**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.)

We have

In the third line, we used the fact that and swapped and in the second term. The fourth line uses 6 and the last line uses 2.

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