de Sitter's Binary Apparitions
Maxwell's equations are very successful at describing the propagation
of light based on the model of electromagnetic waves, not only in
material media but also in a vacuum. According to this model, light
propagates in a vacuum at a speed c = 1/sqrt(e0 u0), where u0 is
the permeability constant and e0 is the permittivity of the vacuum,
defined in terms of Coulomb's Law for electrostatic force
1 q1 q2
F = ------ ------
4pi e0 r^2
The SI system of units is defined so that the permeability constant
takes on the value u0 = 4pi 10^-7 tesla*meter/ampere, and we can
measure the value of the permittivity (typically by measuring the
capacitance C between parallel plates of area A separated by a
distance d, using the relation e0 = Cd/A) to have the value e0 =
8.854187818*10^-12 Coulombs^2 /(Newtons*meters^2). This leads to
the familiar value
c = 1 / sqrt(e0 u0) = 299792458 meters/second
for the speed of light in a vacuum. Of course, if we place some
substance between our capacitors when determining e0 we will generally
get a different value, so the speed of light is different in various
media. This leads to the index of refraction of various transparent
media, defined as n = c_vacuum / c_medium. Thus Maxwell's theory of
electromagnetism (based on the work of Ampere, Coulomb, and especially
Faraday) clearly implies that the speed of propagation of electro-
magnetic waves depends only on the medium, and is independent of the
speed of the source.
On the other hand, it also suggests that the speed of light depends
on the motion of the medium, which is easy to imagine in the case
of a material medium like glass, but not so easy if the "medium" is
the vacuum, i.e., empty space. How can we even assign a state of
motion to the vacuum? In struggling to answer this question, some
people tried to imagine that even the vacuum is permeated with some
material-like substance, called the ether, to which a definite state
of motion could be assigned. On this basis, they imagined that
Maxwell's equations were strictly applicable (and the speed of light
was exactly c) ONLY with respect to the absolute rest frame of the
ether. With respect to other frames of reference they expected to
find that the speed of light differed, depending on the direction
of travel. Likewise we would expect to find corresponding differences
and anisotropies in the capacitance of the vacuum when measured with
plates moving at high speed relative to the ether.
However, when extremely precise interferometer measurements were
carried out to find a directional variation in the speed of light
on the Earth's surface (presumably moving through the ether at a
fairly high speed due to the Earth's rotation and its orbital motion
around the Sun), they found essentially no directional variation in
light speed that could be attributed to the motion of the apparatus
through the ether. Of course, it had occurred to people that the
ether might be "dragged along" by the earth, and so objects on the
Earth's surface are essentially at rest in the local ether. However,
these "convection" hypotheses are inconsistent with other observed
pheneoma, notably the aberration of starlight, which can only be
explained in an ether theory if it is assumed that an observer on
the Earth's surface is NOT at rest with respect to the local ether.
Also, careful terrestial measurements of the paths of light near
rapidly moving massive objects showed no sign of any "convection".
So, considering all this, the situation was considered to be quite
puzzling.
There is a completely different approach that could be taken to
modeling the phenomena of light, provided we're willing to reject
Maxwell's theory of electromagnetic waves, and adopt instead a model
similar to the one that Newton often seemed to have in mind, namely,
an "emission theory". One advocate of such a theory in the early
1900's was Walter Ritz, who rejected Maxwell's equations on the
grounds that the advanced potentials allowed by those equations were
unrealistic. Ritz debated this point with Albert Einstein, who argued
that the observed asymmetry between advanced and retarded waves is
essentially statistical in origin, due to the improbability of
conditions needed to produce coherent advanced waves. Neither man
persuaded the other. (Ironically, Einstein himself had already
posited that Maxwell's equations were inadquate to fully represent
the behavior of light, and suggested something a model that contains
certain attributes of an emissions theory, to account for the photo-
electric effect, but this challenge to Maxwell's equations was on a
more subtle and profound level than Ritz's objection to advanced
potentials.)
Having discarded Maxwell's equations and the electro-magnetic wave
model of light, the advocates of Ritzian emission theories assume a
Galilean or Newtonian spacetime, and postulate that light is emitted
and propagates away from the source (perhaps like Newtonian corpuscles)
at a speed of c relative to the source. Thus, according to emission
theories, if the source is moving directly toward or away from us with
a speed v, then the light from that source is approaching us with a
speed c+v or c-v respectively. Naturally this class of theories is
compatible with experiments such as the one performed by Michelson
and Morley, since the source of the light is moving along with the
rest of the apparatus, so we wouldn't expect any directional variation
in the speed of light for such experiments. Also, an emission theory
of light is compatible with stellar aberration, at least up to the
limits of observational resolution. In fact, James Bradley (the
discoverer of aberration) originally explained it on this very
basis.
Of course, even an emission theory must account for the variations
in light speed in different media, wich means it can't simply say
that the speed of light depends ONLY on the speed of the source.
The speed must also be dependent on the medium through which it is
traveling, and presumably it must have a "terminal velocity" in each
medium, i.e., a certain characteristic speed that it can maintain
indefinitely as it propagates through the medium. (Obviously we
never see light come to rest, nor even do we observe noticeably
"slow" light, so it must always exhibit a characteristic speed.)
Furthermore, based on the principles of an emission theory, the
medium-dependent speed must be defined relative to the rest frame
of the medium.
For example, if the characteristic speed of light in water is c',
and a body of water is moving relative to us with a speed v, then
(according to an emission theory) the light must move with a speed
c' + v relative to us when it travels for some significance distance
through that water, so that it has reached its "steady-state" speed
in the water. In optics this distance is called the "extinction
distance", and it is known to be proportional to 1/(rho lambda),
where rho is the density of the medium and lambda is the wavelength
of light. The extinction distance for most common media for optical
light is extremely small, so essentially the light reaches it's
steady-state speed as soon as it enters the medium.
It happens that an experiment performed by Fizeau in 1851 to test
for optical "convection" also sheds light on the viability of emission
theories. Fizeau sent beams of light in both directions through a
pipe of rapidly moving water to determine if the light was "dragged
along" by the water. We know that, since the refractive index of
water is about n = c/C = 1.33 where C is the speed of light in water,
so C is c/1.33, which is about 75% of the speed of light in a vacuum.
The question is, if the water is in motion relative to us, what is
the speed of light (relative to us) in the water?
If light propagated in an absolutely fixed background ether, and
wasn't dragged along by the water at all, we would expect the light
speed to stil be C relative to the fixed ether, regardless of how
the water moves. In this case the difference in travel times for
the two directions would be proportional to
1 1
- - - = 0
C C
which implies no phase shift in the interferometer. On the other
hand, if emission theories are right, the speed of the light in the
water (which is moving at the speed v) should be C+v in the direction
of the water's motion, and C-v in the opposite direction. On this
basis the difference in travel times would be proportional to
1 1 2v
--- - --- = ---------
C-v C+v C^2 - v^2
This is a very small amount (remembering that C is about 75% of the
speed of light in a vacuum), but it is large enough that it would be
measureable with delicate interferometry techniques.
However, the results of Fizeau's experiment turned out to be
consistent with NEITHER of the above predictions. Instead, he found
that the time difference (proportional to the phase shift) was a bit
less than 43.5% of the prediction for an emission theory (i.e., 43.5%
of the prediction based on the assumption of complete convection).
By varying the density of the fluid we can vary the refractive index
and therefore C, and we find that the measured phase shift always
indicates a time difference of (1-C^2) times the prediction of the
emission theory. For water we have C=0.7518, which explains why the
time lag is 1 - C^2 = 0.4346 times the emission theory prediction.
This implies that if we let S(C,v) and S(C,-v) denote the speeds of
light in the two directions, we have
1 1 2v
------- - ------ = --------- (1 - C^2)
S(C,-v) S(C,v) C^2 - v^2
From this (plus symmetry, etc.) it's easy to infer that the function
S is given by
u + v
S(u,v) = --------
1 + uv
which of course is the relativistic formula for the composition of
velocities. So, even if we were to reject Maxwell's equations, it
would still appear that emission theories cannot be reconciled with
Fizeau's experimental results.
Nevertheless, it's still of interest to consider what other evidence
is available to rule out simplistic emission theories. Around 1919
de Sitter was thinking about such theories, and trying to think of
observations that could distinguish between them and the theory of
relativity. In other words, he wondered whether, if we assumed the
speed of light in a vacuum is always c with respect to the source,
and we assume a Galilean spacetime, we notice anything different in
the appearances of things. One interesting idea was to consider the
appearance of binary star systems, i.e., two stars that orbit around
each other. More than half of all the visible stars in the night
sky are actually double stars, which can be discerned with powerful
telescopes. de Sitter's basic idea was that if two stars are orbiting
each other and we are observing them from the plane of their mutual
orbit, the stars will be sometimes moving toward the Earth rapidly,
and sometimes away. According to an emission theory this orbital
component of velocity should be added to or subtracted from the speed
of light. As a result, over the hundreds or thousands of years that
it takes the light to reach the Earth, the arrival times of the light
from approaching and receding sources would be very different.
Now, before we go any further, we should point out a potential
difficulty for this kind of observation. The "vacuum" of empty space
is not really a perfect vacuum, but contains small and sparse particles
of dust and gas. Consequently it is actually a material medium and,
as noted above, light will reach it's steady-state velocity with
respect to that interstellar dust after having traveled beyond the
extinction distance. For visible light the extinction distance is
quite short, so almost the light will be moving at essentially c for
almost its entire travel time, regardless of the original speed. For
this reason, it's questionable whether visual observations of
celestial objects can provide any good test of the emission theory
predictions.
In fact, it might seem that we can't possibly evaluate the speed of
light in a vacuum, because no vacuum exists! However, this is a bit
too pessimistic. Remember that the extinction distance is proportional
to 1/(rho*lambda) where lambda is the wavelength. If we focus on
light in the frequency range of, say, x-rays and gamma rays, the
wavelengths are so small that the extinction distance is enormous,
even extending to thousands of lightyears. Hence we can carry out
deSitter's proposed observation if we use light from the high-energy
end of the spectrum (e.g., x-rays). This has actually been done by
Brecher in 1977.
So, with the proviso that we will be focusing on light whose extinction
distance is much greater than the distance from the binary star system
to Earth (so that the speed of the light is simply c plus the speed
of the star at the time of emission) how should we expect a binary
star system to appear?
Let's consider one of the stars in the binary system, and write it's
coordinates and their derivatives as
x(t) = D + R cos(wt) dx/dt = -Rw sin(wt)
y(t) = R sin(wt) dy/dt = Rw cos(wt)
where D is the distance from the Earth to the center of the binary
star system, R is the radius of the star's orbit about the system's
center, and w is the angular speed of the star. We also have the
components of the emissive light speed
c^2 = (c_x)^2 + (c_y)^2
In these terms we can write the components of the absolute speed
of the light emitted from the star at time t:
x' = -Rw sin(wt) + c_x
y' = Rw cos(wt) + c_y
Now, in order to reach the Earth at time T the light emitted at
time t must travel from in the x direction from x(t) to 0 at a speed
of x' for a time dt = T-t, and similarly for the y direction. Hence
we have
x(t) + dt[-Rw sin(wt) + c_x] = 0
y(t) + dt[ Rw cos(wt) + c_y] = 0
Substituting for x, y, and the light speed derivatives x', y', we
have
D + R cos(wt) - dt Rw sin(wt) = - dt c_x
R sin(wt) + dt Rw cos(wt) = - dt c_y
Squaring both sides of both equations, and adding the resulting
equations together, gives
2 2 2 2 2 2 2
D + R + dt R w + 2DR[cos(wt) - dt w sin(wt)] = dt c
Re-arranging terms gives the quadratic in dt
2 2 2 2 2 2
(c - R w ) dt + [2DRw sin(wt)] dt - [D + 2DR cos(wt) + R ] = 0
If we define the normalized parameters
v = Rw/c d = D/c r = R/c
then the quadratic in dt becomes
2 2 2 2
(1 - v ) dt + [2vd sin(wt)] dt - [r + 2rd cos(wt) + r ] = 0
Solving this qadratic for dt = T-t and then adding t to both sides
gives the arrival time T on Earth as a function of the emission time
t on the star
vd sin(wt)
T = t - ----------
1 - v^2
__________________________________________________
/ [vd sin(wt)]^2 + (1-v^2)(d^2 + 2rd cos(wt) + r^2)
+ -----------------------------------------------------
1 - v^2
If the star's velocity v is much less than the speed of light, this
can be expressed very nearly as
T = t - vd sin(wt) + d + r cos(wt)
The derivative of T with respect to t is
dT
-- = 1 - v d w cos(wt) - r w sin(wt)
dt
and this takes it's minimum value when t=0, where we have
/ dT \ d
( -- ) = 1 - v d w = 1 - --- v^2
\ dt /min r
Consequently we find the deSitter effect, i.e., dT/dt goes negative
if d > r / v^2. Now, we know from Kepler's third law (which also
applies in relativistic gravity with the appropriate choice of
coordinates) that m = r^3 w^2 = r v^2, so we can substitute m/r
for v^2 in our inequality to give the condition d > r^2 / m. Thus
if the distance of the binary star system from Earth exceeds the
square of the system's orbital radius divided by the system's
mass (in geometric units) we will have the deSitter effect.
As an example, for a binary star system a distance of d = 20000
light-years away, with an orbitral radius of r = 0.00001 light-years,
and an orbital speed of v = 0.00005, the arrival time of the light
as a function of the emission time is as shown below:
This corresponds to a star system with only about 1/6 solar mass,
and an orbital radius of about 1.5 million kilometers. At any given
reception time on Earth we can typically "see" at least three separate
emission events from the same star at different points in its orbit.
These ghostly apparitions are the effect that deSitter tried to find
in photographs of many binary star systems, but none exhibited this
effect. Of course, he was looking in the frequency range of visible
light, which we've noted is subject to extinction. However, in the
x-ray range we can (in principle) perform the same basic test, and yet
we still find no evidence of these ghostly apparitions in binary stars,
nor do we ever see the stellar components going in "reverse time"
as we would according to the above profile. This supports the idea
than the speed of light in empty space is essentially independent of
the speed of the source.
In comparison, if we take the relativistic approach with constant
light speed c, independent of the speed of the source, an analysis
similar to the above gives the approximate result
T = t + d + r cos(wt)
whose derivative is
dT
-- = 1 - v sin(wt)
dt
which is always positive for any v less than 1. This means we can't
possibly have images arriving in reverse time, nor can we have any
multiple appearances of the components of the binary star system.
On the subject, Robert Shankland recalled Einstein telling him (in
1950) that he had himself considered an emission theory of light,
similar to Ritz's theory, during the years before 1905, but he
abandoned it because
"he could think of no form of differential equation
which could have solutions representing waves whose
velocity depended on the motion of the source. In
this case the emission theory would lead to phase
relations such that the propagated light would be
all badly "mixed up" and might even "back up on
itself". He asked me, "Do you understand that?"
I said no, and he carefully repeated it all. When
he came to the "mixed up" part, he waved his hands
before his face and laughed, an open hearty laugh
at the idea!"
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