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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

  1. How to motivate the definition of this tensor.
  2. 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.)

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    $\begingroup$ What exactly were you not able to understand after reading the Wikipedia article, looking up relevant definitions in whatever textbooks you have available, and searching for other websites? $\endgroup$
    – David Z
    Commented Mar 31, 2014 at 6:09
  • $\begingroup$ @DavidZ For example, how do you put everything - stress, energy, momentum - together and realize that it is a tensor? $\endgroup$
    – shrinklemma
    Commented Mar 31, 2014 at 6:23

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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.

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  • $\begingroup$ So is there a simple explanation for that (density, current density) makes a quadrivector? $\endgroup$
    – shrinklemma
    Commented Mar 31, 2014 at 17:00
  • $\begingroup$ Very roughly: The current density tells you how much of the quantity you are considering is traveling in each direction of space. The density tells you how much is traveling in the time direction i.e. towards the future. $\endgroup$
    – Edgar Bonet
    Commented Apr 1, 2014 at 10:37
  • $\begingroup$ By the way, is this conservation law equivalent to the vanishing of the divergence of this tensor? $\endgroup$
    – Sylvain JULIEN
    Commented Jul 18, 2021 at 9:19
  • $\begingroup$ @SylvainJULIEN: Yes, that's how I understand it. $\endgroup$
    – Edgar Bonet
    Commented Jul 18, 2021 at 10:12

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