentz’s theory and his retention of absolute simultaneity and Galilean transformation equations for the absolute time and absolute rest frame.

The nearest to an accurate clock could be any ideally periodic motion or simple harmonic oscillation on a body, a clock, at rest which is relative to absolute space. But in (1998) I argued there is no justification to think that it is necessarily true that the metric of time is either identical with or isomorphic with a light clock or harmonic oscillators or other such ‘‘clock’’ devices. We are immediately acquainted with presence and this is how we identify the absolute frame and absolute time. It cannot be the case, as a critic of the tensed theory of time may suggest that we falsely belief a hyperplane 15 billion years or so later than the big bang is present, whereas in reality a hyperplane near the big bang is present. If a hyperplane near the big bang is present, that is the only hyperplane that exists, since ‘‘is present’’ means ‘‘exists’’ in the present tense sense of ‘‘exists’’. Accordingly, we would not exist and have a false belief about what is present. This argument against the tensed theory seems to be presupposing the ‘‘Moving Now’’ and the A-series formulated by McTaggart (and which he later rejected), which posits that each event has a primitive tenseless existence and that the A-properties of presentness, pastness and futurity have no existential import but are possessed successively by beings that have a primitive and irreducible tenseless existence. It is very difficult to find a tensed theorist who holds this theory. Schlesinger (1981) and Gale (1968) might seem to, but a closer analysis makes it rather ambiguous; in Schlesinger’s case, it is largely due to the vagueness and ambiguity of his theoretical formulations. Tensed theorists, as far as I know, believe that what is present exists, what is no longer present existed (i.e. no longer exists) and that what is not yet present will exist (i.e. does not yet exist). For our critic’s scenario to work, he has to reject this and identify existence with an irreducible, primitive tenseless existence. But then he is no longer talking about theories held by contemporary (or former) tensed theorists of time. Contemporary philosophers of time do not engage in ordinary language analysis, e.g. of the sort popular in the 1940s and 1950s in England and no one would infer an ontological conclusion about the nature of time from ‘‘how we ordinarily talk’’. The issue is of the isomorphism between the deep structure of all possible natural and artificial languages (including the language of the sciences) and reality; the isomorphism is that the deep structure of language is irreducibly tensed and is isomorphic to the tensed structure of reality (Smith 1993: 215– 24; Ludlow 1999). Equally important arguments are from experience of presence (e.g. Craig 2000a; 2000b), the physical sciences (Smith 1985, Craig 2001a; Smith, 1993: 18–24, 225–250), and the formal logical invalidity that is implicit in a tenseless theory of time.

My so-called ‘‘watch’’ (movements of bosons and fermions, or large aggregates of them moving relative to another large aggregate that is relatively and phenomenally at rest to the moving bosons and fermions) may not be similar to somebody’s who is moving at a high velocity relative to me, but what has that to do with time? Both of us perceive presence or the present. This is absolute presentness. If we want to measure accurately intervals of time, we can wait until a ‘‘Valentini clock’’ (2005) is invented, one that uses instantaneous signals, so that the signals and the absolutely present instant t are all simultaneous. An accurate clock is not only physically possible but technologically possible, even though we are a long way away from making this ‘‘technological possibility’’ into a probability or actuality.

Our Bohmian-neo-Lorentz cosmology suggests some implications regarding metaphysical modalities. If and only if the initial conditions of the universe satisfy the precise probability density, ↑ Ψ ↑², are the predictions of Bohmian QM accurate; without them, they are widely inaccurate if Bohm QM or some other theory is possible at all. In other words, Bohmian QM implies the initial conditions that satisfy this probability density. This is another way of saying that the probability density, ↑ Ψ ↑² is not a brute or contingent fact and that Bohmian QM could be true if there were some other initial conditions. Rather it is a way of saying that the probability density is nomically necessary. There is another fundamental nomic necessity about the initial conditions of the universe, this one following from the Bohm-neo-Lorentz cosmology. The basic cosmological law is the distance/ velocity law and this law requires precise initial conditions at the origin t = 0 of the distance scale. We can say that at the big bang or infinitesimally close to the big bang the initial conditions had to be exactly as they are to produce a material universe that satisfied the velocity-distance law. They are nomically necessary and since this initial distribution is also necessary for the probability density, ↑ Ψ ↑² this probability density and the distance/ velocity initial conditions law indicate there is a very strong form of necessitarionism in Bohm-neo-Lorentz QG. It seems to me very hard to avoid the conclusion that some other philosophers have reached for other reasons, namely, that metaphysical possibility and necessity (or ‘‘broadly logical possibility and necessity’’) is identical with Bohm-neo-Lorentz nomic possibility and necessity. It seems to me that this suggests that the only modal logic system that is a valid simpliciter, i.e. that describes the modal logical form of reality, is Triv (Hughes and Cresswell). This modal system implies that what actually exists necessarily exists, and it implies that what possibly exists actually exists.

Clearly a new theory of counterfactuals is needed, especially since laws are distinguished from accidental generalizations by virtue of entailing counterfactuals and modal necessitarionism implies counterfactuals are all necessarily false. There are various ways to re-interpret counterfactuals, probabilities, possibilities and the like. For example, counterfactuals about the particulars will be epistemic in nature, consistent with the Bohmian thesis that we are ignorant of the particular trajectories. ‘‘Possibility’’ in modal talk can be replaced by ‘‘conceivable’’ which can obey some modal system other than Triv, e.g. the modal system T or S5 (except now we should call these systems of epistemic modal logic or, better, systems of the logic of conceivability relations). A number of ways of dealing with this issue are available, but these issues should be discussed in a different work on the metaphysics of modality.

The debate about whether metaphysical necessity (what can and cannot exist) is identical with physical or nomological necessity is tilted towards an affirmative answer if two of our scientific theories, independent of each other, imply that the initial conditions of the universe, and not merely its laws, are nomically necessary. We see that both the Density Postulate of Bohm’s QM and the initial conditions required by the basic cosmological law of our neo-Lorentz cosmology, the distance-velocity relation, require that the number, distribution, state of motion or rest, and kind of particle, of the initial conditions, necessarily be what they actually are, licensing an inference from ‘‘x is actual’’ to ‘‘x is necessary’’. Our postulate is that if inferences from actuality to necessity, regarding all of the precise initial conditions of the universe, and all the precise and completely described careers or histories of the particles that form the material universe, are logically entailed by two logically independent theories (Bohmian QG and the velocity-distance cosmology, their unification belonging to our Bohm-neo-Lorentz QG), both theories of which are observationally confirmed, then the modal system Triv is the one we are most justified in believing is validsimpliciter (and not merely valid-in-Triv) and that in an appropriate sense the modal logical laws of Triv formulate the ‘‘the one and only modal form of the universe. The only valid modal logic system states that, for any true proposition p, actually p implies possibly p, and possibly p implies necessarily p, and necessarily p implies actually p. Equivalently, possibly p implies p, and p implies necessarily p. There is only one metaphysically possible world, the actual world.

We know the answer to these ultimate why-questions: ‘‘Why is this world actual rather than some other possible world?’’ and ‘‘Why is there something rather than nothing?’’ The answer to the two questions is that it is that there is only one possibility, and nothing at all could have been otherwise. The only possible world is our world, the actual world. It is metaphysically impossible for there to be nothing and it is metaphysically impossible that there be a possible world or any possibility that is not actualized.

The belief that our imagination and our modal intuitions show there are other possible worlds is false. This is even the case for conceiving a Friedman world to be possible. A Friedman universe differs from our Bohm-Lorentz world but only in observationally insignificant ways. But a Friedmann universe is mathematically impossible. Consider that Friedman space-time is a continuum of space-time points, and its radius and density are described in Roberston-Walker metric, which includes two equations, one of which is

Here 3(da/dt)² measures the rate of change of the universe’s radius a with time, G is the gravitational constant and it is multiplied first by 8p, and its product is multiplied is proportional to the density p. p’s magnitude is proportionate to the radius squared a² of the universe, and to the velocity of light squared, and to k, the cosmological constant.

Friedman’s universe seems nomically possible, does it not? In fact, it is mathematically and logically impossible. Friedman’s time, the t that appears in the equation, has values that are proportionate to the radius a of the universe. In an expanding universe, the numerical values for the t variable attain higher and higher values as the radius of the universe continues to expand, e.g., if k = 0. There is a possible light clock, where there is a mirror for the top of the cylinder and another mirror for the bottom of the mirror. This clock measures the Friedman time t in his equation our Bohm-Lorentz material universe on sufficiently large scales. But Einstein has identified time with measured values of his clocks. The light-clock has a mirror at its top and a mirror at its bottom; a light pulse is bouncing back and forth between the top and bottom mirrors. Heisenberg’s uncertainly principle shows that the light clock cannot get extremely small, for at 10^-43 s and briefer the gravitational force or field becomes so strong that it makes the light pulse move in a zigzag and non-uniform and non-repeating motion. If all physically possible clocks, within Friedman’s laws, are broken apart at this level, it is no longer possible to measure time by light-clocks and therefore, it follows from Einstein’s GTR, there is no time.

But this contradicts the fact that Friedman’s equations posit a space-time continuum, that there is time ‘‘all the way down’’ to temporal instants that have zero duration. Thus, Friedman’s theory entails there is both time and there is no time, and thus is a self-contradictory theory. I believe other alleged ‘‘possible worlds’’ can be also shown to be inconsistent when examined closely. Humans construct vague images and think that is evidence for other possible words. They are not evidence. On the Bohm-neo-Lorentz theory, the Heisenberg uncertainty principle is merely epistemic. Further, absolute time is neither a clock, a measured time, or anything involving bodies and movements. It exists and ‘‘flows equably’’ regardless of whether or not it is measured.

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