# Deriving Einstein's Relativity Postulates from Minkowski Spacetime

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• AstroMath

#### AstroMath

Does the constancy of the speed of light for all observers naturally emerge from the Minkowski spacetime metric?

Do Einstein's two postulates of relativity emerge from the Minkowski spacetime metric?

Suppose we begin with Minkowski spacetime and the Minkoswki metric.
https://simple.wikipedia.org/wiki/Minkowski_spacetime

Can we then derive Einstein's two postulates of relativity?

1. First postulate (principle of relativity)

The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion. OR: The laws of physics are the same in all inertial frames of reference.

2. Second postulate (invariance of c)

As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body. OR: The speed of light in free space has the same value c in all inertial frames of reference.

Thanks! :)

AstroMath said:
Does the constancy of the speed of light for all observers naturally emerge from the Minkowski spacetime metric?

The fact that there is a finite speed that is invariant for all observers does. The fact that light--electromagnetic radiation--itself travels at this speed requires additional assumptions, basically Maxwell's Equations.

AstroMath said:
Do Einstein's two postulates of relativity emerge from the Minkowski spacetime metric?

It does if you add the fact that all physical laws must be expressed using geometric objects on Minkowski spacetime: scalars, 4-vectors, 4-tensors, etc. Then you can construct inertial frames out of the appropriate geometric objects and show that all physical laws transform appropriately when you change frames.

PeterDonis said:
The fact that there is a finite speed that is invariant for all observers does. The fact that light--electromagnetic radiation--itself travels at this speed requires additional assumptions, basically Maxwell's Equations.

It does if you add the fact that all physical laws must be expressed using geometric objects on Minkowski spacetime: scalars, 4-vectors, 4-tensors, etc. Then you can construct inertial frames out of the appropriate geometric objects and show that all physical laws transform appropriately when you change frames.

Thanks Peter!

So if given the Minkowski spacetime metric of ds^2=dx^2-(ct)^2, we can then conclude that the velocity of c must be invariant for all observers?

AstroMath said:
So if given the Minkowski spacetime metric of ds^2=dx^2-(ct)^2, we can then conclude that the velocity of c must be invariant for all observers?

Yes, because that's what defines the Minkowski metric; ##c## is defined to be a constant (which can be set to 1 by choosing appropriate units).

PeterDonis said:
Yes, because that's what defines the Minkowski metric; ##c## is defined to be a constant (which can be set to 1 by choosing appropriate units).
Thanks Peter!

Well then it does seem that Einstein's second postulate of relativity naturally emerges from the Minkowski spacetime metric! :) (2. Second postulate (invariance of c). As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body. OR: The speed of light in free space has the same value c in all inertial frames of reference.)

What about the first postulate? Could we say that it also naturally emerges from the Minkowski spacetime metric? (1. First postulate (principle of relativity) The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion. OR: The laws of physics are the same in all inertial frames of reference.)

AstroMath said:
What about the first postulate? Could we say that it also naturally emerges from the Minkowski spacetime metric?

The second part of post #2 refers to the first postulate.

AstroMath said:
What about the first postulate? Could we say that it also naturally emerges from the Minkowski spacetime metric?
You need to say a little more than just "Minkowski spacetime metric" for it to follow naturally. The Minkowski metric is just a mathematical concept, so you have to make some connection to the laws of physics.

As @PeterDonis said, the usual way is to require that the laws of physics be formulated in terms of geometric objects (scalars, vectors, etc) in Minkowski spacetime. Then it follows naturally.

Dale said:
the usual way is to require that the laws of physics be formulated in terms of geometric objects (scalars, vectors, etc) in Minkowski spacetime. Then it follows naturally.

Although adding that requirement is itself a postulate, one that is close to being the first postulate restated in the language of Minkowski space (It's also a very reasonable one).

So overall I prefer to say that Minkowski space doesn't eliminate the need for Einstein's two postulates, but it does give us a very clean and elegant mathematical treatment of those behaviors of the universe that are implied by the postulates.

Nugatory said:
I prefer to say that Minkowski space doesn't eliminate the need for Einstein's two postulates,
I agree. I tend to think of it as a translation of the same concept into different languages. The two postulates are SR expressed in natural language, and the Minkowski stuff is SR expressed in math.

• ComplexVar89 and PeterDonis