Principle of relativity, covariance and physical law.

[center o bass]

Hi! I'm studying special relativity and relativistic dynamics and I'm struggeling a little bit with the concept of 'covariance' of physical equations.

As far as I understand so far 'covariance' is related to the 'form invariance' of the equations of motions in relativity and the concept is important because of the principle of relativity, i.e. that all physical laws are the same in every reference frame.

Maybe a good example here is the rate of change of momentum in an electromagnetic field;
one observer might say that the circular motion of an electron is due to a pure magnetic field, while another observer with a relative velocity experiences it as being due to a mixture of magnetic and electric fields.

Is this then an example of something which is in contradiction to the principle of relativity and thus not a physical law, per definition?

Is the reason that we introduce the field tensor and write
m \frac{d^2 x^\mu}{d \tau^2} = F^{\mu \nu} \frac{d x^\nu}{d \tau}
that this equation is covariant in the sense that ALL observers will agree that the proper acceleration of the space time position is due to the field tensor, so that THIS then is an example of a physical law?

I would really appreciate comments on my current understanding and start a discussion around this to get it straight.

 

[Muphrid]

The main motivation for talking about relativistically covariant objects is to be able to talk about them in a different coordinate system from the one you started out with.

In other words, while one can work with E and B and not F, when one transforms into a new coordinate system, to find E' you need to know both E and B, for example. F is regarded as a complete, covariant object because transforming from F to F' only requires information about the components of F (and the nature of the transformation).

The equation you gave is covariant in the sense that different observers will measure the components of the proper acceleration, EM field tensor, and the velocity (you should've had current here) and get the same answer when the equation is assembled.

In a broader sense, covariant objects obey certain transformation laws--not arbitrary ones, but ones based in the geometry of spacetime, like rotations (boosts), translations, stretches, and dilations.