User:ErkDemon/draft

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The Reutersvärd-Penrose triangle is an example of an "impossible object" – a structure that is locally consistent, but globally pathological. Each corner-piece, and each individual connection between two corner-pieces, is free from errors. However, the overall structure is still contradictory. Analogous structures can be constructed from mathematics.

Mathematics lets us construct "impossible objects" by combining and interconnecting definitions that are mutually exclusive. In this example, each statement appears compatible with its neighbours, but any pair of statements contradicts the third. If either Møller's or Schild's arguments are correct, then Einstein's 1916 general theory is an impossible object.

Einstein's "reduction to SR" argument says that zooming in on a region makes the amount of curvature in view decreases and eventually disappear: a relativistic description of the flat region then gives special relativity. But if the region contains a compact gravitational field-source (a massed particle), "zooming" makes the average curvature gets stronger rather than weaker. Einstein's argument works for regions between massed particles, but fails for regions containing massed particles."Geometrical reduction to SR" fails in the presence of matter.

Classical gravitomagnetism according to Namsrai's QM-based field-derivation. The gravity-well represents the conventional gravitational field, the proximity-dependent tilt is the gravitomagnetic field component.The arrow represents the particle's worldline.

Work In progress.

Problems with Einstein's general theory of relativity

Although Albert Einstein's general theory of relativity (equations presented in 1915 and published in 1916 ), contains some of the most powerful arguments and concepts ever presented in the history of gravitational theory, some aspects of Einstein's attempted implementation of a general theory have been found to be problematic, with some of the major criticisms coming from Einstein himself.

Context

Structure of Einstein's theory

Einstein's 1916 system arguably consists of:

1. The general principle of relativity ("GPoR"), the pragmatic minimalist working assumption that the universe has universal laws, which apply universally, including to the experiences of physical observers that accelerate or rotate.
2. The principle of equivalence of inertia and gravitation ("PoE"), a development of the observation that bodies fall at the same rate in a simple gravitational field regardless of their other properties, which also springs from a desire to "geometricalise" gravity.
3. The special theory of relativity ("SR"), Einstein's previous (1905) theory of inertial physics and light, a distillation of the relationships of Lorentz aether theory, based on the assumption of a globally constant speed of light and a "flat" lightbeam geometry.

While the first two points can be considered definitional properties of any "geometrical" general theory, Einstein used the third, compliance with special relativity, as the foundational basis for the new system, and as the starting-point for his GR project.

Einstein's reasons for using SR as his foundation for GR seemed to be partly pragmatic: although he presented himself as not being inherently opposed to the purely abstract idea of a general theory without assuming SR, if such a thing could be shown to work, without SR we didn't know where to start:

Einstein (1913): " 3. Validity of the theory of relativity (in the narrower sense); ... In my opinion it is absolutely necessary to stick with postulate 3 as long as there are no compelling reasons for not doing so; the moment we abandon this postulate, the manifold of possibilities will become indeterminable. "

Types of criticism

Early criticisms of GR1916 focused on Einstein's decision to have the new curved-spacetime theory incorporate the equations of motion originally developed for flat-spacetime special relativity, rather than "starting over" within the new "curved-spacetime" context suggested by his 1911 gravity-shift paper.

Einstein (1920): " §22 ... it has often been contended by opponents of the theory of relativity that the special theory of relativity is overthrown by the general theory of relativity, ... Not in the least ... No fairer destiny could be allotted to any physical theory, than that it should of itself point out the way to the introduction of a more comprehensive theory, in which it lives on as a limiting case. "

From the late 1920s, GR1916 came under scrutiny for its failure to predict the redshift result of "expanding universe" cosmology, and during 1952-1960 apparent incompatibilities were identified between the SR and GPoR geometries, from the 1960s, it was increasingly recognised that the theory violated classical principles by predicting total gravitational collapse to a singularity, and from the 1970s onwards, it was recognised that Einstein's system did not "mesh" with quantum mechanics.

Einstein's "loose" approach to derivations

Unfamiliar territory and guesswork

Since Einstein was exploring new logical spaces during 1912-1915, the new "general theory" underwent some haphazard changes before its 1915 finalisation, with Einstein, aided by Michelle Besso, asserting one set of equations, then another, apparently at least partly based on Einstein's own evolving sense of aesthetics, intuition, and/or personal conviction. Like Feynman ("first, we guess" ), and Hawking ("I'd rather be right than rigorous" ), Einstein felt that the possibility-space was too large to reduce using just conventional logic, and required human imagination and intuition.

Einstein (1950): " This ... makes it difficult to use our empiri­cal knowledge, however comprehensive, in looking for the fundamental concepts and relations of physics, and it forces us to apply free speculation to a much greater extent than is presently assumed by most physicists. "

Pure mathematics starts with known rules and extrapolates: theoretical physics starts with the final phenomena, and works backwards to reverse-engineer what the most efficient rules might be that can generate them. Finding these rules is generally considered to some extent to to be a matter of trained intuition.

Einstein (1913): " I am well aware of the fact that postulates 2-4 better resemble a scientific declaration of faith than a firm foundation. I am also far from claiming that the two generalizations of Newton's theory to be described in the following are the only ones possible; but I dare say that at the present state of our knowledge they are the most natural ones. "

Einstein (1919): " the researcher … does not find his system of ideas in a methodical, inductive way; rather, he adapts to the facts by intuitive selection among the conceivable theories that are based upon axioms. "

Rindler (2012): " The definition of physical quantities, and the laws governing them, are in the nature of axioms; their formulation and adoption are matters of judgement rather than proof. "

, and as a result, even though Einstein uses a lot of mathematics, it is not always clear how much of the work is based on mathematical derivation and how much on personal conviction.

Chandrasekhar (1980): “ The element of controversy and doubt, that have continued to shroud the general theory of relativity to this day, derives precisely from this fact, namely that in the formulation of his theory Einstein incorporates aesthetic criteria; and every critic feels that he is entitled to his own differing aesthetic and philosophic criteria. ”

Einstein was also expert in constructing narratives, in which previous and competing theories had assumed impossible or wrong things, alternative possibilities were expertly skipped over, and his conclusions then seemed to be the only ones possible.

According to Lee Smolin, what made Einstein different to his contemporaries was that:

Smolin (1980): “ Einstein was a storyteller. ... Einstein succeeded when he was able to formulate a principle or hypothesis about nature, which he, or sometimes others, later expressed in mathematical terms; he failed when he attempted to use mathematics as a substitute for insight into nature. "

In cases below, Einstein knew how he thought the universe should work, but was not always able to devise legitimate derivations to support his positions.

Background to the general theory: Approach, development, publication, and subsequent changes

Basis in Special Relativity

Einstein's project to develop a general theory assumed that a larger gravitational theory would necessarily be an extension of the inertial physics described by his earlier 1905 theory of electrodynamics (which he started referring to as the "special" theory), and whose correctness was assumed to be a "given". The new geometrical properties would then be an extension of Minkowski spacetime (MTW Box 6.1: "General Relativity is built on Special Relativity").

Einstein (1920): " §22: ... the general theory of relativity enables us to derive theoretically the influence of a gravitational field on the course of natural processes, the laws of which are already known when a gravitational field is absent. "

Further,

Einstein (1915): " ... the postulate of general relativity cannot reveal to us anything new and different about the essence of the various processes in nature than what the special theory of relativity taught us already. The opinions I recently voiced here in this regard have been in error. Every physical theory that complies with the special theory of relativity can, by means of the absolute differential calculus, be integrated into the system of general relativity theory - without the latter providing any criteria about the admissibility of such physical theory. "

The approach of Einstein's GR project was to be incremental rather than recursive: GR was to be a complete physical superset of the laws and relationships of the special theory, and was not allowed to introduce new arguments that overrode or excluded its predecessor.

Special and general theories

The concept of a "general" theory of the subject echoes Wilhelm Ostwald's textbook on "General Chemistry", which Einstein cited in 1901. Einstein's earliest papers also referred to "general" laws of thermodynamics, and a "general" molecular theory of heat. The distinction between a "special" and a "general" or "generalized" theory of relativity appears in Einstein's notes in 1912, and in his published works from 1913 onwards.

Einstein's papers and letters on the subject of "general relativity" before the 1915/1916 formulation are considered experimental and provisional, and contain errors and radical reversals of opinion, as Einstein "felt his way" towards the shape of what would hopefully be a final theory. "The theory" of general relativity (with "the theory" used as a singular noun rather than referring to as a more indistinct body of theoretical work) is normally considered to commence properly with the content encapsulated in the 1916 paper.

Any inconsistencies specific to the period before 1916 ("Einstein’s first systematic exposition of the foundations of general relativity" ) are not generally considered to be inconsistencies of "the theory" itself, but as representing transitional stages in its development.

Successive approximation, and forced publication

In late 1915, Einstein's hand was forced by the impending publication of a paper by David Hilbert declaring a theory based on equations that Einstein had previously presented, rejected, and was now reconsidering. In order to assert his own priority, Einstein finalised and presented a revised reference version of his equations in November 1915, with submission and publication of a larger paper giving the overall theory following in March and May 1916. This necessary "rush to publication" at the very end of 1915 seems to have resulted in Einstein having to commit to a theory that was still not as finished as he would have liked, as (according to his letter to Hendrik Lorentz in January 1916) it was still derivationally bad:

Einstein (1916): " My series of gravitation papers are a chain of wrong tracks, which nevertheless did gradually lead closer to the objective. That is why now finally the basic formulas are good, but the derivations abominable; this deficiency must still be eliminated. "

Einstein continued to experiment with modifications to the supposedly completed 1916 general theory until his death in 1955. During this time, the identity and definitions of "the theory" continued to shift and change.

General inconsistency of principle, definitions, and structure

After having started with the GPoR and the PoE, by 1921 Einstein was also stressing the importance of gravitomagnetism, Mach's principle and the principle of the relativity of inertia to the theory. The relativity of inertia seemed a clear-cut concept, but it was not obvious whether the theory fully implemented it, while "Mach's principle" was a slightly vague concept that Einstein invoked, and explained the results of, but which seemed to be defined differently depending on context. While Einstein was able to explain the concepts that he was using, the relationships between these ideas and their attempted mathematical implementations was sometimes erratic and not always valid.

Sotiriou, Faraoni, and Liberati (2008): " The axiomatic formulation of general relativity (or gravitational theories in general) seems to resemble the myth of the Holy Grail. Serious attempts have been made to find it and everybody seems to be interested in it, but nobody actually knows where to look for it. "

Reviews of Einstein's system from colleagues after his death tended to emphasise the amazing feat that Einstein had pulled off, his great sense of intuition, and the beauty of the result, rather than discipline, mathematical rigor or scientific correctness, with Max Born praising it as "... a great work of art, to be enjoyed and admired at a distance". However, comments about "artistry" in some compliments may have been double-edged, as "art" is commonly considered to be (a) personal, (b) not science, (c) not a literal representation of reality, (d) not derivable and (e) not reproducable. Some comments seem reminiscent of the reviews of John Dee's tour of the Continent as an infant mathematical prodigy, comparing the sight of an English mathematician to that of a talking dog – what amazed was the idea that such a thing was even possible: one did not care so much about what it actually said.

While the simpler 1905 theory could be analysed as a geometry, and subjected to rigorous forensic mathematical analysis to assess its internal congruity, the same could not be done with the 1916 system – The lack of a clear definitional path, incremental logic or stable definitions made it impossible to certify the theory as logically consistent. Before one could subject the theory to a forensic analysis to establish whether the structure met consistency requirements, one would have to decide whose version of Einstein's theory to analyse, and if Einstein's, at what date, and with which definitions.

1950 abandonment of the two-stage approach

Einstein seems to have been unable to find a rigorous set of arguments that could show the necessity of SR appearing as a physical subset of a general theory, and in 1950 he announced that he no longer considered "non-gravitational" physics to be a legitimate concept. A two-stage architecture – with gravitational physics overlaid on a non-gravitational foundation - could no longer be defended. A general theory's components (wrote Einstein) needed to be compatible with the GPoR from the outset.

Einstein (1950): " ... all attempts to obtain a deeper knowledge of the foundations of physics seem doomed to me unless the basic concepts are in accordance with general relativity from the beginning. "

Einstein's new 1950 concept of a single-stage, self-contained general theory did not seem to yield a new version of GR before his death in 1955, and the multiple unanswered questions regarding the theory that he did produce have led to it being described as "Einstein's unfinished masterpiece".

Overview of Einstein-era and post-Einstein-era problems

During Einstein's lifetime, doubts were expressed regarding the relationship between GR and Hubble shifts (which had not featured in the original theory), the status of absolute event horizons around collapsed bodies, the apparent impossibility of gravitational waves under the system, and how one could reconcile the belief that massed particles always had associated curvature (principle of equivalence) with the belief that a region containing such particles, with relative velocities that were a significant fraction of the speed of light, could be described by a geometry that was intrinsically flat (special relativity).

Between 1952 and 1960, new arguments emerged for the incompatibility of special relativity with the relativities of acceleration and rotation. The 1960s and 1970s saw arguments suggesting incompatibility between Einstein's GR and quantum mechanics, and incompatibility with classical theory due to the generation of singularities.

These are all discussed in more detail below.

Conflict between SR and the principle of equivalence

Definition of the original principle of equivalence ("PoE")

The principle of equivalence of inertia and gravitation (Einstein, 1918: "Inertia and gravity are phenomena identical in nature") says that the fixed proportionality of inertial and gravitational mass that allows all objects to fall at the rate in a simple gravitational field (Eotvos principle), is due to a shared "essential identity": if we tie a rock to the end of a piece of string and whirl it around our head, we can explain the tension in the string as being due to the rock's inertial resistance to being deflected from travelling in a straight line (its inertial mass) ... but if we co-rotate with the rock, it appears to us to be stationary, suspended by the string in an apparent outward-pointing gravitational field that exists in the rotating frame – the same string tension can then be explained as a consequence of the rock's gravitational mass.

Since the PoE says that we cannot have inertial mass without gravitational mass (and its associated curvature) on principle, it is not obvious how a fully PoE-compliant set of geometry can also exactly support the physics of special relativity, which is inertial physics in the absence of gravitation, in apparent violation of the PoE. If we start with a gravitational theory supporting the PoE and try to "switch off" gravity to obtain SR, we also "switch off" inertia.

Criticism

A disagreement with Friedrich Kottler over whether the PoE should apply only to freefall, only to resistance to freefall, or to both, led Kottler in 1916 to accuse Einstein of abandoning the PoE. Einstein's response is more interesting than Kottler's (trivial) objection, as it offers an insight into his thought processes in 1916.

Einstein (1914): " ... in my opinion my theory rests exclusively upon this principle. ... Starting from the limiting case of the special theory of relativity, ... "

" ... this accusation cannot be raised against my theory of generally covariant equations, because ... The postulate for general covariance of the equations embraces the principle of equivalence as a special case. "

Einstein first establishes special relativity as a prior context for the PoE, presuming that the two must be compatible, models the motion of a "material point" (a point-mass) in flat spacetime, and applies covariance arguments to conclude that his system is provably PoE-compliant ... without considering that the PoE requires a massed particle to have associated curvature, meaning that the "moving massed particle in flat spacetime" scenario already violates the PoE.

The Einstein Equivalence Principle ("EEP")

Einstein avoided the problem of the apparent clash between the PoE and special relativity by redefining "the equivalence principle", to avoid mentioning the equivalence of inertia and gravity. The new version of "the principle of equivalence" was now the principle that a small freefalling laboratory was unaffected by a background environmental field gradient and obeyed the rules of "normal" inertial physics ... which Einstein then declared (without a supplied derivation) as being those of special relativity. This further variant on the equivalence principle (the "EEP") now said that GR must reduce to SR physics.

"MTW" (1973) " §38.6: Of all the principles at work in gravitation, none is more central than the equivalence principle. As enunciated in §16.2, it states: “In any and every local Lorentz frame, anywhere and anytime in the universe, all the (nongravitational) laws of physics must take on their familiar special-relativistic forms. ”

The weakness in this argument was that proving a flat background against which inertial physics could play out, was not the same as proving that the physics itself was flat. Although zooming in on the small laboratory did indeed eliminate the larger-scale background field gradient, it did not eliminate any fields that might be associated with the structure of the laboratory itself, or with the experiments taking place within it.

This switching of definitions in mid-theory allowed Einstein to use one definition to engage readers with the idea that "the principle of equivalence" was fundamental, and then use the second SR-centric definition to argue that it led inevitably to special relativity. This mismatch in definitions identified with the same name introduced logical conflicts and confusion:

Di Casola, Liberati, and Sonego (2013): " Non-Equivalence of Equivalence Principles: ... often the same name, "equivalence principle", is associated with statements having a quite different physical meaning. "

General relativity now had at least two different mutually-exclusive rules sharing a single name, "the principle of equivalence" ... one ("no inertia without gravitation") that made compliance with special relativity appear impossible, and another that made it compulsory. Together, Einstein's different versions of "the principle" allow proofs that inertial physics both must, and must not exactly correspond to SR's geometry. A theory that lets us prove two opposite conclusions (both "A" and "NOT-A") can be used with further chains of logic to simultaneously prove and disprove almost anything. Such structures can be classed as pathological.

Failure to implement Mach's principle / the relativity of inertia

Relative acceleration and rotation

Ernst Mach argued that one could "relativise" non-inertial motion by redescribing physics in an accelerated or rotating frame, treating fictitious forces and fields as real, and blaming these fields on the relative motions of background matter.

Mach (1919): " For me, only relative motions exist, ... When a body moves relatively to the fixed stars, centrifugal forces are produced. I have no objection to calling the first rotation "absolute" rotation, if it be remembered that nothing is meant by such a designation except relative rotation with respect to the fixed stars. "

For Mach, there was only one physical reality, supporting multiple descriptions ... but the different descriptions had to agree on a scenario's physical outcome.

Mach (1919): " The universe is not twice given, with an earth at rest and an earth in motion; but only once, with its relative motions, alone determinable. "

This idea is an old one, going back at least as far as George Berkeley.

Berkeley ("De Motu", 1721): " §64. ... for determining true motion and true rest, by which means ambiguity is eliminated and the mechanics of those philosophers who contemplate a wider system of things is furthered, it would suffice to take the relative space enclosed by the fixed stars, regarded as at rest, instead of absolute space. Indeed motion and rest defined by such a relative space can conveniently be applied in place of the absolutes, which cannot be discerned by any mark. ... "

Although the Berkeley/Mach idea was sometimes criticised as an empty hypothesis that could not be tested, Mach's Principle does have testable physical consequences, in that if our physical laws must describe the rotation or acceleration of distant masses as producing fields, then all rotating or forcibly accelerated masses must produce similar field-effects (albeit, normally on a smaller scale). An example of this is the dragging effect of the rotating Earth's gravitomagnetic field, as described by John Wheeler's democratic principle, and demonstrated experimentally by Gravity Probe B.

Relativity of inertia

The relativity of inertia ("Matter there co-determines inertia here") is based on the observation that in a relativistic framework, the force required to accelerate a grain of sand relative to the outside universe must be identical to the force required to accelerate the outside universe (a much larger mass) relative to the sand-grain, as these are merely two different descriptions of the same situation.

A system's resistance to acceleration must then be a function not only of the quantity of matter in the system, but also the characteristics of the background environment that it is being accelerated with respect to.

Einstein (1913): " The theory sketched here overcomes an epistemological defect that attaches ... to the original theory of relativity, ... It is obvious that one cannot ascribe an absolute meaning to the concept of acceleration of a material point, no more so than one can ascribe it to the concept of velocity. Acceleration can only be defined as relative acceleration of a point with respect to other bodies ... the occurrence of an inertial resistance [must] be linked to the relative acceleration of the body under consideration with respect to other bodies. ... this behavior of inertial resistance, which we may call relativity of inertia ... constitutes one of the strongest pillars of the theory sketched. "

In a field description, it was natural to describe this interaction locally as the coupling effect of a body's own field with the surrounding field: if we increased the local background field by increasing the local concentration of matter, the inertia of a body would then increase, slowing the rate at which it responded to applied forces, giving an alternative "Machian" argument for gravitational time dilation.

Einstein (1921): " What is to be expected along the lines of Mach's thought? ... 1. The inertia of a body must increase when ponderable masses are piled up in its neighbourhood ... "

Einstein also ran the same argument backwards, starting with conventional gravitational time dilation, and pointing out that this was equivalent to saying that the inertia of a body was affected by surrounding matter as per Mach.

Einstein (1913): [] " ... the equations entail the conception that the inertia of bodies is not a property of individual accelerated bodies by themselves, but rather an interaction, i.e., a resistance to the relative acceleration of bodies with respect to other bodies – a conception that has already been advanced by Mach and others on epistemological grounds. "

Einstein's 1921 lectures then go on to also present matching "Machian" descriptions of accelerative and rotational gravitomagnetic effects.

Successive failures

Even before 1916, Einstein assumed that his equations already implemented Mach's principle and the relativity of inertia. However, when he found himself unable to specify boundary conditions that allowed Mach's principle to operate, Einstein resolved the situation in 1917 by ''removing the boundary'', and making the universe spatially finite and (hyper)spherical ...

Einstein (1917): " In a consistent theory of relativity there can be no inertia relatively to "space," but only an inertia of masses relatively to one another. "

Einstein (1921): " ... this idea of Mach’s corresponds only to a finite universe, bounded in space, and not to a quasi-Euclidean, infinite universe. "

... while also introducing the Gravitational Constant, "Lambda" ("Λ"). This 1917 revision was not entirely successful, not least because Einstein's "balanced" universe was locally unstable.

Norton (1993): " In 1916, Einstein assumed that his generally covariant theory would satisfy the relativity of inertia, although no proof had been given. ... By 1917, Einstein had found that a simple reading of the relativity of inertia was incompatible with his theory. "

By 1921, Einstein was defending Machian logic by insisting that, qualitatively at least, the idea was contained at least broadly in his theory's equations.

Einstein (1921): " The idea that Mach expressed, that inertia depends upon the mutual action of bodies, is contained, to a first approximation, in the equations of the theory of relativity; it follows from these equations that inertia depends, at least in part, upon mutual actions between masses. Thereby Mach’s idea gains in probability, as it is an unsatisfactory assumption to make that inertia depends in part upon mutual actions, and in part upon an independent property of space. "

By 1924 he had given up on the idea altogether.

Einstein (1924): " Mach ... sought to ... attribute inertia to an unmediated interaction between the considered mass and all the rest of the masses of the universe. This conception is logically possible; however, as action-at-a-distance, it no longer comes seriously under consideration for us today. "

Hence a principle that was supposed to have been built into Einstein's 1916 field equations (but wasn't), which motivated Einstein's 1917 hyperspherical cosmology, that was supposed to now support it (but didn't), and which was said in 1921 to be only "strongly supported" by GR, was now in 1924 apparently abandoned.

If Einstein's system really couldn't support the relativity of inertia in the sense of the "grain of sand" argument above, then judged by his 1917 viewpoint, it had still not achieved the status of being "a consistent theory of relativity".

DeSitter's "empty universe" solution

Einstein's claim that his field equations were Machian because they now described the properties of space as being entirely dictated by the distribution of matter (within a 1917 hyperspherical universe),

Einstein (1920): " According to the general theory of relativity, the geometrical properties of space are not independent, but they are determined by matter. "

, had also been undermined in 1917 when de Sitter produced a solution of the Einstein field equations that seemed to be valid despite describing a universe completely devoid of matter.

SR considered as a limiting case of GR

SR as an idealised solution

Special relativity was built on the assumption, borrowed from Lorentz aether theory, that the speed of light was globally constant with respect to all possible inertial observers, real or hypothetical. This distinguished it from Heinrich Hertz' 1890 theory of relativity, in which light was fully dragged by all moving matter, and lightspeed was only locally constant, and only for real, physical observer-masses.

Einstein (1910): " The simplest [aether] hypothesis is to assume that moving bodies carry along completely the ether they contain. It is on the basis of this hypothesis that Hertz developed an electrodynamics of moving bodies that is free of contradictions. "

Lightspeed constancy was "only local" with Hertz and "global" with Lorentz. Einstein's 1911 paper on light-bending acknowledged that the gravitational deflection of light counted as evidence that lightspeeds were variable, and that "global c" was not a law of nature after all:

Einstein (1911): " The principle of the constancy of the velocity of light does not hold in this theory in the formulation in which it is normally used as the basis of the ordinary theory of relativity. "

Einstein (1920): " ... according to the general theory of relativity, the law of the [global] constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity ... cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position. Now we might think that as a consequence of this, the special theory of relativity and with it the whole theory of relativity would be laid in the dust. But in reality this is not the case. We can only conclude that the special theory of relativity cannot claim an unlimited domain of validity; its results hold only so long as we are able to disregard the influences of gravitational fields on the phenomena (e.g. of light). "

Since the existence of a gravitational field could be defined by the presence of light-beam curvature, special relativity was only valid in regions containing nothing that could cause a deflection of light-beams and/or a change in the velocity of light.

Rather than switch to a Hertz-style system with different equations, Einstein retained special relativity and argued that special relativity's "global c"-based structure was still valid ... but only applied over regions of vanishingly small size.

Einstein (1914): " It is the essence of the theory we derived here that the original theory of relativity holds in the infinitesimally small. "

Einstein (1920): " ... the law of the constancy of the speed of light no longer holds, according to the general theory of relativity, in spaces that have gravitational fields. As a simple geometric consideration shows, the curvature of light rays occurs only in spaces where the speed of light is spatially variable. From this it follows that the entire conceptual system of the theory of special relativity can claim rigorous validity only for those space-time domains where gravitational fields (under appropriately chosen coordinate systems) are absent. The theory of special relativity, therefore, applies only to a limiting case that is nowhere precisely realized in the real world.

Einstein (1921): " There is, therefore, no choice of co-ordinates for which the metrical relations of the special theory of relativity hold in a finite region. "

The special theory was then still nominally correct, but only in regions too small for us to perform physics with moving bodies, or carry out testing.

SR as a null solution?

Einstein's geometrical argument for any gravitational theory reducing to special relativity was that, just as the curved lines of classical geometry became indistinguishable from segments of straight line as we zoomed in on them, so a classical curved spacetime theory of physics (general relativity) should reduce to a flat-spacetime theory (special relativity) over vanishingly small regions. Even if we didn't yet know anything else about a geometrical theory of gravity, we knew that it must reduce to SR.

Einstein (1914): " ... if, without knowing the generally covariant equations of the gravitational field, we specialize the reference system and set up the field equations of gravitation for the special reference system only, then the sole objection that can be raised against the theory is that the equations we have set up might, perhaps, be void of any physical content. But no one is likely to think in earnest that this objection is justified in the present case. ... "

The flaw in this argument was, as Einstein acknowledged, the logical possibility that a "flat limit" of a physical theory might not be a different physical theory, but a limit at which meaningful physics could no longer be said to exist. If matter has curvature, the price of achieving flatness might be the absence of matter. SR might be a theory of physics that only applied when the number of massed particles present was zero, and when no meaningful matter-physics was taking place, making it a null theory.

Failure of SR in the presence of realistic matter

Einstein later revisited this objection, and ceded that it might be reasonable:

Einstein (1919): " It is by no means settled a priori that a limiting transition of this kind has any possible meaning. For if gravitational fields do play an essential part in the structure of the particles of matter, the transition to the limiting case of constant g_uv would, for them, lose its justification, for indeed, with constant g_uv there could not be any particles of matter. So if we wish to contemplate the possibility that gravitation may take part in the structure of the fields which constitute the corpuscles, we cannot regard equation (1) as confirmed. "

Einstein's argument for the possible invalidity of SR also works if we ignore questions of structure and merely accept the weaker condition that matter is always associated with curvature (as it must be according to the PoE): SR can then only be an exact solution in the absence of matter (Taylor and Wheeler, Box 3.1: "The Principle of Relativity Rests on Emptiness!" ). Special relativity, derived for flat empty spacetime, might only be valid for flat empty spacetime.

The "Hole argument" (pre-1916)

Einstein's "hole argument" emerged during the development of general relativity, when Einstein found that the extrapolation of SR coordinate systems into a void surrounded by matter (a "hole") produced inconsistent results. Einstein's eventual solution in 1915 was to declare that these coordinates lacked physical meaning if a region lacked any identifying physical markers for them to be attached to. Physical law had to apply to the intersections of worldlines of real objects. It was not obliged to also apply to fictitious relationships of objects that did not exist.

Einstein (1916): " Taking the place of the hole argument is the following consideration. Nothing is real physically except for the entirety of the spatio-temporal point coincidences. If, for ex., physical events were to be constructed out of the motions of mass-points alone, then the meeting of the points, i.e., the intersection points of their world lines, would be the only real, that is, principally observable, things. ... It is thus most natural to demand that the laws not determine more than the spatio-temporal coincidences as a whole. "

Einstein reprised this idea that "physics" was only required to describe the physical world in his 1923 Nobel address, where he reintroduced it as the reality postulate.

Einstein (1923): " In mechanics, when one speaks of motion per se, one means the motion relative to the system of coordinates. This interpretation does not comply with the reality postulate, though, when the coordinate system is regarded as merely a thing of thought. "

The difficulty for Einstein of the hole argument is that if the properties of coordinate systems in empty space are to be dismissed as irrelevant to physics theory on the grounds of "not being physics", then we have to explain why special relativity and Minkowski spacetime ... which are also built on the properties of coordinate systems in empty space ... aren't to be dismissed for the same reason.

Embracing curved-spacetime arguments

If we use the existence of associated curvature to distinguish between "physical" and "unphysical" classes of observer, then Einstein's "flat" inertial physics only makes predictions for the "unphysical" class, and leaves "physical" observers unmodelled. Einstein tried to correct this deficiency in the 1930s by trying to rederive the equations of motion in the new curved-spacetime GR context, from scratch, without presupposing the validity of SR, by approximating moving massed particles as moving singularities of the field.

Einstein, Infeld, Hoffmann (1938): " Previous attacks on this problem have been based on gravitational equations in which some specific energy-momentum tensor for matter has been assumed. Such energy-momentum tensors, however, must be regarded as purely temporary and more or less phenomenological devices for representing the structure of matter, and their entry into the equations makes it impossible to determine how far the results obtained are independent of the particular assumption made concerning the constitution of matter. "

"Actually, the only equations of gravitation which follow without ambiguity from the fundamental assumptions of the general theory of relativity are the equations for empty space ... "

Einstein was ceding that, prior to 1938, and more than two decades after the publication of the 1916 paper, his general theory still hadn't managed an unambiguous derivation of the equations of motion within the context of curved spacetime. The 1938 attempt suffered from similar shortcomings to the earlier attempts: a mass-singularity represents a point of nominally-infinite field strength: before we reach this point we will encounter a critical surface at which gravity is not yet infinite, but still strong enough to cause a curvature horizon (which conveniently prevents us from having to confront whatever the source of the field really is ).

Moving horizons are expected to fully drag light: Conventionally, outward-aimed signals generated at r=2M are considered frozen into the horizon surface; these signals are then fixed with respect to the hole's position, and "move" if the hole moves. For a rotating hole,

Thorne (1994): " At the horizon, space is locked tightly onto the horizon: It rotates at precisely the same rate as the horizon spins. "

... so if we are in the rotating hole's equatorial plane, patches of horizon profile surface that are moving directly towards or away from us at ± v, will be dragging light directly towards or away from us at ± v. Horizon-bounded masses should then show complete velocity-dependent dragging.

Einstein's 1938 exercise, allowing arbitrarily-strong gravity at arbitrarily-small distances from idealised massed particles, should then assign curvature horizons to the particles. With curvature horizons acting as the default interaction surfaces of massed particles, we will expect these moving particles to be associated with the complete dragging of light ("Hertzian" relativity), rather than the "no-dragging" Lorentz equations.

Einstein's "moving singularities" exercises should then describe "maximally strong-gravity" physics, and generate non-SR equations.

Universality of Doppler equations

Gravitomagnetic frame-dragging effects cause a receding mass to pull light away from us, reducing the momentum of the signals that it aims in our direction, so that we perceive a redshift, while the corresponding effects due to an approaching body drag light towards us, increasing the signal's detected momentum and creating a blueshift (momentum exchange).

• If this gravitomagnetic motion shift is "dual" with the conventional Doppler shift, then standard Doppler shifts must take into account velocity-dependent distortions of the light-metric, and must be different to the "flat" SR predictions.
• If the gravitomagnetic shift acts in addition to the conventional Doppler shift, then the motion-shift relationship must be different to the SR predictions for strong-gravity bodies.

The conventional argument is that SR has the correct equations for weak-gravity physics where gravitomagnetism can safely be ignored, and where strong or extremal gravity is involved and cannot be ignored, we then switch to full-blown general relativity. This gives us a transitional, two-tier ("stratified") physics for "non-gravitational" and "gravitational" problems.

Unfortunately, since the distinction between "gravitomagnetic" and "non-gravitomagnetic" physics affects the form of the Doppler relationships, and relativity and metric principles only allow a single Doppler relationship, which must apply everywhere and to everything, any gravitomagnetic modification of the equations of motion that applies to a strong-gravity scenario then needs to apply identically to all other masses in the universe. A gravitomagnetic deviation from SR must either be identically maximal for all moving masses (invalidating SR), or identically non-existent for all moving bodies (SR solution, invalidating gravitomagnetism).

We cannot transition between two different sets of equations of motion for different sorts of object. If gravitomagnetism applies anywhere, the resulting non-SR equations must apply everywhere, and if SR is exact for any type of real object, it must apply exactly for every type of real object, ruling out gravitomagnetic effects.

A relativistic model is not allowed to "compartmentalise" or "stratify" physics as a way of supporting two different behaviours in the same theory, it cannot support both gravitomagnetism and the SR relationships – Einstein's impossible attempt to support both is an example of cakeism.

Required universality of the relativity principle

The relativity principle requires that our laws of physics be universal – they must predict precisely the same outcome for a "moving" atom or grain of sand as they do for the atom or grain being "stationary" and the entire remaining outside universe moving, instead. This principle of universality means that a relativistic theory's equations, if correct, must apply everywhere without reservation or qualification, to all objects and all physical systems, from the subatomic to the cosmological, subsuming the laws of quantum mechanics, fluid dynamics, gravitational theory, cosmology, and all other physics, including any additional physical laws, effects and fields that may not yet have been discovered.

Einstein (1913): " ... the customary theory of relativity provides only an approximation to reality; it should apply only in the limit case where differences in the gravitational potential in the space-time region under consideration are not too great.

Relativistic laws must, by their nature, and by the requirement that they must apply identically to everything, have unlimited validity. They are either exact everywhere, or they are wrong.

Perfection of Minkowski spacetime

As Einstein's work towards a general theory progressed, he felt more comfortable criticising the absolute nature of Minkowski spacetime.

Einstein (1914): " These privileged reference systems are postulated as those with respect to which the principle of the constancy of the velocity of light in a vacuum is to be valid. There can be no doubt that this principle is of far-reaching significance; and yet, I cannot believe in its exact validity. It seems to me unbelievable that the course of any process (e.g., that of the propagation of light in a vacuum) could be conceived of as independent of all other events in the world. "

Einstein restated the argument in 1921: Minkowski spacetime broke the interaction principle, that if one participant in an interaction comes away changed by the encounter, then so should the other.

Einstein (1921): " ... it is contrary to the mode of thinking in science to conceive of a thing (the space-time continuum) which acts itself but cannot be acted upon. "

In Einstein's 1921 narrative, the recognition of the "unscientific" nature of Minkowski spacetime sets the scene for the introduction of the new, interactive, dynamic, Machian spacetime of general relativity.

Einstein (1920): " Mach’s ether not only conditions the behaviour of inert masses, but is also conditioned in its state by them. Mach’s idea finds its full development in the ether of the general theory of relativity. "

Wheeler (1990): " Spacetime grips mass, telling it how to move ... mass grips spacetime, telling it how to curve. "

However, when EInstein then includes the SR laws of physics and Minkowski spacetime in GR as an assumed physical limiting case, the problematic behaviour "contrary to the mode of thinking in science" has not been replaced in GR, but formalised and made part of the definitions.

The awkwardness in adopting SR equations for macroscopic regions in a generally-relativistic model is that the Minkowski geometry is specific to non-GR assumptions. Special relativity is, in a sense, an already-perfect system, a perfect solution to a specific question ("how can we construct a theory of relativity that works in flat spacetime"), that does not accept further extensions. As Feynman said in the context of comparing Newtonian theory with GR,

Feynman (1964): " ... these are so simple and so perfect, they produce definite results: in order to get something to produce a little different result [it] has to be completely different, you can't make imperfections on a perfect thing, you have to have another perfect thing. "

Einstein's "reduction" argument does not apply to regions containing inverse-square law field sources

Einstein's "geometrical reduction" argument does not work for regions that include sources of gravitational fields. Taking the example of a one-centimetre cube of space containing two fundamental massed particles with relative motion, exchanging signals, surrounded by cubic light-years of otherwise empty space, it is reasonable to argue that the overall multi-lightyear region is "effectively flat". As we progressively zoom in on the larger region to obtain the smaller one, the average curvature of the region under study does not decrease, as in Einstein's argument ... it increases. Zooming in on just one of the particles, approximated (if we know nothing about atomic physics) as a pointlike mass: the curvature continues to increase without apparent limit as we zoom further on the centre-of-mass, until we either find ourselves looking at the real structure of the particle (which might be topologically complex), or our view is blocked by a censoring horizon.

Einstein's argument works if we zoom in on an empty region between particles, but not if the target zoom region contains particles. It is difficult (in a GR context) to say that we know that gravitational theory must yield to SR for simple cases of pairs of interacting particles, if SR can only apply when there are no particles present in our field of view.

Conflict between SR and gravitomagnetism (and the GPoR)

GR's requirement for gravitomagnetism

The general principle of relativity ("GPoR") extends the principle of the relativity of inertial motion to cover all forms of motion, including acceleration and rotation. Machian arguments allow accelerated and rotating observers to blame the apparent gravitational effects that they experience (gee-forces) not on absolute motion, but on real gravitational fields associated with the relative motions of background environmental matter. If the purely relative motion of stars' masses causes distortions in spacetime, then similar effects must arise when any other masses rotate or are forcibly accelerated. Any non-inertial motion of matter must physically deform the metric in a similar way, causing forces on, or deflections of, nearby bodies and light. ("accelerative induction").

Einstein (1921): " A body must experience an accelerating force when neighbouring masses are accelerated, and, in fact, the force must be in the same direction as that acceleration. ... A rotating hollow body must generate inside of itself a “Coriolis field,” which deflects moving bodies in the sense of the rotation, and a radial centrifugal field as well. ... ”

Einstein (1923): Yet another factor speaking in favor of the Machian stipulation in general relativity is that acceleration induction does indeed exist according to the gravitational field equations; even though the effect is so small that direct detection by mechanical experiments is out of the question. "

When motion is associated with forces, the back-reaction of the metric to these forces causes physical distortions of the metric, which in turn cause nearby objects and light to be "dragged". Accelerative and rotational gravitomagnetic effects are necessary to any general theory of relativity.

SR's disproof of gravitomagnetism

In the standard application of SR to acceleration problems (e.g. MTW chapter 6), the curved path of a forcibly-accelerated mass can be broken down into an arbitrarily-large number of arbitrarily-small velocity-differentials, and the total intrinsic distortion associated with the body's acceleration is then the aggregate of the smaller distortions associated with these velocity-differentials. Since the SR equations associate zero curvature with the simple relative motion of masses, the total acceleration curvature is then zero, too. If the equations of motion (and Doppler relationships) are those of SR, then regardless of how a massed body moves, its motion cannot warp the metric.

The "gravitational" effects experienced by the accelerated observer are then described by SR as being due to their curved path through flat spacetime, and no corresponding distortion exists for an inertial bystander – an asymmetrical situation that makes Einstein's 1921 description of "acceleration induction" (gravitomagnetic drag) impossible.

A velocity-dependent GM effect is also present in rotational gravitomagnetism, which presents an obvious velocity component – the receding edge of a rotating star pulls more strongly than the approaching edge, and the passing side drags in its direction of motion. This violates the SR condition that the motion of bodies should have no effect on the shape of the light-metric.

Conflict between SR and the relativity of acceleration

Christian Møller's 1952 book on general relativity declares that Einstein was (slightly) wrong about the Principle of Equivalence. While Møller accepts Einstein's assertion that the apparent ("fictitious") fields experienced by the accelerated observer appear to them to be in all ways identical to "real" fields, and must even obey the same equations, Møller says that the two classes of field are not wholly interchangeable, and reintroduces the C19^th distinction between "real" and "fictitious" gravitational fields, referring to the two types as "permanent" and "non-permanent" fields. For Møller, A "permanent" field is a real, conventional field that represents an intrinsic curvature of the metric and exists for everyone, while a non-permanent ("fictitious") field can be eliminated by a convenient alternative choice of coordinate system.

At this point, Einstein's general theory has fractured into two major systems.

• Einstein, using the "gravitational" side of the theory and Mach's principle proves that an accelerated mass must physically warp the metric, creating a dragging effect on nearby bystanders. This eliminates the original sharp distinction between inertial and non-inertial physics (Einstein (1954): " It is the essential achievement of the general theory of relativity that it freed physics from the necessity of introducing the 'inertial system' (or inertial systems) "

• Møller, on the other hand, using the "SR" side of the theory proves that an accelerated mass must not physically warp the metric, contradicting Einstein's 1921 phenomenology, and reintroducing the physical distinction between inertial and non-inertial physics. If Einstein is correct that his dragging effects are implicit in his 1921 gravitational equations, then these must then presumably be wrong, too. While Møller's presentation of GR appears to support general relativity, it breaks the GPoR in that it has the relative acceleration of background matter being seen to be associated with distortion-fields, but the relative acceleration of other matter not being seen to be associated with distortion-fields.

If the SR equations are valid, a general theory of relativity cannot exist. Conversely, if we want a general theory and "real" accelerative curvature, we must associate curvature with the relative velocity of matter, after which our equations are not those of SR, invalidating both of Einstein's classical theories.

Conflict between SR and the relativity of rotation

In 1959/1960, two experimental teams were in competition to be the first to obtain a credible confirmation of the existence of gravitational shifts. The Harvard Group in the US measured the actual gravitational shift between some floors of a university building, while the Harwell Group in the UK invoked the principle of equivalence and measured the "effective" gravitational shift that existed across the rotating frame of a centrifuge.

With a potential Nobel Prize at stake, the principle of equivalence was scrutinised again more carefully, with the result that it now seemed that the "intrinsic curvature" explanation for the shift in the co-rotating frame could not be reconciled with the "flat" description given by special relativity. Alfred Schild's 1960 paper documents the community's reaction and an apparently unavoidable conclusion:

Schild (1960): " ... Special relativity and the equivalence principle do not form a consistent theoretical system. "

Schild argues that, if we are forced to make a choice between SR and GR, SR has far more supporting experimental evidence. To protect SR, the PoE must be suspended in rotating-body problems.

Schild (1960): " The question arises whether, by virtue of the equivalence principle, [redshift] effects in accelerated systems are to be regarded as verifications of general relativity. There seems to be some confusion on this point and even some lack of unanimity among theoretical physicists. ... within the framework of the theory of relativity the answer is simple and definite. It is 'no!'. "

We can then salvage general relativity by agreeing that it is broadly correct, but that Einstein's statements about the general principle are not to be taken literally.

Schild (1960): " Within the framework of general relativity theory, the statement that acceleration and gravitation are equivalent and physically indistinguishable is only true to a certain approximation. "

Schild's position agrees with Møller's, that the theory presented by Einstein doesn't work if we treat Einstein's statements of principle literally, but that the situation can be rescued by treating SR as exact and the GPoR as approximate.

The "patched" general theory is then no longer quite the same as the version presented by Einstein, which was supposed to to be a purist "principle-based" theory:

Einstein (1919): " The advantages of the ... principle theory are logical perfection and security of the foundations. ... The theory of relativity belongs to the latter class. "

Until 1960, Einstein's theory was supposed to be an exact implementation of the general principle of relativity, but was logically inconsistent. After 1960 (and the "Schild override"), it became internally consistent, but was no longer technically a general theory of relativity.

Incompatibility with gravitational waves

From the perspective of the C21^st it may appear obvious that, if all mass-systems have gravitational fields, any "significant" change-of-state in a system that results in a redistribution of massenergy (such as one atom emitting a photon and another receiving it) should produce a change in the shape of the system's external field. If the speed of gravitational signals is finite, this geometrical change will need to propagate outward as a gravitational wave, carrying information and energy.

To Einstein, this scenario was obviously wrong. Einstein produced two main objections to the existence of gravitational waves: the first was that thermal systems would need to be continually emitting g-waves, and losing energy (compared to the predictions of the "lossless" SR equations). If g-waves were fundamental, then the SR relationships would need to be modified to accommodate the energy-loss per energy transaction.

Einstein (1915): " 4. The Emission of Gravitational Waves by Mechanical Systems ... It has already been emphasized in a previous paper that the end result of this investigation – which would require a loss of energy of bodies due to the thermal agitation – must raise doubts as to the general validity of the theory. It seems that a more complete quantum theory would also have to bring about a modification of the theory of gravitation. "

Einstein's second objection was based on his conviction that physical law should be symmetrical with respect to time, and that gravitational physics should work identically both "forwards" and "backwards", as it did with special relativity. Gravitational waves appeared to break macroscopic time-symmetry, in that a thermal system would radiate g-waves and lose energy in forward time, but absorb them and absorb energy in reversed time.

Einstein (1938): " Since such equations as those of the gravitational and of the electromagnetic field are actually invariant under a reversal of the sign of time ... Our method, in which the time direction is not distinguished, ... cannot lead to the conclusion that in the circular motion of two point masses energy is radiated to infinity in the form of waves. "

By invoking a critical characteristic of the underlying SR relationships – that they were time-symmetrical – Einstein could use his general theory to prove that a binary star system could not possibly emit gravitational waves, contra subsequent observations of the Hulse-Taylor binary pulsar (which is constantly losing orbital energy), and reported LIGO events.

Einstein was extending his SR-centric 1916 framework with ever-more complex mathematics to "prove" wrong results.

Thermal redshifts

With the "balanced" 1905 Doppler equations, a signal sent through a region containing moving transponders always emerges with the same energy it started with (apart from the effect of recoil redshifts). This perfect cancellation of Doppler effects en route allows SR to describe the behaviour of lightbeams with a fixed lightbeam geometry independent of the motions of any matter in a region – by contrast, "Newtonian" Doppler relationships do not cancel ( (c-v)/c × (c+v)/c = 1 - v²/c² ), and even with "recoilless" Mössbauer systems will predict a residual thermal redshift. In Einstein's universe, we can reject the Newtonian equations because they describe energy disappearing from a thermal system without any indication of where it is going to, and can reject gravitational waves because they represent energy leaving a thermal system with no indication of where it is coming from. If both "anomalous" behaviours are causally connected, then both difficulties disappear, leaving only Einstein's objection that physics equations need to be time-symmetrical.

Multiple aspects of Einstein's worldview can then be tested by confirming the non-existence of thermal redshifts in Mössbauer hardware.

Unfortunately, during the setup phase of the famous Pound-Rebka-Snider gravity-shift tests at Harvard the researchers found " the unanticipated effect of temperature as causing relativistic time dilation " ... a residual (time-asymmetrical) thermal redshift that would be expected if micro-gravitational waves were being emitted, and/or if some other, "lossy" set of Doppler equations was in operation rather than Einstein's. This thermal redshift that had not been supposed to exist according to special relativity, and forced the experimenters to improvise a system of crygenic cooling before they could carry out their experiment.

Incompatibility with quantum mechanics

Gravitomagnetism from QM

Ideally, classical and quantum theory should be dual (correspondence principle): Classical field theory should quantise to give quantum mechanics, and QM's statistical mechanics should conspire to build towards an arbitrarily-close approximation of our classical field theory.

Khavtain Namsrai has used quantum mechanics to stochastically reconstruct the expected shape of classical spacetime around a moving particle, converting QM's probability-fields for mass and momentum into classical mass- and momentum-distribution fields. Namsrai's "tilted hat" sketch shows the mass-field as a tilted gravitational "well", with the momentum field component (gravitomagnetic field component) appearing as the shape's proximity-dependent tilt.

This exercise suggests that velocity-dependent gravitomagnetic effects may be an essential feature of any classical theory that wants to be "dual" with QM. Such a gravitomagnetic/momentum-field component would invalidate both SR, and the use of exact SR equations within GR.

Failure to mesh with QM

Section 39 of Misner, Thorne and Wheeler's textbook Gravitation supplies three criteria that any potential competitor to Einstein's general theory must meet, the second being that it must "mesh" with a range of other structures, including QM.

MTW (1973): " §39.1. ... Completeness. To be complete a theory of gravity must be capable of analyzing from 'first principles' the outcome of every experiment of interest. It must therefore mesh with and incorporate a consistent set of laws for electromagnetism, quantum mechanics, and all other physics. "

Shortly after the book's publication, the growing acceptance of Stephen Hawking's 1974 prediction of black hole radiation meant that GR1916 was known to fail criterion #2. Under the MTW rules being applied to competing gravitational theories, Einstein's system would have to be rejected.

QM requirement for relative horizons

Hawking has suggested (2014 ) that QM-compatible radiation effects through a curvature horizon can be achieved by making classical horizons "relative" (observer-dependent) rather than absolute. Relative observational horizons would also have been a feature of the Revd. John Michell's 1784 "dark stars", which, although presenting the same horizon radius to a distant observer ( r=2M G/c² ) as a barrier to direct observation, would have allowed an observer to see deeper into them the closer they were. The indirect radiation of massenergy and information outward through r=2M via interactions with intermediate matter can then be modelled statistically as Hawking radiation. The similarity of the exterior physics of Newtonian dark stars to the predictions of modern QM can be understood visually by comparing the "Newtonian" figure 6.8 (p.252) and the "QM" figure 12.3 (p.443) in Kip Thorne's 1994 book on the history of black holes.

However, GR1916's adoption of the SR shift relationships, expressed via the Schwarzschild solution, makes relative horizons impossible and absolute horizons compulsory. Gravitational theories incorporating the SR relationships are not compatible with quantum mechanics, and since the incompatibility exists at the level of the Doppler equations, the two blocks of theory cannot coexist as part of a larger system (such as a theory of quantum gravity) without contradictions. Any reconciliation of classical and quantum theory requires either the elimination of Hawking radiation, or the elimination of special relativity.

Incompatibility with classical field theory

Formation of event horizons

Einstein's equations lead to the formation of absolute horizons. Since an event taking place within an absolute horizon has no possibility of interacting with the world outside, either directly or indirectly, absolute horizons are event horizons. Einstein's belief in the concept of mutuality prevented him from accepting the existence of a surface with one-way causal characteristics. If the equations led to results that he considered wrong, then it was not the fault of the equations: rather, they had been misapplied to situations that could not physically happen in real life.

Einstein (1939): " Of course, these paradoxical results are not represented by anything in physical nature. ... The "Schwarzschild singularity" does not appear for the reason that matter cannot be concentrated arbitrarily. "

However, since the amount of matter enclosed by a spherical volume in a homogenous universe increases with the cube of the radius, but the surface area only increases with the square of the radius, any positive density of matter can in theory produce a horizon, given a large enough volume.

Formation of singularities

The Schwarzschild solution and event horizons of Einstein's 1916 system forbid the existence of any outward-applied forces that could oppose a complete unresisted freefall collapse to a point-singularity. Classical field theory requires that modelled field-properties vary across space continuously. Since a line straddling a point-singularity encounters an infinitely sharp discontinuity, Einstein's system generates a violation of classical field principles.

Møller (1978): " It is now generally believed that Einstein's beautiful theory of gravitation under special circumstances leads to inconsistent results. In fact, according to this theory a well-defined physical system may after a finite time pass over into an unphysical state, where the metric is singular and consequently the notions of space and time lose their physical meaning. ... For a long time many physicists (including myself) did not believe that Einstein's otherwise so successful theory had such disastrous consequences; ..."

"MTW" (1973): " §44.1. Gravitational Collapse as the Greatest Crisis in Physics of All Time ... today gravitational collapse confronts physics with its greatest crisis ever ... "

The event horizons that leads to total unresisted collapse are a consequence of GR1916's adoption of the relationships of special relativity.

Incompatibility with momentum fields

A velocity-dependent gravitomagnetic field component, describing how moving matter exchanges momentum with nearby masses and light, can be described as a "momentum field", with the g-field acting as an intermediary for momentum exchange ("collision by proxy"). Considered as a "dragging field", this entity will exist in a Hertzian relativity, but is incompatible with Lorentzian relativity.

A static field can be considered as a "spatial extension" of a charge:

Einstein (1952): " Physical objects are not in space, but these objects are spatially extended. In this way the concept 'empty space' loses its meaning. "

If a body's gravitational field can be considered the spatial extension of its mass ("mass-field"), then if moving mass-field carries momentum, by default we have an associated momentum-field. The resulting associated momentum field can be used to model momentum exchange in the case of the slingshot effect, outside the time domain. It would be strange for a moving mass to have a classical field distribution in space but its associated momentum to remain localised and undistributed.

We can find GR texts that do discuss the equivalent of a momentum-field associated with gravitomagnetic side-effects (gravitomagnetic behaviour associated with mass-energy currents ... i.e. moving matter).

Wald (1952): p.78 " ... mass-energy current density 4-vector. Thus, linearised gravity predicts that the motion of masses produces magnetic gravitational effects very similar to those of electromagnetism. "

The "fieldification" of matter generates gravitomagnetism: when a moving massed particle's properties are "blurred" or "smeared out" into the surrounding region of space, its "electric" charge becomes an electric field with the motion of the field becoming a magnetic field component, and its "gravitational" charge ("mass") to becomes a gravitational field, with the motion of the field becoming a gravitomagnetic field component, or momentum field. This simple (and apparently unavoidable) classical field generalisation is incompatible with special relativity.

Incompatibility with modern cosmology

Failure to predict Hubble shifts, expansion, energy-loss and the cosmological arrow of time

The full Schwarzschild solution maintains compatibility with special relativity by having a gravitational shift relationship that exactly inverts when we reverse the direction of the signal (from Wald ):

${\frac {\omega _{1}}{\omega _{2}}}$ = ${\frac {(1-2M/r_{2})^{1/2}}{(1-2M/r_{1})^{1/2}}}$

This has the result of (a) ensuring that a signal passing through a gravity-well returns to its original height with precisely the same energy it started with (route-independence of gravitational shifts), and (b) ensuring that gravitational shifts are time-reversible physics. The gravitational redshift on an outward-aimed signal in forward time translates under time-reversal to a gravitational blueshift on an infalling signal in reversed time). Point (a) means that there are no cumulative gravitational redshifts for a signal traveling cosmological distances through populated space. The natural cosmology for Einstein's 1916 system is therefore a spatially and temporally infinite universe with quasi-Euclidean geometry and time-symmetry, which looks the same in forward and reversed time, is energetically balanced, and and has nothing corresponding to a Hubble shift.

Lambda, the Cosmological Constant.

Einstein's followup cosmological paper of 1917 tried to correct two problems with the 1916 paper in apparently contradictory ways:

The "balanced" nature of the Schwarzschild solution suggested that large-scale geometry of the universe should be effectively flat, but curvature-based arguments suggested that positive curvature needed to be cumulative. A region populated by positive curvatures cannot have an overall zero curvature (cannot be quasi-Euclidean) unless it also contains compensating negative curvature(s), and this may have encouraged Einstein to experiment with the idea of a compensating negatively-gravitating effect to explain how his non-cumulative shift predictions (required for compatibility with SR) could be correct. The 1917 paper also addressed the issue that the universe seemed to need to be hyperspherical in order to support Mach's Principle, and the negatively-gravitating effect was then invoked in the shape of the Cosmological Constant, Lambda ("Λ"), whose presence was used to to keep the size of the universe constant over time (again, compensating for the long-range cumulative effects of gravitation).

Once we have a "spherical" universe, the distance-dependent tilt in spacetime coordinates with location produces an apparent distance-dependent spatial contraction in projected coordinates, and allows us to predict a distance-dependent redshift. This distance-dependent redshift, if interpreted as a velocity effect, tells us that the universe is expanding. Alternatively, if interpreted instead as a gravitational shift, it tells us that the older universe was denser than ours, and that ... once again, the universe is expanding. An expanding universe violated Einstein's conviction that our universe's physics should be time-symmetrical, and he dismissed Georges Lemaître's 1927 suggestion of an asymmetrical expanding-universe cosmology, to Lemaître's face, as "abominable". Invoking the Cosmological Constant allowed Einstein to explain why his spatially-"spherical" universe had a constant radius with time ("cylindrical" spacetime with regards to x₄ ), why there was no distance-dependent redshift, and why the universe was not violating energy-conservation or time-symmetry.

With the publication of Hubble's result in 1929, it started to become apparent that distance-dependent effects were real, and that time-symmetry was violated leading to a cosmological arrow of time – distance-dependent redshifts were real, the universe was losing energy in forward time, and large-scale physics was time-asymmetrical. Einstein's "unerring sense for mathematical elegance and simplicity." (Chandrasekhar ) had led him to the wrong answers.

Later, George Gamow related that Einstein had referred to the introduction of the cosmological constant as the "biggest blunder" of his life, and Einstein took to setting its value to zero (in effect, deleting it).

By 1958, the Solvay Conference was defining "Einstein's theory", or "Einstein’s version of general relativity" (for the purposes of cosmology) by three conditions: (1), the Einstein Field Equations, (2), a spatially hyperspherical universe, and (3), the condition that Λ=0.

While the 1916 paper failed to support the relativity of inertia, the global geometry and reasoning behind the corrective 1917 paper on cosmology (which was supposed to fix things) also seems functionally incoherent: if "Lambda" was supposed to support the idea of a static spherical universe by compensating for the effects of long-distance cumulative curvature, then it was eliminating the very long-range cumulative physical properties that would cause us to expect the universe to be hyperspherical in the first place. A "spherical" universe implies expansion so strongly that it now seems quite natural that the universe is spatially finite and expanding ... but accepting the idea that gravitational shifts (like positive curvatures) are cumulative takes us further away from the perfectly-cancelling time-symmetrical gravitational and Doppler equations given by Schwarzschild, and by the 1905 paper.

The "Lambda" episode (and the correspondence and disagreements between Einstein and de Sitter) illustrate the degree to which the use of general relativity to make broad predictions, while clothed in quantities of advanced mathematics, often came down to matters of personal interpretation, conviction, and aesthetics by experts – Einstein insisted that a single particle in an otherwise empty universe would not have inertia, de Sitter insisted that it would, and both rejected each other's proofs as unphysical. Both parties selected supporting proofs and mathematical arguments depending on what they already believed should be true. General relativity was a collection of reasoned arguments, and perhaps a broad world-view, but not all arguments attributed to it agreed, and the collection did not form a single consistent deterministic theory, in which one could always derive agreed structures and outcomes objectively.

The contradictory nature of Einstein's framework allowed different practitioners to seize upon different aspects of the theory, generating different physics, depending on their pre-existing personal convictions: it allowed for opinions.

Failure to mesh with cosmological equations, duplication of entities

In an expanding universe, a signal that takes cosmological amount of time to reach its destination will start in part of a smaller, denser universe, and end in a region of a larger, less-dense one. There is then an "uphill" field-density gradient along its path, leading us to expect a gravitational redshift. The Hubble shift therefore needs to be describable either as the result of expansion, or as a gravitational shift due to the density-differential between these two spacetime locations. Gravitational and cosmological shifts must then obey the same equations.

This required equivalence of "gravitational shift" and "expansion shift" arguments doesn't work in Einstein's system.

According to Einstein, the Hubble redshift ...

Einstein (1954) " ... can be interpreted in regard to our present knowledge only in the sense of Doppler's principle, as an expansive motion of the system of stars in the large ... "

, but if so, the Doppler equations that apply here cannot be those of special relativity, as the SR recession redshift formula generates an absolute horizon, and cosmological horizons need to be relative and observer-dependent. In a hybrid Einstein-Hubble model, the recession velocity-shift of a body due to basic relative motion or gravitational differential takes one form, and the recession velocity-shift due to expansion takes another.

Similarly, if we try to calculate the Hubble shift as a gravitational shift, we find that Einstein's gravitational shift due to a density-differential in space takes one (SR-compatible) form, and the shift due to a Hubble density-differential in time is required to take another (non-SR) form. Curvature horizons are either absolute (if the curvature is due to gravity) or relative (if the curvature is due to expansion). The rules of geometry for cosmological curvature do not agree with those for Einstein's gravity. To avoid admitting that Einstein's system does not "mesh" with modern cosmology, we invent a second parallel set of geometrical physics in which shifts due to curvature or density differentials, or relative velocities, follow different laws depending on whether or not the cause is cosmological. In effect, we have dual metrics: SR and Schwarzschild metrics for relative motion and gravity, and a separate (more "acoustic" style) metric for cosmology.

A further complication is that if we change our own state of motion, our concept of the alignment of the space and time axes changes, so that the time-aligned Hubble expansion vector takes on some of the properties of a space-aligned gravitational gradient, and vice versa. It is difficult to see how this could work if the equations for cosmological and gravitational effects were inherently different. If a single set of equations must apply to both cosmology and gravitation, the requirement that cosmological horizons be relative and observer-dependent means that this universal set of equations cannot be Einstein's.

Failures of locality, classical field theory, and topology

A classical geometrical theory of physics should let us calculate the behaviour of a signal from just the properties of a long, narrow cylinder of spacetime surrounding the signal path. Knowing just the local curvature properties along the path should be sufficient to let us predict the outcome without knowing (or caring) why that curvature exists, whether it is due to a time-aligned or a space-aligned density gradient, or what is happening elsewhere, outside the cylinder.

For Einstein's system to generate different results from an identical section of metric depending on whether its curvatures are due to gravitation or expansion suggests a failure of the principle that physics is local, and with non-local causality, also a further failure of classical field theory and metric theory principles. If the 1917 description lets the same local geometry have different consequences depending on wider geometry then the system fails to be a conventional geometrical theory, and if local geometry is no longer sufficient to predict local physics, we are probably limited in how we can safely apply coordinate transforms and topological arguments to derive physical law.

Insufficient specificity

25 years after Alexander Friedmann died, Einstein tried to claim that Friedman had shown that the Hubble result had been part of Einstein's theory all along.

Einstein (1952): " Hubble showed ... a red shift which increased regularly with the distance of the nebulae. ... – as required, according to Friedman, by the field equations of gravitation. Hubble's discovery can therefore, be considered to some extent as a confirmation of the theory. "

This seems untrue. In reality, Friedman produced multiple solutions of Einstein's equations, covering almost every possibility, including positive and negative curvature, and expansion and contraction. Einstein's field equations did not tell us which of these were correct. Selecting a specific model required additional auxiliary arguments, and if these were correct, it was the auxiliary arguments that deserved the credit for the prediction, not Einstein's initial framework.

Further, the expanding universe solution seemed to have been obtained, not by extending Einstein's theory, but by overriding its previous predictions. Based on the SR and Schwarzschild equations (which were still part of the theory), we could still "prove" that there could not be any Hubble-style long-distance cumulative gravitational redshifts.

GR from cosmology

Modern cosmology forces Einstein's updated system to try to support two different parallel sets of geometrical rules, for gravitational and cosmological curvature, that do not seem to be reconcilable. If general relativity had been devised after the discovery of the Hubble redshift, its genesis might have been easier: it could have been argued that since cosmological horizons and shifts must be observer-dependent and non-SR, topological consistency requires gravitational horizons and gravitational physics (and by extension, also the basic laws of inertial physics) to be non-SR, too. A Hubble-centric general theory would then have to be Hertzian rather than Lorentzian.

The simplest interpretation of these disagreements is that perhaps Einstein's gravitational theory, whose principles and rules were tailored for a static universe with time-symmetry, reversible equations, and conventional energy-conservation, may simply not work in an expanding universe where none of these laws are valid.

Second thoughts (GR1950?)

Einstein appears never to have found a satisfactory derivational argument for GR's definitional incorporation of SR physics.

According to Einstein's original vision for his general theory,

Einstein (1919): " ... the theory of relativity resembles a building consisting of two separate stories, the special theory and the general theory. The special theory, on which the general theory rests, applies to all physical phenomena with the exception of gravitation; the general theory provides the law of gravitation and its relations to the other forces of nature. "

Special relativity provided a known block of existing non-gravitational theory as a foundation for adding further gravitational arguments.

By 1950, Einstein appeared to reject the validity of this two-stage "SR, then GR" architecture, writing that it was no longer defensible.

Einstein (1950): " I do not see any reason to assume that the heuristic significance of the principle of general relativ­ity is restricted to gravitation and that the rest of physics can be dealt with separately on the basis of special relativity, with the hope that later on the whole may be fitted consistently into a general relativistic scheme. I do not think that such an attitude, although his­torically understandable, can be objectively justified. ... In other words, I do not believe that it is justifiable to ask: What would physics look like without gravitation? "

Since the answer to what physics would look like without gravitation was "special relativity", Einstein appeared to be suggesting that his 1905 theory might have been the answer to an illegitimate question.

Alternative general theories

Parallels to GR1916

John Wheeler's geometrodynamics project (physics as the dynamics of geometry) aimed to reinvent Einstein's GR concept using new tools and methods.

Wheeler (1973): " – the vision of Riemann, Clifford and Einstein, of a purely geometrical basis for physics, today has come to a higher state of development, and offers richer prospects — and presents deeper problems — than ever before. "

Wheeler's project underwent a series of redefinitions in response to successive failures, moving from being a theory of classical geometrodynamics to quantum geometrodynamics, to pregeometry, before eventually concluding that with Einstein's system, nothing (quite literally nothing) worked. While the "modern", explicitly SR-centric, "fixed" (1960) version of Einstein's system seemed to manage to violate the GPoR, the PoE and both classical and quantum field theory, Wheeler's reworking went further in claiming to violate every law known to exist.

Wheeler (1973): " ... However, displacements in flat spacetime, or even in a curved spacetime that is asymptotically flat, make no sense in the closed universe of Einstein's general relativity. In that universe there is no global law of conservation of momentum and energy. More startling, such a universe undergoes gravitational collapse. In that collapse, classical space and time themselves come to an end. With their end, the framework falls down for everything that one has ever called a law of physics. ”

Wheeler concludes not that Einstein's system is wrong, but that our universe ultimately has no laws other than the law that "There are no laws" ("Law without Law") – our search for the ultimate laws governing the universe will fail because no such laws exist.

Wheeler (1973): " The golden trail of science is surely not to end in nothingness. There may be no such thing as 'the glittering central mechanism of the universe' to be seen behind a glass wall at the end of the trail. Not machinery but magic may be the better description of the treasure that is waiting. "

While Wheeler's position was that Einstein's universe did not "break" physical laws but transcended them, total failure of all logical rules is also a feature of pathological systems, and is normally an indicator that we are using a framework that is logically broken. Wheeler's "final transcendence", in its current state, seems functionally indistinguishable from pathologicity.

Supersets of GR1916

According to the PPN (Parameterised Post-Newtonian) formalism for classifying alternative gravitational theories, the principle of equivalence must be implemented as the Einstein Equivalence Principle ("EEP"), which is then in turn defined as incorporating SR. This makes SR-compliance compulsory for any "credible" gravitational theory. MTW also suggests that only metric theories need be considered, and defines a metric theory by the conditions that:

MTW (1973): 39.2 Metric Theories of Gravity, p.1067 "... (1) Spacetime possesses a metric; and (2) that metric satisfies the equivalence principle (the standard special relativistic laws of physics are valid in each local Lorentz frame). Theories of gravity that incorporate these two principles are called metric theories. "

Systems describable using PPN are SR-centric, and therefore tend to be supersets of Einstein's theory (e.g. "GR1916 plus torsion"), and can be characterised as "GR1916 with extras". In this population of similar theories, Einstein's is the most minimalist, and therefore the most preferred.

These alternative theories of gravity must either not claim to be general theories of relativity, or must have the same internal conflicts regarding the GPoR and SR as the 1916 theory.

Subsets of GR1916

GR without SR

Richard Feynman, lecturing more generally on "Scientific Method", says:

Feynman (1964): "The first problem is how to start. 'Ah, start with all the known principles!' But the principles that are all known are inconsistent with each other. So something has to be removed.'"

If Einstein's system is defined as a combination of (1) the GPoR (by definition), (2) the PoE (for geometricalisation) and (3) SR (for continuity with previous theory). then we cannot very well lose either (1) or (2). This leaves open the possibility of eliminating (3), and losing full support for special relativity. Where Møller and Schild solved general relativity's internal contradictions by giving SR priority over the GPoR ... keeping (3) and demoting (1) and (2) ... we might instead choose to give the GPoR priority, and treat the special theory as a first approximation rather than as a foundation.

This sort of more minimalist and more purist (but more ambitious) general theory without a separate SR foundation, as suggested by Einstein in 1950, is logically conceivable, as illustrated in Wolfgang Rindler's 1994 hypothetical "alternative history" in which Riemann is still able to produce a general theory despite not knowing about special relativity:

Rindler (1994): " General relativity before special relativity: ... It is suggested how Bernhard Riemann might have discovered General Relativity soon after 1854 and how today's undergraduate students can be given a glimpse of this before, or independently of, their study of Special Relativity. "

"Cliffordian" universes

Logic does not require a general theory to incorporate SR physics: we can also consider W.K. Clifford's 1876 talk, "On the Space-Theory of Matter" which suggested a universe in which all matter-physics could be described as the result of the interplay of curvature effects.

Clifford (1876): " ... I hold in fact (1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them. (2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave. (3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial. (4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity. "

A "Cliffordian" universe is the logical counterexample to Einstein's argument that a gravitational theory must reduce to "flat spacetime physics" (and SR) over small regions. If a universe is "Cliffordian", then there is no such thing as "flat-spacetime physics" (for matter) for the theory to reduce to. It is therefore a matter of some importance to know whether or not our universe appears to be Cliffordian.

Acoustic metrics and analogue gravity

Just as Einstein accepted in 1910 that the logical alternative to SR and Lorentz aether theory was a fully-dragged aether model, and Lorentzian relativity "geometricalises" to Minkowski spacetime, so Hertzian relativity "geometricalises" to a relativistic acoustic metric whose application to gravitational theory is commonly referred to as analogue gravity. If Hertzian proximity-dependent dragging effects on light-velocities are treated as field effects, Hertzian theory becomes gravitomagnetic theory. We then have a choice of a general theory either reducing to the inertial physics of Lorentz-Einstein-Minkowski, or to that of Hertz-Newton-Visser.

An "HNV" model has the interesting property that (a) it generates observer-dependent acoustic horizons, suggesting a mesh with similarly observer-dependent cosmological horizons and modern cosmology, and (b) that the resulting so-called sonic or acoustic black hole generates the classical counterpart of Hawking radiation,

Visser (2005): " ... an acoustic ... horizon will emit Hawking radiation in the form of a thermal bath of phonons ... "

, suggesting a mesh with QM. This should not come as a surprise, as the development of acoustic metrics was partly motivated by their possible use as a toy model or better for quantum gravity:

Barceló, Liberati and Visser (2005): " Fluid mechanics is a guide to the mathematical possibilities, not an end in itself. The parts of the analogy that do work well are precisely the steps where the standard approaches to quantum gravity have the most difficulty, ...

Secondary reasons include the rather speculative suggestion that there may be more going on than just analogy – it is conceivable (though perhaps unlikely) that one or more of these analogue models could suggest a relatively simple and useful way of quantizing gravity that side-steps much of the technical machinery currently employed in such efforts. "

The "leaky" horizons generated by acoustic models cannot result from the basic equations of special relativity or the 1916 theory. While a field approach to gravity based on fluid dynamics suggests commonality with the statistical mechanics of QM's particle-based approach, it also implies functional similarities with Hertz's "dragged aether", and also to the areas of general relativity incompatible with SR, since the acoustic metric's fluid velocity field corresponds to the momentum field and to gravitomagnetism.

Discussion of "Lorentzian geometries" in the context of acoustic metrics can be misleading, as acoustic metric physics does not correspond to Lorentzian physics, in the sense of the relationships of special relativity or Lorentz aether theory. We can see this in the case of acoustic Hawking radiation, which allows signals, energy and matter to migrate outward through a distant observer's horizon along noninertial paths. Under Einstein's SR-based system, the inward gravitational blueshift between two heights is the inverse of the outward redshift (Wald ), so for an observer hovering at r=2M, where the outward shifted frequency is E'/E = zero, the inward blueshifted frequency and energy is E'/E = 1/0 = infinity. In Einstein's universe, an observer cannot even be stationary at r=2M without being subjected to infinite temperature and infinite inward radiation pressure, much less escape – any system of physics that supports acoustic Hawking radiation therefore relies on the most basic relationships of physics being different to Einstein's, even down to the identity of the Doppler equations and equations of motion.

Hawking and relative horizons (2014)

Hawking has also suggested a switch to relative horizons in the context of solving the black hole information paradox, bringing general relativity's predictions more into line with QM.

Hawking (2014): " A different resolution of the paradox is proposed, namely that gravitational collapse produces apparent horizons but no event horizons behind which information is lost. This proposal is supported by ADS-CFT and is the only resolution of the paradox compatible with CPT. "

The success of this approach again depends on relative, "acoustic" horizons rather than Wheeler's absolute event horizons. Again, since the SR and Schwarzschild equations (and the resulting causal structure) make event horizons provably unavoidable, Hawking's 2014 approach also depends on the basic equations of physics being non-SR.

Developmental difficulties

While the deletion of SR from GR may seem to "fix" general relativity, allowing duality with QM, eliminating the SR barrier to the full implementation of the GPoR, and allowing unification with the equations of cosmology, further development of the concept into a full-blown gravitational theory is difficult, as this would involve making the implicitly non-SR nature of this class of model explicit, and would also involve stating the forms of the replacement equations that would need to supplant the SR set. Pursuing this path would mean suggesting that both of Einstein's classical theories – both special relativity, and the SR-centric 1916 general theory – were wrong.

Since the theoretical community has already standardised on special relativity as being considered correct "beyond a shadow of a doubt". SR-compliance is now built into many of our definitions for assessing new theories.

Will (1988): " Special relativity is so much a part not only of physics but of everyday life, that it is no longer appropriate to view it as the special "theory" of relativity. It is a fact ... "

This public stance makes it difficult for professional researchers to pursue explicitly non-SR solutions.

Emphasis on Covariance

Einstein's position on covariance before 1917 seems to have been that, if a theory's equations were generally covariant, it automatically conformed to the general principle of relativity, satisfying key features such as the principle of equivalence of inertia and gravitation.

Einstein (1914): " The postulate for general covariance of the equations embraces the principle of equivalence as a special case. "

Einstein (1921): " We shall be true to the principle of relativity in its broadest sense if we give such a form to the laws that they are valid in every such four-dimensional system of co-ordinates, that is, if the equations expressing the laws are co-variant with respect to arbitrary transformations. "

This did not logically follow: if we are searching for an elephant, it may help to know that an elephant must be a mammal. But if we find a mammal, it is not automatically an elephant, it merely meets one necessary criterion. Einstein meets the "necessary condition" of being a mammal: but Einstein is not an elephant.

Kretchmann pointed out in 1917 that Einstein's apparent use of "general covariance compliance" to signify "GPoR-compliance" was wrong, as almost any conceivable theory could be described in a covariant way.

Einstein (1918): " ... Herr Kretschmann notes that the principle of relativity, phrased in this manner, is not a statement about physical reality, i.e., not about the content of nature’s laws, but it is rather a demand with respect to their mathematical formulation. "

Norton (1992): " Since these generally-covariant theories include the versions of Newtonian space-time theory that unequivocally violate the usual relativity principles, any interesting connection between general covariance and relativity principles seems extremely dubious. ... "

Einstein (1918) explains that although Kretschmann is technically correct, he (Einstein) had personally found covariance heuristically useful in allowing him to eliminate equations whose covariant expression was cumbersome and therefore unlikely to be right. Einstein has at this point dropped the condition that his general theory conform physically to the general principle of relativity, replacing it with the condition that the theory's mathematical formulation be efficient, under a particular type of description (when associated with certain auxiliary assumptions). Once various selection criteria have been applied, the equations that look simplest in a covariant description are considered to be the right answer.

Physical behaviours such as gravitomagnetism wer now instead covered by the related condition that the theory must comply with Mach's principle.

Norton (1992): " I argue ... that the relativity-principle-like character of Einstein's covariance principles is a peculiarity, ... a conclusion that provides no support to the view that general relativity, freed from the peculiarities of specificic, known formulations, has effected a generalisation of the principle of relativity. " " ... a literal reading of Einstein's claims should appear incoherent to modern readers. "

Einstein's system no longer came with a mathematical test or guarantee that it conformed to the GPoR and was a general principle of relativity as originally advertised.

Summary

Failure to conform to the relativity principle

Although general relativity was one of the great ideas of the Twentieth Century, Einstein never managed to get an implementation of the general principle of relativity to work properly. The 1916 theory failed to implement the relativity of inertia, the 1917 update introduced sphericality but also added Einstein's Cosmological Constant to explain why the universe wasn't varying in size or showing Hubble-like redshift effects. By 1921, Einstein was only able to to claim a partial implementation of the relativity of inertia, and by 1924, he'd given up trying to get this side of the theory to work altogether. By 1950, when "the general theory" had been in play for 35-40 years, Einstein disavowed the two-stage "first SR, then GR" architecture of his own theory, declaring that he no longer believed in the concept of a flat-spacetime physics underlying GR.

General incompatibility with the principle of equivalence, and the general principle of relativity

In 1952 Møller discovered that SR could not coexist with the relativity of acceleration, and in 1960 Schild published the result that SR also couldn't coexist with the principle of equivalence in rotating-body problems. The foundation chosen by Einstein for implementing the GPoR turned out to be incompatible with the GPoR. With fundamental internal conflicts between structure and principle, the 1916 theory failed consistency tests (MTW §39's first criterion, "Self–consistency ..." ), meaning that it was not valid as theory, and technically "not even wrong".

Failure as geometry and mathematics

The SR component inherited by the 1916 theory, which is a perfect fit for flat spacetime, does not work with momentum fields, relativistic gravitation, or accelerative or rotational gravitomagnetism, and its very concept of "inertial physics without gravitational physics" immediately violates the principle of equivalence (of inertia and gravity), a cornerstone of general relativity. Its adoption breaks Einstein's "reality principle". It also violates both classical field theory and quantum field theory, and generates event horizons and Wheeler black holes leading to total gravitational collapse. It cannot model Hawking radiation classically, or explain Hubble shifts or thermal redshifts in Fe⁵⁷, and does not allow gravitational waves. It does not work for cosmological recession velocities, cosmological gravitational differentials, or cosmological horizons. The use of different rules for cosmological and gravitational curvature breaks topology, the principle of locality, metric theory, and arguably breaks geometry itself. The 1905 relationships are fundamentally incompatible with relativistic gravitation, the relativity of inertia, the general principle of relativity and quantum mechanics.

Special treatment?

The degree to which Einstein's system has been "indulged" is apparent when we look at Einstein's own harsh dismissal of other competing theories: Hertz's system of relativity was dismissed by Einstein as "not acceptable" despite being "free of contradictions" for not agreeing well enough with experimental data, and Einstein also dismissed George Stokes' similar earlier idea on the grounds that since it included a contradiction it was not to be considered (along with, perhaps a slight "barb" relating to the degree of authority that we should allow someone who presents an internally-contradictory theory).

Einstein (1950): " In consideration of the subject matter and the authority of Stokes, I deem it necessary to point out that this theory is untenable because it is based upon contradictory assumptions. "

If Einstein has been as strict with his own theory as he was with those of his his competitors, then his own general theory would have had to be discarded, too.

When it was realised that Einstein's GR failed MTW's second criterion for acceptability by failing to "mesh" with quantum mechanics, then, instead of dismissing Einstein's system, we "accommodated" it by suspending the duality of classical and quantum systems, and deciding that there must be some fundamental difference between classical and quantum theory, which we would not fully understand until we had a theory of quantum gravity. When Einstein's system failed a key test, we "moved the goalposts" and continued to claim that the theory had a perfect record.

Falsifiability

The feature of the 1916 theory that impressed Karl Popper so much that he made it his exemplar of a "good" scientific theory, was it's falsifiability. Einstein's system (before 1917) was eminently falsifiable in that it had no free parameters – it was either right, or it was wrong.

Einstein (1919): " The chief attraction of the theory lies in its logical completeness. If a single one of the conclusions drawn from it proves wrong, it must be given up; to modify it without destroying the whole structure seems to be impossible. "

Popper would not have been aware that the framework was pathological, with this pathologicity allowing different human operators a degree of freedom to choose which of its contradicting principles and predictions were correct, and to arrive at different conclusions. Since the conclusions drawn from the GPoR were disproved by the SR side of the theory, and the conclusions of the SR side were disproved by the GPoR, Einstein's general theory should, according to his 1919 rules, have been "given up". Similarly, the more explicitly SR-centric version of GR introduced by Schild would have needed to be "given up" when we confirmed that gravitational waves and gravitomagnetic dragging were real. "GR1916" was self-falsifying, "GR1960" (Schild) is falsified by the available experimental evidence.

Cheerleading

While some researchers accept the historical record for Einstein's general theory, perhaps rationalising its failure to conform to the GPoR by suggesting that what is important to "modern GR" is instead the principle of covariance, with the GPoR and PoE being heuristically useful but "more what you'd call guidelines than actual rules", others insist that Einstein's theory of gravity does not have, and never has had, any logical problems at all:

MTW (1973): " No inconsistency of principle has ever been found in Einstein’s geometric theory of gravity. "

It is difficult to consider these sorts of statements as being made in good faith: Misner, Thorne and Wheeler presumably could not have failed to miss the Møller and Schild episodes, or been unaware of Einstein's own personal struggles with the general theory, and their own book, some 130 pages later, describes total collapse as "the greatest crisis in physics of all time".

Current status

As of September 2024, after having had over a century to work on the problems with Einstein's system, we still do not have a working general theory of relativity, fully compliant with the general principle of relativity and free from serious internal contradictions.

Since the Møller logic makes the SR relationships incompatible with gravitomagnetism (required to implement the GPoR), it does not seem to be geometrically possible to write a GPoR-compliant theory without contradicting special relativity.

If the general principle of relativity is not geometrically reconcilable with the equations of special relativity, any consistent general theory must necessarily be "non-SR". As theory that contradicts special relativity is not currently considered to be acceptable, "Einstein's unfinished masterpiece" must, for now, remain unfinished.

References

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