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 in a vacuum when measured by someone situated right next to it.  But let's approach the question by considering its various meanings (0) .

## 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 (a) .

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 entry 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 (b) .

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 (ba) .

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 1/86,400 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 (bb) In principle, we could get a very different answer using measurements based on laboratory experiments, from the one we get using astronomical observations (bc) .  (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? (bd)

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 (be) .  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 (bf) .

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? (bg)

(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 (c) .  This can be broken down into two parts:

• The speed of light is independent of the motion of the observer.
• The speed of light does not vary with time or place.

To state that the speed of light is independent of the velocity of the observer is very counterintuitive (1) .  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 (1a) .  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 (1b) .

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 (1c) .

### 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.  This is known as the Sagnac Effect.  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 (1d) .

You will sometimes find discussions that insist the only correct way to describe the Sagnac Effect is by reference to an inertial frame: they will say that the only concept with meaning is the locally measured speed of light, which is c, and that what the non-inertial observer sitting on the loop says about the motions of two light rays has no physical meaning.  Whilst the Sagnac effect is easy to calculate using an inertial frame—because then we can use the simple equations of adding velocities in special relativity—it doesn't follow that any non-inertial description of it is invalid.  Those who insist that non-inertial descriptions are invalid are like the man whose house is about to be picked up by a cyclone: they will shout "Don't worry folks!  The wind isn't really circulating at 300 km/h.  It's really Earth that's rotating in an inertial frame, and the resulting differential motions give rise to the illusion that the wind is about to shred this house."  Yes, it's certainly valid to analyse the situation using Newton's laws in an inertial frame.  But you might want to hang on to your house while doing so (1e) .

You might also find it said that the Sagnac Effect is somehow not measuring the speed of the two light beams sent around the loop, but "merely" their times of flight, as if that's somehow different to measuring their (average) speed.  But the simple fact is that if you send two horses in opposite directions around the same race track, then the horse that crosses the finish line first must have run faster. The different arrival times of the two light beams have nothing to do with anything strange going on with "the geometry of spacetime": this discussion holds in the absence of any gravity, in which case spacetime can be flat, and if it's flat for one observer, it's flat for all, including those sitting on rotating loops.  The observer sitting on the rotating loop concludes that the beams simply move at different speeds.  And that's all right, because it's only either an inertial observer who must measure their speeds to be both c, or an observer sitting right next to the light beams.  But the observer on the loop is neither inertial nor sitting right next to each beam at all times of its flight (1f) .

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 measure their weight as constant (1g) .  (That's a very natural definition of uniform acceleration.  I am using what I called the "contact camp" definition of weight in the FAQ entry What is Weight?.)  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 (1h) .

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.  In particular, we'll need the all-important topic of the relativity of simultaneity, for which you can find the expression vL/c2 in most textbooks that discuss the fundamentals of special relativity (1i) .  This quantity is the amount of time by which the clock on the tail of a train reads ahead of the driver's clock when the train has rest length L, approaches us at velocity v (positive for approach, negative for recession), and whose clocks are synchronised in its rest frame.  Suppose the train is at rest and extends from here to the Andromeda galaxy, so that its driver is right next to us and its tail sits in that galaxy, which we'll suppose isn't moving relative to us.  Our standard of simultaneity says that right now on a particular planet in the Andromeda galaxy at the tail of the train, some clock reads zero just as ours reads zero, and that clock clicks at the same rate as ours.  Now start moving and walk towards the galaxy at 1 m/s.  Suddenly the space between here and Andromeda has become like the train mentioned above: that "train" is approaching us at v = 1 m/s with L = 2 million light-years, so that the clock on that particular planet has suddenly jumped ahead of our clock by vL/c2 = about 2 days.  Once we stop accelerating and maintain 1 m/s, the distant clock will run slightly slowly compared to ours (by time dilation), but in the arbitrarily short period of time during which we accelerated, it jumped 2 days ahead (1j) .

Imagine that two planets in that galaxy are 2 light-days apart, and one sends a pulse of light to the other.  During the period that we accelerated and clocks in Andromeda jumped 2 days ahead of us, that light pulse travelled from one planet to the other.  But we can accelerate however quickly we like, so we'll conclude that during our brief period of acceleration, the light passing between those two planets travelled much much faster than c.  So while you accelerate towards Andromeda, both light and clocks (i.e. the flow of time itself) speed up in Andromeda—but only while you accelerate (1k) .

None of the preceding discussion actually depends on the distances being large; it's just easier to visualise if we use such large distances.  So now transfer that discussion to a rocket you are sitting in, far from any gravity and uniformly accelerated, meaning you feel a constant weight pulling you to the floor.  "Above" you (in the direction of your acceleration), time speeds up and light travels faster than c, arbitrarily faster the higher up you want to consider (1l) Now use the Equivalence Principle to infer that in the room you are sitting in right now on Earth, where real gravity is present and you aren't really accelerating (we'll neglect Earth's rotation!), light and time must behave in the same way to a high approximation: light speeds up as it ascends from floor to ceiling, and it slows down as it descends from ceiling to floor; it's not like a ball that slows on the way up and goes faster on the way down.  Light travels faster near the ceiling than near the floor (1m) .  But where you are, you always measure it to travel at c; no matter where you place yourself, the mechanism that runs the clock you're using to measure the light's speed will speed up or slow down precisely in step with what the light is doing.  If you're fixed to the ceiling, you measure light that is right next to you to travel at c.  And if you're fixed to the floor, you measure light that is right next to you to travel at c.  But if you are on the floor, you maintain that light travels faster than c near the ceiling.  And if you're on the ceiling, you maintain that light travels slower than c near the floor.

You can also infer that as a distant wavefront travels transversely to your "up" direction, the more distant parts of it will be travelling faster than the nearer parts.  So, just as light bends when it enters glass at an angle, you won't be surprised to see the distant light bend toward you.  And, of course, bending light is something you'll find in textbooks that illustrate the Equivalence Principle with a picture of a guy in an elevator encountering a beam of light (1n) .

So imagine again that the room in which you're sitting is an accelerating rocket far from gravity, and your weight is due to its acceleration upwards.  Your 1-g acceleration means you infer that light and time flow faster above you and slower below you.  About one light-year below you is a plane parallel to the floor on which light and time slow to a stop, the horizon mentioned a few lines back.  Below that plane time flows backwards, but you can never receive a signal from below that plane—a fact that you can prove easily with a quick sketch on the spacetime diagram of an inertial observer, where you'll notice that you'll forever outrun a light signal that was sent to chase you from that far away, even though an inertial observer says that the light is travelling (at c) faster than you are.  So you'll never see any weird breakage of causality occurring beyond the horizon (1p) .

Saying that light and time have stopped on this horizon is a consequence of your changing standard of simultaneity as you accelerate.  Anyone sitting on or beyond the horizon just continues life as usual; they can't be influenced by your state of motion.  Although you maintain that they have stopped ageing, they themselves notice nothing unusual.  In that sense, what we say about the flow of time and the speed of light is all about the coordinates that we have used to describe the world of our accelerated frame.  But those coordinates are not silly and arbitrary, because they reflect the fact that we can build our accelerated frame by using the standard mechanism of making measurements in special relativity: we construct a rigid lattice of observers whose clocks always agree with ours, and who don't move relative to us.  This construction is precisely what a uniformly accelerated frame is, and it's by no means obvious that it's possible to do: for example, an inertial observer will measure the accelerations of those other accelerated observers to differ from our own acceleration—even though we and all the accelerated observers say that they remain a fixed distance from us and from each other.  That might sound odd, and to see why it's true, you have to follow the special-relativistic ideas of simultaneity, timing, and length very carefully.  So although this changing standard of simultaneity might be referred to by some as just some kind of coordinate artifact, we shouldn't trivialise the use of such coordinates.  They are what our world is built on (1q) .

(This changing standard of simultaneity of an accelerating observer is the real kernel behind resolving the Twin Paradox.  Most discussions of the Twin Paradox try to simplify things by having the space traveller maintain constant speed on both the outbound and inbound leg, necessitating an infinitesimal period of infinite acceleration at the start of the return trip.  In so doing, these discussions throw the baby out with the bath water by producing an analysis that contains an awkward gap in the timing at the moment the space traveller changes direction.  If those analyses were to have the traveller accelerate in a more realistic way, what would result would be a very much more difficult, yet far more complete, analysis of the Twin Paradox that has no weird timing gaps.) (1r)

You can see that as you go about your daily life, accelerating every which way as you walk around, your standard of simultaneity is see-sawing madly all around you.  Playing around with lines of simultaneity on a spacetime diagram and maintaining that time is doing weird things are we accelerate might seem like a departure from good common sense.  We must appeal to experiment to keep from straying into an abstract fairy world that has nothing to do with reality.  But via the Equivalence Principle, these special-relativistic ideas of changing simultaneity feed into general relativity, and in this day and age we do have the luxury of experiments that daily confirm that more advanced theory.  If general relativity didn't work, then the GPS satellite system would fail dismally at telling you where you are and what the time is (1s) .

One note: how can you measure the speed of light if it's not right next to you?  You do that through the standard mechanism, mentioned above, of employing a lattice of observers whose clocks always agree with yours, and who don't move relative to you.  You then use the measurement of the observer who was right next to the light whose speed you wanted to measure.  And that's fine, because that observer in not moving relative to you, and their clock always agrees with yours.  That's the standard way that all measurements are done in the context of special relativity (1t) .

Making observations from an inertial frame (and using its coordinates) produces a speed of light that is always c (1u) .  In that respect the inertial frame's coordinates are better for some analyses; but the accelerated frame is more natural to our description of the world around us.  After all, we don't live our lives in free fall.

## 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 (2) .  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 (3) .  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 (4) .

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 (5) .  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 (6) .

Einstein talked about the speed of light changing in his new theory.  In the English translation of 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 [Einstein clearly means speed here, since velocity (a vector) is not in keeping with the rest of his sentence] of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity [...] (7) cannot claim any unlimited validity.  A curvature of rays of light can only take place when the velocity [speed] 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 (8) (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 (9) .

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 (10) .  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.

When all is said and done, to insist that a non-c speed of light is nothing more than an artifact of a "nonphysical" choice of coordinates is to make a wrong over-simplification.  When we wave goodbye to an astronaut who is about to make a high-speed return journey to the nearest star, it would be wrong to maintain that the slowing of his clock is nothing more than an artifact of a coordinate choice.  It isn't: when the astronaut returns, he will have aged less than we have, and there's nothing illusory about that (11) .

### Reference:

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

For an in-depth analysis of the speed of light in an accelerated frame, see Chapter 7 of "Explorations in Mathematical Physics" by D. Koks (Springer, 2006).

* 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.)

### Reflection by Nicolaas Vroom

When writing something about The speed of light certain issues are very important:
• Considering the subject there are two questions: (1) Is the speed of light constant? (2) What is the speed of light?. That means you want to quantify the parameter.
Both questions are important but you should answer the first before the second.
• The issue: How do you measure the speed of light. This should be clearly explained, because all measurements should be done in the same manner. Related issues are the definition of: Length and time (or duration)
• The issue is that we should clearly make a difference between a one way speed of light versus a two way speed of light. In the last case, at least, we should speak about an average speed of light. This is also the case when large distances are involved.
• In the whole document at least 10 times c is used, but it is not always clear what is meant.
• It should be remembered that the behaviour of light (its speed) is a physical phenomena and has nothing to do with any law or human behaviour.
We humans are surrounded by an endless sea full of photons which in general we do not see. It is the same for a fish swimming in an "empty" ocean, who does not recognize that he swims in water.
• Theories which include the speed of light as an integrated concept should be studied with great care, because the speed of light is maybe not so constant as we "think".

• (0) The short answer is that the speed of light is a functie of gravity. The problem is that the speed of light is difficult to measure. The most practical value is the speed of light in a vacuum which is 299,792,458 m/s. The actual value of the speed of light depends about the numerical method used to calculate the speed.
• (a) The definition defines a meter (in vacuum) but not the speed of light. The speed of vacuum is indirectly assumed to be 299,792,457 m/s in vacuum. The definition also does not assume that the speed of light is constant. It is true that the concept of inertial frame is not defined. Neither in this document.
My definition of inertial frame is a frame which has a constant speed and which is not influenced by gravity.
IMO in general those frames do exist. IMO the only practical purpose are frames linked to one object i.e. the earth, the sun, the center of our galaxy.
• (b) This is an interesting discussion because photons have energy which implies that they have mass. This is in agreement with the law: conservation of energy. The implication that photons have a mass, means, that when a photon travels through a gravitational field, the path is not straigth and the speed is not constant.
• (ba) It is wrong to start from the assumption that the speed of light is a physical constant. In order to declare the speed of light constant you have to measure this speed under as much physical conditions as possible. To declare the speed of light constant for practical purposes, is a whole different subject.
• (bb) The most important issue that generally speaking for all measurements of the speed of light we should use the same method. In that case we can easy compare the results.
• (bc) The words: "In principe" are wrong. The issue is that when you measure the speed of light at two different locations the result can be different. There is nothing wrong with this, assuming you use the same method.
• (bd) Generally speaking it is very difficult to make any physical claim about quantities if different units and measurement techniques are used. IMO this type of discussion is not relevant.
• (be) "The frequencies and wavelengths depend etc on the speed of light" of what? I expect of the reddish-orange light and radiation. The sentence is not clear.
• (bf) In this paragraph there is a discussion about units which are not clearly defined. Of course if you change the definitions the calculations, using these units, change. Such a discussion is confusing within the context "Is the speed of light constant" which should start with a clear definition of all the concepts (words) used.
• (bg) Here the same comments as above. What you can discuss in a separate chapter is: "Was the speed of light (in vacuum) always the same, starting from the Big Bang". The problem is how do you demonstrate that.
• (c) That the speed of light is constant in an inertial frame is correct because no gravity is involved. The problem is that in outerspace there is always gravity. As a consequence light rays are bended and the speed of light in general is not constant.
• (1) The issue is that the speed of light is a physical parameter and has nothing to do with the speed of the observer. The problem is when you want to measure the speed of light. The distance between the source and destination should at least be constant. For the clocks involved there are different rules.
• (1a) Specific in this paragraph it is very important if the speed of light is measured in one way or two way. It should be one way otherwise you can never detect if the spead of light is the same in al directions from your observation point.
• (1b) In this paragraph something should be mentioned exactly what is a inertial frame, a non-inertial frame, a uniform accelerated frame and an accelerated frame.
Starting from the definition that inertial frames have a uniform speed and are not subject to acceleration than in reality all frames in some way or another are non-inertial frames, because they under go accelerations as influenced by gravity.
In that sense a frame linked to the surface of the earth is non-inertial and the same for a frame linked to the center of the earth. The same for any star in our Galaxy and also for the center of our Galaxy.
• (1c) It should be remembered that moving clocks, not running at a constant rate, is nothing strange, because it is a physical effect. It is easy to understand that a clock which working is based on light signals, will run slower, because the extra distance the lightsignal has to take in the direction of movement.
• (1d) The first problem with this paragraph is that it is written in a much to popular fashion. There is no observer involved. It is all done electronical.
A laser gyroscope does not measure the speed of light. Neither it does not detect that the speed of light is constant. In order to decide if the speed of light is constant you either must measure the speed of light at two different locations or under two different conditions.
The purpose of a gyroscope is generally speaking to stay still (should not rotate) in space. This is the case when the arriving time in the two arms is the same. When this is not the case, the electronics within the gyroscope detects this and will take corrective action.
• (1e) Also this paragraph is written in a much to popular fashion. To mention a cyclone and a house should be removed.
• (1f) Only the first sentence is correct i.e. that a gyroscope does not measure the speed of light. It measures the difference in arriving time of to signals and the ellectronics takes corrective actions when there is a difference.
Most of what is written in that paragraph can be removed.
• (1g) How do you measure your own weight in a space ship?
Inside a space ship all loose objects move with the same speed in the same direction. In a sense they are weight less.
• (1h) The issue is if the speed of a light close (being lower) to the surface of the earth is higher or lower than the speed of light above (being higher) the surface. The direction is parellel to the surface.
To answer this question experimentally is very difficult.
• (1i) In the text two parameters are used: c and v. How are these two parameters measured? How is v measured relative to c, the speed of light?
Suppose the object with velocity v changes direction and now has a speed v in opposite direction relative to c. How is that measured?
• (1j) Why such a difficult scenario to explain what?
• (1k) The same. This is a very difficult paragraph to understand. Clocks don't jump ahead of you.
• (1l) The same. Again difficult to understand. When you consider rockets, gravity becomes an issue. All discussions about gravity belong in the next section.
• (1m) And why does a light ray not behave the same as a ball? Remember we are discussing gravity. When you drop a ball its speed increases. When a light ray approaches an object towards the surface the light ray is bended. When you use a gun in horizontal direction, the vertical speed of the bullet increases.
• (1n) Explaining the Equivalence principle should be done in a more professional matter.
• (1o) This text is much to popular.
• (1p) IMO a lot of the text should be made simpler. Using a concept like "Horizon" is a very human concept and is tricky.
• (1q) The concept of simultaneity is a very important concept, because by adhering to it we recognize that there are simulteneous events. In fact at the same time with any event "here" there are millions of other simulateneous events anywhere in the universe. To decide which events are simultaneous you need to synchronise the clocks "in play".
The situation if events are simultaneous or not, has nothing to do with the behaviour of any human being. The whole point is that once the clocks are synchronised, they should not be moved.
• (1r) In order to explain the twin paradox or the behaviour of moving clocks you do need the concept: "changing standard of simultaneity". For an explanation select Example
• (1s) What is written in this paragraph should either be moved to the section relativity or be eliminated.
• (1t) What is written in this paragraph is important for Newton's Law, special relativity and general relativity. Using a lattice or the concept of "one reference frame" is the only way to compare the results if different models are used.
• (1u) It is not true that all experiments will always produce the same result and that the speed of light is constant. See the experiment below which chalenges the concept if the speed of light "is" or "is not" influenced by gravity.
• (2) The text is not very clear but I think what they mean is that it is easy to measure the speed of light when no accelerations are involved and difficult when this is not the case. The problem is in reality always accelerations are involved. IMO the only way to solve that is to use only one reference frame encompassing all objects involved and no moving clocks. For example one reference frame for our own galaxy, the Milky way.
• (3) If that is the case we really have to try a different strategy. The issue is not only the speed of light but speed in general for all objects studied. Again this requires one reference frame ane one set of clocks, which are all synchronised in the same way.
• (4) What the text acknowledges is that the speed of light is very difficult to measure but what is more important also to agree with all about the speed v of each observer.
To claim that each observer is going to measure that the speed of light is c is his vincinity does not make much sense. In fact each observer should measure the speed of light at least twice in order to decide if the speed in his neighbourhood is constant.

In the book "Gravitation" at page 18 we read:
"Rule out both observers to make physics look simple. Instead travel aboard the freely moving spaceship. Nothing could be more natural than what one sees: every free object moves in a straight line with uniform velocity. This is the way to do physics! Work in a very special coordinate system: a coordinate frame in which one is weightless; a local inertial frame of reference."
The question is does this strategy allows you to calculate the movement of objects (in our galaxy) including how lightrays behave? I doubt it. Specific physics should not be based on what one SEES. How do we know that an object (lightray) moves in a straight line? (towards infinity?)
• (5) What the text does not mention is that even in the case where gravity is strong the speed of light (photons) has a distinct physical value.
• (6) The concepts of floor and ceiling speed of light are very unsatisfactory. What is more meaning full is a sentence like: The speed of light near an object (towards and away) is larger than the average speed of light. The average speed is the speed measured from source to destination.
• (7) In this text Einstein discusses General Relativity and explains / acknowledges that the speed of light is not constant.
• (8) In the text a free falling rest frame is mentioned. What does this mean. What does that mean for the earth when we observe star light from all different directions?
In practice the gravitational influence will be minimal, but will be large for stars.
• (9) What is the frame at rest "relative to the source of gravity" when we consider the solar system? The best one is the one linked to the center of the Sun. The issue is that in this reference frame, no moving clocks should be used and all the clocks (based on one clock) should be synchronised in the same manner (continously)
• (10) To answer the question if the speed of light is constant depents on the manner measurements are done to understand the trajectories of objects. Specific these measurements should be done at different locations under different conditions. Only when that is done you can get a global idea what the answer is. Otherwise your answer is only of local significance.
• (11) IMO this text should be modified or removed because it has almost no added value.

One of the most importance issues of the document is that it should highlight that in general the speed of light is not constant and is affected by gravity i.e. the gravitational field. To define the speed of light constant is of practical importance.

In order to show that the speed of light is not constant you have to calculate or to compare the speed of light at two different places. An experiment which compares the speed of light over two different distances is described here: An experiment in a gravitational field. The idea behind the experiment is to compare two "logical" clocks which are of different length. When gravity has no influece than when the difference in length is 1 to 2 than the ticking rate of the clocks also will be 1 to 2. When gravity influences the behaviour of the clocks than this is not the case. The clock with the longest length ticks faster as "expected".
The problem is such an experiment can not be performed in practice (I expect).

A second issue is that if you want to do physics you should start from one reference frame and one clock on which all your observations are based. What is important that all the clocks used should be connected to this frame. That means no moving clocks are involved and one clock should be selected to synchronise all clocks in the same way. Using one reference frame allows you to define the positions of all the objects involved at a sequence of simultaneous events. Such a strategy is the easiest way to develop the laws of nature which describe the movements of the objects involved and to calculate the future.
In the above text uses the wording inertial frame, non-inertial frame, accelerated frame and uniformely accelerated frame.
The most important issue is the origin of the frame.
The problem is that any point in the Universe is subject of gravity and in one way or another moves through space. Some point more, some point less. It is the challenge of physics to decide a point that does "not" move and to consider that point at rest.

The worst is (in most cases or always?) to use more reference frames because then you need transformation equations to go from one origin to the other forward and backward.

### Example of two different reference frames

Consider two trains: one train at rest at a platform as one reference frame and a moving train as a different reference frame. The problem is that the moving train requires accelerations which influences the speed of the clocks on the moving train. The clock on the moving train runs behind.
 ``` 3 | /| b . | / | \ | / . \ . |/ \ 2 . a |\ | / | \ .| /. | \ / | .\|/ . | 1 | . /| . | / | |./ |. |/ | B F ---> ``` The picture at the left shows this behaviour. The train is identified with the letters "F" = Front and "B" = Back. The train moves at a constant speed towards the right. The two vertical straight lines to the points "B" and "F" identify the position of the train at rest at the platform. The two dotted lines starting at the points "B" and "F" identify the position of the moving train. The clock is at the "Back" of the train. The letters B, 1, 2 and 3 identify the path of a lightray which is reflected in a train "at rest" on the platform. The point "1" is the moment when the light signal is reflected at the front etc. The letters B,a and b identify the path of a lightray on a moving train. The point "a" is the moment when the light signal is reflected at the front of the moving train. etc. The point "2" is the event when an observer at the back of the train will observe the first reflection. The point "b" is the same for a moving observer. Comparing the time of the events (the distance on the verical axis shows that the clock on the moving train runs slower.
The actual fact that the clock on the moving train runs slower will be detected when the train returns, by comparing the time of the two clocks
From the point of view of an observer on the train the stay at home clock runs too fast. As a rule the reference frame with the fastest clock is the preferred frame.

The fact that moving clocks run slower than a clock at rest (than means that the moving clock moves from A to B and back to A) has nothing to do with "changing standard of simultaneity". The whole explanation is that the moving clock is accelerated and deaccelerated which changes the physical internal behaviour of the clock, assuming that the clocks working is based on light signals.