Learning the stress-energy tensor I am learning dynamics in special relativity and come across the stress-energy tensor. I have real trouble understanding it. I would love answers on 


*

*How to motivate the definition of this tensor. 

*How to deduce the equation of motion, i.e. the divergence of the stress-energy tensor is 0.


A toy model would be best for a demonstration. 
(You may use the language of Riemannian Geometry freely.)
 A: Here is a handwaving argument on why you can put all these things
together and make a tensor:
You may know that in special relativity there are some (scalar, vector)
pairs that can be combined together to make a quadrivector. For example,
energy and momentum combine to make the (energy, momentum) quadrivector.
A less well-known combination is (density, current-density). For
example, if ρ is the density of some quantity (whatever space-extended
quantity) and j is its current-density, then (ρ, j) is a
quadrivector. You can guess that from the way the conservation equation
$$
\frac{\partial \rho}{\partial t} + \nabla \mathbf{j} = 0
$$
can be rewritten with quadrivectors as an orthogonality relation
$$
\left(-\frac{\partial}{\partial t}, \nabla \right)
    \cdot (\rho, \mathbf{j}) = 0
$$
(I am using c = 1 here). Now, if you accept that (density,
current-density) of whatever makes a quadrivector, you can think that,
since the energy-momentum of the electromagnetic field is extended over
space, you can take the density-current-density of it. But since
energy-momentum is a quadrivector quantity, you end up with a tensor
like
$$
\begin{pmatrix}
\text{energy density} & \text{energy current density} \\
\text{momentum density} & \text{momentum current density}
\end{pmatrix}
$$
If you realize that stress is just momentum flow, you may see that
this is just the stress-energy tensor.
