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

• A
• Luai
In summary: Lulu Press, 2008.In summary, the Cauchy Stress Tensor and the Energy Momentum Tensor have the same units because stress and energy density have the same units. Adding time as a dimension does not change the units of the Energy Momentum Tensor. Einstein did not start with the Cauchy Tensor when working on the right hand side of the field equations of GR. The transformation from the 3D Cauchy Tensor to the 4D Energy Momentum Tensor can be done using tensor operations, specifically the Bianchi identities. This allows for a consistent solution of the Einstein equations and the mechanics equations of motion for the matter. A detailed explanation can be found in "Classical Field Theory" by D. E. Soper.
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?

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

Luai said:
Is there a mathematical operation that transforms the Cauchy Stress Tensor to the Energy Momentum Tensor?
No. They are two different tensors.

Luai said:
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.

Luai said:
Shouldn't adding time as a dimension changes the Energy Momentum Tensor's units?
No. Why would it?

Luai said:
Did Einstein start with the Cauchy Tensor when he started working on the right hand side of the field equations of GR?
No.

topsquark and Luai
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

Luai

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