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I have two questions about SR postulates.
So first question, I heard two formulations of first postulate:

"If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K".

And the second formulation is

" Laws of physics are the same in all inertial frames".

I think there is a problem with the second formulation, because we haven't established definition of inertial frame of reference, we can say that inertial frame is the one where body with no forces acting on it or with forces acting on it compensate each other, this body will move with constant speed, but then the problem is that with such a definition this postulates will not be equivalent, because the first one will be "stronger" (from axiomatic point of view) than the second one, because first one talks about non-inertial frame too. And another problem is that with second formulation we need to establish definition of inertial frame. So what formulation is more rigorous? Or what formulation is right?

The second question: Does the second postulate talk about invariance of speed of light in all frames or only inertial one ? If inertial one, what definition SR establishes for inertial frame of reference, is it one I established earlier ?

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This is a very good thought, and you are realizing why Einstein felt the need for GR after establishing SR.

You are asking: "Does the second postulate talk about invariance of speed of light in all frames or only inertial one ?"

The second postulate says:

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.

The answer to your question is that it is about inertial reference frames.

An inertial frame of reference in classical physics and special relativity is a frame of reference in which a body with zero net force acting upon it is not accelerating; that is, such a body is at rest or it is moving at a constant speed in a straight line.

The definition of inertial frame of reference is of importance here, because SR is usually talking about inertial reference frames, and is the best way to describe things in a non-accelerating reference frame, or when the effects of the gravitational field are not dominant.

Now a little bit about why non-inertial reference frames are important and why it is good to learn about GR:

Now it would be very useful for you to learn about the Shapiro delay. It is a very good way to learn GR time dilation, and the varying speed of light in different gravitational fields.

Now as per GR, the speed of light in vacuum is c, when measured locally.

But if you want to measure the speed of light traveling towards Earth and passing near the Sun, you would see that, when measured from Earth, the speed of light near the Sun is less then c. The reason is that in this case you are talking about non-inertial reference frames, because light is in a different gravitational field near the Sun, then the observer, you. You are in Earth's gravitational field, and the strength of its gravity is different from the Sun's.

These are non-inertial reference frames. The problem with gravitational fields is, that (when an object is inside them) they are non-inertial reference frames.

General relativity is based upon the principle of equivalence: There is no experiment observers can perform to distinguish whether an acceleration arises because of a gravitational force or because their reference frame is accelerating.

Now back to your question:

So in this case the answer to your question is that SR is a very good way to work with inertial reference frames, or non-accelerating reference frames, or objects that are not in gravitational fields, or when the effects of the gravitational fields are not important.

Now to your question:

"" Laws of physics are the same in all inertial frames". "

This is the definition of the first postulate and is more rigorous then the one you posted before that.

You are saying :"because the first one will be "stronger" (from axiomatic point of view) than the second one, because first one talks about non-inertial frame too."

Now the first one you posted does not talk about non-inertial reference frames. it says that one frame is "K' moving in uniform translation relatively to K".

This means that they are indeed inertial reference frames. None of them is accelerating relative to the other one. So both definitions will be used for SR.

As per wikipedia:

All inertial frames are in a state of constant, rectilinear motion with respect to one another; an accelerometer moving with any of them would detect zero acceleration.

So the definition of inertial frames is not only about acceleration, it has to be rectilinear too. The definition of rectilinear motion is:

Rectilinear motion is another name for straight-line motion. This type of motion describes the movement of a particle or a body. A body is said to experience rectilinear motion if any two particles of the body travel the same distance along two parallel straight lines.

So you see that the definition with inertial frames defines the two objects (or the two frames) to move so that none of them accelerates relative to the other and they have rectilinear motion relative to each other.

Now in your other definition, it says "moving in uniform translation relatively"

This is exactly the definition for inertial frames in Newtonian gravitation:

Absolute space Main article: Absolute space and time Newton posited an absolute space considered well approximated by a frame of reference stationary relative to the fixed stars. An inertial frame was then one in uniform translation relative to absolute space. However, some scientists (called "relativists" by Mach[24]), even at the time of Newton, felt that absolute space was a defect of the formulation, and should be replaced. Indeed, the expression inertial frame of reference (German: Inertialsystem) was coined by Ludwig Lange in 1885, to replace Newton's definitions of "absolute space and time" by a more operational definition.[25][26] As translated by Iro, Lange proposed the following definition:[27] A reference frame in which a mass point thrown from the same point in three different (non co-planar) directions follows rectilinear paths each time it is thrown, is called an inertial frame.

In Newton's time the fixed stars were invoked as a reference frame, supposedly at rest relative to absolute space. In reference frames that were either at rest with respect to the fixed stars or in uniform translation relative to these stars, Newton's laws of motion were supposed to hold. In contrast, in frames accelerating with respect to the fixed stars, an important case being frames rotating relative to the fixed stars, the laws of motion did not hold in their simplest form, but had to be supplemented by the addition of fictitious forces, for example, the Coriolis force and the centrifugal force.

The main thing here to understand is simplicity. This simplicity manifests, that inertial frames have self contained physics without the need for external causes.

Non-inertial frames have external causes. This principle of simplicity can be used for Newtonian and SR too.

So basically the first definition, where "moving in uniform translation relatively to" is used, is the same rigorous as where you use inertial frames.

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    $\begingroup$ Maybe I misunderstood you but it in the end you say that 2 frames of reference are inertial if they move with constant speed with respect to each other. But the definition you established, does not seem to be equivalent to it. $\endgroup$ – Юрій Ярош Jul 21 '18 at 19:25
  • $\begingroup$ Acceleration is just one aspect. "K' moving in uniform translation relatively to K". This would mean they are inertial frames with respect to each other. Now the definition of inertial frames is "All inertial frames are in a state of constant, rectilinear motion with respect to one another; an accelerometer moving with any of them would detect zero acceleration." $\endgroup$ – Árpád Szendrei Jul 21 '18 at 22:28
  • $\begingroup$ So not accelerating is just one aspect, it has to be rectilinear motion. The definition of rectilinear motion is "Rectilinear motion is another name for straight-line motion. This type of motion describes the movement of a particle or a body. A body is said to experience rectilinear motion if any two particles of the body travel the same distance along two parallel straight lines." $\endgroup$ – Árpád Szendrei Jul 21 '18 at 22:32
  • $\begingroup$ So to answer your question, both definitions are same rigorous, "moving in uniform translation relatively to", and inertial frames. $\endgroup$ – Árpád Szendrei Jul 21 '18 at 22:49
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I believe Einstein's original (1905) version of the first postulate (Principle of Relativity) was:

“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 co-ordinates in uniform translatory motion”

This is the equivalent of your second formulation.

I believe your first formulation is one where Einstein elaborates on the first principle with an example of two reference frames K and K' that are in uniform translatory motion with repsect to each other. This in effect defines reference frames K and K' as inertial reference frames.

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