The Perihelion
Advance Formula
Tom Van Flandern
Meta Research / Washington,
DC
There are three well-known tests of
general relativity (GR) proposed by Einstein: the extra bending of light rays passing
near the Sun, the slowing of clocks immersed in a gravitational potential, and
the advance of the perihelion of planet Mercury. In the 1960s, Irwin Shapiro
added a fourth test, the slowing of radar signals through the Sun's field. [See
Shapiro et al., 1971.] It was well
known even to Sir Arthur Eddington (1920) that some of these results were also
predicted by aether theories. However, that was not known to be true of the
advance of the perihelion of Mercury, which seemed by far the most impressive prediction
of the theory.
Imagine a tangible but presently undetectable
light-carrying-medium (LCM) surrounding the Sun, whose density varies with the
Sun's gravitational potential. As a light wave passes through such a medium, it
will be bent by refraction away from a radial vector outward from the Sun. This
is because the wave's propagation speed is slower where the medium is denser.
In standard GR, the Einstein curvature tensor is interpreted to indicate the
curvature of space-time. If it is instead interpreted as the density of a
medium varying with the Sun's gravitational potential, then the mathematical
formalism is identical, and the correct light bending amount by refraction is
predicted. (See Eddington, 1920.)
In a similar way, all electromagnetic phenomena must
propagate through the LCM with speeds that slow where the medium is denser.
Hence, the slowing of radar signals and of atomic clocks in a gravitational
potential are predicted with equal ease. (See Hatch, 1999.) Our task here will be to determine if the
same principles apply to predicting perihelion advance, or if that phenomenon
is unique to GR.
"Perihelion", of course, refers to the
point in an elliptical orbit that is closest to the Sun. Our developments here
will apply more generally than just the solar system, so, strictly speaking, we
should use the generic expression "pericenter". However, since we
intend to focus on applications to the planet Mercury, we will continue to
speak of the "perihelion" advance formula.
For Mercury, and in general for all orbits that are
nearly ellipses, the basic form of a formula for perihelion motion is always
the same, being governed by the properties of orbital motion on an ellipse.
Perturbations that are themselves modulated by the size and shape of the
elliptical orbit and by the speed of a body traveling along that ellipse
generally change perihelion motion by simple integer multiples of the basic
form. This basic form (
)
is:
,
where
is the orbital mean motion of the planet, 
is its orbital period, 
is the product of the gravitational constant
and the mass of the Sun, 
is the semi-major axis (mean distance) of the
orbit, 
is orbital eccentricity, and 
= speed of light. We will show the derivation
of this basic form below.
The nature of this basic form is
such that parameter-free perturbations are nearly all constrained to produce
perihelion motions that are integer (or at worst, half-integer) multiples of
it. (This is to be contrasted with parameterized perturbations, for example
those resulting from some undiscovered mass or force, that are proportional to
some free parameter.) The observed perihelion advance for Mercury is three
multiples of the basic form, 
,
to within the error of observations.
In GR, the correct multiplier of is arrived at by combining three
contributions. The first is the effect of "time dilation", which
contributes 
.
The second is the effect of "space contraction", which contributes 
.
The third is the effect of mass or momentum increase with speed, which
contributes 
.
The sum of these three contributions gives the observed amount, .
Einstein, of course, knew the observed amount in advance, and had the
opportunity to combine various effects in various ways until the correct answer
was obtained. It is curious that Einstein required a combination of three
effects, with one of them canceling 40% of the contribution of the other two.
Clearly, there was ample room to argue for a different combination if Mercury's
actual perihelion motion had been different.
By contrast, aether theories have no
such freedom. There is no "time dilation", only clock slowing; and
that has no direct effect on the perihelion motion. There is no "space
contraction", only meter stick shrinkage; and that has no effect on the
perihelion motion. Likewise, just as Mercury's own mass makes no significant
contribution to its own perihelion motion, any mass or momentum change for
Mercury should have no effect on its perihelion motion. Indeed, if matter could
always be treated as an ensemble of particles with purely ballistic motion,
aether theories would not contribute anything to perihelion advance.
However, electrons are not particles
with purely ballistic motion. Electrons have primarily wave-like properties, and
are the main reason that light is considered an "electromagnetic"
phenomenon. Indeed, one of Louis de Broglie's chief contributions to physics
was demonstrating that ordinary matter has wave properties too. We are
therefore obliged to consider that orbiting bodies will be influenced by the
density of the LCM that they travel through because of the influence of the LCM
on their electrons.
Understanding the effect of the LCM
on perihelion motion geometrically is simple. The elliptical motion of orbiting
bodies is slowed most by the LCM near perihelion, where that medium is densest;
and is slowed least near aphelion, where the LCM is sparsest. This velocity
imbalance (slower at perihelion, faster at aphelion) rotates the ellipse
forward, which is what an advance of perihelion means. See Figure 1.
For a more detailed understanding,
we begin by noting that the velocity of anything propagating through a medium
is inversely proportional to the square root of medium density -- a general
wave property derived in many elementary physics texts. When comparing
observable phenomena for a stationary body at different gravitational
potentials, the relativistic clock-slowing, meter-stick-contraction factor is 
,
where 
and the gravitational potential, 
(where 
is distance from the Sun), is proportional to
changes in medium density.
Figure 1. The weaker-than-average light-carrying medium
(gray) near aphelion adds a small velocity (white arrow) to the orbital
velocity vector (black arrow). The denser medium near perihelion creates a net loss
of orbital velocity. The combination rotates the ellipse (advances the
perihelion). Copyright ©1999 by Boris Starosta,
boris@starosta.com.
Quoting the principles outlined by Eddington (1920):
"Light moves more slowly in the material medium then in a vacuum, the
velocity being inversely proportional to the refractive index of the medium.
The phenomenon of refraction is in fact caused by a slowing of the wave-front
in passing into a region of smaller velocity. We can thus imitate the
gravitational effect on light precisely, if we imagine the space around the Sun
filled with a refracting medium which gives the appropriate velocity of light.
To give the velocity 
,
the refractive index must be 
,
or, very approximately, 
."
Note that Eddington's factor 
in our notation.
These principles work well for purely wave
phenomena. But for predominately ballistic motion, the effect of the LCM is to
slightly slow the velocity that the orbiting body would otherwise have. The
relativistic clock-slowing, meter-stick-contraction factor is 
,
where 
,
,
and 
is velocity. Analogous to wave velocities
being slowed by the factor 
,
ballistic velocities are slowed by the factor 
.
These two velocity-slowing effects are sufficient to explain all relativistic
phenomena. To the accuracy we care about here, we can neglect the dependence of
on ,
and assume that 
is independent of the density of the LCM.
A quantitative derivation of
perihelion motion using celestial mechanics follows. Let be the orbital velocity of a massless body.
Hence, if 
is the orbital velocity the body would have in
a Newtonian force field, the actual velocity of the massless body will be 
.
This slower-than-Newtonian velocity will give the false impression that the
value of for the Sun is smaller than it really is. That
by itself does not disturb the elliptical orbit of the massless body. But because
an orbiting body moves faster at perihelion than at aphelion, the factor by
which it is slowed, ,
is smaller at perihelion than at aphelion. And this imbalance is solely
responsible for the perihelion advance.
The extra slowing and speed-up of an orbiting body
due to varying as the body itself slows down and
speeds up simulates an extra force acting on the body along the velocity
vector. The formula for the change in Newtonian velocity is: 
.
The derivative of this with respect to time is the extra acceleration directed
tangent to the orbital velocity vector, 
,
needed to produce the given change in velocity. We will compute ,
treat it as a perturbation, and plug it into standard celestial mechanics
formulas to learn the effect it generates on the motion of the perihelion of
the orbiting body.
Differentiating 
with respect to time to get an expression for ,
and expanding into a series and retaining only the first
term, we get
.
In
this first-order approximation, it is safe to make no distinction between and because that would only lead to higher-order
terms. This leads to:
.
To make the derivative easier, we can substitute
from the energy equation for orbits,
.
The
needed derivative is
.
The
new derivative for can be found, for example, in equation
(6.3.21) of Danby (1988), with the aid of the definition in equation (6.2.10):
,
where
is orbital true anomaly (the angle at the Sun
between the perihelion and the orbiting body) . Making all these substitutions,
we get our new expression for :
.
Danby's equation (1.7.3) shows how
to compute perihelion motion (
= longitude of perihelion) from a perturbation
along the velocity component:
.
This
has a secular and a periodic part. The periodic part gives rise only to small
periodic variations. The secular part, of interest here because it builds
progressively with time, comes from the time average value of 
,
which is 
by using Danby's equation (6.3.3). We will
also need Kepler's law, 
.
This gives the final form of the expression:
,
which
is the same as Danby's equation (4.5.7).
We can apply this formula to Mercury
by substituting the values from Table I, obtaining the result shown in the same
table. This agrees with the observed rate of advance of Mercury's perihelion to
within the ~ 1% errors of observation.
Because this formula for the
perihelion advance of a massless orbiting body is the same as Einstein's
formula, the most significant point of this derivation is that we have arrived
at it from a single principle. We needed only ballistic velocity slowing by the
factor ,
which immediately gave the correct multiple ;
whereas Einstein needed to combine three components to get the correct formula.
Moreover, our method invokes a tangential pseudo-force directed along the
velocity vector; whereas two of Einstein's three components corresponded to
radial pseudo-forces.
If the orbiting body had appreciable
mass (as for binary pulsars), repeating the above derivation for each mass and
combining the results yields an additional factor in the tangential force and in the final pericenter motion formula of
,
where
and 
are the separate masses. is then the product of the gravitational
constant times the sum of the masses.[1]
This factor reaches its minimum value of 0.5 when
the two masses are equal. So the pericenter motion of equal-mass binary
pulsars, for example, will be half what it would be if the same total mass were
in a single body orbited at the same distance by a massless body. This is easy
to understand using our single principle that masses propagate more slowly in a
denser gravitational medium. If the two masses are equal, each moves with half
the total relative velocity. So the velocity-slowing factor 
will have one-fourth the size it would have
for a massless body orbiting a single star. Because there are two masses each
contributing one-fourth as much to the total perihelion advance, the net
advance is raised back to one-half what it would be for the massless orbiter.
Although GR also predicts approximately one-half the
pericenter advance for equal-mass stars, we might well raise an eyebrow at the
unrelativistic nature of this mass factor. In effect, the velocity of each star
with respect to the combined local gravity field affects the overall motion.
The relativity principle would lead us to expect that only the relative
velocity between the two stars would matter; but that is clearly not the case.
Indeed, the predicted pericenter
motion is not identical for our velocity-slowing principle and for GR. For the
Hulse-Taylor binary pulsar PSR 1913+16, the most extensively studied, general
relativity predicts a pericenter advance rate of 4.23°/year. Our formula
predicts 3.55°/year for the same orbital parameters.[2] This is
a 16% difference. Such a large difference would certainly be observable.
However, the matter is not so simple. We do not know the orbital parameters of
the system, such as the star masses, the semi-major axis, or the orbit plane
inclination. These must be solved for along with many other parameters, using
the observations. Although the observations constrain certain parameter combinations
very well, the individual parameters are often not so well constrained.
In the case in point, Damour &
Taylor (1992) have remarked that the gravitational constant could be changed by
35% and still lead to solutions that satisfy all constraints. Using dots to
denote time derivatives and 
for orbital period, their exact words are: "However, the 
test is a mixed test which combines
strong-field and radiative effects in an indistinct way, so that one cannot
logically conclude, when the test is satisfied, that both the specific
strong-field and radiative predictions of general relativity have been
independently confirmed. In fact, examples of theoretically well-motivated
theories have recently been constructed … which have the same post-Newtonian
limit as general relativity, and can pass the test without fine-tuning, while still
differing markedly from Einstein's theory because of the strong self-gravity
effects in the pulsar and its companion. In extreme cases the three curves
defined by 
,
,
and 
can still meet within the observational
precision, while the effective gravitational constant between the pulsar and
its companion differs by as much as 35% from the usual Newtonian value."
This is equivalent to allowing the total system mass
to change by 35%. A change of that size would change the pericenter rate by a
comparable percentage. In short, if the system mass or other correlated
parameters were changed within allowable limits set by the observations, the
predicted pericenter motion using our velocity-slowing principle would rise to
a value also consistent with observations. However, this indistinguishability
of predictions will not last indefinitely. Within a few years, the accuracy of
observations will be sufficient to decide between the two interpretations of
GR.
I'm betting on the simpler
interpretation.
References
Damour, T. and Taylor, J.H. (1992), "Strong-field tests of relativistic gravity and binary
pulsars", Phys.Rev.D 45:1840-1868. (See p. 1841.)
Danby, J.M.A. (1988), Fundamentals
of Celestial Mechanics, 2nd edition, Willmann-Bell, Richmond.
Eddington, A.E. (1920), Space,
Time and Gravitation, reprinted 1987 by Cambridge Univ. Press, Cambridge,
109.
Hatch, R.R. (1999),
"Gravitation: Revising both Einstein and Newton", Galilean Electrodynamics
10#4:69-75.
Shapiro, I. et al. (1971), "Fourth test of
general relativity: New radar results", Phys.Rev.Lett. 26:1132-1135.
Tom Van Flandern
Meta Research / Washington, DC
Our 1999 March 15 issue contained an
article showing how general relativistic effects arise in flat-space-time
theories. Regarding the classical tests of general relativity as it affects
electromagnetic phenomena, we wish to cite the work of Fernando de Felice, “On
the gravitational field acting as an optical medium”, Gen.Rel.&Grav.
2 #4, 347-357 (1971). The author notes that Einstein himself first suggested the
idea that gravitation is equivalent to an optical medium. From the abstract, “…
Maxwell’s equations may be written as if they were valid in a flat space-time
in which there is an optical medium … this medium turns out to be equivalent to
the gravitational field. … we find that the language of classical optics for
the ‘equivalent medium’ is as suitable as that of Riemannian geometry.” Nine
earlier authors who have worked on this problem are cited in the text.
In our MRB article, we placed special
emphasis on the rotation of elliptical orbits (“perihelion motion”) because
that particular effect had been much neglected in the prior literature dealing
with flat-space-time treatments of this subject. We noted how Einstein’s own
approach to getting the correct perihelion motion involved the combination of
three effects, one of which cancels part of the contribution from the other
two; and that it probably involved a trial-and-error approach to get the theory
to correctly produce the known perihelion motion rate for Mercury. Certainly,
Einstein’s choice is not unique.
By contrast, we argued that the
simple, flat-space-time picture used by the Meta Model cosmology and inherent
in LeSage-type (“pushing-particle”) models of gravitation implies a perihelion
motion formula that comes out correctly from a single contribution – the
velocity-slowing effect on masses that must be produced by any underlying
light-carrying medium (LCM). In response to that article, we received reader
feedback from two of you. [For this follow-up note, we will denote velocity by rather than .
In the earlier article, was orbital true anomaly.]
Esko Lyytinen
<esko.lyytinen@minedu.fi> wrote:
“This brings up a point that sometimes has been on my mind while reading
about models using a refracting medium. The main point is that, if the change
of the density of the medium must be ‘tuned’ by the bending of light, then this
is a small negative point for the theory. I am no proponent of GR (and am far
from expert on it), especially as applied to the ‘inside’ of black holes.
However, to be fair to GR, there the space-time curvature is considered to
explain both light and what you call ‘ballistic motion’. For a refracting
medium, it is intuitive to think about the density of the LCM changing with
gravitational potential; but by how much exactly? If this is determined by
observations of how light seems to behave, then there actually is another
observable parameter in the theory. In the MRB article, on page 12, the
light behavior is modeled using a potential-dependent factor 
,
while the ballistic case uses the velocity dependent relativistic factor .
Can these be linked or derived from the same principle? (They are both somehow
related to the density changes of the LCM.)”
and
Victor Slabinski <slabinsk@patriot.net> wrote:
“The recent paper ‘The Perihelion
Advance Formula’ (MRB for March 15, 1999) starts off with a review of
the LCM and how it can explain three different tests of GR (General
Relativity). Then near the bottom of page 11, de Broglie matter waves are
mentioned, in order to suggest that at least for electrons, the LCM index of
refraction will affect their motion. [That notion suggests that LCM effects
depend on the test-body composition; electrons constitute a larger fraction of
the mass of hydrogen (one electron for each proton) than for helium (one
electron for each proton and neutron).] But no further use is made of de
Broglie waves; that paragraph could be omitted without affecting the argument
or the readability.
“Near
the top of page 12, the author notes that ‘the velocity of anything propagating
through a medium is proportional to the square root of the medium density.’ So
what? No further use is made of this fact. Next the LCM index of refraction is
discussed. But in the middle of paragraph 3, out of nowhere, we have the
statement “The relativistic clock-slowing ... is ...”. What does this have to do with the LCM?
Nothing. The author himself says at the end of the paragraph ‘... and assume
that is
independent of the density of the LCM.’ So why is LCM discussed as part of
perihelion advance?
“Finally on page 12, paragraph 4, the author
assumes that 
. The whole derivation is really based on
this formula. This is a velocity-dependence that has nothing to do with
gravitational potential. On what ‘classical, flat-space-time principles’ [claim
made inside front cover] is this formula based?”
My thanks to both readers for
flagging these problems, which all stem from a single cause – a missing step in
the logical development of ideas in my article. In the middle of the third
paragraph on p. 12, we find these sentences: “But for the predominately ballistic motion of a planet or other body,
the effect of the LCM is to slightly slow the ballistic velocity that the
orbiting body would otherwise have because of its wave properties. The
relativistic clock-slowing, meter-stick-contraction factor is ,
where 
,
,
and is the ballistic velocity of a body. Analogous
to wave velocities being slowed by the factor ,
ballistic velocities are slowed by the factor ”
After the first and third sentences in this excerpt, one might well ask “why?”
And in the answer to that lie the answers to all the other questions raised by
these two readers.
In the early 19th century, it was widely
realized that various transparent media influenced the speed of propagation of
light. If it were not so, prisms and lenses would not function as they do. The
optical theory of refraction depended for its successes on the speed of propagation
of light being changed by optical media such as glass or water. When such speed
measurements became possible, they confirmed this prediction of the theory.
How does this influence on the speed of light take place?
At one extreme, the optical medium might temporarily become the light-carrying
medium while light is in transit through it. In that case, the speed of light
in a vacuum, ,
would be reduced to the speed of light in the optical medium, 
.
Also, the speed of any motion of the optical medium, ,
would then add to or subtract from the speed of light through the medium, ,
to yield an observed speed of 
relative to the non-moving laboratory. In
other words, Galilean velocity addition rules would apply.
Alternatively, the LCM might be only somewhat affected, or entirely
unaffected, by the motion of the matter which it permeates. We still assume (as
observed) that a stationary optical medium slows the net light propagation
speed from to ,
an effect classically referred to as “drag”. But when the medium is moving, the
influence of the speed of the optical medium, ,
would be reduced by the influence factor 
,
where 
,
so that the measured speed of light relative to the laboratory would be 
.
On principle alone, we might be inclined to guess that 
in special relativity (SR), thereby assuring
that the speed of light remains the same in all directions and independent of
the speed of the optical medium. However, that is not in fact the prediction
made by Einstein.
The correct formula for 
was first predicted by Fresnel in 1817, and
hence the phenomenon is known as “Fresnel drag”. It was confirmed
experimentally using an interferometer by Fizeau in 1851. And the same formula
was offered by Einstein for SR, derived from the Lorentz transformation for
velocities. That correct formula is 
.
The upper sign applies when the optical medium and wave are moving in the same
direction, and the lower sign when moving in opposite directions. For small
values of ,
we see that 
,
which is a small quantity (i.e., near zero) except for relatively dense optical
media. This original prediction of Fresnel assumes an aether completely
unaffected by the motion of the matter it permeates. In the sense that
“space-time” is like an aether, Einstein’s prediction therefore has this same
characteristic.
To apply this knowledge to
the case at hand (LeSage-type models of gravity with a light-carrying medium),
recall Feynman’s explanation of why light travels slower in a medium. A photon
(whether particle or wave) always travels at speed ,
its vacuum speed, when it is propagating. But in an optical medium such as
water or glass, the photon gets repeatedly absorbed and re-emitted by atoms
encountered along its path. Each such event delays the photon’s forward
progress because, while absorbed, the photon is moving at the atom’s mean speed
instead of its own propagation speed .
The accumulated effect of these absorption delays is what slows the net speed
of the photon relative to the medium from down to .
then is the fraction of the time the wave
spends absorbed by atoms instead of propagating. The mean delay between
absorption and re-emission is typically a few nanoseconds. But since the speed
of light is about a foot per nanosecond, photons in a medium can spend
comparable time absorbed and carried along at speed as they spend propagating at speed .
Still, during the very brief spurts between absorptions, light makes most of
its progress because it travels so fast.
The “drag” effect is on the wave propagation speed, not on the underlying
aether. In effect, whenever the wave is propagating, it’s speed is in
a vacuum or in a stationary optical medium, where 
.
(We assume that the wave makes no progress while it is absorbed by an atom when
the medium is stationary.) But if the optical medium moves with speed ,
then the wave moves forward also at speed during the moments while it is absorbed by an
atom. So the net speed is somewhat larger than when the medium moves in the same direction as
the wave. Moreover, 
is the fraction of the time the wave spends
propagating, and is naturally inversely proportional to medium density when
density is large because absorptions are then more frequent. Its
proportionality to 
therefore implies that wave speed should vary
inversely with the square root of medium density, as has long been known
empirically. It further implies that, for sparse media, would be directly proportional to medium
density, which shows why the empirical rule breaks down for sparse media.
(Strictly, the net speed of
the wave when not absorbed cannot always be simply c, as Feynman assumed.
"Instantaneous" absorptions and discontinuities in speed are not very
plausible, but are simply expressions of our ignorance of the details of the
propagation and absorption processes. But gradual slowing of photons as they
enter the strong electrostatic field of an atom would make more sense,
especially in the LCM model. Whatever the details of the speed of propagation
while a photon is between absorptions, the relation is the one that must hold for consistency with
the model being developed: However, no other use is made of the propagation
speed while non-absorbed, so this detail is
unimportant to the overall development.)
Now that we have refreshed on the concept of Fresnel
drag, we are in a position to understand my allusions to the wave nature of
matter. Focus on a single atom of matter. It is composed of a matter part
moving ballistically (the protons and neutrons), and a wave part propagating
through the LCM (the electrons). If initially the atom has no net speed relative to the LCM, then the electrons simply
propagate back and forth with respect to the atomic nucleus at a wave
propagation speed that is determined by the density of the LCM.
We have already seen (in the
previous article) how that propagation speed changes if gravitational potential
changes: The LCM gets denser near matter because of gravitational compression
of the LCM. Equipotential surfaces become equi-density surfaces in the LCM. We
can neglect an arbitrary additive constant for both potential at infinity and
for LCM density at infinity. Only changes in potential or density can alter
propagation speed and bend or delay the path of a propagating wave. So we have
no need to “calibrate” this derivation using the Einstein light-bending effect.
Now imagine what would
happen to the nucleus of an atom if its electrons stopped propagating and
remained fixed in the aether. The atom would be forced to stop its motion and
remain there also because of the strong electrostatic force between the
electrons and the nucleus. For this reason, if forward electron propagation is
slowed, the forward ballistic motion of the nucleus is slowed by a like amount.
This, then, is the required linkage between the ballistic motion of matter and
propagating wave motion through the aether. Note that this electrostatic
interaction is entirely governed by charge, and is therefore independent of the
masses of the particles involved.
But why do electrons slow their propagation speeds
through the aether whenever the matter containing them moves? If the atom has a
net speed relative to the LCM, it encounters more LCM
per unit time than if it were stationary. It is just as if the LCM were denser.
If the wave propagation speed is and the matter speed is ,
the increased density encountered per unit time is proportional to 
.
This effect is first-order in ,
making it a much more significant effect than the aether density changes due to
gravitational potential, which depend on 
.
Specifically, electrons are
sometimes propagating in the same direction as the ballistic motion of the
nucleus, in which case they encounter more LCM per unit time; and sometimes
propagating in the opposite direction from the nucleus, in which case they
encounter less LCM per unit time. The more LCM encountered, the greater its
effective density, and the slower the electron propagates. And conversely. But
the electron influences the nucleus because of the electrostatic force binding
the two. So when something changes the speed of the propagating electron, the
ballistic speed of the nucleus changes too.
It remains to comment on why the slowed propagation speed
is proportional to .
To answer fully requires a re-derivation of the Lorentz transformations for the
LCM. Here, I will simply remark on the essence of that derivation. If light
propagates at speed 
relative to a source moving at speed over some fixed distance, and then travels the
reverse path over the same distance at speed 
relative to the same source, we can easily
calculate the total distance traveled and the total elapsed time. From there,
total distance divided by total time gives average speed, which is 
.
This is in fact how Einstein explained Fresnel drag. Physics might have taken a
quite different course if Einstein had not then been set on eliminating the
need for an aether.
In summary, electrons in matter propagate through the LCM
at speed determined by the local mean density of the
LCM. The forward speed of the matter c
