pervect

I'm actually interested in a bizarre physics point, so the theoretical answer is fine.

I wanted to see if the intergal of the trace of the classical stress-energy tensor

(T_11+T_22+T_33)*dV

(dV = volume element) was zero for the entire pressure vessel (interior + exterior). It looks like it is.

For the interior of the pressure vessel

T_11 = T_22 = T_33 = P

so we have

volume*3P = 4/3 * Pi * (d/2)^3 * 3*P

For the shell

T_11 = T_22 = P*d/4*t

so we have volume*2*(P*d/4*t) = 4*Pi*(d/2)^2*t*2*(P*d/4*t)

both of which simplify to Pi*d^3*P/2, and since one is pressure and the other is tension, the sum is zero.