# 4.1. Gravitation and Electromagnetism as a Field Theory¶

The action for macroscopic gravity, electromagnetism and (possibly) charged relativistic dust is:

where:

where is the field of the matter, is the electromagnetic field and is the gravitational field. We vary with respect to each of them to obtain (interacting) equations of motion. is the speed of light, is the gravitational constant, the permeability of vacuum. is the mass density of the dust, is the charge density of the dust, is 4-velocity of the dust, is the electromagnetic field tensor, is the Ricci scalar.

## 4.1.1. Gravitation¶

We vary with respect to . By changing the metric, we also change the invariant volume element (thus also ), so we need to be careful to vary properly. We start with :

Variation of is:

The variation of is:

The variation of .

The equations of motion are:

We rearrange:

We define the stress energy tensor as:

(4.1.1.1)¶

(4.1.1.2)¶

And we get:

(4.1.1.3)¶

The equations (4.1.1.1) are called Einstein’s equations and the equations (4.1.1.3) are stress energy tensors for the relativistic dust and electromagnetism. The equation (4.1.1.2) is the stress energy tensor corresponding to the given action. Sometimes it is not possible to write an action for more complex matter (perfect fluid, Navier-Stokes equations for fluid, …) in which case we cannot use (4.1.1.2), but we can still specify the stress energy tensor directly and (4.1.1.1) are the equations of motion.

## 4.1.2. Electromagnetism¶

We vary with respect to . The variation of . The variation of . The variation of is:

The variation of is:

The equation of motion is:

Rearranging:

## 4.1.3. Relativistic Dust¶

We vary the whole action with respect to . The variation of . The variation of is:

The variation of . The variation of is:

The equation of motion is:

Rearranging:

This is the geodesic equation with Lorentz force.

## 4.1.4. Equations of Motion¶

All together, the equations of motion are:

The first equation determines from the given sources (the stress energy tensors) on the right hand side, that depend on , , and . The second equation determines from the sources ( and ) and from (through the covariant derivative). Finally, the last equation determines and from the given fields (through the electromagnetic field tensor) and (through the covariant derivative).

### Conservation¶

We apply covariant 4-divergence and use Bianci identities on the first equation:

So the total stress energy tensor is conserved. This fact makes the equations of motion (that follow from the action principle) not all independent. The third equation can be derived from the fist two as follows.

We calculate:

and we get:

The first term vanishes, because:

where we used (follows from differentiating ) and (contracting symmetric and antisymmetric tensors). We are left with:

Which is the third equation.