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ResearchGate, 12th February 2021

Gravitomagnetic horizons and the comprehensive

failure of Einstein’s 1916 general theory

Eric Baird

A receding horizon-bounded body generates an “effective”, gravitomagnetic horizon

between the observer and the usual r=2M horizon surface. This purely observer-

dependent horizon does not obey SR concepts of causality or the SR shift

relationships, and its classical behaviours instead correspond to those of acoustic

metrics and of quantum mechanics (including the emission of classical Hawking

radiation). Since the horizon position alters with the relative state of motion of the

observer, who can be arbitrarily distant, non-SR “acoustic metric” physics then

effectively operates across the entire external visible universe. Adjusting the body’s

gravitational shift equations to match those of its surroundings then also converts

the r=2M surface into an observer-dependent “acoustic” horizon, solving a number of

long-standing problems in standard theory, including the black hole information

paradox. We conclude that the definitions and relationships of special relativity,

originally derived for flat spacetime, do not function properly in a gravitational

universe, and should not be thought of as valid components of a general theory of

relativity.

Table of Contents

1. Introduction………………………………………………………………………………………………………………………………….3

2. e unavoidability of gravitomagnetic eects…………………………………………………………………………….4

2.1. Gravitomagnetic effects of moving masses.................................................................................................................4

2.2. Secondary horizons...........................................................................................................................................................4

3. Properties of the secondary horizon…………………………………………………………………………………………….5

3.1. Overview of the problem.................................................................................................................................................5

3.2. Non-Wheeler behaviour...................................................................................................................................................5

3.3. Gravitomagnetism of a moving black hole:.................................................................................................................5

3.4. Gravitomagnetism of a moving final observer...........................................................................................................6

3.5. Gravitomagnetism of an intermediate observer.........................................................................................................6

4. Experiences of local and distant observers………………………………………………………………………………….8

4.1. Observerspace.....................................................................................................................................................................8

4.2. Direct and indirect observation......................................................................................................................................9

5. Compatibility with quantum mechanics……………………………………………………………………………………10

5.1. “Real” and “virtual” particles.........................................................................................................................................10

5.2. Generation of Hawking radiation................................................................................................................................10

5.3. The QM pair-production description, from classical geometry............................................................................11

6. What happened to special relativity?…………………………………………………………………………………………12

6.1. Incompatibility of effective horizons with the SR shift equations.......................................................................12

6.2. Replacing the relativistic shift equations to allow Hawking radiation...............................................................12

6.3. Gravitomagnetic shifts are also motion shifts..........................................................................................................13

6.4. All bodies are “strong-gravity” bodies at sufficiently small distances................................................................13

6.5. Goodbye, special relativity............................................................................................................................................14

7. Other notes………………………………………………………………………………………………………………………………….15

8. Summary……………………………………………………………………………………………………………………………………..16

9. Conclusions…………………………………………………………………………………………………………………………………17

References……………………………………………………………………………………………………………………………………….17

page 1 of 19

Gravitomagnetic horizons vs. Einstein’s GR, Eric Baird, February 2021

1. Introduction

The subject of gravitationally “cloaked” stars is surprisingly old, and dates at least as far back as

John Michell’s 1983 letter to the Royal Society (published 1784), [1] [2] [3] [4] which explored the

hypothetical behaviour of stars whose gravity was so strong that their escape velocity exceeded

the speed of light. In Michell’s description, the critical surface where vESCAPE=c appeared at a radius

of r=2MG/c2, (often abbreviated as “r=2M”). Any light or subluminal matter emitted below this

radius would be turned around by gravity, and could not reach an arbitrarily-distant onlooker

along a “ballistic” or inertial path – to us, such stars would appear dark. Particles from below

r=2M could visit the outside region for a limited time, [4] and these “visiting” particles could then

be knocked free from the star’s gravity by random collisions (with passing matter, or each other)

and escape, [5] so these “dark stars” were still able to radiate indirectly, along accelerated paths. i

Einstein’s flat-spacetime special theory of relativity (“SR”, 1905) [6] modified the C19th Newtonian

equations and Doppler relationships, and Einstein’s incorporation of these new relationships into

his larger curved-spacetime general theory of relativity (“GR”, 1916) [7] [8] changed the predicted

behaviour of super-dense gravitational bodies. Under the new system, if a distant observer could

not see light emitted by events occurring below r=2M, then for these observers, the associated

emission events did not take place [9] (or were assigned to the more-than-infinitely-distant future).

With this new, more literal version of perceived causality, where what we can see defines the

physics, [10] the region below r=2M was unable to communicate with the outside world in any

way, there was no longer a mechanism for indirect radiation, and the horizon became an absolute

event horizon. While Einstein argued that GR’s predictions for these dense objects must be

unphysical, [11] John Wheeler championed the concept, [12] and … since the new horizons had a

zero temperature and exerted zero outward radiation pressure ... referred to them as black holes.

The 1960s saw a healthy discussion about the legality of absolute horizons, complicated by the fact

that many of our standard coordinate-system tools broke down for regions straddling these

horizons. ii These debates eventually produced a general consensus that, although local physics

appeared unremarkable at the horizon to infalling observers, [13] total collapse and the formation of

absolute event horizons were unavoidable for bodies smaller than r=2M. [14]

In this paper, we instead look outside r=2M, at the gravitomagnetic effect of a retreating GR1916

black hole on its wider environment, and find an external, observer-dependent gravitomagnetic

horizon. Since its position depends on the state of motion of an (arbitrarily distant) observer,

gravitomagnetic behaviour then applies to all moving observer-masses outside r=2M, despite the

SR assumption that small masses can move however they like without affecting the propagation

of light. These arguments tell us that, in effect, the entire outside universe must be operating in

accordance with non-SR physics and non-SR shift equations, after which we lose the original

justification for having a Wheeler horizon. The idea that a general theory can successfully

incorporate SR physics, and the associated idea of an absolute horizon, are self-invalidating.

We conclude that Einstein’s decision to use equations derived from flat spacetime in the context

of solving an inherently curved-spacetime problem (gravity) was mistaken. More encouragingly,

the appearance of classical Hawking radiation across the external “effective” horizon appears to

limit our options for a relativistic replacement for the SR relationships to a single solution. [15] The

1916 theory does not just self-invalidate, the nature of the breakdown tells us the identity of the

replacement equations that must appear in a revised and corrected general theory.

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page 2 of 19

Gravitomagnetic horizons vs. Einstein’s GR, Eric Baird, February 2021

2. The unavoidability of gravitomagnetic effects

2.1. Gravitomagnetic effects of moving masses

Gravitomagnetism and its analogues exist in almost every area of physics, apart from special

relativity. If the speed of gravitational interactions is finite i then a moving body’s gravitational

field will be slightly “smeared” by its motion, with this smearing effect appearing in a field

description as an additional gravitomagnetic field component (weakening the attraction of the

body’s default “rest field” forwards and strengthening it rearwards), or, in a geometrical

description, as a tilting of the throat of the body’s associated gravity-well, to align with the

moving body’s tilted worldline (weakening the curvature ahead of the moving well, and

increasing it behind). [5] The result is a net pull on nearby objects and light in the direction of a

mass’ motion – a gravitomagnetic dragging effect – in the rotation plane of a rotating star, the

receding edge should pull more strongly than the approaching edge, and the star will tend to

drag matter and light around with it. Gravitomagnetic effects are compulsory in a general theory

of relativity, with rotational dragging [18] (and therefore a velocity-dependent gravitomagnetic

component) an essential consequence of the relativity of rotation. Carlip’s argument [19] also

makes velocity-dependent gravitomagnetism necessary for the emergence of Newton’s First Law.

The strength of the expected dragging effect can be calculated from the relative velocities of the

bodies involved, and their relative contributions to the total field intensity at the location of a

test body (Wheeler [20]). For the case of a moving black hole event horizon, where external

physics is not allowed to modify the apparent frozen state of the horizon, the dragging effect is

usually treated as absolute (Thorne [21] [22]). “Fossil light”, emitted at r=2M by a body that has

long-ago fallen into a black hole, rotates at the horizon in lockstep with the hole’s rotation.

This same “total dragging” is then also required for the horizon of a black hole moving in a

simple straight line: if outward-aimed light emitted at r=2M is considered to be frozen into the

horizon surface, then, if we subsequently decide to move relative to the hole, the “frozen” light,

still trapped at r=2M, must seem to move with the rest of the hole. Certainly, if the hole recedes

from us at v m/s, the trapped light cannot recede from us at any less than v m/s, or else the

horizon would recede from us faster than the frozen signal, which would then find itself left

behind and exposed outside the absolute horizon, and be free to escape and be seen by us.

Absolute horizons therefore require total light-dragging.

2.2. Secondary horizons

We can also define an effective horizon, as being the critical surface at which light aimed in our

direction neither approaches or recedes. If a black hole moves away from us, then any light

trapped in the surface of its r=2M horizon must also be moving away from us, and is therefore

(by definition) no longer at our effective horizon, but somewhere behind it.

Since light aimed towards us far outside the hole moves towards us, and similarly-aimed light at

or below the horizon moves away from us, a classical gravitational model has to predict an

effective horizon somewhere between these two locations, somewhere outside r=2M. For us, the

standard Wheeler horizon must necessarily be concealed behind an additional secondary,

gravitomagnetic horizon that can be blamed on the receding hole’s increased attraction. ii

We will now briefly look at some of this secondary horizon’s properties.

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page 3 of 19

Gravitomagnetic horizons vs. Einstein’s GR, Eric Baird, February 2021

3. Properties of the secondary horizon

3.1. Overview of the problem

We will define O as an arbitrarily-distant observer for whom the hole is receding at v m/s, and X

as the centre of the black hole. On the line O-X, H1 is the intersection point of the absolute

Wheeler horizon at r=2M, and H2 is the corresponding point on the additional, secondary

external horizon. i

O … . . . ……………………… H2 ……. H1 ……… (X)

O' … . . . …………………………….…… H1 ……… (X)

The secondary horizon’s exact position and properties depend on our observer’s circumstances:

if the recession velocity of O was zero (giving O'), H2 would coincide with H1 … increasing the

recession velocity moves H2 further away from H1 . ii iii

3.2. Non-Wheeler behaviour

While the region below the H1 surface cannot be seen by O, and the region outside H2 is fully

visible to our distant onlooker, the status and rules of causality of the region between H1 and H2

– which we might refer to as “the twilight zone” – are rather more ambiguous.

Signals created by events generated within this “ambiguous” zone cannot reach our onlooker O

directly, iv but can be seen by the onlooker’s colleagues in a research vessel moving with the

hole, some distance above H2 – these observers can see all the way down to H1, and there is

nothing to prevent them communicating what they see back to their distant colleague, O.

The horizon H2 is therefore not an event horizon. It does not exist for all observers, and it does

not mark the location of an absolute causal barrier. While it separates the regions that O can and

cannot see directly, events occurring behind H2 (but outside H1) can still influence O indirectly.

3.3. Gravitomagnetism of a moving black hole:

The receding hole’s increased attraction can also be derived from a range of other arguments: a

redshifted receding body’s apparent slowed timeflow combined with Huygens’ principle results

in light being deflected more towards it; shift equivalence allows the recession redshift to be

interpreted as a gravitational redshift; the apparent field intensity value can be considered as

being timelagged; the body can couple with surrounding bodies via its field, deflecting light and

matter in its direction of motion through indirect collision; momentum exchange; “gravitational

slingshot”-style time domain arguments; classical field effects for a moving “gravitational

charge” analogous those associated with a moving electrical charge; [23] negative virtual particle

pressure; and/or a generalisation of the Fizeau effect. [24]

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page 4 of 19

Gravitomagnetic horizons vs. Einstein’s GR, Eric Baird, February 2021

3.4. Gravitomagnetism of a moving final observer

If our distant region contains two almost-adjacent observers, O and O', one for whom the hole is

receding and one for whom it is stationary (O'), then although O cannot see all the way down to

the Schwarzschild horizon, their neighbour O' can. O' can tell O what they see, send O a

photograph or video clip, or aim an angled mirror so that O can see things from their viewpoint.

From an SR-based point of view, this description is nonsense: when O and O' are adjacent,

a signal that can reach O' must also reach O – signal behaviour is defined by the geometry of the

background space, not by any observer-properties. In our description, however, the different

properties of observers O and O' must be physically altering the behaviour of light in their region

i ii – the retreat of O must somehow be contributing to the total curvature between O and X. In

other words, if we (reasonably) assign gravitomagnetic effects to the moving black hole, we also

have to assign them to all other masses that might exchange signals with the hole, and to all

potential observers. iii

Since the motion of hypothetical, or purely mathematical SR-style observers cannot physically

distort spacetime, we must reject this class of observer as “unphysical”, [25] and only accept

“material” observers as valid, possessing both inertial and gravitational mass in accordance with

the principle of equivalence. iv In a valid general theory, “observers” are no longer passive

placeholders in a predetermined spacetime, but active participants in a dynamic geometry. v

3.5. Gravitomagnetism of an intermediate observer

These non-SR behaviours become even more pronounced when we consider the case of

intermediate observers:

If we drop off an “intermediate” observer “OI”, somewhere between O and H2, initially with the

same state of motion as the hole, then since they do not (initially) see the hole to be moving, they

will be able to see all the way down to H1, and since they are outside H2, they can (initially)

relay what they see back to us.

We then have a bizarre situation in which the “twilight zone” H2-H1 region, which is normally

not visible to us, becomes visible though the addition of an intermediate mass (such as a simple

sheet of glass) placed in the signal path – we should then be able to see into an initially-

forbidden region by viewing the region through the glass. At this point, the laws of physics are

quite clearly not obeying the rules of Minkowski spacetime: the presence and relative motion of

a mass in the signal path – any mass at all – is modifying the region’s spacetime geometry. vi

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page 5 of 19

Gravitomagnetic horizons vs. Einstein’s GR, Eric Baird, February 2021

DISCUSSION

Sections 1-3 show that the physics outside a black hole must be non-SR.

Sections 4 onwards explore how this can work, the required modifications to some basic

concepts, and the apparent agreement with quantum theory.

page 6 of 19

Gravitomagnetic horizons vs. Einstein’s GR, Eric Baird, February 2021

4. Experiences of local and distant observers

4.1. Observerspace

“Observerspace” logic can be characterised as the literal, “no interpretation” interpretation of

data (a “no theory” theory). According to observerspace principles, reality is whatever impinges

on our senses or is experienced by an atom, and there is no such thing as an observational

“artefact”. How a massed particle senses, and how it is sensed, is its physics.

Special relativity is well known for placing great store by “observer” arguments, and … as it

assumes flat spacetime and global lightspeed constancy, then, … apart from simple signal

timelags as a function of distance ... under SR the universe is exactly as we “observe” it to be, and

observation is reality. Light from events below r=2M never travels outside r=2M, because

according to the distant observer’s projected coordinate system, the emission events never

actually happen. If the light is never emitted, it cannot possibly have indirect consequences. [9]

Special relativity, however, is not quite a pure observerspace theory … it applies the

observation principle rather selectively, since Einstein instructs his “SR observers” not to be

“perfect observers” reporting back what they see without interpretation, but to report what they

observe, which to Einstein means correcting the raw data according to the belief that space is flat

and the speed of light is globally fixed. i For Einstein, what an SR entity observes is the result of a

calibrated measurement process, and does not necessarily describe their direct experience. ii iii

Observerspace can be less reliable for more distant observers. If a gravitational-lensing

body moves across our field of view in front of a distant galaxy, we may see that galaxy break up

into multiple smaller copies, which swirl around before magically re-coalescing back into their

original form after the “lens” has passed. We do not treat this view of reality literally and assume

that the lensed galaxy’s component stars really do split and re-merge – we assume that there is a

simpler spatial underlying reality that we would see if only we were close enough.

Similarly with distant causality – we cannot always assume that what happens lightyears away

can be simply extrapolated directly from what we see, without interpretation. The apparent time-

ordering of events can be “scrambled”, just as our lensed galaxy’s spatial locations appear

scrambled, but we can, again, assume a more rational underlying reality, that applies locally, and

that we would expect to see if only we were close enough to have a proper view.

This conflict between literal (“what we can see defines reality”) and interpreted theory (“reality

decides what we can see”), eventually caused Einstein to change his views on observerspace:

Heisenberg: [10] “ … And since it is but rational to introduce into a theory only such quantities as can be

directly observed, the concept of electron paths ought not, in fact, to figure in the theory.

To my astonishment, Einstein was not at all satisfied with this argument. He thought that every theory in

fact contains unobservable quantities. The principle of employing only observable quantities simply cannot

be consistently carried out. And when I objected that in this I had merely been applying the type of

philosophy that he, too, had made the basis of his special theory of relativity, he answered simply: ‘Perhaps

I did use such philosophy earlier, and also wrote it, but it is nonsense all the same … it is theory which first

determines what can be observed.’ ”

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page 7 of 19

Gravitomagnetic horizons vs. Einstein’s GR, Eric Baird, February 2021

Acoustic metrics (whose predictions appear to be “dual” to those of quantum mechanics) [27] adopt

this later position, i and are capable of modelling more sophisticated and complex (and sometimes

frustratingly chaotic!) behaviours. It is one of the features of an acoustic metric that, while local

causality is everywhere preserved, a distant onlooker can see things that leave them confused.

Physical reality must generate observed reality in a simple and deterministic way … but the

process is under no obligation to work as simply or easily in reverse.

4.2. Direct and indirect observation

The idea that reality must only be defined by what we can directly observe was one of the

defining features of the Copenhagen Interpretation of quantum mechanics. Having decided

that light could only be emitted or absorbed by atoms in discrete quanta, we were not obliged to

assume that the light also moved from emitter to absorber as quanta. Light transmission might still

be wavelike, and perhaps, since any attempt to intercept the light with a detector to measure

whether it really was a wave or particle was doomed to always answer “particle!” because of the

quantised nature of our measuring equipment … perhaps the “particle” answer had no deeper

significance. If we chose to decide that light was transmitted as waves, there was no obvious

independent way of showing us that we were wrong.

The Copenhagen Interpretation disagreed. If light was always created and detected as quanta,

then this was reality, and it was philosophically wrong to hypothesise some other underlying

level of reality whose physical existence could only be inferred indirectly. Einstein vehemently

disagreed with the idea that QM statistics represented fundamental reality without a further

underlying explanation [35] (“Quantum mechanics is very worthy of respect. But an inner voice tells

me that this is not the genuine article after all. The theory delivers much, but does not really bring

us any closer to the secrets of the Old One. … I am convinced that He is not playing dice” [36]), but at

the time, subscribers to “hidden variable” interpretations of QM were in the minority.

QM applied to curvature horizons has since reintroduced and re-legitimised ( via the concept of

“real” and “virtual” particles) the idea of a level of reality that cannot be directly measured (by a

given observer), but whose existence can be deduced indirectly, and which does have a physical

effect on an observer’s other experiences (see: section 5). Hawking radiation has also prompted a

reassessment of some of our ideas about causality. While some QM researchers had believed that

its “unpredictable” processes were truly random, if this was true, the particle-pair-production

description of Hawking radiation implied that the radiation process was producing new,

“random” information from nowhere, [37] and since quantum-level events can be scaled up to

have macroscopic consequences (as in Schrödinger’s “cat” example), we then also lost

macroscopic causality. The universe would be allowed to do things for no reason.

On the other hand, if there was a persistent classical causality underlying and underpinning

quantum statistics (“microcausality”), then the information encoded in Hawking radiation had

to originate behind the horizon, and QM was doing a suspiciously good job of mimicking the

statistics of a universe in which classical horizons were effective rather than absolute, the motion

of massed particles warped spacetime, and SR-based classical theory was wrong. ii

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page 8 of 19

Gravitomagnetic horizons vs. Einstein’s GR, Eric Baird, February 2021

5. Compatibility with quantum mechanics

5.1. “Real” and “virtual” particles

The gravitomagnetic behaviours in our section 1-3 descriptions, and the resulting distinction

between “directly-observable” and “only-indirectly-observable” events (which makes no sense

under SR-based conventions), may be recognised by some readers as having direct counterparts

in modern quantum-mechanical descriptions of black hole behaviour – modern QM descriptions

support a distinction between particles that can directly reach the observer along unaccelerated

paths (“real” particles), and those that can only affect the observer indirectly, or via acceleration

(“virtual” particles). [39] i

As in our classical description, QM allows “real” particles to be converted to “virtual” particles by

acceleration; [40] [41] [42] ii iii particles deemed “real” for one observer can be “virtual” for another;

[42] a scoop dragged through a supposedly empty region can emerge full, as the scoop’s acceleration

converts virtual particles into real ones: lowering a fibre-optic cable into a hidden region lets us see

particles via light pulled out of the region by the acceleration forces in the cable; and a spaceship

passing by the horizon can be seen illuminated by light that would otherwise not be described as

existing there, with the collisions turning virtual photons into real photons.

In other words, every counterintuitive non-SR effect that results from our purely classical

arguments in sections 1-3 already seems to have a ready-made counterpart behaviour under

quantum mechanics. It would seem that QM already “makes a fit” to these behaviours (at least

qualitatively) in a way that can’t be achieved in an SR-based system.

5.2. Generation of Hawking radiation

Within an acoustic metric, horizon surfaces can be considered a form of observer-dependent

projective geometry. As a result, if a body behind a horizon undergoes a forced acceleration

towards the observer, the resulting spacetime distortion (Einstein 1921 [18]) may make the

projected horizon surface jump discontinuously from a location in front of the body to behind it,

making that body suddenly visible. If we were to mistakenly take the horizon as a fixed

reference, we might think that the object (and a surrounding region of space) had jumped

acausally into existence outside the horizon, perhaps as a quantum-tunnelling event. [43]

Like Hawking radiation under QM, acoustic horizons fluctuate and radiate and leak information.

[27] This ability to generate classical versions of QM effects that had no counterparts under

SR/GR1916 iv provided a strong motivation for the investigation of acoustic metrics, and the

subject started to be studied intensively from the late 1990s onwards. [27] [28] [29] v

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page 9 of 19

Gravitomagnetic horizons vs. Einstein’s GR, Eric Baird, February 2021

5.3. The QM pair-production description, from classical geometry

As previously mentioned, descriptions of Hawking radiation traditionally involve non-

classical pair-production effects occurring outside the horizon. Although the QM description

seems at first sight to defy classical interpretation, it also turns out to be the result of starting

with the “acoustic” classical description, and then inappropriately projecting SR-style

coordinates and causal conventions onto the final outcome:

If a particle undergoes a continuous acceleration or a series of accelerations that take it

outward through an effective horizon, far and fast enough to completely escape and reach a

distant observer, this observer’s attempt to back-calculate an inappropriate inertial history

for the particle will generate the “naive” 1970s description of Hawking radiation –

When we receive the escaped particle, we will know its final trajectory, but we will not

necessarily know anything about the accelerations and changes in trajectory that are part of

the particle’s history, and that are the reason it was able to escape. Working only from the

information available to us, a back-extrapolation of the particle’s supposed path from its final

state creates a false description in which a supposedly non-accelerated particle would have

needed to initially be travelling at more than lightspeed to escape. Since a particle

approaching at v>c is seen with its sequence of events and chiral characteristics reversed, an

escaped electron generates an artificial projected history in which the first, faster,

supposedly-superluminal part of its path appears to us to belong to a positron, moving away

from us. We then have a composite description of an (entirely fictitious!) escape path in

which the electron originated as one half of a particle-antiparticle pair created somewhere

outside the horizon, with the particle escaping and its antiparticle being swallowed.

A classical, continuous, and entirely causal description of radiation effects associated with

physical accelerations, when we try to explain the end results inertially, generates the

“traditional” 1970s non-classical, discontinuous, QM explanation of Hawking radiation. i

A final objection might be that these two descriptions cannot be dual, because if they were,

every observer stationed alongside the fictitious escape path would have to describe the pair-

production event as happening somewhere lower than their own position: different

observers would then assign the supposed event to different positions along the combined

path, potentially all the way down to a Planck distance above the r=2M horizon. ii This

description appears to be contradictory – however, the observer-dependency of the assumed

particle-creation point is also part of the QM description. iii

It appears that we have no obvious way of distinguishing between between the final

predictions of the non-SR “acoustic” description and those of the QM description. QM-

compatibility requires classical inertial physics to appear to be operating according to non-

SR rules and non-SR Doppler relationships.

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page 10 of 19

Gravitomagnetic horizons vs. Einstein’s GR, Eric Baird, February 2021

6. What happened to special relativity?

6.1. Incompatibility of effective horizons with the SR shift equations

The mechanism for indirect escape though a horizon does not work with SR-based systems.

Under special relativity, the Doppler equation for a body receding at v is: [6]

E'/E =

√

c−v

c+v

Swapping the sign of the velocity inverts the equation, so in SR-based systems, the gravitational

blueshift, E'/E, is the inverse of the corresponding gravitational redshift. For a hypothetical

stationary emitter-observer at the Schwarzschild horizon, the outward redshift on signals

exchanged with a distant observer would be E'/E=zero, and the inward blueshift, E'/E=infinity.

Such a body would be instantaneously vaporised by the infinitely-blueshifted infalling radiation,

and unable to resist the infinite associated inward radiation pressure. Not only can a massed

particle at r=2M not move outwards, it cannot even remain stationary. The r=2M horizon is then

an absolute barrier, analogous to the SR lightspeed barrier in standard C20th inertial physics.

If the gravitomagnetic H2 horizon obeyed the SR shift relationships and the associated shift was a

function of velocity as predicted by SR, then since the H2 horizon for O has a shift of E'/E=0, O

should presumably again expect the blueshift seen by a body at H2 to be infinite. H2 would need

to be another absolute horizon, and it should be impossible for O to accept that a body could

hover legally there. But in reality, a spaceship at H2 is quite capable of firing its engines and

hovering or passing outward through H2 into the visible zone.

We therefore have to conclude that the H2 horizon’s defining Doppler equation, as judged by O,

with v as the recession velocity of the hole, cannot be the one supplied by special relativity.

6.2. Replacing the relativistic shift equations to allow Hawking radiation

The fact that bodies must be allowed to move outward through H2 then puts severe limits on the

possible Doppler law candidates.

Assuming that the principle of relativity still holds, any departure from the SR shift equations

must be “Lorentzlike”, of the form E'/E = [1 – v2/c2]x . If we plot a graph of the inward blueshift

seen at the horizon as a function of x, only one solution is both non-infinite and non-zero [15] – a

“cliff-edge” solution, that only appears when the predicted shifts are exactly one full Lorentz

factor redder than SR’s. i

We therefore conclude that in order for a relativistic theory to allow classical Hawking radiation

(and agree with our gravitomagnetic description), the Doppler equations that apply in the

horizon region must be redder than those of SR by an additional gamma factor, representing the

additional effect of geometry-change due to gravitomagnetism. ii

This, coincidentally, turns out to be the same modification already required to bring Einstein’s

GR (which was originally designed around a static universe [45]) into line with modern

expanding-universe cosmology, [46] and to function consistently with gravitomagnetism.

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page 11 of 19

Gravitomagnetic horizons vs. Einstein’s GR, Eric Baird, February 2021

6.3. Gravitomagnetic shifts are also motion shifts

Gravitomagnetic shifts, like conventional motion shifts, are red for recession and blue for

approach. For a black hole, they have the same sign and magnitude as a conventional Doppler

shift, and take the same form as a conventional (non-SR) Doppler shift. If the gravitomagnetic

shift acts in addition to the SR motion shift, then the total motion shift must be non-SR and

around twice as strong as expected. A more credible option is that the gravitomagnetic shift is

the motion shift, calculated outside the time domain from the curvature associated with relative

motion. i In this case, the motion shift on a black hole is still non-SR. If a signal moving between

two gravitational bodies with relative velocity v also changes velocity en route by v, ii then light

has already adjusted to the speed of the receiver as it arrives – the gravitomagnetic effect

extinguishes and replaces the conventional motion shift.

We then also have gravitomagnetically-regulated local lightspeed constancy for all observer-

masses and no further need for special relativity.

6.4. All bodies are “strong-gravity” bodies at sufficiently small distances

At this point we might be tempted to create separate “domains” for the strong-gravity physics of

our moving black hole, and the negligible-gravity, effectively-flat physics of special relativity.

This is not possible (see page 5, footnote iii). Metric theories require the signals from a cluster of

objects with different macroscopic field strengths to shift with velocity by precisely the same

amount. The principle of relativity agrees – if weak-gravity bodies obeyed the SR Doppler

relationships, and strong-gravity bodies obeyed a different relationship, observers would be able

to tell who was “really” moving, and how fast, by exchanging and comparing signals. The

principle of relativity rejects the possibility of any difference between the Doppler shifts of

strong-gravity bodies and those of other masses – if the gravitomagnetic motion-shift of a

moving black hole is forced to obey a different equation to SR’s, then so must that of every other

mass, and we must live in a wholly non-SR universe. iii

The same geometrical tyranny that says that SR must be correct in a relativistic universe

supporting global c says that SR must not be correct in a relativistic universe with gravitation. We

may or may not like these results: the geometry does not care.

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page 12 of 19

Gravitomagnetic horizons vs. Einstein’s GR, Eric Baird, February 2021

6.5. Goodbye, special relativity

Special relativity is, of course, the provably-correct solution to restricted relativity in flat

spacetime. What is more challenging is establishing that the same flat equations should still hold

when we move on to a more advanced, general theory, in which all observer-masses have

associated curvature (via the principle of equivalence of inertia and gravitation), and in which

the motion of observer-masses must be accompanied by complicating gravitomagnetic effects

that the special theory does not attempt to model. Once we have multiple masses with relative

motion, all associated with their own velocity-dependent curvatures, we have a geometrically

dynamic system instead of the fixed flat geometry of Minkowski spacetime.

As pointed out by Moreau (1994 [50]), Minkowski spacetime and special relativity can be derived

from just the relativistic aberration formula and the SR Doppler predictions for wavelength-

change (which assume flat spacetime). Since all relativistic models share the same aberration

formula, any deviation from Minkowski geometry that still obeys the principle of relativity can

must be expressible as a Lorentzlike deviation from the SR Doppler relationships, after which, we

are no longer doing “special” relativity. i ii

A general theory of relativity must also be a theory of gravitomagnetism, and gravitomagnetism

is fundamentally incompatible with SR. As a matter of geometrical principle, the SR-style

simplifications and the resulting “flat” SR equations do not “carry over” into a gravitomagnetic

theory, in which any relative motion of matter must be associated with curvature side-effects. iii

GR and SR are two distinct solutions for two different scenarios, existing within two different

and logically-separate universes. It is therefore quite legitimate to defend the validity of SR in flat

spacetime, while rejecting its inclusion in a more advanced general theory. iv

&4)… 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).*[52]6!!

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page 13 of 19

Gravitomagnetic horizons vs. Einstein’s GR, Eric Baird, February 2021

7. Other notes

… i ii iii iv v vi

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Davies-Unruh (FDU) effect can be rigorously derived and extended to nonlinear quantum fields from the general Bisognano

and Wichmann’s theorem, the technicalities involved and probably its ‘paradoxical appearance’ has kept part of the community

quite skeptical up to now … Many physicists, thus, have decided to ‘leave the case to the experiments’.*[42]1

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page 14 of 19

Gravitomagnetic horizons vs. Einstein’s GR, Eric Baird, February 2021

8. Summary

Sections 1-3 of this paper show that Einstein’s 1916 general theory is, in its current

configuration, essentially unworkable.

Once we realise that distant observers can see different depths into a gravity-well depending on

their states of motion, the principle of relativity requires the relative motion of all masses to be

associated with strong gravitomagnetic curvature; once we accept the existence of velocity-

dependent curvature, we have a nonlinear, gravitomagnetic, “acoustic” metric rather than the

Minkowski metric, and a different set of equations to SR; once we realise that inertial physics is

not a flat-spacetime problem and is not correctly described under GR by flat-spacetime SR, then it

becomes clear that current “textbook” general relativity has been partly founded on inappropriate

geometry and principles, and also has the wrong gravitational shift equations.

Sections 4 onwards exist to reassure the reader that there is a way forwards: that the scary

“new” behaviours described are not new, i are not irrational or inexplicable, can be described

geometrically with the help of an acoustic metric, and are already predicted by quantum

mechanics. However, even if the reader does not accept the suggested solution, or acoustic

metrics, or Hawking radiation, or quantum mechanics, sections 1-3 alone are enough to put an

end to Einstein’s 1916 general theory. A researcher finding a problem in a current theory is not

obliged to offer a solution: if one is offered (as here), and the reader does not like it, then they are

cordially invited to try to construct their own – but regardless of whether or not “acceptable”

alternatives are available, the existing theory is still wrong. ii

It should be stressed that the non-SR behaviours described in sections 1-3 are not some tentative

proposed “hypothesis” or “theory” based on any personal convictions or beliefs as to how physics

might or ought to behave, and are not a matter of personal judgement or opinion: they are pure

geometry. Once we accept the existence of the secondary horizon in the case of a receding black

hole, all these non-SR physical behaviours are already in play.

While it was quite understandable that Einstein would want his initial theory of relativity to live

on as part of his general theory, [52] geometrical considerations make this impossible, leaving the

1916 theory dead in the water. At best we can consider it a transitional theory, with one foot in the

past and one foot in the future, a temporary “bridge” attempting to include aspects of two different

and incompatible systems, that might prepare the way for a “proper” general theory at some later

date. Unfortunately, a more appropriately-designed general theory then failed to materialise. iii iv

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page 15 of 19

Gravitomagnetic horizons vs. Einstein’s GR, Eric Baird, February 2021

9. Conclusions

It is a damning measure of the degree of failure of the 1916 theory that it cannot consistently

describe the external appearance of the simplest possible gravitational object, moving in a straight

line at constant velocity, as seen by a distant massed observer in otherwise-flat spacetime.

While the Mach-Einstein concept of a general theory was visionary, Einstein’s attempted

implementation of a general theory, as proposed in 1916, [8] does not work either as a physics or as

a geometry. This is due to the 1916 theory’s inappropriate adoption of relationships and

conventions derived from the 1905 theory’s assumption of flat spacetime, which fail in a curved-

spacetime context, and make the theory internally inconsistent and incompatible with modern

cosmology, [46] gravitomagnetism, and quantum mechanics.

A general theory of relativity is also a theory of relativistic gravitation and a theory of

gravitomagnetism, and relativistic gravitomagnetism turns out to require a very specific Doppler

relationship that is not the one given by special relativity. Constructing a valid general theory to

Einstein’s specifications, in which gravitational theory reduces to SR physics over small regions

for small bodies, is therefore a geometrical impossibility. A genuine general theory is an entirely

separate class of theory, and has to reject the 1905 theory’s “flat” philosophy, definitions and

relationships, and re-invent relativity theory from first principles, in the new context of metric-

distorting observers, gravitomagnetic lightspeed regulation, QM-style observer-dependency, and

curved spacetime. It must be designed as an iteration of relativity theory, not as an incremental

upgrade that tries to retain all the features of the previous system. What Einstein delivered in

1915/1916 was in many ways still an evolutionary step, when what was required was something

much more revolutionary.

While these problems can all be fixed – this paper mentions the necessary additional Lorentz

modification that needs to be made to the SR/GR1916 Doppler relationships, and existing peer-

reviewed work on acoustic metrics provides an analytical and definitional framework for an

“acoustic” relativistic replacement to Minkowski spacetime [27] – the biggest problem preventing

further scientific advance is not technical, but psychological and social:

It requires us to understand and appreciate, in the face of a century’sworth of teaching to the

contrary, that special relativity is not to be thought of as a proper or credible foundation theory

for gravitational physics.

---<=>---

page 16 of 19

Gravitomagnetic horizons vs. Einstein’s GR, Eric Baird, February 2021

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,,,---,,,

page 19 of 19