Let's take a look at the stress-energy-momentum tensor:
We can distinguish various components.
The 00 component represents the mass-energy density as seen in a local frame moving along with the masses or in the case of radiation the energy contained in the EM field if it's present.
The 0k and k0 components represent the fluxes of energy or equivalently the densities of linear momentum in each direction.
The diagonal components show us the flux of linear momentum in each direction. The normal stress. If all these components are the same it's called pressure.
Finally there is the shear stress represented by the other six components.
So in general, there are ten independent components.
To keep things simple, let's consider two massive objects moving anti-parallel. I wish to write down the components of the SEM-tensor. It would be easy if the objects were moving at a collision course. In the COM frame the non-diagonal components vanish. There is only a 00 and one kk component, which we can choose to be the 11-component.
If we now displace one trajectory in the the x-direction, things obviously change. There will non-diagonal components, as a momentum component in the y-direction appears.
We can look at the situation from one of the masses (let's suppose they are identical, spherical masses m, with a relative velocity, v.
It seems obvious that the y-component of the momentum is time dependent, as seen from one of the masses (I'm not sure if this is easier to calculate in the COM, which has the advantage that it endows the situation with a symmetry).
Do these off-diagonal components lead to framedragging effects when solving the Einstein field equations? In other words, will the metric contain (time-dependent) off-diagonal effects and how will this show in the motion of the masses?
