The text below contains my findings which may be of interest to
people who have inquisitive minds and want to discover how Nature
really works. (This text is taken from my book "Foundational Flaws
in Modern Physics", which can be found at [Only registered users see links. ])
Summary of Chapter 4:
SPACETIME CURVATURE AND GRAVITATION
The 'bending of light' is usually considered by relativists as a
confirmation of spacetime curvature, even though it can also be
explained by a variation of the speed of light alone. Many
prominent physicists, like Max Born, admit that flat space has not
been disproved experimentally. The fact is that there is really no
experimental basis for saying that our space is curved and not flat.
Some physicists, like B. S. DeWitt, believe that: "Both the
geometrical and the flat space-time points of view have the same real
physical content" (DeWitt: [R#1] p.267). R. H. Dicke pointed out
that "a units transformation can be used to redefine the Riemannian
geometry of general relativity in such a way that the resulting
geometry is flat" (Dicke: [R#2] p.2163).
According to a well-known relativist and a coauthor of Gravitation,
Kip S. Thorne, "physicists can and do use the two viewpoints
interchangeably when trying to deduce the predictions of general
relativity" (Thorne: [R#3] p.400). K. S. Thorn writes that the two
paradigms "are mathematically equivalent". However, even if this
were true, they no doubt differ from the physical and epistemological
point of view. One of the most important differences is that the flat
paradigm not only does not need spacetime curvature, but it does not
even need the four-dimensional spacetime. This was maybe not
important for Thorne's considerations, but it is of tremendous
importance for the unification of physics. As we know, quantum
mechanics does not seem to be compatible with the four-dimensional
"reality". Besides, as we've already seen, the concept of time as the
fourth dimension is just an ad hoc hypothesis which leads to
predictions contradicting evidence.
In Einstein's days one could've justifiably said that experimental
evidence did not let us positively decide in favour either of flatness or
of small curvature; but this is no longer true. Experiments
conducted decades ago have proved beyond any doubt that the curved
spacetime paradigm is flawed. Hence, the only correct and adequate
paradigm for explaining reality (and not just for a mathematical
description of it) is the flat space paradigm.
As is explained in §4.2 ([Only registered users see links. ]),
a clock on the surface of the Earth is affected more by the
gravitational field of the Sun than by the Earth's field (roughly 14
times more). Because of the huge distance from the Sun, its field
around us is almost homogeneous. Therefore, clocks anywhere on
the Earth are slowed down by this field by almost the same amount.
A clock on the side of the Earth which faces the Sun is slower (due
to the Sun's field) than a similar clock on the opposite side of the
Earth, by an insignificant amount. However, this would not be so if
clocks were affected by spacetime curvature. A clock on the side
facing the Sun would be affected by the curvature of the Sun
diminished by the oppositely directed curvature produced by the
Earth. At the same time, a clock on the opposite side of the Earth
would be affected by the curvature of the Sun increased by the
Earth's gravitational field. This clock would have to be slower than
the one facing the Sun, by a significant amount. With the precision
of modern atomic clocks this difference could not have been
overlooked. If Nature worked like that - GPS would be very
inaccurate, if its clocks were not adjusted to account for this effect.
But this adjustment is not needed; and this proves that it's not the
spacetime curvature which affects clocks.
In summary: Spacetime curvature cannot be considered as an
explanation or a cause for the gravitational effects on clocks. Even
worse, it leads to incorrect predictions of the value of these effects.
Hence, the claim - that the flat paradigm is equivalent to the
spacetime curvature paradigm - is incorrect. Spacetime curvature or
tensor approaches fail miserably in situations in which more than one
gravitating body are involved - whereas the flat space paradigm
considerations always give correct solutions.
In any case, since the gravitational red-shift does not depend on
space-time curvature - it cannot prove the existence of curved
space-time as C. Will and others seem to imply.
Some physicists seem to believe that "Einstein thus reduced gravity
to pure geometry" (Davis: [R#4] p.150). But, as P. G. Bergmann -
who was one of Einstein's collaborators - pointed out: "Einstein has
stated repeatedly that he did not consider geometrization of physics a
foremost, or even a meaningful objective, and I believe that his
comments remain valid today" (Bergmann: [R#5] p.16).
Evidently, Einstein didn't consider spacetime geometry as an
explanation of gravitation; but only as a way to describe gravitational
interactions. This is quite clear from the fact that he did not count
his general relativity as a constructive theory.
But even more telling is the fact that in 1928 Einstein started
working on a theory in which spacetime curvature is abandoned and
distant parallelism is adopted (for more details see section §4.3 [Only registered users see links. ]). Einstein's work on distant
parallelism showed that he did not believe that gravitation could be
explained by the curved spacetime.
In 1911 Einstein predicted the bending of light phenomenon, but his
prediction amounted to half of the correct value. Many claim that
this prediction was incorrect because he didn't take into account the
curvature of spacetime; but this is misleading because he could have
arrived at the correct prediction without assuming spacetime curvature.
When we compare the value of the effect of gravitational field on
the value of the speed of light, which Einstein assumed in 1911, to
the one assumed by Einstein later - we see that they are not the
same. In Einstein's later assumption the effect of gravitational field
on the velocity of light doubled, so no wonder that his numerical
prediction of light bending also doubled (for more details see §4.4 [Only registered users see links. ]).
If Einstein used this latter value of the light velocity in 1911, he
would've gotten the correct value of light deflection. This may be a
surprise to most who believe that the so called spacetime curvature is
needed to fully account for observed light deflection by the Sun. Not
so. The curvature is not only not needed, but actually if one wanted
to add any effect of curvature to the result which Einstein should
have arrived at in 1911, one would get an incorrect prediction. The
curvature effect has to be considered as zero. And hence, the result
of the 1919 experiment should not have been taken as proving that
there is a space or spacetime curvature!
There are implications that "the speed of light is not only observed
to be constant; in the light of well tested theories of physics it does
not even make any sense to say that it varies" (Gibbs & Carlip:
We know beyond any doubt that clocks in gravitational fields slow
down. If an observer in a given gravitational field measures the
speed of light with the help of his slower clock, she/he can get the
result c only if either his rulers expand or light slows down.
Moreover, the above does not depend on what theory one uses. And
because there is no theory which suggests that rulers in gravitational
fields get longer, we must assume that it's the speed of light which
compensates for the lower rates of local clocks. The speed of light
is observed to be constant, but it's just because of the effects of
gravity on rods and clocks. However, the coordinate speed of light
differs, as measurements unambiguously show.
As we know, electric and magnetic fields, and wavefronts of
electromagnetic radiation can only be perpendicular to the direction of
motion. Hence, if each side of a wave front travels at the same
speed, the direction of propagation of this electromagnetic wave
cannot change. Even if the wavefront were by some means shifted
sideways (along a direction parallel to the wavefront) - the direction
of propagation of this electromagnetic wave would stay the same. So
if something, a gravitational field or whatever, doesn't cause one side
of the wavefront to slow down (or speed up) relative to the other side
- it cannot cause a change of direction of propagation of the wave
As is explained in §4.4 ([Only registered users see links. ]),
even if some kind of curvature of space caused the energy of
electromagnetic wave to shift sideways near the Sun, where this field
is strong, this would not affect the direction of propagation of this
wave at a greater distance from the Sun. So curvature of spacetime
cannot change the direction of propagation of electromagnetic waves.
Hence, we must conclude that in contradiction to the common belief,
the bending of light experiments do not prove that spacetime is
curved, because the observed effect cannot be caused by a spacetime
curvature near the sun as it should be expected. So there is no
basis to believe in some spacetime curvature. Only the variation of
the coordinate speed of light can explain the bending of light.
Evidently, Einstein was right in stating that "A curvature of rays of
light can only take place when the velocity of propagation of light
varies with position" (Einstein: [R#7] p.76).
As is pointed out in §4.5 ([Only registered users see links. ]),
the time delay of light experiments show that the coordinate speed
of light is changed by gravitational fields. This change of the speed
of light explains the observed deflection of light totally and there is
no basis for claims that the bending of light experiments prove the
existence of some spacetime curvature.
There is still a lot of literature which contains erroneous implications
that even the time delay experiments need spacetime curvature for
their explanation. However, from geometric considerations one can
easily deduce that the length of the path cannot increase by much if
the curvature is at least comparable to that of the path of light, as
observed in the bending of light experiments. The curvature would
have to be a few orders of magnitude greater to be able to explain
half of the time delay of the signal. Moreover, we would have to
assume that light follows some weirdly curved paths, which are not
permitted by evidence.
It is evident that the spacetime curvature does not solve or explain
anything - in fact - it only fools us into believing that it does.
Students of physics should be aware of the fact that some texts create
a false impression that the opposite is true (for more details see
§4.5.1 - [Only registered users see links. ]).
Rejecting Einstein's belief, that the speed of light is affected by
gravitational fields, would leave us without any real explanation of the
time delay of light experiments and it leads to paradoxes; as is
explained in §4.5.2 ([Only registered users see links. ]).
The belief that between a given pair of points, A and B, there is
only one "straightest possible" line (a geodesic) which any body
moving from A to B must follow as "space-time geometry" dictates -
is false. We believe in it despite the fact that we clearly see that in
our three-dimensional realm it is incorrect.
The paths of water and of light, as is discussed in section 4.6
([Only registered users see links. ]), are not the same line; it
doesn't matter how we call them, geodesics or whatever. If we
exclude two points at which these paths cross (see Figure 55 on
p.417 ([Only registered users see links. ])), none of the points which
are crossed by the light ray are crossed by water at the same time.
Therefore, the points in space which are on the line of motion of
photons cannot, at the same time, belong to the line along which the
water particles move. Hence, they cannot have the same spacetime
coordinates. No fiddling can help - two different lines in three
dimensions can't become one line in four dimensions, at least
according to any known geometry. Implying that the path of light
and of water is the same geodesic line - is implying a falsehood.
The four-dimensional curvature alone does not explain the path along
which a body must move and it doesn't explain effects of gravitation
on moving bodies. And hence, as we discussed earlier, geometry
alone can't be said to dictate or explain motion of bodies in a
gravitational field. Some other factor is needed. So general
relativity cannot be considered as a theory explaining gravitation and
it would be misleading to imply that the spacetime curvature explains
In a gravitational field, there is no single unique straight line
connecting any 2 given points which a test body must follow. There
are many of them. Which one the body must follow, depends on its
velocity. Even if we consider this velocity only as a "direction" in
four-dimensional spacetime, it changes nothing. Just imagine that the
Earth is an ideal globe. We know that there are many "straightest"
possible lines connecting the North and South poles. They have
exactly the same curvature and length. Can we say that they are all
just one line which passes through Greenwich? Only the direction,
in which one starts to move, dictates whether following a straightest
line connecting the North and South poles - will make one travel
through Greenwich or Tokyo. And it would be foolish to say that it
is geometry alone that dictates whether one travels through Tokyo or
Greenwich. This makes it clear that four-dimensional geometry by
itself doesn't explain why bodies move as they do.
We must stop interpreting general relativity as a constructive theory
and its mathematical concepts as physical entities. Let's stop
confusing mathematics with physics.
Non-linearity of Einstein's equations is not an asset, as some
believe; it is a liability, as has been explained in section 4.7
([Only registered users see links. ]).
Because it is generally believed that General Relativity (GR) is an
extension of special relativity (SR), many imagine that GR is also
compatible with electromagnetism. Yet, it is not; even though many
tried to make them compatible.
But the nonlinearity made general relativity incompatible not only
with Maxwell's theory and SR, soon it was discovered that it was
also incompatible with newly developed Quantum Mechanics (QM).
And all attempts to make GR and QM compatible also failed.
The problem is that working on changing or rejecting QM's
interpretation is futile, because QM does not depend on one.
Changing or rejecting its formalism, which gives us so precise a
description of natural phenomena, would be equivalent to rejecting
experimental data; hence it also seems to be totally pointless. So
what can we do to make QM compatible with nonlinear GR? It
seems that we can do nothing!
It is commonly believed that the equations of General Relativity
(GR) precisely describe the gravitational interactions. However, what
may surprise many, for more than three quarters of a century we've
had evidence which clearly shows that this is not always true. We
have solutions of Einstein's nonlinear equations which "permit for
curvature even when no matter or energy is around" (Peat: [R#8]
p.17). These solutions prove beyond any doubt that Einstein's
nonlinear equations are not always correct. This means that they are
at best only approximations and not the exact representations of the
true laws of nature.
The correctness of the linear formalisms of Maxwell and quantum
mechanics has been confirmed by a multitude of experiments.
However, the correctness of the assumption of nonlinearity of the
gravitational laws - has very little evidence in support. In fact, as
Walter E. Thirring wrote: "It turns out that only the perihelion shift
of planetary orbits is sensitive to the nonlinearities of the theory"
(Thirring: [R#9] p.112).
Obviously, now we have more evidence of this kind coming from
observations of binary stars etc., but it is still the same type of
evidence. So should we try to change the rest of physics on the
basis of one kind of evidence and in disregard to all other evidence,
which seems to indicate that the laws of nature are linear? Should
we also ignore the fact that there is one kind of evidence which seems
to indicate, as was mentioned earlier, that the nonlinear gravitational
equations don't always give correct results?
This is what has been done for more than half a century, but
without any success. However, it looks that there is a simple
resolution of the issue. It looks that we don't need to assume that
gravitational laws are nonlinear. The need for nonlinear equations
disappears when we appreciate the fact that there is an additional
linear effect which interferes with the linear Newtonian gravitational
As we know, a particle's velocity has a limit - it cannot exceed c.
We also know that it is not just an abstract law. The impossibility of
a matter particle to reach the velocity of light is directly related to
what the equation m = m_0/sqrt(1-v²/c²) conveys. When, in the case
of a given particle traveling at velocity v, the ratio v/c approaches 1,
the total mass-energy of the particle approaches infinity.
Experiments leave no doubt that it is the ratio v/c which dictates the
ratio of rest energy of a particle to its total energy. Simply, it looks
that nature doesn't allow particles to travel faster than EM fields.
As we have learned, when light enters a strong gravitational field, its
coordinate velocity decreases. It is not hard to figure out that the
same must happen with matter particles which travel at subluminal
velocities. If it were not so, they would travel faster than light.
The speed of light we measure is no doubt diminished significantly by
the gravitational potential of the matter of the universe. Yet, we
don't observe any particles moving faster than this 'slower' light. In
this context - the experimental evidence which confirms that the total
energy of a particle depends on its velocity according to eq.: m =
m_0/sqrt(1-v²/c²), lets us understand the equation in more precise
terms. This evidence teaches us that it's the local velocity of light
(and in local units) that stands for c in this equation. And the same
must also apply to the variable v. What really seems to matter is
the ratio v/c, in which v and c are as measured locally.
The above effect must have an influence on orbits of planets like
Mercury. When the planet gets closer to the Sun, its coordinate
velocity is not exactly as we would predict on the basis of Newton's
equations. Its value must be multiplied by (1-2GM/Rc²) and hence,
the closer to the Sun it is, the greater the discrepancy. This means
that Mercury stays in the area where the gravitational field is
stronger for a little bit longer time than we would expect; and hence,
its orbit is curved more than Newton's equation would make us
predict. This effect explains 1/3 of the perihelion shift.
There is one more phenomenon, related to the above, which we
have to consider. As we know, light always moves in a direction
perpendicular to its wave-front. It cannot move 'sideways'. In a
medium which is inhomogeneous along the direction perpendicular to
the direction of motion of light, different parts of a given wave-front
move at different velocities. The wave-front changes its orientation.
As we said, the light can't move sideways - hence as the orientation
of the wave-front changes, the direction of the light's motion must
Now, experiments clearly show that matter in motion exhibits
wavelike properties. The question arises: is the propagation of matter
waves affected by the gravitational field being inhomogeneous, in a
similar manner as the propagation of light waves? This should be
the case if the matter waves have some physical reality.
Experiments seem to leave us no choice in this matter. For this
reason - as some say - "the entire discipline of quantum mechanics is
based on a wave description of matter" (Fromhold: [R#10] p.503).
Calculations show that taking the above mentioned facts into
consideration leads to the expectation that Mercury's perihelion will
shift exactly by the amount which is observed. This cannot be a
coincidence. And according to me, this result leads to more than
just the same correct prediction as arrived at by using Einstein's
equation; it leads to understanding why the effect takes place. And
the explanation of the effect is easy to understand because it is in
three-dimensional terms. Acceptance of this explanation is virtually
forced on us by experimental evidence from the field of wave
mechanics, and from observed variable properties of Einstein's ether
or physical space.
As we know General Relativity (GR) is only a phenomenological
theory. The amended Newton's theory is not only simpler than GR,
but it is also the only theory which explains the effect. Moreover,
GR's "machinery of curved space-time" is not only more complicated,
but in fact not very usable.
There is no doubt that "the Mercury prediction" played a very
important role concerning the acceptance of GR. It also led to
creating a legend about the accuracy and usefulness of Einstein's GR
equations. Reality is sobering. There are very few situations in
which the exact correctness of these equations can be checked. In
fact, Einstein's equations cannot be solved for the case where there
are two or more gravitating bodies (for more details see §4.8.1
([Only registered users see links. ])). So they cannot be solved
in the case of our planetary system, and this solution is needed for
calculating Mercury's perihelion shift; because 92% of this shift is
caused by the gravitational attraction of other planets.
In practice: we calculate the influences of other bodies on the path
of Mercury by using the Newtonian method which is assumed not
exactly correct. On top of this, we superpose these influences; which
is incorrect according to General Relativity (GR). Moreover, we
have no means to know how the predictions of GR and of the
Newtonian theory compare in this case because there is no way to
solve Einstein's equation for multi-body cases. Hence the only thing
we know is that GR would give a different prediction as to the
magnitude of the perturbation, which is exerted by all other planets on
Mercury's orbit. Only if the difference were small could GR be
considered as correct. If the difference weren't small, GR would
have to be considered as incorrect. This already shows that we need
a dose of faith to believe that the shift of Mercury's perihelion proves
that GR correctly describes gravitational interactions. And this is
not the only problem here.
This "proof" of the accuracy of Einstein's equations which looks so
convincing at first sight, is not proof at all. It is based on our faith
in the correctness of the assumptions we used, and not on real
calculations according to Einstein's equation, because these cannot
As is evident, if gravitational fields of more than one body need
to be taken into account, the amended Newtonian scalar theory is
the only choice; and it is the only choice not just in the case of
Mercury's perihelion shift. Moreover, it is the only constructive
theory of gravitation which can explain gravitational interactions,
and not just predict them.
[R#1] Editor: Louis Witten (RIAS) - "Gravitation: an Introduction to
Current Research" - John Wiley & Sons, Inc. - New York ·
London - 1963.
[R#2] R. H. Dicke - "Mach's Principle and Invariance under
Transformation of Units" - Physical Review 125 (1962) p. 2163.
[R#3] Kip S. Thorne - "Black Holes & Time Warps - Einstein's
Outrageous Legacy" - W. W. Norton & Company - New York ·
London - 1994.
[R#4] Paul Davis - "Superforce - The Search for a Grand Unified
Theory of Nature" - Simon and Schuster - New York 1984.
[R#5] Editor: Yuval Ne'eman - "To Fulfill a Vision" - Addison-Wesley
Publishing Company, Inc. - London · Amsterdam · Don Mills,
Ontario · Sydney · Tokyo - 1981.
[R#6] Philip Gibbs "Is the Speed of Light Constant?" - 1 July 1996,
updated 8 August 1997 by Steve Carlip - Usenet Physics FAQ - [Only registered users see links. ]
[R#7] Albert Einstein - "Relativity - The Special and General Theory"
- translated by Robert W. Lawson - 15th edition - Crown
Publishers, Inc. - New York - 1952.
[R#8] F. David Peat - "Superstrings and the Search for the Theory
of Everything" - Contemporary Books - Chicago · New York -
[R#9] Walter E. Thirring - "An Alternative Approach to the Theory
of Gravitation" - Annals of Physics 16 (1961), pp. 96-117.
[R#10] Editor: Robert A. Meyers - "Encyclopedia of Modern Physics"
- Academic Press, Inc. - Harcourt Brace Jovanovich, Publishers