A Compute Normal Forces on Box Sides via Stress-Energy Tensor

[pervect]

Suppose one has a box moving through flat space-time with a stress energy tensor $T^{ab}$ that's non-zero inside the box and zero outside the box. How does one compute the normal forces on the faces of the box associated with it's motion? I am assuming that the normal forces are measured via a spring scale attached to the appropriate box face. I'm also interested in what other boundary conditions the stress-energy tensor must satisfy (if there are any).

If the metric at a box face is diag (-1,1,1,1) I'm pretty sure the answer is that the area of the box face multiplied by the pressure term $T^{ii}$ gives the normal force on the box wall. But this doesn't seem to be true in general.

 

[Paul Colby]

I can highly recommend "The Linearized Theory of Elasticity" by William S. Slaughter. What you seek are called traction forces. For a box made of a real material the traction forces must sum to zero for an inertial box. While an applied stress can be arbitrary any real situation is the result of solving the equations of motion.

$\rho\ddot{u}_m = \partial^n T_{nm} + f_m$

For a free standing box not under the influence of external forces, $f_m = 0$, the stress must vanish at the boundary. In the above $n$ and $m$ run over the space components only.

 

[PeterDonis]

Suppose one has a box moving through flat space-time with a stress energy tensor $T^{ab}$ that's non-zero inside the box and zero outside the box.

Strictly speaking, this isn't possible; if the SET is nonzero anywhere, the spacetime can't be flat. Are you basically considering an approximation where the SET is so small that its effect on the spacetime geometry is negligible?

 

[pervect]

Strictly speaking, this isn't possible; if the SET is nonzero anywhere, the spacetime can't be flat. Are you basically considering an approximation where the SET is so small that its effect on the spacetime geometry is negligible?

Yes - more precisely, I'm using the stress-energy tensor in the context of special relativity rather than general relativity. This is equivalent to the approximation you mentioned.

Basically I envision a box of idealized matter of constant density in the Rindler metric. The box is imagined to be dangling from a spring scale connected to the top of the box, and the goal is to plot the force reading from the spring scale vs the proper acceleration, both the force and proper acceleration being measured at the same point, the top of the box (the location matters).

Essentially, one "weighs" the box, plotting the weight (interpreted as a force) vs the acceleration.

The interesting result I'm getting and the goal of the exercise is to demonstrate that the force-acceleration curve for the box is not linear, unlike the case of a point particle.

The problem setup is basically specifying the boundary conditions (including how to compute the force at the box top, and setting the forces to zero on the other sides), and the vanishing of the divergence of the stress-energy tensor, $\nabla_a T^{ab} = 0$.

I've got some preliminary calculations, but I'm trying to look more closely at justifying them, in particular the boundary conditions, particularly at the top of the box.