Updated 1997 by Steve Carlip.
Original by Philip Gibbs 1996.
There are a number of senses to the meaning of this question and so there are a number of different answers. Firstly . . .
Yes. Light is slowed down in transparent media such as air, water and glass. The ratio by which it is slowed is called the refractive index of the medium and is always greater than one*. This was discovered by Jean Foucault in 1850.
When people talk about "the speed of light" in a general context they usually mean "the speed of light in a vacuum". This quantity is also referred to as c.
At the 1983 Conference Generale des Poids et Mesures the following SI (Systeme International) definition of the metre was adopted:
The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.
This defines the speed of light in vacuum to be exactly 299,792,458 m/s. This provides a very short answer to the question "Is c constant": Yes, c is constant by definition!
However, this is not the end of the matter. The SI is based on very practical considerations. Definitions are adopted according to the most accurate known measurement techniques of the day and are constantly revised. At the moment you can measure macroscopic distances most accurately by sending out laser light pulses and timing how long they take to travel using a very accurate atomic clock. (The best atomic clocks are accurate to about one part in 1013.) It therefore makes sense to define the metre unit in such a way as to minimise errors in such a measurement.
The SI definition makes certain assumptions about the laws of physics. For example, they assume that the particle of light, the photon, is massless. If the photon had a small rest mass, the SI definition of the metre would become meaningless because the speed of light would change as a function of its wavelength. They could not just define it to be constant. They would have to fix the definition of the metre by stating which colour of light was being used. Experiments have shown that the mass of the photon must be very small if it is not zero (see the FAQ: What is the mass of the photon?). It is certainly too small to have any practical significance for the definition of the metre in the foreseeable future but it cannot be shown to be exactly zero even though currently accepted theories indicate that it is. If it wasn't zero, the speed of light would not be constant but from a theoretical point of view we would then take c to be the upper limit of the speed of light in vacuum so that we can continue to ask if c is constant.
Previously the metre and second have been defined in various different ways according to the measurement techniques of the time. They could change again in the future. If we look back to 1939, the second was defined as 1/84,600 of a mean solar day and the metre as the distance between two scratches on a bar of platinum-iridium alloy held in France. We now know that there are variations in the length of a mean solar day as measured by atomic clocks. Standard time is adjusted by adding or subtracting a leap second from time to time. There is also an overall slowing down of the Earth's rotation by about 1/100,000 of a second per year due to tidal forces between the Earth, Sun and Moon. There may have been even larger variations in the length or the metre standard caused by metal shrinkage. The net result is that the value of the speed of light as measured in m/s was slowly changing at that time. Obviously it would be more natural to attribute those changes to variations in the units of measurement than to changes in the speed of light itself, but by the same token it is nonsense to say that the speed of light is now constant just because the SI definitions of units define its numerical value to be constant.
But the SI definition highlights the point that we need to first be very clear what we mean by constancy of the speed of light before we answer our question. We have to state what we are going to use as our standard ruler and our standard clock when we measure c. In principle we could get a very different answer using measurements based on laboratory experiments from the one we get using astronomical observations. (One of the first measurement of the speed of light was derived from observed changes in the length of the eclipses of Jupiter's moons by Olaus Roemer in 1676.) We could, for example, take the definitions of the units as they stood between 1967 and 1983. Then the metre was defined as 1,650,763.73 wavelengths of the reddish-orange light from a krypton-86 source, and the second was defined (then as now) as 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of cesium-133. Unlike the previous definitions these depend on absolute physical quantities which apply everywhere and at any time. Can we tell if the speed of light is constant in those units?
From the quantum theory of the atom we know that these frequencies and wavelengths depend chiefly on the values of Planck's constant, the electronic charge and the masses of the electron and nucleons as well as the speed of light. By eliminating the dimensions of units from the parameters we can derive a few dimensionless quantities such as the fine structure constant and the electron to proton mass ratio. These values are independent of the definition of units so it makes much more sense to ask if these values change. If they did change it would not just be the speed of light which was affected. The whole of chemistry is dependent on their values and significant changes would alter the chemical and mechanical properties of all substances. Furthermore, the speed of light itself would change by different amounts according to which definition of units you used. In that case it would make more sense to attribute the changes to variations in the charge on the electron or the particle masses than to changes in the speed of light.
In any case, there is good observational evidence to indicate that those parameters have not changed over most of the lifetime of the universe. See the FAQ article Have physical constants changed with time?
[Note that the fine structure constant does change with energy scale but I am referring to the constancy of its low energy limit.]
Another assumption on the laws of physics made by the SI definition of the metre is that the theory of relativity is correct. It is a basic postulate of the theory of relativity that the speed of light is constant. This can be broken down into two parts:
To state that the speed of light is independent of the velocity of the observer is very counterintuitive. Some people even refuse to accept this as a logically consistent possibility but in 1905 Einstein was able to show that it is perfectly consistent if you are prepared to give up assumptions about the absolute nature of space and time.
In 1879 it was thought that light must propagate through a medium in space just as sound propagates through the air and other substances. Two scientists Michelson and Morley set up an experiment to attempt to detect the ether by observing relative changes in the speed of light as the Earth changed its direction of travel relative to the sun during the year. To their surprise they failed to detect any change in the speed of light.
Fitzgerald then suggested that this might be because the experimental apparatus contracted as it passed through the ether in such a way as to countermand the attempt to detect the change in velocity. Lorentz extended this to changes in the rates of clocks to ensure complete undetectability of the ether. Einstein then argued that those transformations should be understood as transformation of space and time rather than physical objects and that the absoluteness of space and time introduced by Newton should be discarded. Just after that Minkowski, a mathematician showed that Einstein's theory of relativity could be understood in terms of a 4 dimensional non-euclidean geometry of space-time.
The theory is not only mathematically consistent, it is in agreement with countless direct experiments. The Michelson-Morley experiment was repeated with greater accuracy in the years that followed. In 1925 Dayton Miller announced that he had detected a change in velocity of the speed of light and was even awarded prizes for the discovery, but a 1950's appraisal of his work indicated that the most likely origin of his erroneous results lay with diurnal and seasonal variations in the temperature of his equipment.
Modern instruments could easily detect any ether drift if it existed. The Earth moves around the sun at a speed of about 30 km/sec so if velocities added vectorially as Newtonian mechanics requires, the last 5 digits in the value of the speed of light now used in the SI definition of the metre would be meaningless. Today high energy physicists at CERN and Fermilab routinely accelerate particles to within a whisper of the speed of light. A dependence of the speed of light on reference frames would have shown up long ago unless it is very slight indeed.
But what if we pursued the original theory of Fitzgerald and Lorentz who proposed that the ether is there but undetectable because of physical changes in the lengths of material objects and the rates of clocks, rather than transformations of space-time? For such a theory to be consistent with observation the ether would have to be completely undetectable using clocks and rulers. Everything, including the observer, would have to contract and slow down by the right amounts. Such a theory could make exactly the same prediction in all experiments as the theory of relativity but in that case the ether would be no more than a meta-physical construct unless there was some other way of seeing it which nobody has found. Such a construct would be an unnecessary complication to be eliminated from the theory in the view of Einstein.
Einstein went on to discover a more general theory of relativity which explained gravity in terms of curved space-time and he talked about the speed of light changing in this new theory. In the 1920 book "Relativity: the special and general theory" he wrote: . . . according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity . . . cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position. Since Einstein talks of velocity (a vector quantity) rather than speed it is not clear that he meant the speed will change but the reference to special relativity suggests he did mean so. This interpretation is perfectly valid but a more modern interpretation is that the speed of light is constant in general relativity.
The problem here comes from the fact that speed is a coordinate-dependent quantity, and is therefore somewhat ambiguous. To determine speed (distance/time) you must first choose some standards of distance and time, and different choices can give different answers. This is already true in special relativity: if you measure the speed of light in an accelerating reference frame, the answer will, in general, differ from c.
In special relativity, the speed of light is constant when measured in any inertial frame. In general relativity, the appropriate generalization is that the speed of light is constant in any freely falling reference frame (in a region small enough that tidal effects can be neglected). In this passage, Einstein is not talking about a freely falling frame, but rather about a frame at rest relative to a source of gravity. In such a frame, the speed of light can differ from c, basically because of the effect of gravity (spacetime curvature) on clocks and rulers.
If general relativity is correct then the constancy of the speed of light, in inertial frames is a tautology from the geometry of space-time. The causal structure of the universe is determined by the geometry of null vectors. Travelling at the speed c means following world-lines tangent to these null vectors. The use of c as a conversion between units of metres and seconds, as in the SI definition of the metre, is fully justified on theoretical grounds as well as practical terms because c is not merely the speed of light, it is a fundamental feature of the geometry of space-time.
Like special relativity, the predictions of general relativity have been confirmed in many different observations. The book by Clifford Will is an excellent reference for further details.
Finally we come to the conclusion 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.
Reference:
C.M. Will, "Was Einstein Right?" (Basic Books, 1986)
* Strictly speaking, the refractive index is not always greater than one. Indeed, it is almost always less than one for X-rays. This is because the phase velocity of X-rays in a medium is faster than light and the refractive index is the ratio of phase velocities. The speed of photons is the group velocity which is always slower than c (except when it isn't :-). For simplicity we ignore the distinction in this article. See the Relativity FAQ article on faster than light (phase velocity) for an explanation. [Thanks to Pieter Kuiper for pointing this out.]