# Relation between coordinates and frames of reference

I always get a little uneasy that all the theories I can think of (at least since Newton) are constructed in a way such that they would be true in heaven and on earth ... but we can never go everywhere and test it out.

So here is the question:

Is there some good justification to implement something like the principle of relativity in scientific theories other than it turned out to work good so far?

Some more motivation:

We have an understanding of different places in space (and time) and what different velocities are. Like imagine me and my droogs cruising our skateboards down the neighborhood and there is a truck driving in the other direction. I see a cactus on the roadside and I wonder how the trucker in his ride sees it.

Now in the maths, space $\vec x$ and spacetime $t$ represent physical space and physical time. And if I know my coordinates, the form of the plant and its location and orientation in space, I can find out what I see and also what the trucker from his position sees. A coordinate transformation (replacing some letter on a piece of paper with some other letters in a systematic way) is conventually interpreted as taking the data from one "perspective" and transforming it into "another perspecitive".

It's supposed to be a fruitful approach to physics to consider only the observable quantities. Maybe I interpret the principle of relativity the wrong way, but I find it funny that a theory tells me there are spacetime events where I can never get to (outside the light cone). And simultaneously I'm guaranteed that if I'm there I would also be able to physics and come to the right conclusions. At the very least, I feel this is somewhat redundant - why not drop it?

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It's nice when theories are based on a simple idea. I don't know of any God given reason why the universe should be basically simple, but I find it an immensely appealing idea that it is.

To take your specific example, the truckers view of the cactus is obviously different to yours. OK, so what is his point of view? To explain it the only thing you need to know is that the line element:

$$ds^2 = -c^2t^2 + dx^2 + dy^2 + dz^2$$

is an invarient i.e. every observer will calculate the same value of $ds^2$ for the spacing between any two points $(t, x, y, z)$ and $(t + dt, x + dx, y + dy, z + dz)$. This single fact tells you everything about special relativity, and knowing it tells you not only how the truckers view differs from yours, but how every observer anywhere in the universe will see the cactus.

It's certainly true that when we look more deeply into the equation for the line element we discover odd things like the breakdown of simultaneity, time dilation, length contraction, twin paradox and so on. But we don't choose to believe SR because of all the weird stuff. We choose to believe it because it's so beautifully simple (and of course because it works :-).

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Is there some good justification to implement something like the principle of relativity in scientific theories other than it turned out to work good so far?

This is not so much an answer as it is an observation: if there were some good justification (for the principle of relativity in scientific theories) other than "it works", would not that justification then be the actual principle at work and thus also subject to your question?

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Only under the condition I'd know of it. –  NikolajK Jul 31 '12 at 19:08

I think your question belongs more to philosophy than it does to physics.

1) "......but we can never go everywhere and test it out. "

Actually you don't need to. The physics that you get here, will also work there, because you and here is mere accident. You can always understand whether your physics belongs only to here, if here and there are different, physically. It can hardly be that the physics we discover is limited only to where we live and go on to live forever with that, for we would know that it is incomplete or wrong, when we find it not agreeing with some other physics we are sure is not dependent on any here.

2)"Is there some good justification to implement something like the principle of relativity in scientific theories other than it turned out to work good so far?"

Suppose you at once forget there is anything as the principle of relativity, or suppose that you just don't know it. Then you do some physics. Then you do some experiments or go cruising in skateboards with your droogs. You'll find that your physics does not quite describe the happenings in your lab or the neighbouhood. It is not that we first found the principle of relativity and then went around labs and neighborhoods seeing whether it "works" or not. The principle was self emerging. It does not require any justification. And if you really find any justication, it would simply be the actual principle at work as Alfred pointed out, but then you would be back to where you started, the justication will seem to you as something again at "work".

3)".....that a theory tells me there are spacetime events where I can never get to (outside the light cone). And simultaneously I'm guaranteed that if I'm there I would also be able to physics and come to the right conclusions."

The problem is you can't be "there". There is no point in discussing or speculating about the physics "there".

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but I find it funny that a theory tells me there are spacetime events where I can never get to (outside the light cone).At the very least, I feel this is somewhat redundant - why not drop it?

Mathematical theories do not come a la cart, i.e. they are not patched together, cut as you go, constructs. Theories are axiomatic self consistent and sustained. They arise and are accepted because they explain usually a large number of observations to great accuracy. A theory is either invalidated by disagreeing with some data, or is consistent with all known data until further experimental research. Now mathematics being what it is, the theories are extended to non physically accessible regions, and one accepts the conclusions since the theory fits the known regions.

The specific example you use is not a particularly useful one, since calculations are done off the light cone in Feynman diagrams and now even more complicated calculations, which in the end are absolutely consistent with data to high accuracy, since the light cone excursions are virtual. Even if one could construct a theory where only the inside of the light cone were mathematical described , it would be a wrong theory for particle physics data.

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But in standard quantum field theory (like the Standard Model and its extensions) space-time coordinates are not linked to observables, unlike in classical mechanics of point particles or in quantum mechanics (regarding spatial coordinates). However, mean lifetimes and mean free paths are connected to physical measurements of time and length and these respects causality owing to the cluster decomposition principle. (I think Anna v does know this, but maybe the reader doesn't.) –  drake Aug 5 '12 at 17:46

To expand upon Alfred's point:

The scientific method basically involves making assumptions and using them to make testable predictions. The results of your tests on those predictions are taken as evidence for or against your assumptions.

In theoretical physics, many of these fundamental assumptions regard the basic symmetries of spacetime, such as the principle of relativity (Lorentz invariance). So, in that sense, we can't ever hope to derive Lorentz invariance, in the same way as we can't derive F=ma from any deeper principle of classical mechanics. If it were possible to derive Lorentz invariance from a more fundamental assumption, as Alfred says, we would just be moving our problem one level down.

Thus the fundamental assumptions can only be justified by our tests on the predictions that can be made from it. In the case of the principle of relativity, a very small number of assumptions can be used to make a great many rich and interesting predictions, all of which have been found to be true. This is taken as evidence that the principle of relativity is indeed observed in Nature.

This is the reason a lot of time, money and effort is spent looking for violations of symmetries in particle physics. For example, until a few decades ago CP (charge-parity) symmetry was thought to be a fundamental symmetry of nature. When it was discovered that it is not, new physics had to be developed.

So you see, it's entirely possible (though unlikely) that the principle of relativity may turn out not to be an exact symmetry at some point in the future. In that case it will be superseded by a more fundamental theory, in the same way that Einstein's relativity overthrew Galilean relativity, which was a more naive interpretation of the same principle.

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Since you wrote:

Now in the maths, space and spacetime t represent physical space and physical time.

I will assume that you are talking about the Galilean and the Special Principle of Relativity because the hole argument prevents that interpretation in General Relativity and the latter case has almost nothing to do with the formers.

I will just give an answer to the question: why do we impose invariance under Lorentz transformations when we are looking for new physical laws (this is what one really imposes and not the principle of relativity. In principle, it could exist (and they do exist) more transformations that connects inertial observers). It is totally based on a derivation of Lorentz and Galilei transformations begun by W. von Ignatowsky's (see here for references) in 1911; which is unfortunately little-known. Since in this derivation these transformations can be derived without making reference to any specific physical phenomenon or law like Maxwell theory or motion in inclined planes (this the most important property of this derivation) and they mostly rely on properties of space and time, one can claim that implementing Lorentz invariance is equivalent to assume some features of space and time. So that, as usual with physical explanations, this answer simply brings your question to another perhaps deeper question: Why is space-time homogenous and isotropic?

Consider two inertial observers. Assume:

1. Space and time are homogeneous (there is not especial points in space or time).
2. Isotropy of space (no especial directions in space).
3. The transformations have a group law (there is one transformation that connects one observer with himself, if two observers are connected by one transformation, and the second one is connected with a third observer, then first and the third are also connected). This requirement is due to the equivalence of inertial observers.
4. If two events happen at the same place for one observer, their time order must be the same for all the observers.

Then, the most general transformation between observers are Lorentz and Galilei transformations (the latter saturates the former). The relativity or absoluteness of simultaneity permits one to physically distinguish one from the other.

Footnote: Of course, space and time are neither homogenous nor isotropic in the real world, but they are in the domain where the Relativity Principle holds.

Added: Here http://arxiv.org/abs/gr-qc/0107091 you can find more details and references about what I have tried to summarize. You will find that the interpretations of the natural coordinates are readings of clocks and rods. Note that this derivation do not assume that spacetime is a metrical space. Indeed, as it is well-known, Galilei spacetime is not compatible with a four dimensional metric tensor.

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First, it was hard to me get your questions, so likely my answer will be terribly unrelated. By I hope I'll provide some points that will help you.

We have an understanding of different places in space (and time) and what different velocities are.

Maybe I'm too nagging and it doens't really matters here, but there is no such things as places. There bodies (objects or call them as you like) and relations between them. These relations determine whether two given bodies can affect each other or not. Moreover, if the two body's can affect each other this relations come into play to say how much is the effect.

Maybe I interpret the principle of relativity the wrong way, but I find it funny that a theory tells me there are spacetime events where I can never get to (outside the light cone). And simultaneously I'm guaranteed that if I'm there I would also be able to physics and come to the right conclusions. At the very least, I feel this is somewhat redundant - why not drop it?

Suppose you are not in right relations to observe the event, but later (your later) you can observe it consequences. You can pretend you were here and do the physics to model how it was and see how it predicts what you have observed.

That's kind of archeology, you know the rules, you know the results, you reconstruct the history. Though you can try to model the future as well.

UPD I've just remembered the thought that if everything is determined there is in some sense no time. So to the extent one can predict (things are determined) space-time restrictions on our knowledge are weaker.

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