Massive Object's Relativistic Velocity: What Happens to Earth?

[cowmoo32]

If somehow a massive object, let's say 3e5kg, was accelerated to a relativistic velocity and it blows by Earth at 0.9c a few hundred km above the surface of the earth, what happens? At that speed, it clearly has enough energy to warp spacetime; would it have a noticeable effect here on earth? Theoretically, couldn't a relatively small object be traveling with enough energy to throw us off of our oribt? I'm roughly thinking of it like an impulse problem considering the time it would have to influence us would be minimal at best. I guess this is akin to the "a gnat can stop a train with enough speed" kind of question.

 

[DaveC426913]

I don't understand the question. OK, an object the size of an asteroid shoots past very close to Earth.

A larger slow moving asteroid will warp spacetime just as much. So Earth moves a millimeter out of it path. In fact, since it's moving slower, it will have a bigger effect than your small, fast-moving one.

What are you expecting to happen?

 

[PAllen]

There are two important ways to look at this:

1) If you consider total curvature produced (e.g. with some invariant measure like the Kretschman invariant of the curvature tensor), then the speed of the object is completely irrelevant. At least, before it is near earth, you can consider coordinates where the body is at rest. Anything invariant you compute in such coordinates is true for all coordinates. This part of the answer explains why a rapidly moving body has no tendency to form a black hole, just because its total energy is arbitrarily large in some other frame of reference.

2) If you consider the 'force' (speaking loosely, because gravity isn't a force in GR) between two bodies with rapid relative motion, the effective force at closest approach is, indeed, larger than the same bodies moving slowly past each other (the relation is not simply multiplication by gamma, however). On the other hand, the time during which the influence is strong is very short, so the total deviation produced doesn't go up the way you might think.

A final question is 'which moves more'? The smaller rest mass body is the one the moves more. So what this would look like is unspectacular. Ever nearer c asteroids flying by would be deflected by similar amounts because you have more effective force, but more force needed to achieve a given deflection.

 

[pervect]

If your main concern is the Earth being thrown out of orbit, http://adsabs.harvard.edu/abs/1985AmJPh..53..661O looks at the induced velocity on a test-mass after a fly by. I'll quote the abstract below, I'm not sure if the full paper is available online (I think it was not, I haven't checked recently).

If a heavy object with rest mass M moves past you with a velocity comparable to the speed of light, you will be attracted gravitationally towards its path as though it had an increased mass. If the relativistic increase in active gravitational mass is measured by the transverse (and longitudinal) velocities which such a moving mass induces in test particles initially at rest near its path, then we find, with this definition, that Mrel=γ(1+β2)M. Therefore, in the ultrarelativistic limit, the active gravitational mass of a moving body, measured in this way, is not γM but is approximately 2γM.

If your concern is the peak tidal force on the Earth during the flyby, the answer is different. There's an exact solution for a boost that approaches infinity, the Aichelberg-Sexl ultraboost (see for instance http://arxiv.org/abs/gr-qc/0110032, but the limiting form here involves delta functions, so it's not as illuminating as one might hope.

From some past calculations I get get something like

(2GM/r^3) gamma^2 (1 + beta^2/2)

for the tidal force (compare to the Newtonian expression of 2GM/r^3), which is computed by boosting the Riemann. I don't think anyone has ever cross-checked this calculation, though. See for instance

https://www.physicsforums.com/showpost.php?p=689706&postcount=2

beta = v/c and gamma^2 = 1/(1-beta^2)