# Invariance of tensor forlorentz force law under all possible coordinate transformatio

1. Oct 10, 2011

### 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.

2. Oct 15, 2011

### quantum13

Re: invariance of tensor forlorentz force law under all possible coordinate transform

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

3. Oct 15, 2011

### Matterwave

Re: invariance of tensor forlorentz force law under all possible coordinate transform

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).