In Electromagnetism I understand it in terms of the Lorentz force: the E-component of the field causes the charge to respond infinitesimally with a $\vec{v}$ in the E-direction such that the $\vec{v}\times \vec{B}$ term produces a force in the EM-field's $\vec{E}\times \vec{B}$ direction of motion. Is there an analog of this in General Relativity?


In general relativity, instead of momentum, you have local stress-energy tensor, which contains energy density, momentum and stress components. But, these are local - there are in general no conserved integral quantities (such as total energy, momentum,... across a large chunk of space-time) because (loosely speaking), conservation of a quantity from one moment in time to another makes no sense, if you can't define what's "at the same time". That's also why energy contained in a gravitational wave can only be defined if you look at an average over many wavelengths. So... you have to ask yourself "what is momentum". If you say "derivative of position over time", you have a problem "position and time with respect to which inertial frame that stays inertial for long enough to perform the measurement".

What happens in gravitational wave is, that the distance between two objects in space just happens to change all of a sudden. Then, matter does whatever matter would do if the distances between different parts were wrong. Imagine a water surface with two corks on it (it's not a perfect metaphor because water ripple is not longitudinal). A wave comes, the corks move with it (the surface represents the space-time), and after the wave passes, the corks are just where they were before. No momentum transfer, they just happened to be at a place which moved relatively to other points in space. But if you connect the corks with a string, making their surface fixed, then they will feel the force between them through the string and end position won't be the same.

As said... it's hard to answer your question without going deep into the metric theory of space-time and the index notation nightmare... but a lot can be understood by considering it's the space that moves and you just follow it, and unless something around you depends on the distances and angles, you won't even notice something happened.

  • $\begingroup$ I was thinking that since gravitons (in first-order semiclassical QM-GR) transfer momentum, there might be an explanation a bit simpler than invoking the full machinery of GR. But then again, in the photon picture of EM I don't have a conceptual picture of how momentum is transferred other than through the classical picture I described (I don't think a Feynman diagram counts as a conceptual understanding here, because that just pushes the problem down a level to why diagrams are allowed that transfer momentum rather than just causing oscillatory motion). $\endgroup$ – user1247 Jun 27 '16 at 15:17

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