Essay written for the Gravity Research Foundation 2017 Awards for Essays on Gravitation. 29-Aug-2016
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When a black hole moves: The incompatibility between gravitational
theory and special relativity
Eric Baird firstname.lastname@example.org
Abstract: When a gravitational source moves, gravitoelectromagnetic effects alter
the energy and momentum of nearby light, causing the body's motion-shift
characteristics to depart from those of special relativity. Since wave theory requires
identical equations of motion for “gravitational” and “non-gravitational” bodies, a
full gravitational theory cannot reduce perfectly to the physics of special relativity.
The relationship between special relativity and gravitational theory has never been entirely
straightforward. After designing the 1916 general theory of relativity to reduce to the existing 1905
“special” theory as a limiting case, [
] Einstein wrote in 1950 that he no longer believed this approach
to be legitimate, [
] and in 1960 we discovered a fundamental geometrical conflict between special
relativity and the general principle of relativity applied to rotation, [
] the community's response being
to give SR priority, and downgrade the GPoR from a rule to a guideline that was to be suspended
whenever it threatened to disagree with SR-based arguments.  We then found in the 1970s that
the resulting SR-centric theory refused to mesh with quantum mechanics and with basic statistical
laws, apparently as a result of the SR component.
To these difficulties we will now add an even more fundamental problem:–
It appears that the equations of motion for a gravitational body – and by extrapolation, the equations
for any moving body whatsoever – need to obey a different set of equations to those of special
We will start by imagining a region of space containing a Wheeler black hole, and consider only the
rays of light that were aimed directly away from the gravity-source. The approximate behaviour of the
forward lightcones for events [
, while the corresponding sketch for just the outward-aimed rays is:
Figure 1: Forward light-cones around a stationary black hole
Figure 2: Outward-aimed lightrays around a stationary black hole
When a black hole moves … (Eric Baird, 2016)
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The horizon surface at r=2MG/c2 (abbreviated “r=2M”) is commonly interpreted as a critical surface
at which outward-aimed light-rays are expected to be “frozen” at the horizon and stationary with
respect to the hole – pointing parallel to the hole's path in four dimensional spacetime.
We can now examine how the situation appears from the point of view of an observer moving away
from the object:
If the outward-aimed rays originating at r=2M travel with the hole, then these “critical” rays at the
rear of the receding hole must now point away from the rearward observer. With ray-angle a smooth
function of position, the critical surface at which outward-aimed rays appear frozen to our observer
(i.e. appearing vertical in their spacetime diagram) must now lie somewhere outside the originally-
defined surface, with the moving hole's “official” horizon at r=2M trailed by an “effective” horizon
some distance behind.
A functional description of the physics seen by this observer is that the receding black hole appears to
show an enhanced gravitational pull in its direction of motion – it is exerting an asymmetrical, velocity-
dependent gravitoelectromagnetic (“GEM”) effect in addition to its conventional static gravitational
field, changing the energy and momentum (and angle) of nearby light-rays.
If we build a polyhedral framework fitted with detectors and transceivers around the hole some
distance outside r=2M, and the motion-shift of the “empty” framework is correctly described by flat
spacetime and special relativity, then since the gravitoelectromagnetic shift effect is not part of this
calculation, its inclusion as an additional effect when the moving framework encloses a gravity-source
will inevitably produce a departure from the SR motion-shift predictions.
It is tempting to assume here that special relativity is still fundamentally correct, that any such GEM
correction must be tiny, and that a significant divergence from SR can only appear in bodies with
sufficiently strong surface gravitation (“For weak gravity use SR, for strong gravity use GR”). In reality,
the GEM motion-shift effect for a black hole seems to have the same magnitude as the conventional
Doppler effect, and compatibility with wave theory requires the same motion-shift relationships to
apply to all objects regardless of their gravitational properties (principle of universality, below):
– Imagine a remote system far from the Earth containing multiple signal sources with different surface
gravities and different constant states of motion, and further imagine that each signal source emits a
(different) single frequency of light, and that circumstances conspire (somewhat improbably) to make
all these signals arrive at our detector with the same final frequency and phase, along effectively the
same path. We would receive a simple monotone sinewave. If motion-shifts varied with the surface
gravity of the source, and we changed our own state of motion, this sinewave signal would then need
to “split” at our detector into differently-shifted components depending on the surface gravities of
the original distant sources, a result that conflicts with wave theory.
If the recession velocity is sufficiently large, and the framework is sufficiently close to r=2M, the receding
hole's effective horizon may even appear to intersect the framework.
Figure 3: External “effective” horizon behind a receding black hole
When a black hole moves … (Eric Baird, 2016)
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If the “moving black hole” exercise shows any deviation from the equations of special relativity, then
all other moving-body problems need to show precisely the same deviation
– wave theory demands
that “the black hole solution” applies everywhere.
With this in mind, we can now try to resolve our conflict.
Given that a single motion-shift law must apply to both gravitational and non-gravitational bodies, we
have three main options:
Approach #1: SR plus gravitoelectromagnetism
Assuming the validity of the SR approach, and also accepting the existence of additional GEM
motion-shift effects, results in an odd situation in which SR's flat geometry, derivations and
equations are all considered nominally correct, but do not correctly describe physical reality,
Approach #2: A “curvature-compatible” special theory
The correspondence between the polarity and magnitude of the GEM motion-shift effects and
conventional Doppler shifts suggests an alternative approach: What if the GEM effects and
the more familiar Doppler effects are not cumulative, but are dual, alternative descriptions of
the same physics? Inertial physics would then no longer be a flat-spacetime problem, the usual
derivations and proofs of special relativity's Doppler relationships would not apply,
would need a new derivation more compatible with curved spacetime.
Can we rederive special relativity without presuming flatness? Apparently not – given that
SR’s relationships are a perfect expression of the geometrical relationships of flat Minkowski
spacetime, it should not be possible to obtain them all, exactly, from a physically different
Approach #3: The “total rewrite” option
This leaves us with the intriguing possibility of a “freestanding” general theory of relativity
with no imposed reduction to Minkowski spacetime – a “GR without SR” realising the vision
of Clifford in 1870 [
] (“all physics is curvature”), and Einstein in 1950  (“no physics without
The resulting “Cliffordian” general theory would not need to compromise its structure or
principles to accommodate special relativity, would show more strongly nonlinear behaviour
than GR1916, would presumably have to adopt a different set of Lorentzlike equations, and –
since its spacetime would then be described by a relativistic acoustic metric – would “mesh”
more easily with quantum mechanics. [
At this point, the only remaining defence of special relativity would seem to be that if these GEM
effects really are incompatible with SR, then since SR cannot be wrong, the effects cannot exist, on
principle. This is perhaps more of a defensive “theological” position than an investigative and
For instance, by treating fundamental particles with mass as tiny black holes.
In a Cliffordian universe, a general theory is not geometrically required to reduce to flat-spacetime physics –
curved-spacetime physics is self-contained, with no underlying layer.
When a black hole moves … (Eric Baird, 2016)
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The idea of using a black hole problem to undermine special relativity may seem misguided – special
relativity is (after all) explicitly a flat-spacetime theory, and a black hole arguably represents the
maximal departure from flatness. “It should not be considered surprising”, a physicist can respond,
“that the special theory is not up to the job of describing a black hole problem, as it was never designed
for this situation! What is required here is the application of full general relativity!”
However, general geometrical considerations seem to require the relativistic equations of motion of
a body with a significant gravitational field to be something other than those of special relativity, and
the principle of universality then requires the behaviour of all other bodies in the universe to show
analogous behaviour – in other words, in reality there is no such thing as flat-spacetime physics. The
exercise appears not just to invalidate special relativity, but also (by association) Einstein's 1916
general theory, the amended 1960 version, and also every other gravitational theory currently
designed to reduce to the physics of SR as an exact solution.
This is not the first time that incompatibilities between special relativity and other fundamental
principles have been discovered that do not seem to permit a “clean” geometrical resolution. In this
case, though, the incompatibility appears to be truly primal – SR seems to be not just irreconcilable
with the principle of relativity applied to rotation , but also irreconcilable with almost any general
description of gravitational physics in which objects with gravitational fields can support the property
of relative motion.
In conclusion: A fully-functional general theory of relativity, or any other fully consistent gravitational
model, cannot and should not include a reduction to the flat-spacetime equations of the 1905 theory.
] Albert Einstein, “What is the Theory of Relativity?”, The Times newspaper, 28th November 1919.
] Albert Einstein, “On the Generalized Theory of Gravitation”, Scientific American, April 1950, pages 13-17.
] Alfred Schild, “Equivalence Principle and Red-Shift Measurements”, Am. J. Phys. vol. 28, pages 778-780
] Charles W. Misner, Kip S. Thorne and John Archibald Wheeler (“MTW”), Gravitation section 33.2:
“Principal features of Holes”, page 881 (Freeman, New York, 1973). ISBN 0716703440
] W. K. Clifford, “On the space-theory of matter” Proceedings of the Cambridge Philosophical Society
1864-1876, vol. 2, pages 157-158 (1870).
] Carlos Barceló, Stefano Liberati, and Matt Visser, "Analogue Gravity" gr-qc/0505065 (2005-)