Notations used in this Paper

Latin indices range (1,2,3); Greek (0,1,2,3)
$\delta$ refers to a differential form
$d$ refers to an exact differential
$g_{mn}$ is the spatial metric
$G_{\mu \nu }$ is the space-time metric
$\delta s$ is the spatial distance
$\delta \sigma$ is the space-time line element/invariant
$\mathbf{u}$ are the general curvilinear spatial coordinates
$\mathrm{\Sigma }$ is an inertial reference frame
${\rm Y}$ is a general non-inertial reference frame
$\eta$ is the Natural Reference Frame, in which matter is everywhere at rest
The signature of $G_{\mu \nu }$ is taken to be (+, -, -, -)

1. The Mainstream Picture of Reality

The typical popular book on science presents a materialistic, atomistic picture of the world. That is, in the beginning there is a collection of elementary particles (“atoms”) of some kind. In particular, we speak of quarks and gluons that eventually crystallize into atoms of matter and photons of light.

Steven Weinberg himself betrays belief in such an outdated, 19^th-Century picture when he says that, “The more the world seems comprehensible, the more it also seems pointless.” The basis of this remark is a belief that the world is a vast billiards game of particles caroming off each other with no object or rules for determining the “winner.”

One false element of this picture is the presumption that the world is built of atoms in the same way that a house is built of bricks. More fundamentally, it presumes that atoms or “particles” are indeed small “objects” like tiny billiard balls or bricks possessing position, velocity and, above all, identity, even if these quantities can be determined only statistically.

2. Problems with the Definition of the World in Quantum Theory

Most, even popularized, books on quantum theory tell us that elementary particles are not to be visualized as small billiard balls or objects possessing predetermined properties.

Objects possess identity even if they are constitutionally indistinguishable, but the thermodynamic statistics satisfied by quantum particles indicate that they are fundamentally indistinguishable and lack identity.

There are two approaches to the study of quantum many-particle systems.

The first and most elementary approach is to regard them as collections of individual particles. Then their indistinguishability is represented by using wave functions that are completely symmetric (bosons, such as photons) or completely antisymmetric (fermions, such as electrons) in the positions of any two particles. This means provisionally assigning them identities and then wiping out the identities by entangling the various particles. Particles lacking identity, that is, identical particles, are, in this representation of the theory, entangled nonidentical particles with identity. They have identity, but the experimenter cannot tell one from another.

But this is a contrived picture because it ascribes identity to things that have no identity. This is in the same spirit as the anti-relativists who insist that in relativity there really is an absolute space, but that which of the equivalent inertial reference frames it corresponds to cannot be determined experimentally. Such assertions are non-empirical and those making such assertions presume themselves in possession of esoteric knowledge unavailable to the rest of us.

The concept of an “object” (with identity and pre-existing properties that are determined by measurement) is actually an emergent phenomenon. By the term “emergent phenomenon,” we mean the circumstance whereby, for example, individual gold atoms are neither shiny nor ductile, but a macroscopic bar of gold is both: these properties are “emergent.” This is what Philip Anderson means by his famed dictum, “more is different.” In the same way, a gold atom does not possess identity, while a macroscopic bar of gold does. Gold atoms are not objects, while bars of gold are.

Objects first emerge in the mesoscopic world: genes, chromosomes and the cells of a body are objects, and the human body could be said to be built of cells in the same way that a house is built of bricks, but the analogy of the statement, substituting the word atoms for cells and cells for the human body is false.

To represent the idea of indistinguishability in a more fundamental and natural way, we employ the second, more advanced approach to quantum multi-particle theory (quantum field theory) in which individual elementary “particles” are represented simply as quantum numbers with no position, velocity or identity.

I call this picture “Quantum Continuous Medium Mechanics.” It is reminiscent of 19^th Century Continuous Medium Mechanics in which media are represented as continuous functions: mass density, charge density, charge current density, and electric and magnetic field strengths, only these quantities are quantized, that is, they are integral multiples of fundamental units such as electronic or quark mass and charge. The world is fundamentally macroscopic; it is not built, bottom up, of atoms, but is comprised ab ovo of quantized. macroscopic objects This picture is top-down or synthetic rather than bottom-up or analytic.

But this picture is just as antiquated as the more elementary treatment of multi-particle theory by virtue of being materialistic, describing a materialistic, objective world in which the observer is ancillary. Consciousness and subjectivity are regarded as epiphenomena. But these things are the things of whose existence we are most certain and should be regarded as most fundamental. It is matter, not consciousness, that should be regarded as an epiphenomenon. We should speak of “the hard problem of matter” (why does the existence of matter seem so palpable?) rather than “the hard problem of consciousness” (how can the mere motion of inanimate electrons give rise to something as alien as consciousness? Short answer: it can’t). Materialism is a false theory of reality/experience. But writers such as Daniel Dennett, who are not very philosophically astute, are so convinced of the theory of materialism that they deny the existence of consciousness. You must wonder what he is thinking when he writes these words: does he seriously believe that he has no awareness of his words and that they are produced on the page by an elaborate stimulus-response mechanism? What gives him the ambition and drive to go through the challenging process of publishing his books and disseminating his theories or even coming up with his theories, more stimulus response mechanisms?

The trend in modern physics with the advent of relativity and quantum theory is away from an objective, impersonal description of the world towards one based upon the perceptions of observers immersed in the world. The Copenhagen Interpretation of Quantum Theory (invented by Bohr and his student Heisenberg, based in Copenhagen), which remains among the most popular interpretations of Quantum Theory, is just the common-sensed notion that the world one experiences depends upon the details of the material design of the measuring instrument or biological observer. This is the import of Thomas Nagel’s famed essay, “What is it Like to be a Bat?” (For further discussion of these remarks see Appendix I.)

Along these lines, quantum physicists usually believe that material objects do not possess pre-existing properties that are merely determined by measurement, but that measurement is a more interactive process; a measurement is as much a property of the observer or measuring instrument as the thing observed or measured; a measurement is the result of an interaction between the measuring instrument and the thing measured. Measurements are created in the act of measurement. What else is this but Berkeley’s formula, esse est percipi?

A lot of nonsense has been written about quantum entanglement being something intrinsically different than plain old correlation. Entanglement happens when a wave function of two variables cannot be resolved as the product of two single-variable wave functions and correlation happens when the probability density function cannot be resolved as the product of two single-variable probability densities. It can be demonstrated by a quite trivial computation that these two formulations are equivalent (see Appendix II for clarification of these remarks).

Now, if you insist that there is something preternatural about entanglement, then there is no reason for you to resist my conclusion that quantum theory is essentially idealist (in the Berkeleyan sense); why would you insist that quantum theory has a prosaic materialist interpretation if it in some sense possesses preternatural elements? You are mixed up!

3. Problems with the Definition of the World in Relativity

Relativity has idealist notions á propos the definition of time.

The fundamental problem is the definition of synchrony: making two spatially separated clocks read the same time at the same time. Einstein’s fundamental insight was that there is no physically intrinsic meaning to the definition of “the same time” at two different points of space. A definition must be established by some arbitrary convention. More to the point, there really is no such thing in physical reality; each observer has his own local time and no physically real correlation between two of them can be established. Thus a “reference frame” should not be considered as a global entity defined by its velocity at each point, but one should be based upon the perceptions of each individual observer localized at a single position in space.

And with relativity we come to conundrums involving the definition of time, in particular with a global time or foliation of the “world.” The time of a single, localized observer is just the reading on her local clock, but how do we synchronize clocks so that they read the same time at the same time of the world?

In Newtonian Mechanics, this is trivial: there is an “absolute time” shared by the world and all individual observers, and the world of matter is the configuration (positions and velocities) of all the material particles making up the world at that instant. But the picture in relativity is more cloudy, especially in general relativity or even in a non-inertial reference frame in special relativity.

In 1905 with the creation of special relativity, Einstein declared that what is meant by the “same” time at two spatially separated points is not physically defined, but such a definition utilizing some artificial and arbitrary convention is necessary, if at all, only for the convenience of the theoretician.

But my (as yet unpublished) researches indicate, (1) the convention that Einstein hit upon is not artificial and arbitrary but is in fact physically necessary, and, (2) the definition does not extend to non-inertial reference frames (or to general relativity, where, in general, all global reference frames are non-inertial).

That is, the “world” does not exist at a single, unique global instant; there is no global temporal foliation of space.

The definition of a world-instant would need to be well defined and globally space-like. As an example of what I mean by this, the connection between 8am EST in the Big Apple and 8am PST in the City of Angels is time-like; with the use of time zones, the formula t = constant leads to the same clock reading at two spatially separated points being at two different times: the time 8 am in the Big Apple is not the same time as 8am in the City of Angels, despite having identical clock readings. If we could subtract out the difference in time zones (resulting in UTC), then 8am in the Big Apple would be the same time as 8am in the City of Angels. In Newtonian Mechanics this is always trivially possible: we just define UTC to equal the absolute time. But in relativity (as I show in my as yet unpublished research), any definition of UTC (or the “same” time at two spatially separated points) is path dependent. That is, if we slowly translate our clock from the Big Apple to the City of Angels along a direct line between the two cities, we get a different definition of the “same” time than if we, say, take a detour over the North Pole. This is bound up with something called the Sagnac Effect, after a Belgian anti-relativist experimentalist who confirmed the existence of an effect originally predicted by relativists, and then he claimed it disproved relativity! (The history of this is curious.) In short, a unique synchronization between two spatially separated points cannot be defined in a non-inertial reference frame. Each individual observer has a well-defined time (the reading on her local clock), but the world does not. This problem is even more obfuscated in a non-rigid reference frame, and all real global reference frames are non-rigid except maybe in certain local patches.

A case could be made that the only reference frame whose contemplation is theoretically justifiable is the reference frame in which matter is everywhere at rest (I call this the Natural Reference Frame) because it is the only one whose properties are knowable: every observer, test particle, measuring instrument and clock is composed of matter. This reference frame is clearly non-inertial and non-rigid.

In this case, reference frame transformations would be theoretically prohibited. The only two permissible types of transformation would be coordinate transformations of the type u’ = u’(u) and time zone transformations of the type t’ = t + f(u), where t is the time reading on a clock at u and u are the three spatial coordinates.

This reference frame cannot be described in terms of smoothly differentiable functions (consider two contiguous bits of matter travelling at different velocities, such as a leaf blowing down or a person walking down the street or the thermodynamic motions of atoms of air in the atmosphere). One might object that such a reference frame could not be specified in a vacuum where there is no matter, but no vacuum exists anywhere: even in intergalactic space there are about 10 hydrogen atoms per milliliter. Moreover, just as the properties of reference frames besides the Natural one are unknowable, “space” unoccupied by matter does not really exist. Instead of a spatial continuum described by three continuous coordinates, we could designate each particle by a single “serial number,” s, that would take the place of the three spatial coordinates, u, together with the time reading, t, on the local clock associated with the particle. Thus, instead of a continuum, space would be characterized as a discrete manifold possessing a continuous time variable.

Instead of an initial value problem, with data collected at an instant throughout the continuum of space, forecasts of the future behavior of the world might be based upon data collected over a finite interval of time at each discrete point, s, making up the spatial manifold.

These questions considered individually and collectively open up whole new fields of research. For example, is the discretized spatial manifold that we have described metric or not (in the sense of possessing a distance function, d(s1, t1, s2, t2) between the particles s1 at its local time t1 and s2 at its local time t2)? I don’t know the answers; in this paper my only purpose is to provide the questions and open new venues of research.

A Final Note: The well-known light-synchronization that exploits the observed constancy of light in an inertial reference frame in special relativity is equivalent to slow clock transport (see Jammer ^[6]), a process that is more intuitive, general and, therefore, fundamental. (For example, if you and I are on opposite sides of a room and we want to know if our watches are synchronized, one of us simply walks over to the other and makes a direct comparison.)

There is just one problem: in relativity, definition of simultaneity by slow clock transport is path dependent, as discussed above. A clock cannot even be consistently synchronized with itself, for the time differential around a closed loop is nonzero and may be made arbitrarily large by increasing the number of revolutions without bound.

This is puzzling: a clock cannot be synchronized with itself?? Sense can be made of this only if we dissociate the notions of slow clock transport and simultaneity; in other words, there is only slow clock transport (a well-defined physical process), but it cannot be used for synchronization of clocks because there is no such thing as simultaneity! An individual observer has a well-defined time, but the world does not.

This leads to another open problem: that of the spatially extended observer. Any real observer has a consistent sense of existing at an instant of time, but she is also spatially extended, and, as we have just seen, the notion of a synchronized time at two spatially separated points, however close, cannot be defined even approximately (as we saw, a clock at a single point cannot be synchronized with itself and the time differential can be made arbitrarily large; similarly, of course, for two nearby points). So, what accounts for the consistent sense of temporal instants of a spatially extended observer?

I have no clue what the answer to this question is, and I believe that it has no simple theoretical answer, but it depends on the properties and details of the physical construction of the observer, measuring instrument or clock.

4. A New, Provisional Picture of Reality

In my view, the significance of the fact that the world does not have a well-defined time, together with the idealist (in the Berkeleyan sense) implications of quantum theory described above, is that we must complete the trend begun by relativity and quantum theory, but not yet perfectly completed, of transitioning from an objective, bird’s-eye description of the world to the subjective perceptions of observers immersed in the world.

In a pioneering effort to sketch a new picture of the world based upon the perceptions of observers, we note, in the first place, that each observer sees a visual field, an image, consisting of the photons streaming into her eyes. I call this the Light Matter Complex, because not only photons, but electrons, et cetera, are streaming into her eyes and can leave marks on photographic plates and, presumably, retinas. Moreover, there are senses other than sight, and a mature theory would account for them all.

Generally speaking, the photons streaming into one’s eyes provide information about presumably distant events or objects. The events would be tagged not by the source time, t’, the local time of the distant event, but by the field time, t = t’ + r/c (to use a simplified formula), at which the observer saw it.

Moreover, as we noted, the images are formed by the photons entering the observers eyes at her local time. In an environment of muddy water or foggy atmosphere, the correlation with distant events and objects (at least in the visible spectrum) would be lost, and the sense in which one is seeing the image of a distant object would be obscured. There is only a world of images (the Light Matter Complex) at each point, and the sense in which objects or particles or “matter” exist would be moot. The world has some of the features of a kaleidoscope of subjective perceptions rather than an orderly monolith of an external world of objects occupying space and evolving in time.

5. The Relationship Between Light Matter Complex Theory and Idealism

From here it is but a short leap to the full idealism of Berkeley, in which consciousness, not matter, is the basis of reality. Does the moon exist when no one is looking at it? It doesn’t even exist when everyone is looking at it. In the words of Howard Bloom, reality is a shared hallucination. With this observation, the “hard problem of consciousness” becomes replaced by “the hard problem of matter”: why is this hallucination so pervasive and why are there such deep correlations between this matter (e. g., the brain) and consciousness (the mind)?

6. Conclusion: Epistemological Questions Answered by Idealism

Those who find the idealist picture of reality too psychotic should note that it does solve a lot of fundamental epistemological questions. For example, “How is knowledge possible?”: the scientist and her experiment are immersed in the world, themselves natural phenomena qua natural phenomena, just like clouds, rivers or flocks of geese, and the world controls them externally and to some extent deterministically, including how she draws conclusions from the results: could she be mis-lead by impersonal, insentient forces such as gravitational and electromagnetic fields and similar electrochemical processes in her brain? It seems highly likely. Writer Walker Percy ^[9] points out that theories (and the way we associate symbols or words with objects) are not space-time events, unlike we are told that material objects and their interactions are by the materialists who have only a stimulus-response theory to offer for communication and theory (or knowledge) generation. Such a theory makes it impossible to grasp how a conscious event such as language or a theory, which has no location in space, could be correlated with a space time event in the first place, for language, theories, knowledge, matter and space-time events are all of a piece; they are phenomena in consciousness. That is, if the world is idea, the flow from experiment to knowledge and theory (or between event and language) is seamless: they are all events in sentient consciousness and there are no external space time events or objects.

Arthur Schopenhauer once said that the philosopher experiences reality as a waking dream.

7. Acknowledgements

I would like to express my gratitude to the reviewers, whose observations helped me strengthen and improve the paper. Special thanks to the reviewers who requested clarification on details related to my Natural Reference Frame concept and my ideas á propos the definition of synchrony and simultaneity. I thought long and hard and developed new insights, particularly á propos the history-dependence of the transformation laws, the laws of physics and the inadequacy of the current formulation of relativity theory.

Appendix I: Interpretation of Relativity and Quantum Theory as Perceptions of the Observer

a. Relativity

The reference frame is determined by the motion of the observer. As demonstrated in Appendix III, there is no global foliation of time—no “world time,” just the time of individual observers. Therefore, a properly defined reference frame should not be a global entity defined by a velocity field, $\mathbf{v}(\mathbf{u},t)$, where $\mathbf{u}$ are the general curvilinear spatial coordinates and $t$ is the time coordinate defining the reference frame, but should be tied to the position and velocity of a single observer localized at a single point. What was formerly regarded as a single, globally defined reference frame would be regarded as a collection of reference frames tied to a single point, $\mathbf{u}$.

From here, it is but a short leap to replacing the “reference frame” concept to considering a collection of visual images seen by observers at individual points at the time that the observer sees the image.

Let me clarify: under the current conception, we might see the image of an object at distance $r$ at time $t+r/c$, where $t$ is the time at which the distant object appeared at its location. Under the new conception, we would use the time at which the observer saw the image, $t^{\prime }=t+r/c$, as the time of the event (i.e., whatever was going on at distance $r$ at time $t$). That is, the observer located at $\mathbf{u}$ would tag the event by the time, $t^{\prime }$, at which he saw it rather than by the time, $t$, at which it presumably objectively occurred. (Of course, the relation $t^{\prime }=t+r/c$ holds only in an inertial frame in special relativity. In general, the relation is more complicated—in general, $t$ is the “lookback time,” but here we are just speaking in general terms.)

This is a vision that I call the “Light Matter Complex.” I call it a vision rather a theory, because the mathematical details have not yet been worked out, indeed, I am not even sure how to represent the Light Matter Complex as a collection of physical variables. Various speculations are offered in Appendix V.

b. Quantum Theory

In general I am a disciple of the Copenhagen Interpretation, which however means different things to different people. In one version of it, the microscopic world is a black box that can be understood only in terms of its effect on classical, macroscopic measuring instruments. But this interpretation, is at a certain level, fatuous: the macroscopic world is not classical; it is also quantum—the macroscopic world is also a “black box.” There is no clear line of demarcation between the microscopic and macroscopic worlds: macroscopic quantum effects exist, for example, the Josephson junction.

The version of the Copenhagen Interpretation that I focus on is described by statements such as, “the answers the experimentalist gets depend upon the questions she asks.” That is, for example, matter appears to be wave or particle depending upon the experimental set-up, that is, upon the specific details of the mechanical design of the measuring instruments used in the experiment.

Note. In the Madelung formulation, or the closely related Bohm–de Broglie pilot wave theory, matter is described unambiguously as particles that scatter statistically in a wave-like manner. This formulation attempts to answer all quantum paradoxes in a fully materialistic way: there are pre-existing properties of matter that are merely determined by measurement, measurements are not created in the measurement act itself. But there are at least two problems with this scheme that I can think of: (1) there is currently no known way to describe spin in such an objective manner; (2) the multiparticle Madelung formulation has epistemological problems with the description of the indistinguishability of particles as described in the text. Particles not possessing identity are not “objects” in the materialistic sense. There is something intangible about the quantum particle concept that the Madelung formulation leaves unclarified.

To obtain an unambiguous description of the perceptions of a human observer we would need to describe the details of the mechanical construction of the human brain and perceptual organs (at bottom, the entire human body: today researchers do not speak of the brain and the body, but the brain-body, an integrated whole) as they interacted with the rest of the world. A deep problem that present itself here is the description of the human body, which is neither a fixed collection of particles nor a fixed region of space. Technically, this problem presents itself even for inanimate manmade measuring instruments, or, indeed, any real macroscopic object.

This is one version of the description of the Light Matter Complex, discussed in Appendix VI, where we discuss various speculative ideas for how to represent it.

c. More on Quantum Theory and the Copenhagen Interpretation

In Heisenberg ^[5], Heisenberg discusses how a quantum phenomenon can be described either by describing just the quantum system itself or by describing the quantum system + the measuring instrument, and he shows that these two descriptions are consistent. (He considers the case of a beam of electrons illuminating two atoms, the electron beam being the quantum system and the two atoms representing the measuring instrument. It is my understanding that von Neumann formalizes this procedure in a fully general way in von Neumann ^[8], but I have never studied von Neumann’s book.)

Thus the version of the Copenhagen Interpretation that regards the theory as describing the effects of a microscopic system on macroscopic classical measuring instruments would be replaced by nestings at higher and higher levels describing the measuring instruments as well as the microscopic system, then a higher nesting describing how the microscopic system + measuring instrument (which is still a black box) would be observed at a higher level, and so on, until one implicated an infinite nesting involving the whole universe that was observed by who-knows-what, maybe God? This representation is artificially elaborate—using an infinite nesting to redundantly describe what was already described at the lowest level—as well as nonempirical—observers, individually or collectively have no knowledge of conditions existing throughout the entire universe or of any putative God.

An empirical theory would just rely upon the actual observations of real observers and may be underdetermined, permitting multiple solutions (forecasts of future behaviors) consistent with the observations—the condition of uniqueness of solutions may need to be forsaken.

This is consistent with our Light Matter Complex concept.

Appendix II: Analysis of the Bell Inequality

The Bell inequality, said to describe a condition consistent with any classical, local, realistic theory, is nothing but a condition that the properties of the two particles are statistically independent. The claim is that, in a classical world, measurements made at two spatially separated points, assuming there is no “spooky action at a distance,” must be statistically independent, that is, the two-variable probability distribution, $\rho (a,b)$, must be resolvable as the product of two independent probability distributions, $\rho (a,b)={\rho }^{\prime }(a){\rho }^{″}(b)$.

But this condition is fatuous: the Bell EPR experiment is conducted with two particles whose spins at the common origin at the initial time are correlated, and the experiment is conducted in such a way as to minimize external influences and preserve the correlations as the two particles are spatially separated with time, else the correlations are destroyed and the experiment would fail to give conclusive results.

It is the measurement methods that must be independent at the two spatially separated points; there is no a priori reason that the measurement results must be independent: the assumption of the Bell EPR set-up is precisely that they are not. It is very easy to construct a classical Liouville probability distribution, $\rho (p_1,q_1,s_1,p_2,q_2,s_2)$, (where $s$ are the spins of the two particles) that is not of this form even if $q_2\ne q_1$, and it would have never occurred to any classical physicist that there was anything nonrealistic or nonlocal about this.

The proof of the Bell inequality that I am familiar with is given in Schommers ^[10]. Schommers’ proof makes no explicit provision for the separation of the particles with time; thus, the correlations would be forbidden even if the particles were coincident, and no one is making this claim.

This does not mean I am claiming that quantum mechanics can be fully represented by a classical, realistic, local theory, but the true source of the incompatibility of the two theories stems from the fact that the classical probability distributions would need to be negative in certain regions of phase space. For example, the Wigner Quasiprobability Distribution, $\rho (p,q)$, is not globally non-negative. From a classical viewpoint, this result is absurd, because in standard probability theory, probability is defined as the ratio of two non-negative integers: the number of times a certain measurement value, or set of measurement values, occurred, divided by the total number of runs of the experiment as the latter number increased without bound. The source of this difficulty is found in the non-commutativity of the operators $p$ and $q$, that is, in the fact that it is not possible to measure both precisely at the same time. An implicit assumption of standard probability theory is that if we have a probability distribution of two quantities, they can be measured precisely simultaneously and uncertainties in the measurements are represented by variation in the measured values with repeat runs of the experiment. I believe that we can surmise that if a Wigner-like calculation of the probability distribution, $\rho (s_x,s_y,s_z)$, were made, it would also be found to be not globally non-negative since the different components of the spin vector do not commute.

Appendix III: Synchrony and Simultaneity in Relativity

Using a formula of Tolman ^[11] that many authors seem unfamiliar with (for example, Guidry ^[4], in discussing the Kerr vacuum metric, writes $g_{33}=-G_{33}$), viz.,

and choosing $G_{00}$ to equal $c^2$, we can write $\delta {\sigma }^2$ in the form,

Note: The choice $G_{00}=c^2$ can always be made, which ensures that the time coordinate, $u^0=t$, is the reading on a clock at rest in the chosen reference frame. For $\delta {\sigma }^2:=c^2\delta {\tau }^2$, where $\delta \tau$ is the proper time interval, while the proper time of a clock at rest is just its time reading, and, choosing $d{u}^m=0$, we have $c^2\delta {\tau }^2=c^{2}d{t}^2$.

Writing,

where the quadrature is performed treating $\mathbf{u}$ as constants effects the choice. Moreover, in any physically real solution, $G_{00}$ is globally positive, for if there were regions where it was $\le 0$, an intermediate region where it equaled $0$ must exist, and this would be a region where the space time was degenerate, three dimensional, that is, it represents a pole or a terminus of the space time.

Not many in the mainstream are familiar with this principle, so let us illustrate with an example. The Schwarzschild line element is,

Now, it is usually erroneously assumed that $\hat{r}$ ranges $0$ to $\mathrm{\infty }$, but by the principle just cited, it can range only from $2GM/c^2$ to $\mathrm{\infty }$. Note that $\hat{r}$ is not the radial distance from the center; that would be,

which is an elementary quadrature that can be looked up in a standard table of integrals. It is nonsingular and equals zero at $\hat{r}=2GM/c^2$. $G_{00}$ becomes zero at $\hat{r}=0$, which coincides with the mass singularity at the origin.

Define $\delta T=dt-{\lambda }_{m}d{u}^m$, where ${\lambda }_m:=-\frac{G_{0m}}{c^2}$, a quantity I call the “temporal drift” because it represents the variation, or “drift,” with position of synchronized time with spatial position. Then the line element takes the form,

which is reminiscent of an inertial reference frame in special relativity.

A global instant of time presumably defines the conditions existing in a spatial manifold at that instant. That is, all events existing at this instant must have space-like separation, $\delta {\sigma }^2\le 0$. The only potential condition of simultaneity that guarantees this exist throughout the whole space is $\delta T=0$. But because in general $\frac{\partial {\lambda }_m}{\partial u^n}\ne \frac{\partial {\lambda }_n}{\partial u^m}$ and $\frac{\partial {\lambda }_m}{\partial t}\ne 0$, that is, ${\lambda }_m$ is not necessarily of the form $\mu (\mathbf{u})$, where $\mu$ is some function, and $T=\int \delta T$ is not of the form $T=t-\mu (\mathbf{u})$, it is path dependent. Thus, it is not possible to say in an unqualified manner whether the time, $T$, at two spatially separated points, ${\mathbf{u}}_1$ and ${\mathbf{u}}_2$ are synchronized because the determination depends upon the path taken between the two points. Even considering a closed path, $\mathrm{\Delta }T=\oint \delta T$ does not equal $0$ in general, so a clock cannot even be synchronized with itself! Moreover, the mismatch is not necessarily small: it may be made arbitrarily large by increasing the number of revolutions of the closed path without bound.

This result is absurd unless we posit that there is no such thing as simultaneity. This is astonishing: it makes us wonder what is really going on when we are talking to a friend who is standing right in front of us. Moreover, it presents the question, why does a spatially extended observer have the sense that she exists at a distinct instant of time. This is probably answered by psychological considerations, the structural construction of the brain or possibly even related to the hard problem of consciousness—thus physics, the definition of simultaneity, becomes entwined with problems of the mind and takes on a Berkeleyan cast.

One more issue that is tangentially related to the above, though possibly, more properly, it should be relegated to a separate appendix.

The Einstein Field Equations are incomplete, in the sense that they allow non-physical solutions. For example, the energy momentum tensor for empty space (special relativity) is zero and the solution of the field equations could be a space-time metric that was any collection of constants, while the only constant solution that is physically real is $G_{mn}=-{\delta }_{mn}$, $G_{0m}=0$, $G_{00}=c^2$. Therefore, the Einstein Field Equations need to be supplemented by auxiliary physicality conditions.

One such condition might be the existence of inertial frames of reference where the space-time metric was of the form, $G_{mn}=-g_{mn}(\mathbf{u},t)$, $G_{0m}=0$, $G_{00}=c^2$.

Assuming this, if only for the sake of argument, the transformation to another, non-inertial reference frame, characterized by the space-time variables, $({\mathbf{u}}^{\prime },t^{\prime })$ with transformations, ${\mathbf{u}}^{\prime }={\mathbf{u}}^{\prime }(\mathbf{u},t)$, $t^{\prime }=t^{\prime }(\mathbf{u},t)$, which can equivalently be expressed as $\mathbf{u}=\mathbf{u}({\mathbf{u}}^{\prime },t)$, $t^{\prime }=t^{\prime }({\mathbf{u}}^{\prime },t)$.

The differential form of the Lorentz transformation tells us,

where $v^2:=g_{mn}({\mathbf{u}}^{\prime },t)v^m({\mathbf{u}}^{\prime },t)v^n({\mathbf{u}}^{\prime },t)$, where $\mathbf{v}({\mathbf{u}}^{\prime },t):=\frac{\partial \mathbf{u}({\mathbf{u}}^{\prime },t)}{\partial t}$ and $g_{mn}({\mathbf{u}}^{\prime },t)$ can be calculated from $g_{mn}(\mathbf{u},t)$ by making the substitution, $\mathbf{u}({\mathbf{u}}^{\prime },t)$.

Thus, the space-time transformations between the inertial and non-inertial frames of reference are,

where $\mu$ is an arbitrary function. This is not a contact transformation, but depends upon a history-dependent integral.

Thus, space-time is not a manifold, for the definition of a manifold includes the condition that coordinate transformations are contact transformations.

Therefore, the current formulation of general relativity is inadequate.

There is, moreover, no a priori reason to attribute a special rôle to time, so we may speculate that the proper formulation of a relativistic theory would be nonlocal too, involving integrals over spatial regions.

These conclusions are consistent with those we reach in Appendix V, where we discuss our “Light Matter Complex” speculation.

One final point:—

The inability to define a unique global world-time or foliation of time opens up avenues of research into the question of the significance or possibility of defining “initial value problems,” where the material conditions throughout space are specified at a given instant of time and forecasts generated. Perhaps even in a materialistic conception, the idea of “initial conditions” must be replaced by history-dependent correlations among conditions at different points of space over finite intervals of time, as in the Light Matter Complex conception discussed in Appendix V.

Appendix IV: The Metric Structure of the Natural Reference Frame

This speculative hypothesis postulates that only $\eta$ (the natural reference frame) exists. It does not postulate the existence of a space-time manifold with metric $G_{\mu \nu }$ whose covariance properties would demand the existence of general space-time transformations, including reference frame transformations (velocity boosts), which was then artificially restricted to the single reference frame, $\eta$.

We would not speak of “reference frames” at all, but only the “material manifold,” the unique collection of material particles or fluids (which here we use as a synonym for continuous media, be they liquid, gas or solid) that evolves in time.

It would be characterized by the spatial metric, $g_{mn}(\mathbf{u},t)$, and the temporal drift, ${\lambda }_m(\mathbf{u},t)$ (see Appendix III for the definition of this quantity).

As the material manifold is just a spatial manifold in which time appears as a parameter, naturally the only coordinate transformations available are of the form ${\mathbf{u}}^{\prime }={\mathbf{u}}^{\prime }(\mathbf{u})$. As time is embedded as a parameter, we could redefine the time zones by transformations such as $t^{\prime }=t+\mu (\mathbf{u})$, where $\mu$ was some function.

For example, even in Newtonian Mechanics, time zones could be introduced: we can use local solar time or UTC (“absolute time”), and the two quantities are related by the foregoing formula.

As we saw in Appendix III, there is no meaningful way to define the notions of synchrony or simultaneity in relativistic theories, so there is no way to define a unique UTC, and the use of these time zone transformations cannot be avoided.

The speed of light in $\eta$ would be determined by the formula,

where $c^{\prime m}$ are the components of the velocity of light in $\eta$, which is, in general, a non-inertial reference frame where light speed is not the constant, $c$.

Appendix V: The Light Matter Complex

One reviewer requested clarification on how to represent the intersubjective relations of the different observers of the Light Matter Complex without reintroducing a space-time structure.

Here are my thoughts:—

Let us here, for the sake of argument, consider the representation of the Light Matter Complex in terms of the energy flux intensity, $I(\nu ,\theta ,\phi ,t;n)$, where $n$ designates the observer without specifying her spatial position or the region, $\mathcal{R}$, occupied by her perceptual system, which, in turn, is calculated from the conditional probability density distribution, $\rho (p|\mathcal{R})$. (For definition of these quantities, see Appendix VI.)

The goal is to find equations that can be used to generate forecasts of future values of $I(\nu ,\theta ,\phi ,t;n)$ that involve only the $I(\nu ,\theta ,\phi ,t;n)$ themselves, with no explicit or implicit use of space-time information. Because there are only a discrete, or even finite, number of observers, $n$, and we have no access to space-time information, there is a considerable loss of information with respect to the materialistic, space-time theory, and this is only exacerbated if we let $\mathcal{R}$ vary with time to represent the external and internal motions of the biological observer or inanimate measuring instrument.

Some of this information can be recovered by considering the variation in $I(\nu ,\theta ,\phi ,t;n)$ over a finite time interval. Because of the paucity of information, the forecasts might be statistical, or the equations might be underdetermined and permit several possible forecasts, all equally compatible with the “initial conditions,” to which no meaningful statistical weights could be given, and the forecasts could be refined as time evolved and more information was gathered.

Because the resulting laws of physics (the equations used to generate the forecasts) were dependent upon time intervals, the equations would be history-dependent, perhaps involving history-dependent integrals over time intervals.

Appendix VI: Speculations on Possible Representations of the Light Matter Complex

I have no one specific idea how to represent the Light Matter Complex, just the general idea that it is the visual image seen at any given point by, say, the human eye or a camera. We must also have a way to represent material particles (hence, the Light Matter Complex), which, at least classically, might be represented by the Liouville multiparticle probability density, $\rho (p_1,q_1,p_2,q_2,\dots )$, at a given point, $q=q_1=q_2=\dots$. The visual part of the spectrum might, for example, be represented by a Fourier transform of the Poynting vector over a finite region, $\mathcal{R}$, corresponding, say, to the volume of the eyeball or the optical chamber of a camera, that is, ${\int }_{\mathcal{R}}\mathbf{S}dV$. The Liouville probability density could, similarly, be converted into a normalized conditional probability density,

where $[p]$ refers to the whole collection, $p_1,p_2,\dots$.

Similarly, a quantum version of the latter could be represented by the definition,

where $q^{\ast }$ refers to the completely symmetric (Boson) or completely antisymmetric (Fermion) multiparticle wavefunction, $\psi ([q])$ with $q=q_1=q_2=\dots$. Then we would define

Either of these probability densities would need to be converted into an energy flux density, $I(\nu ,\theta ,\phi ,t;\mathcal{R})$, with dimensions energy/unit time/unit area/unit solid angle. That would be the definition of the Light Matter Complex.

Another possible representation of the visual part of the Light Matter Complex would be to consider the image formed by a pinhole camera placed at any point in space and pointed in any direction, as calculated by the usual optical theory.

One of my favorite ideas, compatible with Quantum Field Theory, is the notion of the Elementary Eye.

The Elementary Eye would be a two-state quantum system placed at any point in space in any material situation as defined by specific initial conditions, not just eigenstates of the wave vector.

It would consist of two energy states, $E_0$ and $E_1$, and be characterized by an interaction Hamiltonian of variable functional dependence and strength, able to interact with different kinds of fields or particles.

The exact instant at which the transition $E_0\to E_1$ took place (which, in quantum theory could be predicted only statistically) would define the time at which a “particle” of the relevant type was “detected.”

The proper interpretation would be that there was no “particle” that was “detected,” there would be only the transition events themselves. Nature would be understood not as a collection of things or objects but in terms of transitions or processes—and philosophers have been telling us for a long time, since antiquity in fact, that there are no material objects, just processes. As Heraclitus said, the only permanent thing is change. In the words of Nietzsche, “How much rudimentary psychology is still resident in your atom, my dear physicists!” What Nietzsche meant was that we, as humans, desire something permanent that can be grasped or held, if only temporarily, in this ever-changing world in which we are living, but none such is forthcoming. Life is a bird on the wing. Similarly, in the image-based conceptions of the Light Matter Complex given above, the images are not images of anything, there are just the images themselves. In another of Nietzsche’s quips he said that, to paraphrase, there is no reality deeper than the appearances themselves, no material substratum or earth residuum holding it all up, just the senseless kaleidoscopic stream of images and perceptions.

The concept of the Elementary Eye is a simplified abstraction of the real material molecules responsible for human sensory experience and perception. In a mature theory, the properties of the Elementary Eyes would be replaced by the properties of these real perceptual molecules, which, in turn, would be defined in terms of processes too. Reality is a video camera pointed at its own monitor.