The meaning of constancy of the speed of light

[Killtech]

The constancy of the speed of light is a fundamental assumption in modern physics, built into both relativity and the SI system of measurement. I’ve been wondering: to what extent is this a fundamental property of nature, and to what extent is it a convention tied to our choice of units and measurement definitions? And does our current measurement framework even allow us to establish the possibility of it to vary in the first place?

The Issue of Measurement

• The SI second is defined using atomic clocks based on the frequency of a cesium transition.

• The meter is defined in terms of the speed of light, which is fixed at 299,792,458 m/s by definition.

• Since c is numerically fixed, any potential variation in light’s speed would be hidden within changes in how we measure time and space rather than appearing as an explicit difference in measured speed.

A Need for a Counterhypothesis

• To test for a varying c, we’d need a physical framework where such variation makes sense and is not simply reabsorbed into our measurement definitions.

• But how do we define an operational way to measure a changing c, when our time and length units are already tied to its assumed constancy?

The Variation of c

• The fine-structure constant alpha is given by: e² / (4pi epsilon0 hbar c), meaning that if c varied, so would the constant.

• Since the energy levels of Caesium atoms - and thus atomic clock frequencies - depend on alpha, any variation in c would affect the very clocks we use to define the second.

• This creates a self-referential issue: if we use atomic clocks to measure changes in c, but those clocks themselves change due to variations in c, can we even establish whether c is varying in the first place?

• If c depended on location, it would cause clocks in different regions to tick at different rates.

• Since clocks define proper time, this effectively means that a variation in c would manifest as spacetime curvature.

• How, then, would we distinguish such an effect from gravitational time dilation caused by mass-energy?

Back around 1900, Poincaré already recognized the subtleties in these assumptions and criticized how astronomers arrived at their conclusion that c was constant. In modern physics this consideration is even more interesting to explore. I tried this conversation with ChatGPT, and it was well versed to discuss the topic. Here is a link to that conversation: link deleted; violation of 2.13

I’d love to hear thoughts on whether this is a meaningful issue to explore.

[studiot]

Just now, Killtech said:

I’d love to hear thoughts on whether this is a meaningful issue to explore.

If c varies first and foremost you need to specify what it varies with.

Secondly you then need to rewrite our system of mathematics since constancy is a requirement of Maxwell's equations, and the wave equation in general.

Can you offer this ?

[swansont]

!

Moderator Note

keep ChatGPT out of mainstream discussions. see rule 2.13

 

 

As studiot notes, the invariance of c is embedded in Maxwell’s equations. EM waves wouldn’t be waves if c wasn’t invariant. Yet they are waves even when there is relative motion between source and observer.

 

3 hours ago, Killtech said:

• Since the energy levels of Caesium atoms - and thus atomic clock frequencies - depend on alpha, any variation in c would affect the very clocks we use to define the second.

• This creates a self-referential issue: if we use atomic clocks to measure changes in c, but those clocks themselves change due to variations in c, can we even establish whether c is varying in the first place?

The solution here is straightforward: you measure the stability of the fines structure constant without measuring a duration. Say, by measuring transitions in different elements. If alpha changes, the transition frequencies change by different amounts, and you can measure that 

[KJW]

@Killtech, I think you are on the verge of realising that c (and the other fundamental constants) must be constant because when we measure something, it is relative to the definition of the units that have been used, and therefore, in order to obtain a definite value for a measurement, the units of measurement have to be assumed to be intrinsically constant.

 

[studiot]

Just now, KJW said:

@Killtech, I think you are on the verge of realising that c (and the other fundamental constants) must be constant because when we measure something, it is relative to the definition of the units that have been used, and therefore, in order to obtain a definite value for a measurement, the units of measurement have to be assumed to be intrinsically constant.

 

But in Natural units the speed of light is dimensionless and exactly 1.

https://en.wikipedia.org/wiki/Natural_units

 

So whatever units you measure distance and time or other constants in is irrelevant.

c is still constant.

[Killtech]

3 hours ago, studiot said:

If c varies first and foremost you need to specify what it varies with.

Secondly you then need to rewrite our system of mathematics since constancy is a requirement of Maxwell's equations, and the wave equation in general.

Can you offer this ?

42 minutes ago, swansont said:

As studiot notes, the invariance of c is embedded in Maxwell’s equations. EM waves wouldn’t be waves if c wasn’t invariant. Yet they are waves even when there is relative motion between source and observer.

assume c(x,t), i.e. it may vary by time and region

A simple approach for Maxwell with non constant c derived from current physics: SI second definition explicitly states that corrections due to the local gravity field should not be applied. Now define an alternative time measure and clocks with corrections for the local gravity field such as to enable clock in different regions to tick with the same rate. As clocks define the proper time this condition can be translated as to finding a correction such that curvature due to energy-mass vanishes. 

Defining a new clock and unit of time is so far a change of conventions, therefore does not affect real physics. However it changes measurement (and all SI units depend on the second) and the mathematical model of physics. Now transform maxwell equations from GR proper time tau to new proper time tau' and you get additional time and region dependent terms, one of which describes the wave speed and this we call c(x,t). These new terms reflect the physical degrees of freedom which GR describes via geometry. As a trade-off all physical degrees present in GR are gone, as it now becomes flat by construction.

More specifically you do this:
 

in short, the effect of this transformation is that curvature of space time transforms Maxwell into a wave equation in a medium where the medium is just an equivalent representation of what was previously embedded in the geometry. EM-waves represented via tau' now bend in vacuum because it gets a refractive index replacing curvature. it still the same physics, just a different mathematical model that yields the same predictions.

42 minutes ago, swansont said:

The solution here is straightforward: you measure the stability of the fines structure constant without measuring a duration. Say, by measuring transitions in different elements. If alpha changes, the transition frequencies change by different amounts, and you can measure that 

To measure alpha, you need to do measurements and all those measurements use SI units, all of which depend on the second. While alpha has no unit itself, it still depends on the geometry of space time and since the definition of the second does define proper time, the geometry is a convention in this context. So no, in SI units alpha can never change, but other units that do not inherently assume c to be constant do allow it to change.

33 minutes ago, KJW said:

@Killtech, I think you are on the verge of realising that c (and the other fundamental constants) must be constant because when we measure something, it is relative to the definition of the units that have been used, and therefore, in order to obtain a definite value for a measurement, the units of measurement have to be assumed to be intrinsically constant.

You are exactly right, and so far you are the first person i meet that understand this. It however means that the constancy of c is more of a convention of our physical model and specifically our concept of measurement rather than fundamental law of physical. It would also make us free to change that convention, if there was a use case where a different convention would be more beneficial, no?

Edited Monday at 08:15 PM by Killtech

[KJW]

1 hour ago, Killtech said:

It however means that the constancy of c is more of a convention of our physical model and specifically our concept of measurement rather than fundamental law of physical. It would also make us free to change that convention, if there was a use case where a different convention would be more beneficial, no?

I don't agree with this because one also has to consider that the laws of physics determine the intrinsic size of the units that we define. The constancy of the fundamental constants is implied by the constancy of the laws of physics. For example, suppose you measure the length of some object. You use a steel ruler to measure the object in metres. But you could count the number of iron atoms along the edge of the ruler. Thus, instead of measuring the length of the object in metres, you have measured the object in terms of iron-to-iron interatomic distances. Any change in the laws of physics that alters the iron-to-iron interatomic distance would also alter the length of the object by the same amount, and therefore the length of the object in terms of iron-to-iron interatomic distances would be unchanged. But this invariance implies that the changes of the laws of physics cannot be measured, which justifies the assumption of the constancy of the laws of physics.

When you express the fundamental constants in terms of their dimensions, the result is a system of equations. When this system of equations is inverted, you obtain a definition of the Planck units in terms of the fundamental constants. In principle, you could measure everything in terms of the Planck units. The laws of physics govern the intrinsic size of the Planck units, but you can't actually measure the Planck units because everything is measured relative to the Planck units, and therefore the laws of physics cannot be anything but constant. Also, because the Planck units are expressed in terms of the fundamental constants, the fundamental constants cannot be anything but constant.

 

 

1 hour ago, studiot said:

But in Natural units the speed of light is dimensionless and exactly 1.

https://en.wikipedia.org/wiki/Natural_units

 

So whatever units you measure distance and time or other constants in is irrelevant.

c is still constant.

As I see it, the use of natural units is about making all the ostensibly different dimensions of measurement the same. So, whereas time and length appear to be different, multiplying time by c rescales time so that it is the same as length. And when this is done, c becomes 1 and dimensionless (but only because time and length now have the same dimensions).

 

 

 

Edited Monday at 09:17 PM by KJW

[Killtech]

1 hour ago, KJW said:

I don't agree with this because one also has to consider that the laws of physics determine the intrinsic size of the units that we define. The constancy of the fundamental constants is implied by the constancy of the laws of physics. For example, suppose you measure the length of some object. You use a steel ruler to measure the object in metres. But you could count the number of iron atoms along the edge of the ruler. Thus, instead of measuring the length of the object in metres, you have measured the object in terms of iron-to-iron interatomic distances. Any change in the laws of physics that alters the iron-to-iron interatomic distance would also alter the length of the object by the same amount, and therefore the length of the object in terms of iron-to-iron interatomic distances would be unchanged. But this invariance implies that the changes of the laws of physics cannot be measured, which justifies the assumption of the constancy of the laws of physics.

you forget an important aspect in your example. While the laws of physics will remain invariant under alterations of the iron-to-iron interatomic distance, the geometry of your space won't be. Lets for example assume the alteration only affects the distance in a local region but not around it. In the case we will measure that this region has a different volume then it would have it it were flat space, no? Therefore the effect is measurable even when using this specific distance definition but realize differently then one might naively think.

to make your example more concrete, we can use a simplified temperature model as a factor to alter interatomic distances with the special assumption that our iron ruler always has already taken the environmental temperature before we use it to measure anything (and lets also assume there is no kind of hysteresis effect which would disqualify it from defining a valid distance - as needed for the mathematical definition of a metric space). In this example temperature will change geodesics, i.e. what the shortest path between two points is. Now coincidentally sound waves will follow these geodesics (under ideal assumptions)... therefore the sound wave equation would suddenly become invariant under local changes of temperature. 

Edited Monday at 09:54 PM by Killtech

[swansont]

2 hours ago, Killtech said:

A simple approach for Maxwell with non constant c derived from current physics: SI second definition explicitly states that corrections due to the local gravity field should not be applied. Now define an alternative time measure and clocks with corrections for the local gravity field such as to enable clock in different regions to tick with the same rate. As clocks define the proper time this condition can be translated as to finding a correction such that curvature due to energy-mass vanishes. 

Defining a new clock and unit of time is so far a change of conventions, therefore does not affect real physics. However it changes measurement (and all SI units depend on the second) and the mathematical model of physics. Now transform maxwell equations from GR proper time tau to new proper time tau' and you get additional time and region dependent terms, one of which describes the wave speed and this we call c(x,t). These new terms reflect the physical degrees of freedom which GR describes via geometry. As a trade-off all physical degrees present in GR are gone, as it now becomes flat by construction

This ignores what I said completely 

As far as Maxwell’s equations go, it’s the wave equation. If c isn’t invariant, you no longer have a wave equation. You can’t transform to another inertial frame and recover the formula. There’s no time measurement involved at all.

For the fine structure, there’s no time measurement either. It’s a comparison of two measurements - any time dependence drops out. 

[Killtech]

13 minutes ago, swansont said:

This ignores what I said completely 

As far as Maxwell’s equations go, it’s the wave equation. If c isn’t invariant, you no longer have a wave equation. You can’t transform to another inertial frame and recover the formula. There’s no time measurement involved at all.

For the fine structure, there’s no time measurement either. It’s a comparison of two measurements - any time dependence drops out. 

Why would it not be a wave equation? How are Maxwell equations in a medium not wave equations since the speed of light in mediums does vary and is related to the refractive index? How are sound wave equations not wave equations? I fail to understand what you are trying to say.

Also i am not transforming between frames. This has nothing to do with coordinates. It is far more fundamental then that. We are transforming between two concepts of proper time. Look at the formula. That is far more involved then a change of coordinates or something as trivial as switching between SI and natural units. Natural units still use the very same definitions of time and space as SI does but just scale them differently with some factors. But a change of proper time is like a local transformation of units. It forces you to rewrite all laws of physics. It changes what is constant and what isn't. Look at the equation and understand how massively it impacts the whole model but not the physics.

Edited Monday at 10:42 PM by Killtech

[swansont]

6 minutes ago, Killtech said:

Why would it not be a wave equation?

Math has rules. The wave equation has a specific form to it.

If you’re going to dive into a discussion on the subject, it’s good to have a grasp of the basics.

6 minutes ago, Killtech said:

How are Maxwell equations in a medium not wave equations since the speed of light in mediums does vary and is related to the refractive index? How are sound wave equations not wave equations? I fail to understand what you are trying to say.

We’re talking about Maxwell’s equations in a vacuum, since that’s a condition for c being invariant. In a medium, light doesn’t travel at c, so the wave equation uses c/n (n is the index of refraction). Sound waves don’t travel at c, either (or at all, in a vacuum) If you are discussing the invariance of c, one of the stipulations is that it’s a vacuum, since c is the speed of light in a vacuum.

[studiot]

Just now, Killtech said:

I fail to understand what you are trying to say.

This is very clearly the case.

Just now, Killtech said:

assume c(x,t), i.e. it may vary by time and region

A simple approach for Maxwell with non constant c derived from current physics: SI second definition explicitly states that corrections due to the local gravity field should not be applied. Now define an alternative time measure and clocks with corrections for the local gravity field such as to enable clock in different regions to tick with the same rate. As clocks define the proper time this condition can be translated as to finding a correction such that curvature due to energy-mass vanishes. 

 

Just now, Killtech said:

Why would it not be a wave equation? How are Maxwell equations in a medium not wave equations since the speed of light in mediums does vary and is related to the refractive index? How are sound wave equations not wave equations?

This is not at all what I would expect from someone who understands what they are claiming.

I would expect replacement equations for those of Maxwell.

Then I would expect to see some math deriving the conditions for wave motion from the equations and finally ending with a wave equation.

What I would not expect to see is the assumption of that which was to be derived, namely assume c(x, t).

I would also expect the claimant to understand that the definitions and derivations of Maxwell work on the basis of a vacuum with no gravity fields or anything else.

c does not even appear until the last line of the standard derivation.

The onus is entirely on the claimant to derive the claim.

So show us your maths please.

 

Edit

Gosh I keep x posting with swansont.

Just now, KJW said:

As I see it, the use of natural units is about making all the ostensibly different dimensions of measurement the same. So, whereas time and length appear to be different, multiplying time by c rescales time so that it is the same as length. And when this is done, c becomes 1 and dimensionless (but only because time and length now have the same dimensions).

Multiplying by c does not rescale time at all.
In conventional (SI) units it transforms the dimensions so that the product has the dimensions of length.
This is in much the same way as mass (which can be variable) transforms the dimensions of acceleration to yield Force, in Newtonian theory.

As I understand matters the choice of c and other natural constants is because they are constant in a universe where the dimensional quantities are variable according to circumstance and observer.
This then avoids the issue of having to refer back to standards of mass, length and time etc.

Edited Monday at 11:02 PM by studiot

[KJW]

1 hour ago, Killtech said:

Lets for example assume the alteration only affects the distance in a local region but not around it. In the case we will measure that this region has a different volume then it would have it it were flat space, no? Therefore the effect is measurable even when using this specific distance definition but realize differently then one might naively think.

This fails to distinguish between a change caused by a change of the laws of physics and a change caused by some physical field. The problem with changing the laws of physics is that there needs to be a basis for that change. If the change is physically real, then that implies the existence of some field that gives rise to the measured change. And there also has to be underlying laws of physics that govern the basis for that change. So, the original set of changing laws of physics become replaced by a new set of constant laws of physics. In general relativity, this gives rise to covariance.

 

[Killtech]

48 minutes ago, KJW said:

This fails to distinguish between a change caused by a change of the laws of physics and a change caused by some physical field. The problem with changing the laws of physics is that there needs to be a basis for that change. If the change is physically real, then that implies the existence of some field that gives rise to the measured change. And there also has to be underlying laws of physics that govern the basis for that change. So, the original set of changing laws of physics become replaced by a new set of constant laws of physics. In general relativity, this gives rise to covariance.

In my case the basis for the change of laws of physics is the consequence of switching between different clocks concepts with which observations and measurements are performed. Thus the change is not physically real, but is a pure result of change of conventions / definitions.

Lets consider two particles Pa and Pb modelled in GR with different initial conditions but starting at the same space time location. Lets call this event S. Each particle follows a trajectory dictated by the laws of physics. Lets assume the trajectories are such that they eventually collide. Lets call this event E.

Now lets consider two different clocks being placed on each particle, one is the regular SI clock type L1, the other a clock type L2 that corrects for local gravity field. the first clocks describes the regular proper time of GR for each particle, the other clocks provide another proper time which will disagree with the former. Let's assume that the flight time between S and E as measured by L1 clocks shows that Pa needed less time then Pb. Now L2 clocks tick rate will locally differ from L1, specifically let's assume Pb passes through a region where its L1 clock runs slower like a large mass. L2 clocks will adjust for the effect thus lets assume according to L2 clocks Pb needed just as long as Pa. Since L1 and L2 are different measures, there is no contradiction in that observation.

In terms of Riemann geometry we can describe the equation of motion for each particle in either L1 or L2 proper time. However, L1 and L2 come with two different (1,3)-Riemann manifolds, metrics, and hence a different connections. These are different geometries. Expressing the very same physics in terms of two different connections will make the same laws of physics look entirely different for each. And they need to be in order to yield the seemingly disagreeing observations of L1 and L2. Yet their predictions, like the resulting trajectories and where they collide will still perfectly agree. 

This is what the transformation in the quoted post below does

5 hours ago, Killtech said:

 

[Markus Hanke]

3 hours ago, Killtech said:

However, L1 and L2 come with two different (1,3)-Riemann manifolds, metrics, and hence a different connections.

No. In your example you’re just using different coordinates on the same manifold, this isn’t a change in “geometry”. You are free to use any definition of time and space you like, so long as these remain valid solutions to the field equations along with the same boundary conditions.

8 hours ago, Killtech said:

Now define an alternative time measure and clocks with corrections for the local gravity field such as to enable clock in different regions to tick with the same rate.

Curvature is not a single scalar quantity, and in general there is no global notion of simultaneity in a curved spacetime, regardless of coordinate choices.

8 hours ago, Killtech said:

As clocks define the proper time this condition can be translated as to finding a correction such that curvature due to energy-mass vanishes. 

Locally this simply means you’re in free fall. Globally this isn’t possible, since such a global transformation would by definition not be a diffeomorphism, and thus not the same physical situation.

[Markus Hanke]

Just to elaborate a bit more. When we speak of the invariance (not constancy!) of the speed of light, what this physically means is that the outcome of experiments is always the same in all inertial frames, ie uniform relative motion has no bearing on the outcome of experiments. This has nothing much to do with units or numerical values.

Yes, it is always possible to describe the same physical situation in terms of different “geometries”, if you so will. You can eg forego any reference to curvature completely by choosing a different connection on your spacetime - the geometry is now curvature-flat, and instead contains all information about gravity in the form of torsion. But all this is saying is that one can draw different types of maps over the same territory, like having a topographical map vs a road map over the same region. That way you emphasise different information, but the actual experience of physically crossing that terrain is always the same, irrespective of what map you use to navigate. This is not revolutionary or mysterious, and reveals nothing new about the world. It’s “kind of trivial” as the poster in your screenshot correctly said.

So I think if you put enough thought into it, it may perhaps be possible to come up with a mathematical description of spacetime in which c is explicitly a function of something. The reason why no one uses such a description is that any measurements of space and time obtained from this description won’t directly correspond to what clocks and rulers physically measure in the real world - you’d have to first map them into real-world measurements, which means additional work and complications without any discernible benefit. Irrespective of what description you use, the outcome of experiments will still be the same in all inertial frames, and this is what we actually observe in the real world.

[Killtech]

11 hours ago, Markus Hanke said:

Just to elaborate a bit more. When we speak of the invariance (not constancy!) of the speed of light, what this physically means is that the outcome of experiments is always the same in all inertial frames, ie uniform relative motion has no bearing on the outcome of experiments. This has nothing much to do with units or numerical values.

Yes, it is always possible to describe the same physical situation in terms of different “geometries”, if you so will. You can eg forego any reference to curvature completely by choosing a different connection on your spacetime - the geometry is now curvature-flat, and instead contains all information about gravity in the form of torsion. But all this is saying is that one can draw different types of maps over the same territory, like having a topographical map vs a road map over the same region. That way you emphasise different information, but the actual experience of physically crossing that terrain is always the same, irrespective of what map you use to navigate. This is not revolutionary or mysterious, and reveals nothing new about the world. It’s “kind of trivial” as the poster in your screenshot correctly said.

So I think if you put enough thought into it, it may perhaps be possible to come up with a mathematical description of spacetime in which c is explicitly a function of something. The reason why no one uses such a description is that any measurements of space and time obtained from this description won’t directly correspond to what clocks and rulers physically measure in the real world - you’d have to first map them into real-world measurements, which means additional work and complications without any discernible benefit. Irrespective of what description you use, the outcome of experiments will still be the same in all inertial frames, and this is what we actually observe in the real world.

Okay, i see a lot of misunderstandings here. I admit what i am suggesting here is so far from the usual of what is done in physics thus is quite natural to happen. But i understand where you are coming from and think we can work it out, if we both try to understand each other.

You mention choosing a different connection - and that's what i am doing here. however in GR all such things are done only on a single Riemann manifold, and not between two different ones. This is why you state that we can exchange curvature for torsion and that is correct. In my case i utilize the very same underlying smooth manifold for both geometries (hence coordinates remain the same in both), each is equipped with a different metric tensor. And i do not mean rewritten in different coordinates. L1 and L2 are two disagreeing concept of clocks. I know the concept of coordinate time but this is not what i am referring to.

In my case both L1 and L2 are associated each with the temporal part or their own metric tensor g and g' (g_mu,nu  != g'_mu_nu in general) - since they represent the proper time of their geometry. For now i left out the discussion of what happens to the remaining complements of the L2 metric tensor - just assume it is constructed suitably. Given a metric tensor, there is a unique torsion free connection associated with it, the Levi-Civita one. We have two metrics, hence two manifolds, hence two LC-connections and each is torsion free on its manifold. On the other hand, since both connections are defined on the same smooth manifold, we can look at the L2 LC-connection from L1 geometry and will notice that it neither won't be torsion free nor more importantly it won't even preserve the L1 metric. The latter renders it quite easy to make c a function of something. I am not sure you are familiar with dealing with such kind of connections. L2 will break the usual parallel transport. It will yield different geodesics. It is entirely a different geometry. Yes, yes i hear all your alarm bells ringing - i know such a thing would normally break all laws of physics. But it is though to be used in accordance to constructing clock and ruler devices that conform to the L2 metric. And if we have both, then it is in fact just like you said, draw different types of maps over the same territory. To stay in your analogy, our maps are good for those that navigate via vision, but for bats that navigate via acoustics, a very different type of map may be needed. Not very revolutionary at all.

And sure, so far mathematically this is still kind of trivial, even if it is used in an unfamiliar way. Indeed such a construction appears anything but practical for now. However, we are talking of what would happen if we were to exchange the clocks and rulers we are using in the real world for something else that is not equivalent. I do not mean just using different units. I mean changing such fundamental relations like A longer then B becomes A shorter then B - simple change of units cannot do that. A change of geometry can. 

Before you object that only one of the statements can be 'real', consider that in math we do often use different metrics or norms on the same space and they will disagree with each other without there being an issue. The reality is that there is more then one way to compare things. The gist of it the definition of the SI second chooses a very specific way to compare time, yet it is by no means a unique choice.

[swansont]

1 hour ago, Killtech said:

The gist of it the definition of the SI second chooses a very specific way to compare time, yet it is by no means a unique choice.

Yes, it’s defined a particular way, in order to be useful to us. The realization of some constants relies on c being invariant, since we can then pick c to be a defined value. But realizing these constants don’t generally rely on measuring durations, since that would be an intermediate step which would reduce precision. 

Quote

The reality is that there is more then one way to compare things

Indeed. And smart people work on these problems, and find ways to do comparisons that don’t have extra biases and errors

[studiot]

One of the most fundamental principles in Physics is that of the isotropy and homogeneity of space / spacetime.

In order to theorise the observation that it does not matter where or when we look, the physics of light and physics in general appears much the same.

The principle of relativity follows directly from this, as does the principle of equivalence.

In other words there is no preferred origin for empty space or time.

 

It is a requirement of every cubic metre of space and every second of time be indistinguishable from every other.

This leads directly to the constancy of c since  c² = ε₀μ₀

If you want to consider non constant c then you must have non isotropic space  /spacetime then epsilon and mu become tensors to reflect the non isotropy / homegeneity and c² becomes a tensor product.
In short you have either a non vacuum or an aether.

Any maths you use must also reflect this.