How do you prove that the laws of physics are the same in all inertial frames of reference?
A quick comment first: the laws of physics are the same in all inertial frames of reference of the same handedness.
Now to answer your question. The rough approach is:
-- Define an inertial coordinate system. Deduce the most general coordinate transformations between such frames allowed by the definition. Now choose an inertial coordinate system to map out your surroundings and experimentally determine the laws of physics in that frame. Now mathematically check if applying the general inertial coordinate transformation to these laws of physics leaves the laws the same.
Historically, this actually happenned the other way around. We had a rough idea of an inertial frame, but found the physics had a different symmetry. This caused a lot of confusion at first, but eventually realized our approach to inertial coordinate systems was incorrect (the Newtonian concept of absolute time). Now that we have the concept of a spacetime, most definitions of an inertial frame essentialy are just defining what the metric for flat spacetime should be in such a coordinate system. We can then deduce the most general coordinate transformations preserving those metric components (lorentz boosts, rotations, translations, etc.) and check if the physics we measure is invariant under these coordinant transformations.
So when you here about an experimentalist seaching for "lorentz violating dynamics/processes/etc", they are testing the principle you asked about.
Length contraction, Time dilation, and other Relativistic effects go a long way towards to showing the invariance of the laws of physics across IRFs. Essentially, every proof for SR/GR is a proof of that postulate.
You don't is the answer, it's a postulate. Special relatvity is designed to preserve the laws of physics in different inertial frames, so any argument that it proves this statement is circular. If you like an inertial frame by definition is a frame belonging to a certain class of frames where the laws of physics and the speed of light are invariant (and constant) under transfomations between frames belonging to this class.
Postulates are generally speaking inferred emprically, i.e. from experiment. The success of special relatvity for describing certain physical phenomena can be taken as 'proof' of it's postulates.
You could use a different set of postulates which allow you to derive the postulates of special relativity. I still think that's a bit circular as essentiallly you're still setting out to create a theory which includes the postulates of SR.
But it would certainly be possible to falsify the postulate experimentally, so it is testable in that sense. Basically I think the idea is that you can determine the correct equations that govern the dynamical behavior of particles/fields in the lab frame (like finding that charged particles obey Maxwell's laws in the lab-frame), and then if the equations are written in terms of x,y,z,t, you can see what happens when you use the Lorentz transformation to perform a substitution in your equations:
x' = gamma*(x - vt)
t' = gamma*(t - vx/c^2)
y' = y
z' = z
(with gamma = 1/sqrt(1 - v^2/c^2))
If the equations after the substitution can be reduced to a form that looks identical to the original equations before the substitution (but with x' in place of x, t' in place of t, etc.), then the equations are Lorentz-invariant. It's logically possible that we might instead discover that the equations that give correct predictions in the lab frame do not have this property, which would falsify relativity unless we could show that the equations could be viewed as approximations to some other Lorentz-invariant equations which were consistent with all our experimental results.
As I said, support for Relativity = support for its postulates, but theories aren't "proven" in physics; they're used, improved, and eventually discarded. I haven't seen any wrong answers to the OP's question in this thread, just differing approaches.
To prove that it's true, you would need to start with a set of assumptions and then carry out some logical reasoning. What set of assumptions do you have in mind?
To prove that it's not true, you would need to find an experiment that comes out different when you do it in different inertial frames of reference. For example, if the Michelson-Morley experiment had come out with a positive result instead of a negative one, it would have disproved your statement.
The two possibilities --- proof and disproof --- are totally asymmetrical. One experiment can disprove a theory. No number of experiments can prove a theory.
Can I choose FOR freely?
An experiment, let's take a satellite orbiting around the Earth as FOR. The Earth moves so time on Earth should go slower (SR, time dilation).
In reality time goes slower for satellites.
This is due to the difference in gravitational potential. You need to take it in consideration in your calculations. See http://relativity.livingreviews.org/Articles/lrr-2003-1/ [Broken].
Myslius, please explain what you think "inertial" means in a SR context.
I guess this refers to parity violation?
Classically it makes no difference, does it?
My understanding about "inertial" was wrong. Inertial in SR context means that is it not affected by any force, and goes at a constant speed in the straight line.
Satellite is affected by Earth's gravity. So it can't be FOR. Right?
Actually, none object with mass can be inertial FOR.
What observations conflict with relativity? As far as i know, relativity does not work at extremums: blackholes, beggining of the big bang etc.
In the context of SR, right.
In this context, you'd find not a single exact inertial frame. But there are many situations where you can simply neglect the deviations, and where SR calculations are sufficient.
For example, the lab frame in a circular particle accelerator is inertial enough for all practical purposes.
The frame of a particle there can be approximated by an inertial frame, but only for a few meters - as long as their path looks almost straight.
Things that go in circles definitely do not qualify as inertial in SR.
Satellite clocks go faster not slower with respect to a clock on earth.
If you're considering only gravitational time dilation that'd be true, but for satellites in orbit around Earth I think velocity-based time dilation would have a larger effect, causing them to have elapsed less time on each successive orbit when they pass near an Earth-based clock (and if you're imagining a purely SR analysis of satellites where they are traveling in a circle in flat spacetime, which is what Ich was talking about with the comment 'In the context of SR', there would be no gravitational time dilation)
That is not my impression, I thought that the gravitational "part" was stronger than the SR "part".
But we should consult the literature. I have a paper by Richard Shiffman with the exact calculations that seem to agree with me. But I am not sure if this paper is published in a serious magazine and peer reviewed. Wikipedia also seems to agree with me but Wikipedia cannot always be relied on. There is Neil Ashby's document in Living Reviews but I browsed it and could not find any place where "the rubber meets the road" where it said unequivocally that one is slower or faster than the other one.
By the way, contexts or not, do you agree there is only one valid answer whether the clocks go faster of slower?
Edited: I checked the http://relativity.livingreviews.org/Articles/lrr-2003-1/" [Broken] article again and I think you want to take a look at Eq. 35, the satellite clock goes faster.
No, the effect is about 6 times smaller.
The gravitational effects heavily dominate the effects due to relative motion by a factor of +46us/day vs -8us/day, so, the terrestrial clocks lag the satellite clocks by a net of +38us/day. See here..
In order to compensate for this effect, the frequency of the atomic clocks is adjusted down at launch, making it one of the most direct tests for relativistic time dilation.
Yes, you were correct all along, by 38us day. This is why the frequency is adjusted down at launch (to make it count the same amount of units of time as the terrestrial clocks) The gravitational effects dominate the speed effects by a factor of 6.
Neither JesseM nor Myslius were talking especially about GPS satellites. Satellite clocks "go slower" for low earth orbits, like Space shuttles, ISS, and most satellites. Mostly communication satellites (incuding GPS) are in higher orbits, with "faster" clocks.
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