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Some physical theories such as special theory of relativity are based on the invariance of the speed of light.
However, is the invariance of the speed of light, in the first place, logically possible?

I'm talking about this version or its equivalence: the speed of light in vacuum is always measured to be the same value from any uniformly moving observer even if at different (but constant) velocities.

Suppose there are two uniformly moving observers at different velocities:
An observer O moving at constant velocity v with respect to some observer
Another observer (light in vacuum) L moving at constant velocity c with respect to the same observer
where c = k * v for some real number k except 1

Also consider the velocity of L with respect to each observers:
The velocity of L with respect to O = c - v
The velocity of L with respect to L itself = c - c

If the invariance of the speed of light is true, those values should be identical for O and L are defined as uniformly moving observers.
Then, we can obtain this equation:
c - v = c - c
But the solution v = c contradicts our definition v and c are different.

Therefore, I believe the invariance of the speed of light is untrue.
Or is there any error in my disproof above?

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    $\begingroup$ Your observer L cannot exist in special relativity. No observer can have a speed of c relative to any other observer. $\endgroup$ – PM 2Ring Dec 23 '19 at 13:09
  • $\begingroup$ The way to see that special relativity is consistent is to study the basics of Minkowski spacetime. This structure is clearly consistent (assuming arithmetic is). The two questions left are - what does it say? and does this agree with experiment? $\endgroup$ – Ponder Stibbons Dec 24 '19 at 0:34
  • $\begingroup$ Thank you for all answers. I tried to calculate the values using the Lorenz transformation. I got c as the velocity of L with respect to O since it was the very intention of that transformation. But it was impossible to get the value of the velocity of L with respect to L itself, because the denominator is 0 when the observer is moving at the speed of light, i.e., v = c in the Lorenz transformation. $\endgroup$ – stackexchange_k Dec 25 '19 at 3:13
  • $\begingroup$ Even if it was calculated also as c, we cannot treat that result as a proof for the invariance of the speed of light. Generally, although we can make up a linear transformation to make the “transformed” speed of something invariant, it doesn’t alter its “real” speed. The invariance of the speed of light seems to be, after all, mere a postulate that we can neither prove nor disprove. $\endgroup$ – stackexchange_k Dec 25 '19 at 3:14
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Your argument is faulty. You set out to prove that the speed of light cannot be invariant, and you do so by assuming it is not invariant. Specifically, you say that the velocity of L with respect to O is c-v, which means that the velocity of L is not invariant. If L is light in a vacuum then its velocity is c with respect to any other observer.

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You're correct that you've reached a contradiction but it is because you have used the non-relativistic velocity addition formula. The relativistic version is needed to keep the speed of light invariant.

Also, how can the speed of light be different but always be measured to be the same? The value of a number can only be determined by measurement so it makes no sense to speak of a real, unmeasurable value as it will have no physcal meaning.

However, it is good to question theories to help build an understanding of the underlying physics, so keep it up :)

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Also consider the velocity of L with respect to each observers

What seems obvious to us now wasn't as obvious to people at the time of Galileo. Although we are used to Galilean transformation so much that we consider it trivial and obvious, it is not the only logically consistent way to transform positions and velocities. The usual formula for addition of velocities simply as vectors, ubiquitous in classical mechanics, is a corollary of Galilean transformation, which is only an approximation to the more precise Lorentz transformation used in relativistic mechanics.

When talking about speeds comparable to speed of light (or when combining any speeds with the speed of light), Galilean transformation is inadequate, and the full Lorentz transformation should be used. After you do this switch, your paradox will be resolved.

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There are two related but distinct assumptions in your argument that are culprits:

  1. An object moving at the speed of light can be an observer.
  2. If the velocity of an object is $\vec{u}$ w.r.t. an observer $O$ who itself moves at velocity $-\vec{v}$ w.r.t. another observer $O'$ then the velocity of the object w.r.t. $O' $ is given by $\vec{u}+\vec{v}$.

Notice that these assumptions are indeed physical assumptions. They don't follow simply out of principles of logic. Let me elaborate a bit more on both of these assumptions.

  1. Logically, there are two possibilities: either the speed of light is invariant among all inertial observers or not. If it is invariant then any object moving at the speed of light cannot be an observer because an observer must be at rest w.r.t. itself which cannot be achieved without violating the condition of the invariance of the speed of light. If it isn't invariant then clearly, an object moving at the speed of light can be an observer--there is no contradiction. So, both these scenarios are internally consistent possibilities. Which of these two internally consistent possibilities are realized in nature is an empirical question. Turns out, the speed of light is invariant and correspondingly, an object moving at the speed of light cannot be an observer.
  2. This assumption is true only if the Galilean transformation of coordinates holds true between inertial observers. Whether or not this is the case is an empirical question. Turns out, it doesn't hold true. The transformation of coordinates that does hold true between inertial observers is the Lorentz transformation. Under the Lorentz transformation, the law of transforming velocities between inertial observers is precisely such that the speed of light remains invariant.

Finally, following this wonderful paper by Pal, I would add this clear logical framing in which one can address this topic: If one assumes the following simple but physical conditions to hold

  • homogeneity of space
  • homogeneity of time
  • isotropy of space
  • principle of relativity among inertial observers

then it mathematically follows that there must exist an invariant speed which should also be the same as the maximum speed limit--however, this speed can either be finite or infinite. Galilean relativity corresponds to the case of this invariant speed being infinite (i.e., the case of no physically meaningful invariant speed) whereas Einsteinian relativity corresponds to the case of this invariant speed being finite. Turns out that nature is, in fact, Einsteinian.

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