Principle of Relativity, from a mathematical perspective

[ralqs]

I didn't know where to post this question, but the general math forum seemed like the best idea.

Newton's Laws are invariant with respect to the Galilean transformations. Maxwell's equations are invariant with respect to the Lorentz transformations. My question is, is it necessary, mathematically, for a law of laws of physics to be invariant with respect to some transformation? Put another way, if we took Maxwell's equations and started adding and removing terms, will the resulting equations have to be invariant with respect to an appropriate transformation rule?

 

[dalcde]

It doesn't HAVE to be invariant under some transformation, but PHYSICALLY, it is better that it is invariant under some transformation.

 

[robphy]

My question is, is it necessary, mathematically, for a law of laws of physics to be invariant with respect to some transformation? Put another way, if we took Maxwell's equations and started adding and removing terms, will the resulting equations have to be invariant with respect to an appropriate transformation rule?

Strictly speaking, there's always the trivial "identity transformation".

As dalcde says, I don't think that the laws of physics (and therefore mathematical encodings of them) has to necessarily have non-trivial symmetries... for example, physics in an Aristotelian-type world.

However, experiment has suggested that there are non-trivial symmetries in the physical world (so, e.g., we don't seem to live in an Aristotelian- or even Galilean-type world). So, it seems that it fruitful to seek out symmetries and express physical laws with mathematics which reflects those symmetries.

You might be interested in a lecture on symmetry by Feynman .

 

[Phrak]

...Put another way, if we took Maxwell's equations and started adding and removing terms, will the resulting equations have to be invariant with respect to an appropriate transformation rule?

No. But why would this be interesting, or are you alluding to a particular variation of maxwell's equations and want to know if the variation still obeys some particular symmetry. If so, I would be interested in such variant.

Symmetry rules are made fragile to better ideas.

 

[HallsofIvy]

The fact is that when I walk from here to there my shape does not change. That is, immediately, an "invariance with respect to some transformation". So, yes, "experimental evidence" (in this case, the experiment is just observing what happens when something moves) says that motion is invarient with respect to some transformation.

 

[ralqs]

Okay, thanks every one.