[Physics FAQ] - [Copyright]

Updated 2014 by Don Koks.  Original by Steve Carlip (1997) and Philip Gibbs 1996.


Is The Speed of Light Everywhere the Same?

The short answer is that it depends on who is doing the measuring: the speed of light is only guaranteed to have a value of 299,792,458 m/s when measured by an inertial observer in a vacuum.  But let's approach the question by considering its various meanings.


Does the speed of light change in air or water?

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 usually 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.  They also usually mean the speed as measured in an inertial frame.  This vacuum-inertial speed is denoted c.


Is c, the speed of light in a vacuum inertial frame, constant?

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.  Unfortunately it doesn't mention anything about inertial frames, but you can consider a measurement in an inertial frame to be implied.

However, this is not the end of the matter.  The SI is based on very practical considerations.  Definitions are adopted according to the most accurately 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, it assumes 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.  The SI Committee could not just define it to be constant; instead, 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?).  Any such possible photon rest mass 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 the mass weren'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 whether c is constant.

The SI definition also assumes that measurements taken in different inertial frames will give the same results for light's speed.  This is actually a postulate of special relativity, discussed below.

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 Earth's rotation by about 1/100,000 of a second per year due to tidal forces between 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's 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 first to be very clear about 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 measurements of the speed of light was derived from observed changes in the timing 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 caesium-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?

The quantum theory of atoms tells us 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 on 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 the units, so it makes much more sense to ask whether these values change.  If they did change, it would not just be the speed of light which was affected.  All of chemistry depends 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 was 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.)


Special Relativity

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 the same in all inertial frames.  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, the ether, just as sound propagates through the air and other substances.  The 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 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 idea to changes in the rates of clocks to ensure complete undetectability of the ether.  Einstein then argued that those transformations should be understood as changes of space and time rather than of physical objects, and that the absoluteness of space and time introduced by Newton should be discarded.  Just after that, the mathematician Minkowski showed that Einstein's theory of relativity could be understood in terms of a four dimensional non-euclidean geometry that considered space and time as one entity, ever after called spacetime.

The theory is not only mathematically consistent, it agrees with many 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 the speed of light and was even awarded prizes for the discovery, but a 1950s appraisal of his work indicated that the most likely origin of his results lay with diurnal and seasonal variations in the temperature of his equipment.

Modern instruments could easily detect any ether drift if it existed.  Earth moves around the Sun at a speed of about 30 km/s, 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 in Geneva and Fermilab in Chicago routinely accelerate particles to within a whisker of the speed of light.  Any dependence of the speed of light on inertial reference frames would have shown up long ago, unless it is very slight indeed.  Their measurements are actually made in a non-inertial frame because gravity is present.  But in the context of the measurements, this non-inertial frame is almost identical to a "uniformly accelerated frame" (this is actually the content of Einstein's Principle of Equivalence).  And it turns out that a measurement of light's speed made in a uniformly accelerated frame directly by someone who is very close to the light will return the inertial value of c——although that observer must be close to the light to measure this value.

But what if we pursued the original theory of Fitzgerald and Lorentz, who proposed that the ether is there, but is undetectable because of physical changes in the lengths of material objects and the rates of clocks, rather than changes in space and time?  For such a theory to be consistent with observation, the ether would need to be completely undetectable using clocks and rulers.  Everything, including the observer, would have to contract and slow down by just the right amount.  Such a theory could make exactly the same prediction in all experiments as the theory of relativity; but it would reduce the ether to essentially no more than a metaphysical construct unless there was some other way of detecting it——which no one has found.  In the view of Einstein, such a construct would be an unnecessary complication, to be best eliminated from the theory.


The Speed of Light as Measured by Non-Inertial Observers

That the speed of light depends on position when measured by a non-inertial observer is a fact routinely used by laser gyroscopes that form the core of some inertial navigation systems.  These gyroscopes send light around a closed loop, and if the loop rotates, an observer riding on the loop will measure light to travel more slowly when it traverses the loop in one direction than when it traverses the loop in the opposite direction.  The gyroscope does employ such an observer: it is the electronics that sits within the gyro.  This electronic observer detects the difference in those light speeds, and attributes that difference to the gyro's not being inertial: it is accelerating within some inertial frame.  That measurement of an acceleration allows the body's orientation to be calculated, which keeps it on track and in the right position as it flies.

Discussing non-inertial observers can be simpler if we consider not the rotating frame of a laser gyroscope, but the uniformly accelerated frame of someone who sits inside a rocket, far from any gravity source, accelerating at a rate that makes them feel their weight is constant.  In fact, the room in which you are sitting right now is a very high approximation to such a frame——as mentioned above, this is the content of Einstein's Principle of Equivalence.

So consider the question: "Can we say that light confined to the vicinity of the ceiling of this room is travelling faster than light confined to the vicinity of the floor?".  For simplicity, let's take Earth as not rotating, because that complicates the question!  The answer is then that (1) an observer stationed on the ceiling measures the light on the ceiling to be travelling with speed c, (2) an observer stationed on the floor measures the light on the floor to be travelling at c, but (3) within the bounds of how well the speed can be defined (discussed below, in the General Relativity section), a "global" observer can say that ceiling light does travel faster than floor light.

That might sound strange, so let's take it in stages.  Begin with the relativity idea that an inertial observer does measure the speed of light to be c.  Suppose I want to measure the speed of light that is distant from me.  Even though I can't physically interact with a distant photon (if I could, it would no longer be distant), I can employ a continuum of inertial observers placed throughout spacetime, and ask them to make measurements for me of everything happening only in their vicinity.  (That's the normal textbook procedure in special relativity.)  I do this not because I'm simply employing other people to do my job, but rather because it's possible to set these observers up in such a way that they all agree with me on the distances between objects, and which events are simultaneous.  That is, we share a "common ruler" and a "common clock".  That way, when I ask myself what "velocity" means (distance travelled over time taken), I have a well-defined set of events to measure, and a well-defined global set of rulers and clocks with which to do it: a job that I can pass on to the band of observers without getting tangled up in ideas of measuring things at a distance, with all the difficulties related to signal-travel times which that would entail.

Only the observer who is close to the photon will interact with it and measure its speed.  Assuming the basic postulates of relativity are correct——which tie in with assuming Maxwell's equations are correct (they predict a speed of c for inertial observers)——the observer nearest the photon will measure a speed of c.  But because all of the accelerated frame's observers share the same standards of distance and simultaneity (i.e. we all agree on when the stopwatches were started and stopped), I can take the measurement made by my distant associate as "the" speed of light.

Now here's the crucial point.  If an observer is sitting up at the ceiling of this room and another is sitting on the floor, and they each have their own identical factory-set clocks and rulers, the ceiling observer measuring light in his vicinity will measure c, and so will the floor observer.  But if I now ask these observers to set their clocks and rulers up so that they agree with me on distances and simultaneity——which can be done, although it's not an obvious thing——then things change.  I collate their measurements and find that in this "global" uniformly accelerated frame, ceiling light travels faster than floor light, because the clocks involved are actually running at different speeds locally in order for their measurements to be agreed upon by all of the observers who make up the frame.  Does this come down simply to an arbitrary setting of clock speeds?  No; the clocks must be set to run at different speeds in order for us all to assign the same times to events, and everything is set up in such a way that we all agree on the lengths of rulers.  The reason for that lies in special relativity itself, and is not just some artificial thing.  The clocks really do reflect the different flows of time on ceiling and floor.  It's a nice set-up, but one that can be accomplished only for inertial and uniformly accelerated frames.


General Relativity

Einstein went on to propose a more general theory of relativity which explained gravity in terms of curved spacetime, and the next level of sophistication of treating our ceiling and floor observers takes real gravity into account.

It's easy to build a continuum of observers in flat spacetime with everyone inertial, who each measure events only in their vicinity.  It's possible but much harder to do the same for a uniformly accelerated frame.  For more complicated frames and also for real gravity, we find that I simply can't populate space with a continuum of observers who all agree with me on distances and simultaneity.  We just won't have a common standard of rulers and clocks.  Each observer is going to measure the speed of light to be c in his vicinity, but I can't accurately talk about the speed of a distant light ray (or anything else), because I can't enlist anyone to make measurements for me in such a way that we all agree on what space and time standards we're using.

Given this situation, in the presence of more complicated frames and/or gravity, relativity generally relinquishes the whole concept of a distant object having a well-defined speed.  As a result, it's often said in relativity that light always has speed c, because only when light is right next to an observer can he measure its speed—— which will then be c.  When light is far away, its speed becomes ill-defined.  But it's not a great idea to say that in this situation "light everywhere has speed c", because that phrase can give the impression that we can always make measurements of distant speeds, with those measurements yielding a value of c.  But no, we generally can't make those measurements.  And the stronger gravity is, the more ill-defined a continuum of observers becomes, and so the more ill-defined it becomes to have any good definition of speed.  Still, we can say that light in the presence of gravity does have a position-dependent "pseudo speed".  In that sense, we could say that the "ceiling" speed of light in the presence of gravity is higher than the "floor" speed of light.

Einstein talked about the speed of light changing in his new theory.  In his 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 [Einstein means speed here] of propagation of light varies with position."  This difference in speeds is precisely that referred to above by ceiling and floor observers.

In special relativity, the speed of light is constant when measured in any inertial frame.  In general relativity, the appropriate generalisation 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 not-quite-well-defined "speed" of light can differ from c, basically because of the effect of gravity (spacetime curvature) on clocks and rulers.

In general relativity, the constancy of the speed of light in inertial frames is built in to the idea of spacetime being a geometric entity.  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 vacuum-inertial speed of light, it is a fundamental feature of spacetime geometry.

Like special relativity, some of the predictions of general relativity have been confirmed in many different observations.  The book listed below by Clifford Will is an excellent reference for further details.


Reference:

C.M. Will, "Was Einstein Right?" (Basic Books, 1986)

* The refractive index can be less than one.  Indeed, it is almost always less than one for X-rays.  This is because the phase speed of X-rays in a medium (i.e. the speed of their wave fronts) is faster than the phase speed of visible light, and the refractive index is the ratio of phase speeds.  The speed of photons is the "group speed", 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 speed) for an explanation.  (Thanks to Pieter Kuiper for pointing this out.)