A Going from Cauchy Stress Tensor to GR's Energy Momentum Tensor

[Luai]

TL;DR Summary

Is there a mathematical operation that transforms the Cauchy Stress Tensor to the Energy Momentum Tensor? If the former lives in 3D and latter lives in 4D, how come they have the same units?

1. Why do the Cauchy Stress Tensor & the Energy Momentum Tensor have the same SI units? Shouldn't adding time as a dimension changes the Energy Momentum Tensor's units?
2. Did Einstein start with the Cauchy Tensor when he started working on the right hand side of the field equations of GR?
3. If so, What tensor operation(s) would transform the 3D Cauchy Tensor into the 4D Energy Momentum Tensor of GR?

 

[PeterDonis]

@Luai I have edited your post to remove the bold. There is no need to put an entire post in bold.

 

[PeterDonis]

Is there a mathematical operation that transforms the Cauchy Stress Tensor to the Energy Momentum Tensor?

No. They are two different tensors.

If the former lives in 3D and latter lives in 4D, how come they have the same units?

The units of stress are the same as the units of energy density. Stress is force per unit area. Energy density is energy per unit volume, i.e., (force x distance) / (area x distance), i.e., the same as force per unit area.

Shouldn't adding time as a dimension changes the Energy Momentum Tensor's units?

No. Why would it?

Did Einstein start with the Cauchy Tensor when he started working on the right hand side of the field equations of GR?

No.

 

[vanhees71]

In relativistic physics, the "Cauchy stress tensor" form the space-space components of the energy-momentum tensor. The time-time component is the energy density and the time-space components are the momentum density (times $c$).

The interesting thing with GR is that when you take the "mechanical energy momentum tensor" (ideal/viscous fluids, elastic bodies,...) on the right-hand side if you have a solution of the Einstein equations, due to the Bianchi identities the equations of motion for the matter, which is given by $\vec{\nabla}_{\mu} T^{\mu \nu}=0$ is automatically fulfilled, i.e., you can get a fully consistent solution of the Einstein equations only if you simultaneously solve the mechanics equations of motion for the matter.

A very nice treatment of all this can be found in

D. E. Soper, Classical Field Theory