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In the first paragraph of Wikipedia's article on special relativity, it states one of the assumptions of special relativity is

the laws of physics are invariant (i.e., identical) in all inertial systems (non-accelerating frames of reference)

What does this mean? I have seen this phrase several times, but it seems very vague. Unlike saying the speed of light is constant, this phrase doesn't specify what laws are invariant or even what it means to be invariant/identical.

My Question

Can someone clarify the meaning of this statement?

(I obviously know what an inertial frame is)

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this phrase doesn't specify what laws are invariant

It doesn't need to since it is a guiding principle, a razor. It is a statement about the nature of physical law.

Put another way, on this principle, an alleged 'physical law' that isn't invariant under inertial coordinate transformations is not a genuine physical law.

or even what it means to be invariant/identical.

Consider, for example

$$\vec F = m \vec a $$

If this equation holds in one coordinate system, it holds in all the coordinate systems related to this one by a Galilean transformation. Thus, it is invariant (unchanged) by this transformation.

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  • $\begingroup$ Alfred Centauri: "['The laws of physics are invariant'] is a statement about the nature of physical law. Put another way, on this principle, an alleged 'physical law' that isn't invariant under inertial coordinate transformations is not a genuine physical law." -- In other words: Any assertion which does not involve or refer to any coordinates at all, does, by all appearances, conform to the "nature" of "a genuine physical law". And, arguably, by content, too. "Consider, for example $\vec F = m~\vec a$. If this equation holds in one coordinate system [...]" -- Does it ?? ... $\endgroup$ – user12262 Jan 31 '15 at 8:48
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The laws of physics are invariant

means slightly different, but (almost) equivalent things depending on what formulation you are working with.

Given a collection of transformations (a symmetry/transformation group) and a Lagrangian formulation, you can check whether the Lagrangian changes when you apply the transformation. If it does not change (or only by a total derivative), then the action is invariant under the transfomation, and using the principle of extremal action will yield the same equations of motion as before in the sense that they extremalize the same action, and hence describe the exact same system.

Given a collection of transformations and a Hamiltonian formulation, it is of course the Hamiltonian that has to be invariant. The Hamiltonian formalism is not manifestly Lorentz invariant, and it is a bit difficult to use it for relativity, but one can do so. Again, a Hamiltonian unchanging under a transformation induces physically equivalent equations of motion describing the exact same system.

In the case of the statement about inertial frames, the corresponding transformations are given by the Lorentz group $\mathrm{SO}(1,3)$.

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  • $\begingroup$ ACuriousMind: "Given a collection of transformations (a symmetry/transformation group)" -- What is thereby supposed to be transformed (what are the "objects of operations")? Surely not "coordinates" (and/or "just subsets of $ \mathbb R^n$") ?? $\endgroup$ – user12262 Jan 31 '15 at 8:55
  • $\begingroup$ @user12262: Every object in the Lagrangian (usually these are fields, but they can be coordinates or operators as well) has to transform in a given representation of the transformation group. Specifying the representations is part of giving the transformation. $\endgroup$ – ACuriousMind Jan 31 '15 at 12:57
  • $\begingroup$ ACuriousMind: "[...] Specifying the representations is part of giving the transformation." -- Then let's look specificly at "representations of the Lorentz group" (since your answer suggests specific relevance to the OP's question about "inertial frames"). Now, Wikipedia seems to have quite an extensive page on that topic. However, the word "event" seems to appear on that entire page but once: in the link to Current events. (Hence: I can rest my case.) ... $\endgroup$ – user12262 Jan 31 '15 at 19:11
  • $\begingroup$ @user12262: "Events" are just points in spacetime, which, in special relativity for which the Lorentz group is relevant, is just $\mathbb{R}^{1,3}$ - the fundamental representation of the Lorentz group. This induces that also the (co)tangent vectors transform in the (anti-)fundamental representation, and this linearly extends to the tensor products of them, so every field/form on spacetime also has a natural notion of transforming under the Lorentz group given by it being a tensor of a certain rank. I don't understand what you want. $\endgroup$ – ACuriousMind Jan 31 '15 at 19:32
  • $\begingroup$ ACuriousMind: ""Events" are just points in spacetime" -- No, not "just", but (also) "spacetime coincidences {such as} encounters between two or more material points". "what you want {?}" -- An explicit description how to assign (subsets of) "the fundamental representation of the Lorentz group; just $\mathbb R^{1,3}$" to given sets of encounters between two or more identified "material points"; or at least appreciation for the difficulties involved, since the OP asked about physics. $\endgroup$ – user12262 Jan 31 '15 at 23:01
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The laws of physics are the same in every inertial frame of reference.

If the laws differed, that difference could distinguish one inertial frame from the others or make one frame somehow more correct than another. Here are two examples:

Suppose you watch two children playing catch with a ball while the three of you are aboard a train moving with constant velocity. Your observations of the motion of the ball, no matter how carefully done, cant tell you how fast (or whether) the train is moving. This is because Newtons laws of motion are the same in every inertial frame.

Another example is the electromotive force (emf) induced in a coil of wire by a nearby moving permanent magnet. In the frame of reference in which the coil is stationary the moving magnet causes a change of magnetic flux through the coil, and this induces an emf. In a different frame of reference in which the magnet is stationary the motion of the coil through a magnetic field induces the emf. According to the principle of relativity, both of these frames of reference are equally valid. Hence the same emf must be induced in both situations. (Examples are taken from the book ,UNIVERSITY PHYSICS).

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In leyman's terms, it just means that the laws of physics are the same everywhere. Here, on the Moon, even in another galaxy, or in a spaceship travelling at near light speed to another galaxy.

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According to Einstein:

All our well-substantiated space-time propositions [and consequently, all of our statements concerning facts and findings in physics] amount to the determination of space-time coincidences. If, for example, the [course of events] consisted in the motion of material points, then [...] nothing else are really observable except the encounters between two or more of these material points.

where "determination of space-time coincidences" is thought, at least in principle, to be unambiguously, definitively and consistently obtained by each individual participant.

The statement that

the laws of physics are invariant (i.e., identical) in all inertial systems (non-accelerating frames of reference)

can be understood as a less precise (possibly circular) and more restrictive formulation of Einstein's maxime quoted above. (It is based on Einstein's earliest, preliminary attempts at trying to express his maxime.)

(I obviously know what an inertial frame is)

Really?!? (cmp. "What determines which frames are inertial frames?", PSE/q/3193)

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  • $\begingroup$ Lol I stand corrected. I might know what an inertial frame is depending on the context. $\endgroup$ – Stan Shunpike Jan 30 '15 at 6:26
  • $\begingroup$ Stan Shunpike: "I might know what an inertial frame is depending on the context." -- Fair enough. But then you ought to make damn sure you know how to recognize and to communicate the context which you want (and which you want anyone else, too) to consider. $\endgroup$ – user12262 Jan 30 '15 at 6:52
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When constructing equations of motion which are the reflection of laws of nature so to speak, we must make them Lorentz invariant and invariant to spacial rotations. This means that they must have the same form under these transformations. One example is construction of a field theory, in which you begin by forming an action which is Lorentz invariant making sure from the very start that you will get it right. Action is a physical quantity with a dimension of Js (joule-second). This quantity is very important for the thing called Hamilton principal of stationary action...So laws of nature same in all inertial reff frames = equations that describe them invariant with form to Lorentz transformations.

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In layman's terms, it just means that the laws of physics are the same everywhere. This means that we are talking about one common set of laws. The fun part is figuring out how one common set of laws can behave the same, while they are taking place within different frames of reference. Thus we have a one, that is shared by a many. How can this be, when each frame of reference is different.

Of course once you fully understand both the cause and structure of Special Relativity, the answer becomes obvious.

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