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You may know I started a thread on this forum asking questions about the statement in General Relativity that "the laws of nature are the same in inertial frames". Guessing about the answers I got, I arrived in the following conclusion. I'd like to know whether these make sense & are correct or not.

On the Newton's laws:

On (Maxwell's) electromagnetism & Newton's laws:

Having these in mind, we could conclude:

I followed a response Dr. Greg gave me and arrived in the following invariant equation:

$$m \frac{d^2}{d \tau^2} x^\alpha = f^\alpha$$

where ##m## is the inertial mass and ##\tau## is the particle proper time (of a given particle, lol). This is invariant because it's the same in two inertial frames, in fact in all frames, since it's the time measured by a clock in the particle's reference frame. The equation above is not the same as Newton's second equation for two things; the first is that the time in the derivative is not the coordinate time and the other thing is that we are now dealing with four-vectors, which also means the force on the RHS is no longer the force on the corresponding RHS of Newton's second law. Note: I considered a Minkowskian framework on the equation above.

Main conclusion (because it motivated me to start the other thread): The statement that the laws of nature are the same in all inertial frames is not quite correct. Better would to say that they are valid as long as we keep ourselves in a theory where a invariance like the one in (*) exists. So I think one should specify in what theory they are going to talk about the laws of nature, because the invariance of laws of nature seems to depend up on the kind of invariance that equations obey.

There remains one question: how can we know whether a given reference frame is inertial? Apart from the obvious way of making an experiment to see whether Newton's first law works.

* What I mean is that all vectors appearing in Maxwell's equations obey the same invariance rule than the one appearing on Newton's second law (the acceleration vector): $$ \vec{A} = (A_1 , A_2 , A_3) \\ (A_1)^2 + (A_2)^2 + (A_3)^2 = \text{constant} \\ $$ Obs. : usual metric above.

On the Newton's laws:

**1**- The validity of the second law implies the validity of the first law;**2**- The second law not being valid does*not*imply that the first law is not valid;**3**- The first law being valid or not does not imply the validity of the second law.On (Maxwell's) electromagnetism & Newton's laws:

**4**- The laws of electromagnetism have the same kind of invariance* than those of Newton.Having these in mind, we could conclude:

__Reason for invariance of light speed in inertial frames__: Suppose there are two reference frames in which Newton's second law applies; from**1**above they are both inertial frames. Furthermore, as Newton's second law is invariant between them, so are Maxwell's equations (from**4**). Conclusion: Speed of light is the same in both of them.__Newton's second law:__Newton's second law is no longer valid in the relativistic theory because it doesn't remain invariant between two frames related by a Lorentz transformation, even that they are inertial frames (from**2**).I followed a response Dr. Greg gave me and arrived in the following invariant equation:

$$m \frac{d^2}{d \tau^2} x^\alpha = f^\alpha$$

where ##m## is the inertial mass and ##\tau## is the particle proper time (of a given particle, lol). This is invariant because it's the same in two inertial frames, in fact in all frames, since it's the time measured by a clock in the particle's reference frame. The equation above is not the same as Newton's second equation for two things; the first is that the time in the derivative is not the coordinate time and the other thing is that we are now dealing with four-vectors, which also means the force on the RHS is no longer the force on the corresponding RHS of Newton's second law. Note: I considered a Minkowskian framework on the equation above.

Main conclusion (because it motivated me to start the other thread): The statement that the laws of nature are the same in all inertial frames is not quite correct. Better would to say that they are valid as long as we keep ourselves in a theory where a invariance like the one in (*) exists. So I think one should specify in what theory they are going to talk about the laws of nature, because the invariance of laws of nature seems to depend up on the kind of invariance that equations obey.

There remains one question: how can we know whether a given reference frame is inertial? Apart from the obvious way of making an experiment to see whether Newton's first law works.

* What I mean is that all vectors appearing in Maxwell's equations obey the same invariance rule than the one appearing on Newton's second law (the acceleration vector): $$ \vec{A} = (A_1 , A_2 , A_3) \\ (A_1)^2 + (A_2)^2 + (A_3)^2 = \text{constant} \\ $$ Obs. : usual metric above.

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