Proving Invariance of Physical Laws Under All Transformations

[marmot]

Hi. So if you have $$\frac{d p_{\alpha}}{ds} = \frac{q}{c} F^{\alpha \beta} u_{\beta}$$ how could you possibly go on proving this its form is invariant under all coordinate transformations? Or any physical law of any form, really? I guess my point is how do you represent "all possible transformations", because a lot of textbooks go about how the form of a certain physical law is invariant but they never prove it for all possible transformations.

 

[quantum13]

You need some machinery about tensors and their transformation properties to prove this generally.

 

[Matterwave]

Everything there is either a tensor, a vector, or a scalar, so, of course it transforms properly. At least under Lorentz transformations, they do. If you have general coordinate transformations, you need to make sure your definitions for those objects are correct so that they are still vectors, scalars, and tensors. For example, the Faraday tensor, as defined on flat space will not be a tensor in a general curved space-time, you have to modify it a little (basically take partial derivatives go to covariant derivatives in the definition).