Looking for a stress analogue in E&M

Hello,

I was wondering if anyone has heard of or seen any work done into looking for what would be the analogue of stress in General Rel. for E&M. Im not talking about actual stress but the analogue to it. In the flat space metric with perturbations, linearized gravity begins to have close resemblance with E&M, save for the stress tensor. the densities and the currents are both analogues, however it doesnt look like E&M has a source for influence on the fields from what would be analogues for stress. Any comments on this?

Answers and Replies

Stingray
Science Advisor
The thing you're looking for is called the Maxwell stress tensor. More generally, the full stress-energy tensor for an electromagnetic field $F_{ab}$ is
$$T_{ab} = \frac{1}{4\pi} ( F_{a}{}^{c} F_{bc} - \frac{1}{4} g_{ab} F_{cd} F^{cd} )$$
Purely spatial components of this do not necessarily vanish.

The thing you're looking for is called the Maxwell stress tensor. More generally, the full stress-energy tensor for an electromagnetic field $F_{ab}$ is
$$T_{ab} = \frac{1}{4\pi} ( F_{a}{}^{c} F_{bc} - \frac{1}{4} g_{ab} F_{cd} F^{cd} )$$
Purely spatial components of this do not necessarily vanish.

Unfortunately I was trying to avoid this confusion, but the Maxwell stress tensor is stress. I am looking for the analogue if there is one. I can see the similarities between GR and E&M from there sources for influence. Mainly energy density vs charge density, and energy flux/momentum density vs current density, but then there is $$T^{ij}$$ which is the stress tensor in GR and which there appears to be no analogue in E&M. To me it seems like it would be a tensor of sort named "Current flux tensor" or "$$\dot{I}$$ density tensor" with the dot being the time derivative.

has anyone seen work related to this or have any suggestions or comments?

Stingray
Science Advisor
Ah, ok. I misunderstood you.

There is no analog of stress that acts as a source in electromagnetism. Gravity is described by a symmetric rank 2 tensor (the metric), and electromagnetism by a rank 1 tensor (the vector potential). The former requires a more complicated source structure than the latter.