# Why does speed of light have to be constant?

My question is not about why speed of light has a particular constant value which has been sufficiently addressed in other questions on SE already. I want to know whether the fact that speed of light has to be constant is a consequence of other fundamental laws of Universe, or is it just a fundamental law itself.

My motivation for asking this question is thinking about Special Theory of Relativity. One way of understanding why time dilation and length contraction happen is to look at it from the perspective of constancy of speed of light. Alice who is on the moving source of light and Bob, a fixed observer see the speed of light as constant. In order for Alice to see the speed of light as a constant value, time slows down and length expands (in forward direction) thus keeping the speed constant.

In the course I'm doing, the Professor tells that this constancy of speed of light could be more generally referred to as, "Principle of wave constancy". That is, the speed of any wave is a property of the medium (and thus it remains constant). So, the speed of a wave in water, for example, remains constant even if the source of the waves (say a paddle keeps moving). Frequency and wavelength may change but the speed remains constant. In a similar fashion, light is an electromagnetic wave and thus its speed is constant. And the constant speed is defined by the medium (which happens to be vaccum i.e., no medium for light).

My questions are twofold now -

1. Is the above reasoning correct to explain why speed of light is constant? (i.e., thinking of it as constancy of any wave in general)
2. Theoretically, could there be other waves that could travel through vaccum at a different speed? If yes, time dilation and length contraction will have to be different.

I'd appreciate it if the answer is kept non-technical (essential math is okay). More importantly, I request you to try and explain without using General Theory of Relativity. Because I'm thinking how the scientific understanding by 1905 had already knew that speed of light has to be constant.

• If you solve the equations of electromagnetism, (Maxwell’s equations) with no charges or currents, you derive an equation for a wave traveling at speed c. This derivation makes no reference to any preferred inertial reference frame, suggesting that the speed c would not change from one inertial reference frame to another (“the speed of light is constant”). Oct 28, 2018 at 3:44
• As I said in a recent answer, it's an accident of history that we first encountered $c$ as the speed of light, but its true significance (according to SR) is that it's the space / time conversion factor. Also see this longer answer of mine. Oct 28, 2018 at 3:49

Is the above reasoning correct to explain why speed of light is constant? (i.e., thinking of it as constancy of any wave in general)

I would say no, although I'm sure you could argue for it if you defined you terms carefully. I just don't think it's good to put light waves and other waves on the same footing. First, light does not need a medium to travel through. Second, according to SR, light behaves much different than waves such as sound in air. I will explain.

Let's say I'm moving near but less than the speed of sound in air relative to the air itself. I emit a sound wave. Since the speed of sound is constant relative to the medium, I will practically be riding right next to this sound wave. In my own frame, I will observe the speed of my sound wave to be very slow, since I am moving just a little less faster than it.

Now let's say I take off in my spaceship near the speed of light. Then I turn on my ship's headlights. According to my own reference frame, I won't be practically riding along with the light I just emitted. I will see it move away from me at the speed of light relative to me.

And this is where SR breaks away from the behavior of sound waves. Sound waves move relative to the air they move through. Their speed is defined relative to the medium. You could argue that the medium serves as an absolute frame. But light does not have this property. There is no absolute frame we can use.

I want to know whether the fact that speed of light has to be constant is a consequence of other fundamental laws of Universe, or is it just a fundamental law itself.

This is held to be a postulate of SR, but whether or not it is a fundamental law is somewhat subjective, and the answer could change based on what we end up discovering about the universe in the future. According to SR it is a fundamental property of the universe. But maybe we will discover more that explains why this happens. Then that explanation will be the fundamental explanation of the universe.

I will note that your title question isn't answerable with physics. Nothing has to be any way. Why does mass have to warp space-time? Why does the universe have to have more than two fundamental forces? There isn't anything saying it has to be this way. But it is.

Theoretically, could there be other waves that could travel through vaccum at a different speed?

I believe the answer here is no, but I have to admit I can't think of a good reason. Even if there was, I'm not sure it would ruin SR. I believe it would just mean that the particle associated with this wave would need to have mass, but then I'm not sure it could propagate without a medium. I'm not sure about all of this though.

• My source of confusion seems to have been assuming "wave constancy" to be constant universally. But it's constant only relative to the medium. Your sound wave example cleared it up. Thanks! Dec 1, 2018 at 4:58
• @yathish glad I could help! Dec 1, 2018 at 5:49
• I can't explain downvotes, as I seems to be a good answer to me. I have a question, though, about the last paragraph. Why do you suspect (if you actually do) that a wave associated to a particle couldn't propagate without a medium? Dec 19, 2018 at 4:45
• @anonymous I'm not entirely sure of it. I was just thinking of a reason as to how a wave could travel without a medium in the vacuum but not propagate at the speed of light. The only reason I could think of is mass, but I didn't want it to seem like I was saying that's for sure. Dec 19, 2018 at 6:09
• I see. I think that that statement is true, as particle fields can have plane wave solutions, and they travel with a speed that is smaller than $c$ just because they are massive. Dec 19, 2018 at 6:17

I want to know whether the fact that speed of light has to be constant is a consequence of other fundamental laws of Universe, or is it just a fundamental law itself.

I know, I'm taking the risk of getting the mark TL;DR. But I wasn't able to say what I judge relevant using few words.

I would begin with one word you wrote: "constant". In physics it must be used with caution, since its meaning is not always the same. We can mean "constant in time", like in "energy of an isolated system is constant". Or "constant in space" like in "according to cosmological principle, matter density is constant in the Universe". Or, to come nearer to our topic: "speed of e.m. waves in vacuum is constant wrt to frequency."

Let me expand this point. When this happens, not necessarily for a light wave but for whichever sort of wave, we say that the medium in non-dispersive. This is what your teacher meant with his "principle of wave constancy". It is a property of the medium, and does not generally hold true, for elastic waves, for sound waves (a kind of elastic waves) etc. It may hold approximately - this happens for sound waves in air - or not hold at all: think of gravity waves we see at sea surface or on a lake, on a pond...

But the principle of wave constancy brings also another meaning, which you quoted: the speed doesn't depend on the source's motion. This is really general, as far as waves are concerned. A wave is a motion of a medium (air, water, rock...) which is originated by an excitation produced by a source. But once the wave leaves the source and propagates in the medium, its behaviour becomes independent of what originated it.

A last, and totally different, meaning of "constant" you also made use of: "independent of reference frame". This meaning is of utmost importance in relativity, so much so that there is a special word you should use in place of the ubiquitous "constant". We say invariant. Speed of light in vacuum is invariant. This is what your question is about.

The above was necessary to establish some points I will rely on in the following. Now my answer can begin.

It's unavoidable to talk a little about the story of these ideas.

• What is light made of? Particles or waves?
• What was known at beginning and at end of 19-th century?
• What did Maxwell teach us?

1) Nature of light concerned physicists from the very beginning of modern physics - since 17-th century. Newton - as already recalled by @annav - adopted the particle model. But I certainly would not say that "before Maxwell, light was not considered a wave" - the opposite is true. Wave nature of light was sustained by Huygens, a contemporary (a little older) of Newton's. It was well known that a particle model encountered several difficulties; the main reason in favour was that a mechanical theory was more understandable at those times, when the theory of waves was in its infancy (but Huygens' principle should not be forgotten).

2) In later times experimental and theoretical arguments in favour of waves accumulated. Famous Young's double-slit experiment goes back to the very beginning of 19-th century. In that century, well before Maxwell's work, evidence for light as a wave became overwhelming and I dare say that in the mid-1800s no physicist believed in the particle theory. Leaving for a moment Maxwell aside, optics was a well-established branch of physics, capable of sophisticated experiments - above all I'm thinking of interferometry. Optics grounded on wave theory served in designing and building instruments (microscope, telescope, spectroscope) which were fundamental for progress of other sciences: astronomy, biology, chemistry. So, light is a kind of waves - said physicists. But waves of what? The only answer could be given was to coin a name for a substance endowed with very special properties and whose vibrations we could see as light - the ether.

3) Then Maxwell came. Around 1870 he put together all what was known about electromagnetic phenomena and made a synthesis of e.m. laws. In doing so he discovered a gap, an inconsistency - the differential form of Biot-Savart law, for magnetic field generated by an electric current, disagreed with other laws (I cannot deepen the subject - I would deviate too much from what has been asked). It is well known how Maxwell solved the problem: by inventing his displacement current - a term to be added into Biot-Savart law.

This step gave rise to "Maxwell equations" for the e.m. field as we know them today. Its relevance for present discussion is in a "side effect". Maxwell was able to show that his equations implied the existence of electromagnetic waves - something nobody had imagined before. Not just this: he computed the speed of these new waves in vacuum and saw its value fitted well with that of light speed, known from optics measurements. He concluded:

It is manifest that the velocity of light and the ratio of the units are quantities of the same order of magnitude. Neither of them can be said to be determined as yet with such a degree of accuracy as to enable us to assert that the one is greater or less than the other. It is to be hoped that, by further experiment, the relation b tween the magnitudes of the two quantities may be more accurately determined.

In the meantime our theory, which asserts that these two quantities are equal, and assigns a physical reason for this equality, is certainly not contradicted by the comparison of these results such as they are.

(Treatise, 3rd ed. 1873, vol. II p. 388)

Few years later experimental evidence for e.m. waves - not light, but of much longer wavelength - was given by Hertz. Practical applications soon arrived (wireless telegraphy - Marconi, at the turn of the century) the most impressive being sea rescue to ships in distress: e.g. Republic 1909, Titanic 1912.

Coming back to Maxwell, it must be noted that in his view e.m. waves were vibrations of a medium: the ether. At that time no physicist had doubted how to answer the question: "In which reference frame do Maxwell equations hold true? In which it is true that e.m. in vacuum (i.e., in absence of polarisable matter) do propagate with speed $$c$$?" (BTW, in those times this symbol had not yet been adopted for light speed.) They all would answer: "In the rest frame of ether." Every physicist would consider obvious that in any other frame light would move at a different speed, furthermore depending on direction.

Then the problem was how to determine such frame. If Earth was moving wrt ether, optics experiments conducted on Earth would reveal that. This was the motivation of Michelson, and I would not insist on this point, to finally come to Einstein. Here there is an unsolved historical problem, as Einstein himself declared that when he wrote his famous paper (1905) he did not know of Michelson-Morley experiment, or at least it had not been his motivation. No doubt that experiment is not quoted in Einstein's paper, which in its first page gives different arguments for his universally known postulates.

And now we are near to answer your question. Einstein's two postulates are:

1. (Principle of relativity.) Expressed in modern words, it states that all physical laws are the same in every inertial reference frame.

Comment. You may find the same principle, in a form related to those times, in a famous page of the Dialogo sui Massimi Sistemi (Dialogue Concerning the Two Chief World Systems) here (Page 107. Read from "Shut yourself up with some friend".)

The difference is that Galileo, when physics was in its infancy, could not talk of physical laws or of inertial frames, whereas Einstein, three centuries later, does this revolutionary step.

1. (Invariance of light speed.) Light propagates in vacuum with the same speed in every inertial frame, irrespective of the source motion.

Note that Einstein's postulates imply the non-existence of ether, which raises an obvious question: if so, e.m. waves lose their medium - how can it be possible? I must leave aside the question.

Now beware: what I'm going to write is a personal view, not necessarily shared by other physicists. I simply assert that second postulate is redundant, as it follows from first.

Let me explain. First of all note that second postulate actually consists of two separate statements:

• light speed is invariant
• it is independent of source's motion.

As to the latter, I already observed that it follows from wave character of light. Once the wave has been emitted from its source, it "forgets" that and propagates according its law, which contains no reference to the source and its motion.

Invariance of light speed follows from first postulate if Maxwell equations are included among accepted physical laws. It is generally stated that is not correct, because of Ignatowsky's theorem (1911). It says that using first postulate alone, together with homogeneity of space and time (i.e. invariance under space and time translations) only two possibilities exist for the trasformation law between intertial frames:

• Galileo's transformation
• Lorentz transformation, but with an undetermined velocity in place of $$c$$.

Then the limiting speed will not be $$c$$ but another (greater) velocity. If this were the case, Maxwell equations couldn't be exactly true but only approximately so. Up to now, no deviation from their validity has been found, but we must always leave the possibility open, as for any physical law.

Following this line of thought there is no reason however to assume the second postulate as a ground postulate of SR. It would be better to look at it as a corollary of physical laws under experimental scrutiny like many others. In my view the principle of relativity keeps a higher rank, even if I don't mean that it is exempt from possible experimental falsification, of course. But I would not delve here into a genuine epistemological discussion.

• It's more like TTL;DR! Thanks for your time and explanation. The historical context was particularly useful to me. I haven't understood everything you said (especially your personal conjecture in the last section). I'll revisit this once I finish my course and comment further, if need be. Dec 1, 2018 at 4:36
• Maxwell's equations only imply invariance of light speed provided invariance of permitivity and permeability of free space. But they don't imply that, and it does not hold for permitivity and permeability of other media. In other medium light moves slower than $c$ and then its speed is no longer invariant and may also not be isotropic. So it is still needed to postulate something. Feb 28, 2019 at 6:49
• You wrote: "speed of e.m. waves in vacuum is constant wrt to frequency." Where did you get that from? Or how can one know? Dec 15, 2019 at 15:49

Is the above reasoning correct to explain why speed of light is constant? (i.e., thinking of it as constancy of any wave in general)

It depends on the meaning of "explain". In physics one uses mathematical models, with extra axioms called postulates, or principles ,or laws, which pick up a consistent subset of the mathematical format to fit experimental observations. These models have to be predictive of future measurements, (with no predictive power it would just be a mathematical mapping).

Classical electromagnetic theory as given by Maxwell's equations is a perfect demonstration of this. The mathematical subset is picked up by the laws that described electricity and magnetism, before Maxwell's equations, because they fitted the data and were predictive.

The constancy of speed of light emerges from the set of equations, which are wave equations. In this sense, as for other wave equations fitting physical observations, the constancy of the speed of light can be "explained" as the constancy of wave propagation in a medium. In reality it is the result of the specific observational laws which lead to the wave equations. Before Maxwell, light was not considered a wave, certainly not by Newton, they worked with optical rays.

Theoretically, could there be other waves that could travel through vaccum at a different speed? If yes, time dilation and length contraction will have to be different.

It would have to be a different mathematical format , i.e. not described by what is called a wave equation presently. (see here also). It would have to have sinusoidal solutions, which are what waves are about, and one would have to see whether it fits data, and whether a velocity of propagation different than c appears in this theory.

It’s a fundamental law in the sense that it has been deduced from EM and then empirically ratified. It’s also been theoretically ratified a postereroi, by being incorporated into QED and the Standard Model. So anyone wanting to posit a varying speed of light will have a lot of explaining to do.

Nevertheless, there are theories that posit a changing velocity of light. For example, Joao Maguiejo has come up with such a theory, called VSL (Varying Speed of Light), and earlier, John Moffat.

The notion here is that c varied in the early cosmological epoch. One innovative explanatory feature of this theory is an alternative mechanism to explain the horizon problem of cosmology which is generally explained by inflation. (Personally, I happen to like it as it explains the horizon problem without positing an additional inflationary field).

I will answer the question that was asked: "In a similar fashion, light is an electromagnetic wave and thus its speed is constant. And the constant speed is defined by the medium (which happens to be vaccum i.e., no medium for light)." "My questions are twofold now -

1. Is the above reasoning correct to explain why speed of light is constant? (i.e., thinking of it as constancy of any wave in general)
2. Theoretically, could there be other waves that could travel through vaccum at a different speed? If yes, time dilation and length contraction will have to be different." (1) Not correct. For light, the speed is constant at c, even if the source and receiver are approaching each other. That it is "constant in the medium" does not mean anything about such cases. (2) I only know of gravitational waves in vacuum. I imagine GR wants them to be at a constant, c, but I don't know of any experiment for their speed properties.

Well, first, it bears pointing out, there's a fundamental problem with calling any dimensioned quantity constant or variable: the infinite regress implied. What about its units? And with respect to which units do you assess their constancy or variableness?

How do you do you even compare the units at one place and time with those at another place and time, except by moving them over from one place to the other and just asserting by decree that it's the "same" unit? In fact, how can you even define derivatives for dimensioned quantities, if you have no way of directly comparing units at different places and times?

So, the very terms "constant" and "variable" are meaningless; and the answer is that light speed is neither constant nor variable, but is a dimensioned quantity.

So, to by-pass this issue it is better to both deconstruct what we're actually trying to say with the central hypothesis of Relativity (both Special and General) and rephrase the question; doing so in a way that will allow us to actually resolve this matter at a much deeper level, and better express what key postulate of Relativity is actually saying.

Relativity: Contrasted With The Newtonian and Carrollian Worlds
The central postulate of Relativity, both Special and General, is that space and time - together - form a chrono-geometry that locally looks like a Minkowski space. The key property of Minkowski space is that at each point in space and time, in each direction, there is a finite non-zero speed that is absolute.

Contrast and compare this with other conceivable possibilities. In the world of Newtonian physics, the assumption is that instead of a finite speed, it is the infinite speed that is absolute. This is the speed of being at different places at the same time. So another way of expressing the same idea, through the back door, is that the quality of being simultaneous is absolute. That's referred to as Simultaneity.

Another, lesser discussed, possibility is a world in which the absolute speed is 0. That's referred to as the Carrollian universe, because in it, things can have momentum, but don't move anywhere.

Another possibility is where all 4 dimensions are spatial and there is no time-like dimension nor any notion of "speed" at all; just a timeless 4D space. There, instead of "speeds", you'd be talking about "angles" with respect to the 4th spatial dimension.

The Temporal Logics Of Newtonian and Minkowski Worlds
In the world espoused by Newton, there is such a thing as a "there and now". For instance, with respect to any place and time on Earth, there is a unique time at any given location on the moon that is "now". All times after it are in our future, and all times before it are in our past. Likewise, with respect to that "now" on the moon, all points after our now are in its future, and all points before our now are in its past. What is past and future of a given time is, thus, independent of your location.

In a world that is locally Minkowski, there is no such thing as a "there and now". In particular, infinite speed is not even absolute, but relative. With respect to a moving frame of reference an infinite speed transforms to a finite faster-than-light speed. Conversely, for every faster than light speed, there is a moving frame of reference in which it is infinite.

Correspondingly, with respect to any point on Earth at any time, since the moon is about 1½ light seconds away (using very round numbers), there is a gap of about 3 seconds' worth of the moon's time line that can be considered to be "simultaneous" in some frame of reference. As such, no part of that gap can be considered to be either before or after here and now on Earth. It is neither in our past nor our future. The term sometimes used is "absolute elsewhere".

Another example: Betelgeuse is (using round numbers again) about 1000 light years from us. If there is more than 1000 years remaining before the view of the explosion of the star reaches us, then the explosion is in our future. If there is under 1000 years before we see its explosion, then it is neither in our past nor our future.

The notion that "it has already happened, but we haven't seen it yet" is wrong. That's Newtonian thinking, and even some astrophysicists (like Tyson, hello Tyson?!) have been sighted talking Newtonian. In the world just described here, the explosion still hasn't happened yet - in our past. There is no "now" or "just now over there, but we haven't seen it yet", because there is no such thing as "there and now" to begin with.

That talk is qualified a bit, further below, with the mention the "co-moving" frame, and that's the only out someone like Tyson could claim. As an aside, I've also seen Sabine Hossenfelder employ the co-moving frame, in one of her videos, as a possible way to define an "absolute now". But this is all pertaining to General Relativity, not Special Relativity.

From 3D Equal-Time Layers To Light Cones
If you were to draw a diagram, suppressing one spatial dimension, showing the everyone's time line in the vertical direction, the locus of all light-speed trajectories emanating from a given point in space and time, would form a conical shape. That's called its "future light cone". Similarly, the locus of all light-speed trajectories arriving at that point in space and time would form its "past light cone".

The past of the point at the given time lies on and within the past light cone. In particular, your visual field - including the sky - is on your past light cone and, for all intents and purposes, is your past light cone.

The future of the point at that time lies on and within the future light cone. The rest of the 4D continuum is the absolute elsewhere.

What's your "past" and "future" depends on both when you are and where.

So, it is not that "light speed is constant" in Relativity, but that it is giving you an infrastructure of a world that's different from that espoused by Newton. Instead of a stack of 3D "equal time" spaces layered up one on top of the other, as is the case in Newtonian physics, you have a 4-dimensional continuum enmeshed with a field of light cones.

Constancy Of Light Speed: Special Relativity
Now, we can address the matter at hand in a much more direct way that gets to the root of the matter. Instead of asking "is light speed "constant", in its place, we can ask: "is the field of light cones constant"?

At bare minimum, this should consist of at least the following:

(1) The field of light cones is position-independent; i.e. spatially homogeneous.

(2) The field of light cones is time-independent; i.e. temporally homogeneous.

In that way, we can assure that it is the same at all places and times.

Second: we also want it to be the same in every direction, so that:

(3) The field of light cones is isotropic; i.e. invariant with respect to "rotations"; i.e. reorientation of spatial axes.

Finally: the key hypothesis of Special Relativity is that light speed be observer-independent. So, we also want:

(4) The field of light cones is invariant with respect to "boosts"; i.e. changes to moving frames of references - those changes, in particular, being identified as "Lorentz transformations".

Generically, a transform to a moving frame of reference is referred to as a "boost". The contrast being drawn here is between the Lorentz transform and the one applicable to the world of Newton - the Galilei transform.

Under Galilei transforms, the layering of the world into 3D "equal time" slices remains fixed, but not the field of light cones (nor any field traced out by finite speed trajectories).

Under Lorentz transforms, it is the light cones that remain fixed, while the layering into "equal time" 3D slices does not.

So, here's the conclusion: a chrono-geometry that possesses a field of light cones, along with a set of transforms for (1) spatial translations, (2) time translations, (3) rotations and (4) boosts that leave it invariant is one and the same as a Minkowski Geometry. It is one way to characterize a Minkowski Geometry.

On a slight technicality, it's not completely true, because you also have conformal Minkowski spaces, but I'm just going to ignore the case, for the sake of expediency.

Non-Constancy: The Generalization To General Relativity
If "constancy" is defined this way, then the transition from Special Relativity to General Relativity - a transition away from Minkowski Geometries to more general "Lorentzian space-times" consists of rescinding the constancy condition.

Instead, the assumption reverts to the weaker condition that the invariant speed everywhere be finite and non-zero, while allowing for fields of light cones that need not be spatially or temporally homogeneous, isotropic or boost-invariant.

Contrast Minkowski geometry to the geometries presented in General Relativity. In particular, for any of the Big Bang geometries (i.e. the instances of the FRWL metrics), while it is true that we have a field of light cones that possess (1) spatial homogeneity and (3) isotropy, they are not invariant under (2) time translation, nor under (4) boosts!

The lack of boost-invariance means that a particular frame is selected out. In cosmology it is referred to as the "co-moving" frame. With respect to it, the Earth has a definite motion ... I think it's in the direction of Sagittarius.

In the co-moving frame, the Cosmic Microwave Background would appear color-uniform. In contrast, in the Earth's frame, there is slight red-shifting on one end, and slight blue-shifting on the other.

Another property of the co-moving frame - far more important - relates to the question of the size of the Universe. When we say "the universe is spatially finite/infinite" - which 3D layering are we referring to? And the answer is: the 3D layering that is associated with the co-moving frame.

Technically, the layering is the locus of all spatial paths that are "orthogonal" to the co-moving trajectories. It is only in the co-moving frame that this layer wraps around to form a connected 3D layer (that is, if the Universe is spatially finite). In all other frames, the corresponding 3D layer (for the spatially finite case) would wrap around and come back to our location at a different time, thereby coiling to make up a infinite 3D helix.

The lack of time invariance means that there is time dependency on the structure of light cones. One way of saying this is that the speed of light is simply variable, and that this is the variability that the FRWL metric is actually describing!

But no matter how you describe or characterize it, the fact remains, the field of light cones for the FRWL metrics do not possess time-translation symmetry.

We can see this more clearly as follows. The FRWL metrics allow for 3 main cases: (1) where the spatial 3D layers are uniformly positively curved and wrap up into hyperspheres - the above-mentioned "spatially finite" case, (2) where the spatial 3D layers are uniformly negative curved and form an infinite hyperbolic geometry and (3) where they are flat and form a 3D Euclidean geometry.

Hence, the origin of the joke "define the Universe and give 3 examples".

To a very high degree of accuracy, the observed universe falls into the the flat case (3). For such cases, the FRWL metric has a form given by the following line element:

$$A(t) \left(dx^2 + dy^2 + dz^2\right) - c^2 dt^2,$$

where the "growth" factor $$A(t) > 0$$ varies with time, in such a way that $$A(t) → 0$$ as $$t → 0$$, where $$t = 0$$ is set as the time of the "Big Bang".

For Minkowski geometry, $$A(t) = 1$$, and $$c$$ is the finite non-zero absolute speed that is the topic of this discussion: the "(in vacuo) speed of light".

However, this way of writing the metric - as a line element for distances - obscures the view of the matter. If you rewrite it as a line element for proper time

$$dt^2 - \frac{A(t)}{c^2} \left(dx^2 + dy^2 + dz^2\right).$$

Now, you can more clearly see that this metric:

$$dt^2 - \frac{1}{c(t)^2} \left(dx^2 + dy^2 + dz^2\right).$$

that is just that for Minkowski space, but with a variable speed of light:

$$c(t) = \frac{c}{\sqrt{A(t)}}.$$

In turn, this also helps make more clear just what the Big Bang "singularity" actually is. The $$t = 0$$ hypersurface of this geometry is actually an envelope of light cones. Another way of saying the same is that it is a "null surface"; and that all paths in the "t = 0" surface are "light-like". In other words, light speed is infinite on it and it's a 3D slice of Newtonian geometry! (That is: of a Newton-Cartan space-time).

Technically, the "t = 0" surface actually violates the "local Minkowski" axiom for General Relativity, if it's included in the geometry. Therefore, most cosmologists exclude the "t = 0" surface and treat only the "t > 0" sub-space as the Big Bang space. There are others, like Hawking & Hartle people, or the people who hang with Mansouri, who consider and treat "signature changing" geometries. Hawking's treatment still excluded the "t = 0" surface, and he had a quantum jump from the "t < 0" subspace, which is purely spatial 4D locally Euclidean and is not a space-time at all (a "timeless space") to the "t > 0" subspace, which is the locally Minkowski space. Others, like Mansouri, also include the "t = 0" null 3D surface and try to address the singularity issue head-on.

So, for this kind of geometry, it is only the "t > 0" subspace that conforms to the "locally Minkowski" axiom of General Relativity, and you don't even have that - much less the constancy of light speed axiom, if considering the entire geometry for all $$t$$.