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I'm pretty sure everyone's probably very tired of this thread by now, but I can't resist One More comment.
It seems to me that the best approach to considering what happens to gravity at high velocities is to go back to the idea of exploring "tidal gravity", the Riemann tensor, at high velocities, due to the aforementioned problems of dealing with the traditional notion of gravity as a "force".
This approach has the definite advantage that it can be done at a point - one does not need any reference to the outside world or dependence on an external global coordinate system to know what tidal forces one is experiencing, one can measure the tidal forces directly. (Except for the problem of rotation, which I'll get into).
The biggest stumbling block I have here is the issue of how to deal with rotation. Some relativly simple calculations can give the tidal forces on a body moving at relativistic velocities relative to a large mass. One needs to compute the Riemann tensor rather than the connection coefficients, then the tidal forces can be neatly summed up by the following matrix:
[tex]E^a{}_b = R^a{}_{bcd} u^b u^d [/tex]
where u^x is the four-velocity of the moving object.
The main difficulty in making this approach rigorous is dealing with eliminating rotation from the coordinate systems used, so that "centrifugal forces" from a rotating coordinate system don't get confused with the actual components of the tidal force.
Thus, we can directly measure the tidal forces we experience due to passing close to a body moving at relativisitic velocities directly if, and only if, we have zero rotational angular momentum - the later is not a very stringent condition, but it does mean we can't quite ignore the rest of the universe, we have to pay enough attention to it to be able to say that we aren't rotating.
Another thing which one can calculate in principle is the total amount of momentum transferred to a body by an object "flying by".
So if one was initially at rest, and an object came whizzing by at ultra-relativistic speeds, then left for infinity again, it's reasonable to ask what velocity one has after the body has left. Space and space-time should be perfectly flat when the massive body has "passed through", so there shouldn't be any ambiguity in this question.
It seems to me that the best approach to considering what happens to gravity at high velocities is to go back to the idea of exploring "tidal gravity", the Riemann tensor, at high velocities, due to the aforementioned problems of dealing with the traditional notion of gravity as a "force".
This approach has the definite advantage that it can be done at a point - one does not need any reference to the outside world or dependence on an external global coordinate system to know what tidal forces one is experiencing, one can measure the tidal forces directly. (Except for the problem of rotation, which I'll get into).
The biggest stumbling block I have here is the issue of how to deal with rotation. Some relativly simple calculations can give the tidal forces on a body moving at relativistic velocities relative to a large mass. One needs to compute the Riemann tensor rather than the connection coefficients, then the tidal forces can be neatly summed up by the following matrix:
[tex]E^a{}_b = R^a{}_{bcd} u^b u^d [/tex]
where u^x is the four-velocity of the moving object.
The main difficulty in making this approach rigorous is dealing with eliminating rotation from the coordinate systems used, so that "centrifugal forces" from a rotating coordinate system don't get confused with the actual components of the tidal force.
Thus, we can directly measure the tidal forces we experience due to passing close to a body moving at relativisitic velocities directly if, and only if, we have zero rotational angular momentum - the later is not a very stringent condition, but it does mean we can't quite ignore the rest of the universe, we have to pay enough attention to it to be able to say that we aren't rotating.
Another thing which one can calculate in principle is the total amount of momentum transferred to a body by an object "flying by".
So if one was initially at rest, and an object came whizzing by at ultra-relativistic speeds, then left for infinity again, it's reasonable to ask what velocity one has after the body has left. Space and space-time should be perfectly flat when the massive body has "passed through", so there shouldn't be any ambiguity in this question.
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