The speed of sound in air is sometimes said to be 343.2 metres per second or 768 miles per hour. But actually, it varies. It usually decreases with altitude up to about 11 kilometres above sea level. That’s about 36,000 feet, which is typical for a passenger jet. At that altitude the speed of sound is circa 295 m/s or 660 mph, which is one reason why passenger jets don’t fly as fast as you might like. Interestingly enough, the speed of sound typically decreases with increasing altitude, so sound tends to get refracted upwards. It’s similar for sound waves in the sea, but there the speed of sound typically decreases with depth. So a horizontal sonar wave tends to get refracted downwards:
Image from FAS and the US Navy, see course ES310 chapter 20
The sonar waves “veer” rather like a car veers when it encounters mud at the side of the road. The mud slows down the wheels on the left, so the car pulls to the left. You steer a tank to the left in a similar fashion, by slowing down the track on the left.
It’s c for celeritas which means speed
See Wikipedia and you can read that the speed of sound “is conventionally represented by c, from the Latin celeritas”. Celeritas means swiftness or speed, which varies according to the properties of the medium. The speed of sound in sea water is circa 1500 m/s or 3,355 mph. In steel it’s circa 6000 m/s or 13,421 mph. In diamond it’s circa 12,000 m/s or 26,843 mph. It’s similar for seismic waves. Wikipedia says the propagation velocity depends on the density and elasticity of the medium, and is up to 13 km/s in the deep mantle. That’s 13,000 m/s or 29,080 mph. Of course in seismology we have P-waves and S-waves. The former are “primary” longitudinal waves like sound waves. The latter are “secondary” transverse waves like light waves. The equation for the speed of a transverse seismic wave is given as csolid,s = √(G/ρ), where G is the shear modulus and ρ (rho) is the density. The equation for the speed of light in the vacuum of space is given as c = 1/√(ε0μ0) where ε0 is vacuum permittivity and μ0 is vacuum permeability. There’s a reciprocal because permittivity is a “how easy” measure rather than a “how hard” measure. Other than that, the equation takes the same general form, and the speed of light in space, in “vacuo”, is said to be 299,792,458 m/s or 670,616,629 mph.
The speed of light in space is said to be constant
Note that the speed of sound in air varies, the speed of sound in water varies, and the speed of seismic waves in rock varies. But it is said that the speed of light in space does not vary. Search the internet and you can find lots of articles saying this. See for example Luke Mastin’s article on the speed of light and the principle of relativity. He talks about the Michelson-Morley experiment, which “unexpectedly demonstrated that light travels at the same speed regardless of whether it was measured in the direction of the Earth’s motion or at right angles to it”. Mastin also says when light moves from one medium to another, its speed “can of course change depending on the new medium’s index of refraction, and this “bending” of light is essentially how lenses work”. However he doesn’t mention gravitational lensing. Instead he says regardless of the speed of a light source and regardless of your speed, the “light still travels at a steady 300,000 km/s, completely contrary to classical physics and common sense”. He also says it was the young Einstein’s genius to explain why. And that in 1905, Einstein realized that “the whole idea of aether as a medium for light to travel in was totally unnecessary”. It sounds good. Especially since “the constant speed of light was to become one of the two main planks of his Special Theory of Relativity”. Unfortunately there’s a problem. A big problem. How big? The size of Texas.
When a clock goes slower
If you had a clockwork clock that was running slow, and I told you it was because time was running slow inside the clock, you’d laugh in my face. You’d know that it was due to some issue with the mechanism. Perhaps the oil was drying out and gumming up the works. Whatever the reason, you’d know that your clock was going slower because the clockwork was going slower. So, what if I showed you an optical clock going slower? An optical clock is typically an atomic clock which employs ytterbium atoms instead of caesium atoms, and visible light instead of microwaves. It isn’t some cosmic gas meter with time flowing through it. Nor is a clockwork clock. When a clockwork clock goes slower it’s because the clockwork goes slower. And when an optical clock goes slower, it’s because the optics goes slower:
ytterbium lattice double clock image from NIST
And when does an optical clock go slower? See the interview with David Wineland of NIST: “if one clock in one lab is 30cm higher than the clock in the other lab, we can see the difference in the rates they run at”. An optical clock goes slower when it’s lower. This is said to be the hard scientific evidence for gravitational time dilation. But it’s really the hard scientific evidence for light goes slower when it’s lower.
Einstein’s postulate only lasted two years
Yes, Einstein said the speed of light is constant in 1905 when he was doing special relativity, but by 1907 he was broadening his horizons and looking into what would become general relativity. That’s when he wrote a paper on the relativity principle and the conclusions drawn from it. He used Φ (phi) to denote gravitational potential, and he said this: “These equations too have the same form as the corresponding equations of the nonaccelerated or gravitation-free space; however, c is here replaced by the value c[1 + γξ/c²] = c[1 + Φ/c²]. From this it follows that those light rays that do not propagate along the ξ-axis are bent by the gravitational field”. Only two years after his special relativity postulate, there’s Einstein talking about a speed of light that varies with gravitational potential. This wasn’t some one-off. He said the same thing in 1911. That’s when he wrote a paper on the influence of gravity on the propagation of light. He said this: “If c₀ denotes the velocity of light at the coordinate origin, then the velocity of light c at a point with a gravitation potential Φ will be given by the relation c = c₀(1 + Φ/c²). The principle of the constancy of the velocity of light does not hold in this theory in the formulation in which it is normally used as the basis of the ordinary theory of relativity”. He said the principle of the constancy of the velocity of light does not hold. And it’s clear from the context that the word velocity is as per “high velocity bullet”. It’s the common usage as opposed to the vector quantity. Einstein was talking about the speed of light, which is why he was referring to c.
Einstein’s VSL attempt in 1911
There’s a Wikipedia article on the Variable Speed of Light, which is often abbreviated to VSL. If you take a look at an old version dating from 2014, you can see a section entitled Einstein’s VSL attempt in 1911. This says Einstein first mentioned a variable speed of light in 1907 and reconsidered the idea more thoroughly in 1911. However it then goes on to say Einstein abandoned the idea in 1912 because it only predicted half the deflection of light by the Sun. However it isn’t true. Einstein didn’t abandon the idea. That’s why you can find him saying the same thing year after year:
1912: “On the other hand I am of the view that the principle of the constancy of the velocity of light can be maintained only insofar as one restricts oneself to spatio-temporal regions of constant gravitational potential”.
1913: “I arrived at the result that the velocity of light is not to be regarded as independent of the gravitational potential. Thus the principle of the constancy of the velocity of light is incompatible with the equivalence hypothesis”.
1914: “In the case where we drop the postulate of the constancy of the velocity of light, there exists, a priori, no privileged coordinate systems.”
1915: “the writer of these lines is of the opinion that the theory of relativity is still in need of generalization, in the sense that the principle of the constancy of the velocity of light is to be abandoned”.
1916: “In the second place our result shows that, 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 and to which we have already frequently referred, cannot claim any unlimited validity”.
1920: “Second, this consequence shows that the law of the constancy of the speed of light no longer holds, according to the general theory of relativity, in spaces that have gravitational fields. As a simple geometric consideration shows, the curvature of light rays occurs only in spaces where the speed of light is spatially variable”.
The last quote is the English translation of what Einstein said in German in 1916: “die Ausbreitungsge-schwindigkeit des Lichtes mit dem Orte variiert”. That translates to “the propagation speed of light with the place varies”. Einstein never did abandon his variable speed of light. The people who tell you that grew up before the Einstein digital papers were online. The general relativity they were taught wasn’t the same as Einstein’s.
A shift in interpretation
As to why, see Clifford M Will’s paper The Confrontation between General Relativity and Experiment. On page 4 he refers to a period of hibernation between 1920 and 1960, and a golden era between 1960 and 1980. On page 9 he says special relativity only became mainstream in the late 1920s. That sounds odd. Something else that sounds odd is something in the Wikipedia history of general relativity article. It says Kip Thorne “identifies the “golden age of general relativity” as the period roughly from 1960 to 1975 during which the study of general relativity, which had previously been regarded as something of a curiosity, entered the mainstream of theoretical physics”. So, special relativity wasn’t mainstream until circa 1930, and general relativity wasn’t mainstream until circa 1960. There’s more, because during this golden age, “many of the concepts and terms which continue to inspire the imagination of gravitation researchers and the general public were introduced”. All in all it would seem that something was changed, and something was lost.
The speed of a light wave depends on the strength of the gravitational potential along its path
As for when I’m not sure. But Irwin Shapiro was talking about the variable speed of light in 1964. His paper was all about what we now call the Shapiro delay. Wikipedia faithfully quotes what Shapiro said, which is that “the speed of a light wave depends on the strength of the gravitational potential along its path”. That’s in line with Einstein and the evidence of optical clocks going slower when they’re lower. And yet when you google it, you find people saying “the Shapiro delay is caused by the distance between source and emitter being longer”. And you find people saying the unchanging speed of light in a vacuum is a foundational fact of relativity. There’s a huge contradiction between what Einstein said and what many physicists say today. So much so that general relativity today sometimes feels like an ersatz imitation of the real thing, authored by people who advocate time travel.
The prevailing opinion is wrong
However with a nudge from yours truly, the Wikipedia Variable Speed of Light article now talks about Einstein’s proposals after 1911. It says this: “there is no other way to interpret the velocity of light in this usage except as a variable scalar speed”. It also says “Peter Bergmann did not agree with Einstein, but left the dispute out of his earlier book in 1942 to get Einstein’s endorsement”. Peter Bergmann was Einstein’s research assistant in Princeton between 1936 and 1941. He wrote the first textbook on general relativity in 1942. But after Einstein died “Bergmann wrote a new book in 1968 claiming that vector light velocity could change direction but not speed. This has become a prevailing opinion in science”. I’m afraid to say that this prevailing opinion is wrong.
Of course, not all physicists say the speed of light in vacuo is constant. Two who don’t are John Moffat and João Magueijo. In their 2007 Comments on “Note on varying speed of light theories” they said this: “Can c vary? Could such a variation be measured? As correctly pointed out by Ellis, within the current protocol for measuring time and space the answer is no. The unit of time is defined by an oscillating system or the frequency of an atomic transition, and the unit of space is defined in terms of the distance travelled by light in the unit of time. We therefore have a situation akin to saying that the speed of light is “one light-year per year”, i.e. its constancy has become a tautology or a definition”. They talked about a tautology. To appreciate this, take a look at the NIST caesium fountain clock:
Image courtesy of NIST
It’s used to define the second as “the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom”. The radiation is microwave radiation, which is light in the wider sense. In essence you count light waves going by, and when you get to 9,192,631,770 you say a second has elapsed. Then you use the second along with the light to define the metre as “the length of the path travelled by light in vacuum during a time interval of 1 ⁄ 299,792,458th of a second”. You use the motion of light to define the second and the metre. And then you use them to measure the motion of light. That’s why you always measure the speed of light to be 299,792,458 m/s. It doesn’t matter how fast the light is moving, you will always end up saying the speed of light is 299,792,458 m/s. That’s the tautology. The speed of light is defined to be constant, and it is absolutely amazing that this is not generally recognised. Can you imagine what you’d think if somebody tried to tell you that the speed of racehorses was constant? And that the winning horse won because time flowed faster for it? You’d think they were crazy, or stupid, or both. But that’s what they tell you about optical clocks, and about light.
The reality that underlies gravitational time dilation
Imagine you repeated your definition of the second and the metre at a lower elevation. Light goes slower when it’s lower. When the light goes slower the second is bigger. Then the slower light and the bigger second cancel each other out such that the metre is unchanged. You will still say the speed of light is 299,792,458 m/s. Then you might be tempted to say 299,792,458 m/s at one elevation is the same as 299,792,458 m/s at another, even though the metres are the same and the seconds aren’t. That’s clearly wrong. It’s like saying 100 metres per second is the same as 100 metres per second and a half. But people will insist that 299,792,458 m/s at one elevation really is the same speed as 299,792,458 m/s at another, even though they know the seconds are different. They insist even though they know that optical clock rates vary and there isn’t any actual time flowing through them. Even though they know that the things that move inside optical clocks are things like light. At times the resistance is dogged. At time it’s almost mindless, as if Morton’s demon is sitting on their shoulder. You have to make things extremely simple to corner the reluctant ego. The simplest thing I’ve found is the parallel-mirror light-clock. It’s employed in the simple inference of time dilation due to relative velocity, and it can also be employed to demonstrate the simple reality that underlies gravitational time dilation. See this gif:
Gif image by Brian McPherson
The two light pulses aren’t going at the same speed. If they were, the parallel-mirror light clocks would stay in time. And if they did, the NIST optical clocks at different elevations would stay in time too. Only they don’t.
The speed of light varies in the room you’re in
See Is The Speed of Light Everywhere the Same? It’s a PhysicsFAQ article by Don Koks. He talks about laser gyroscopes and the Sagnac effect, and later says “Einstein talked about the speed of light changing in his new theory”. This new theory was of course general relativity, which concerns non-inertial reference frames. The article goes on to say the room you’re sitting in right now is a very high approximation to a non-inertial reference frame. And that whilst an observer stationed at the ceiling will measure the speed of light to be c, as will the observer stationed at the floor, the “global” observer will say that light at the ceiling travels faster than light at the floor. You aren’t situated at the ceiling, or at the floor. You’re sitting in your chair. You are that global observer. What this means, is that the speed of light varies in the room you’re in. If it didn’t, your pencil wouldn’t fall down. Pick up your pencil, hold it up at head height, then drop it on your desk. The speed at which it hits the desk depends on the difference between the speed of light at head height, and the speed of light at desktop height. The speed of light varies like the speed of other waves vary. Just like Einstein said.
The ascending photon speeds up
The situation is similar to sound waves in the sea, where the speed of sound typically decreases with depth. That’s why a horizontal sonar wave bends downwards. In similar vein the horizontal light wave bends downwards because the speed of light decreases with elevation. However the vertical sonar wave does not bend downwards, and nor does the vertical light wave. Imagine you had a torch that could emit a single photon. Imagine you aimed it straight up and pressed the button. What happens to the ascending photon? It isn’t like an ascending brick. The speed of light at the ceiling is greater than the speed of light at the floor. So the photon doesn’t slow down as it ascends. Au contraire, the ascending photon speeds up. This is, of course, a matter of some gravity.