Is this scenario simple enough to properly analyse with GR?
Consider two infinite straight thin parallel rods of uniform density $$\rho$$ (mass per unit proper length), in constant relative inertial motion such that they remain parallel and separated by proper distance 2L.
Choose an inertial (x,y,t) reference frame S, such that:
- The upper rod is positioned at y=L, and has velocity v parallel to the x-axis.
- The lower rod is positioned at y=-L, and has velocity -v parallel to the x-axis.
S' (the upper rod rest frame) has velocity v parallel to the x-axis relative to S.
S'' (the lower rod rest frame) has velocity -v parallel to the x-axis relative to S.
In S, the density of both rods is $$\gamma\rho$$.
In S', the upper rod has density $$\rho$$, and the lower rod has density $$\gamma'\rho$$.
In S'', the lower rod has density $$\rho$$, and the upper rod has density $$\gamma'\rho$$.
Where $$\gamma' = (1 + v^2/c^2)/\gamma^2$$
So, three questions:
- What is the acceleration of a test mass A at rest in S at (x,y,t)=(0,0,0)?
- What is the acceleration of a test mass B with velocity v along the x-axis?
- What is the acceleration of a test mass C with velocity -v along the x-axis?
By symmetry,
A must have zero acceleration, and the proper accelerations of
B and
C should be equal and opposite.
Working in the rest frames of the test masses, the
naive use of SR + newtonian gravity suggests paradoxical answers:
- In S, all test masses have zero acceleration
- In S', the test masses accelerate in the negative y direction
- In S'', the test masses accelerate in the positive y direction
Clearly, SR + newtonian gravity isn't an adequate model in this scenario.
So what does GR say?