2.10 The Starry Messenger

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, which is considered to be a region free of material substances. According to this model, light propagates in a vacuum at a speed , where m₀ is the permeability constant and e₀ is the permittivity of the vacuum, defined in terms of Coulombs law for electrostatic force

The SI system of units is defined so that the permeability constant takes on the value m₀ = 4p10^-7 tesla meter per 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 coulombs² per newton meters². This leads to the familiar value

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 electro-magnetism seems to clearly imply that the speed of propagation of such electromagnetic 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 of empty space. How can we even assign a state of motion to the vacuum? In struggling to answer this question, people tried to imagine that even the vacuum is permeated with some material-like substance, the ether, to which a definite state of motion could be assigned. On this basis it was natural to suppose 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 fairly high speed due to the Earth's rotation and its orbital motion around the Sun), essentially no directional variation in light speed was found 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, so that objects on the Earth's surface are essentially at rest in the local ether. However, these "convection" hypotheses are inconsistent with other observed phenomena, 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 terrestrial measurements of the paths of light near rapidly moving massive objects showed no sign of any "convection". 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 early 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.)

In place of Maxwell's equations and the electromagnetic wave model of light, the advocates of "emission theories" generally 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 to find any directional variation in the speed of light in 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, which means it can't simply say that the speed of light depends only on the speed of the source. It 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_w, and a body of water is moving relative to use with a speed v, then (according to an emission theory) the light must move with a speed c_w + 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/(rl), where r is the density of the medium and l 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.

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_w = 1.33 where c_w is the speed of light in water, we know that c_w 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 (relative to us) of the light in the water?

If light propagates in an absolutely fixed background ether, and isn't dragged along by the water at all, we would expect the light speed to still be c_w relative to the fixed ether, regardless of how the water moves. This is admittedly a rather odd hypothesis (i.e., that light has a characteristic speed in water, but that this speed is relative to a fixed background ether, independent of the speed of the water), but it is one possibility that can't be ruled out a priori. In this case the difference in travel times for the two directions would be proportional to

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_w+v in the direction of the water's motion, and c_w-v in the opposite direction. On this basis the difference in travel times would be proportional to

This is a very small amount (remembering that c_w is about 75% of the speed of light in a vacuum), but it is large enough that it would be measurable with delicate interferometry techniques.

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_w, and we find that the measured phase shift always indicates a time difference of (1-c_w²) times the prediction of the emission theory. For water we have c_w = 0.7518, so the time lag is (1-c_w²) = 0.4346 of the emission theory prediction.

This implies that if we let S(c_w,v) and S(c_w,-v) denote the speeds of light in the two directions, we have

By partial fraction decomposition this can be written in the form

where

Also, in view of the symmetry S(u,v) = S(v,u), we can swap c_w with v to give

Solving these last two equations for A and B gives A = 1 - vc_w and B = 1 + vc_w, so the function S is

which of course is the relativistic formula for the composition of velocities. So, even if we rejected Maxwell's equations, it still appears that emission theories cannot be reconciled with Fizeau's experimental results.

Nevertheless, it's of interest to consider what other evidence is available to rule out simple emission theories. One bit of evidence comes from observations of a supernova made by Chinese astronomers in the year 1054 AD. When a star explodes as a supernova, the initial shock wave moves outward through the star's interior in just seconds, and elevates the temperature of the material to such a high level that fusion is initiated, and much of the lighter elements are fused into heavier elements, including some even heavier than iron. (This process yields most of the interesting elements that we find in the world around us.) Material is flung out at high speeds in all directions, and this material emits enormous amounts of radiation over a wide range of frequencies, including x-rays and gamma rays. Based on the broad range of spectral shifts (resulting from the Doppler effect), it's clear that the sources of this radiation have a range of speeds relative to the Earth of over 10000 km/sec. This is because we are receiving light emitted by some material that was flung out from the supernova in the direction away from the Earth, and by other material that was flung out in the direction toward the Earth.

If the supernova was located a distance D from us, then the time for the "light" (i.e., EM radiation of all frequencies) to reach us should be roughly D/c, where c is the speed of light. However, if we postulate that the actual speed of the light as it travels through interstellar space is affected by the speed of the source, and if the source was moving with a speed v relative to the Earth at the time of emission, then we would conclude that the light traveled at a speed of c+v on it's journey to the Earth. Therefore, if the sources of light have velocities ranging from -v to +v, the first light from the initial explosion to reach the Earth would arrive at the time D/(c+v), whereas the last light from the initial explosion to reach the Earth would arrive at D/(c-v). This is illustrated in the figure below.

Hence the arrival times for light from the initial explosion event would be spread out over an interval of length D/(c-v) - D/(c+v), which equals (D/c)(2v/c) / (1-(v/c)²). The denominator is virtually 1, so we can say the interval of arrival times for the light from the explosion event of a supernova at a distance D is about (D/c)(2v/c), where v is the maximum speed at which radiating material is flung out from the supernova.

However, in actual observations of supernovae we do not see this "spreading out" of the event. For example, the Crab supernova was about 6000 light years away, so we had D/c = 6000 years, and with a range of source speeds of 10000 km/sec (meaning v = � 5000) we would expect a range of arrival times of 200 years, whereas in fact the Crab was only bright for less than a year, according to the observations recorded by Chinese astronomers in June-July of 1054 AD. For a few weeks the "guest star" (as they called it) in the constellation Taurus was the brightest star in the sky, and was even visible in the daytime. Within two years it had disappeared completely (to the naked eye). The event was also recorded by Islamic observers, but (oddly) it seems to have passed unnoticed (or at least unrecorded) in Europe. In the time since the star went supernova the debris has expanded to it's present dimensions of about 3 light years, which implies that this material was moving at only (!) about 1/300 the speed of light. Still, even with this value of v, the bright explosion event should have been visible on Earth for about 40 years (if the light really moved through space at c � v). Hence we can conclude that the light actually propagated through space at a speed essentially independent of the speed of the sources.

However, although this source independence of light speed is obviously consistent with Maxwell's equations and special relativity, we should be careful not to read too much into it. In particular, this isn't direct proof that the speed of light in a vacuum is independent of the speed of the source, because for visible light (which is all that was noted on Earth in 1054 AD) the extinction distance in the gas and dust of interstellar space is much less than the 6000 light year distance of the Crab nebula. In other words, for visible light, interstellar space is not a vacuum, at least not over distances of many light years. Hence it's possible to argue that even if the initial speed of light in a vacuum was c+v, it would have slowed to c for most of its journey to Earth. Admittedly, the details of such a counter-factual argument are lacking (because we don't really know the laws of propagation of light in a universe where the speed of light is dependent on the speed of the source, nor how the frequency and wavelength would be altered by interaction with a medium, so we don't know if the extinction distance is even relevant), but it's not totally implausible that the static interstellar dust might affect the propagation of light in such as way as to obscure the source dependence, and the extinction distance seems a reasonable way of quantifying this potential effect.

One way of strengthening the case for source-independence of light speed based on astronomical observations is to use light from the high-energy end of the spectrum. As noted above, the extinction distance is proportional to 1/(rl). For some frequencies of x-rays and gamma rays the extinction distance in interstellar space is about 60000 light years, much greater than the distances to many supernova events, as well as binary stars and other configurations with identifiable properties. By observing these events and objects it has been found that the arrival times of light are essentially independent of frequency, e.g., the x-rays associated with a particular identifiable event arrive at the same time as the visible light for that event, even though the distance to the event is much less than the extinction distance for x-rays. This gives strong evidence that the speed of light in a vacuum is actually invariant and independent of the motion of the source.

With the aid of modern spectroscopy we can now examine supernovae events in detail, and it has been found that they exhibit several characteristic emission lines, particularly the signature of atomic hydrogen at 6563 angstroms. Using this as a marker we can determine the Doppler shift of the radiation, from which we can infer the speed of the source. The energy emitted by a star going supernova is comparable to all the energy that it emitted during millions or even billions of years of stable evolution. Three main categories of supernovae have been identified, depending on the mass of the original star and how much of its "nuclear fuel" remains. In all cases the maximum luminosity occurs within just the first few days, and drops by 2 or 3 magnitudes within a month, and by 5 or 6 magnitudes within a year. Hence we can conclude that the light actually propagated through empty space at a speed essentially independent of the speed of the sources.

Another interesting observation involving the propagation of light was first proposed in 1913 by DeSitter. He wondered whether, if we assumed the speed of light in a vacuum is always c with respect to the source, and if we assume a Galilean spacetime, we would notice anything different in the appearances of things. He considered 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. DeSitter'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 problem (again) is that 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. Since the extinction distance for visible light in interstellar space is quite short, 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. However, once again we are rescued by the high-frequency end of the spectrum. If we focus on light in the frequency range of, say, x-rays and gamma rays, the extinction distance is much larger than the distances to many binary star systems. Hence we can carry out DeSitter's proposed observation (in principle) if we use x-rays, and 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 (making the speed of the light 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

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

c² = c_x² + c_y²

In these terms we can write the components of the absolute speed of the light emitted from the star at time t:

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 for a time Dt = T-t, and similarly for the y direction. Hence we have

Substituting for x, y, and the light speed derivatives , , we have

Squaring both sides of both equations, and adding the resulting equations together, gives

Re-arranging terms gives the quadratic in Dt

If we define the normalized parameters

then the quadratic in Dt becomes

Solving this quadratic 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

If the star's speed 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

and this takes it's minimum value when t = 0, where we have

Consequently we find the DeSitter effect, i.e., dT/dt goes negative if d > r / v². Now, we know from Kepler's third law (which also applies in relativistic gravity with the appropriate choice of coordinates) that m = r³ w² = r v², so we can substitute m/r for v² in our inequality to give the condition d > r² / 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 would expect DeSitter's apparitions - assuming the speed of light is c � v.

As an example, for a binary star system a distance of d = 20000 light-years away, with an orbital 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 traces 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. Hence these observations support the proposition that 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

whose derivative is

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.

Regarding this 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

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