Required math: algebra
Required physics: special relativity
Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 4; Problems 4.6.
As we’ll study in more detail a bit later, the electric and magnetic fields can be combined into a single tensor known as the electromagnetic field tensor :
We can see from its definition that this tensor is anti-symmetric, that is, that . For any anti-symmetric tensor we can show that
In this equation, is the metric tensor in flat space and is the four-velocity, but in fact the formula is valid for any tensors and , provided that is anti-symmetric. The proof involves a bit of index-switching.
In the second line, we swapped the dummy indexes and , and in the third line we swapped and . The result shows that the original quantity is equal to its negative, which means it must be zero.
In terms of , the electric and magnetic (Lorentz) force laws for a charge can be combined into a single equation:
where is the four-velocity.
For example, if we get
In the non-relativistic limit, and this is the component of the force law . We’ll explore some of the other properties of this tensor later.
Since the square of the four-momentum of a particle is the negative of its mass squared (), this should be conserved for a charged particle moving in an electromagnetic field. (Its total momentum is, of course, not conserved since the fields exert a force on the particle.)