# Effect of Compression/Tension on the Rest Mass of an Elastic Solid

If I compress an elastic solid I will strain the material in the direction of the force and therefore do work that is stored as potential energy in the material. The resulting material will have no velocity, but its rest mass will increase, presumably causing increased gravitational bending of spacetime.

Assume I have a cube of an elastic material aligned with some Cartesian coordinates in a natural way. If I compress it along the x direction, placing it under mechanical compression, will the stress mass energy tensor increase its 0,0 component or its 1,1 component?

Now, if the answer is the 0,0 component, and if I then look at the same situation from a coordinate system rotated 45 deg from the original system about an axis perpendicular to the compression force - say about z, then the resulting system will have sheer stress on faces defined by constant x and y values in the new system, at least the force will no longer be normal to them. Are these stresses accounted for in the 0,0 component of the stress mass energy tensor or in the (1,2), (2,1) components.

If in the 0,0 component why is the 1,2 component labeled as sheer stress. In other words, when a sheer stress or any other stress is applied, the velocity of the material remains zero once the stress is established. So how can a sheer stress also be a momentum flux?

Questions are not rhetorical. Terribly confused. Thanks.

• Where did you get the idea that compressing an elastic solid will increase its rest mass? – Bob D Mar 20 at 18:41
• There was an answer to a question in stack exchange that said most of the mass of common objects was binding energy. Another said that electrostatic potential energy between electrons caused rest mass. I was actually typing that question when I stopped because I thought it had been answered. – Justintruth Mar 20 at 21:08
• "In that context it is worth noting that most of the mass of ordinary matter is binding energy due to the strong force, so most of the "ordinary mass" around you is of exactly the kind you are asking about" from answer to the question: "Does potential energy of an object increase its relativistic mass?" – Justintruth Mar 20 at 21:37

## 1 Answer

Special relativity states the equivalence between mass and energy with the celebrated formula $$E = m c^2$$, to be read in the rest frame of a body. That means that the rest mass of a compound object is given by the sum of the rest mass of its constituents plus their kinetic energy plus their interacting energy, as measured in the rest frame of the object (no net momentum). The stress–energy tensor or energy–momentum tensor describes the density and flux of energy and momentum of a system in spacetime. The stress–energy tensor is the source of spacetime curvature in the Einstein field equations of general relativity.

The stress-energy tensor of a perfect fluid is given by:
$$T^{\mu \nu} = (\rho + p) U^\mu U^\nu + p g^{\mu \nu}$$
where:
$$T^{\mu \nu}$$ stress-energy tensor
$$\rho$$ energy density in the rest frame
$$p$$ isotropic pressure in the rest farme
$$U^\mu$$ four-velocity
$$g^{\mu \nu}$$ inverse metric tensor
The isotropy means the stress-energy tensor is diagonal in its rest frame, i.e. no net flux of any component of momentum in an orthogonal direction. In a spatially rotated reference frame the stress-energy tensor will confirm to be diagonal due to the isotropy of the system.

However if you compress an elastic body along a direction the resulting elastic energy will increase the rest mass of the body and an anisotropy in the system will happen. The idealization of a perfect fluid is no more applicable. The stress-energy tensor will show uneven spatial diagonal terms in the rest frame and if you spatially rotate the reference frame off-diagonal terms (shear) occur.

Note: In a fluid in the rest frame the net momentum is zero, but the constituents particles are in relative random motion. That is how pressure and shear are generated.