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If something is moving past me at a relativistic speed, does its gravitational force on me increase as its speed increases? That is, not as if its speed is changing, but if multiple of the same (rest) mass go past me in the same path, is the gravitational attraction at the closest point the same for each, or is it higher the faster it's moving?

As some background, I was recently surprised to see that I should discard the idea of "relativistic mass", so I've been trying to work on that. Please forgive me mixing Newtonian physics with relativity, but I'm trying to get a high level understanding before going into the math too much. I understand now that instead of $F=m_{relativistic}a$ I should use $F=\frac{d}{dt}(\gamma m_0v)$, but what about $F=\frac{Gm_1m_2}{r^2}$? Is that still roughly as it is, or do I need to throw a factor of $\gamma$ in there, or something else entirely?

Yes, I know this is talking about acceleration and so it doesn't really fit into special relativity, and lots of other disclaimers, but I'm hoping for a simple answer to get a general understanding.

Neutronium
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2 Answers2

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To be more correct and more specific, the answer is yes because the source of a gravitational field, which in General Relativity (GR) is spacetime curvature, is the stress energy tensor $T^{\mu\nu}$. The 0i and i0 components of that tensor, $T^{0i}$ and $T^{i0}$ (which are equal because the tensor is symmetric in GR), are actually the $i^{th}$ component of the momentum density, $p^i$, for i = 1-3 the 3 spatial components, and 0 the timelike coordinate.

Since higher velocity is higher momentum there is a stronger source if it is larger. Of course it depends on the reference frame and how things are arranged geometrically.

See the stress energy tensor in Wikipedia at https://en.m.wikipedia.org/wiki/Stress–energy_tensor In GR, the Einstein tensor (including the Ricci curvature) is proportional to the stress energy tensor.

So, you can deal with the effect of higher velocities in a more correct way than dealing with relativistic mass. Velocity, as momentum, is part of the source for GR curvature.

Bob Bee
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Yes. Gravitational attraction in general relativity is based on an object's energy, and an object's energy is greater when its speed is greater.

Travis
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