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In other words, when a particle's mass increases significantly as it approaches the speed of light (c), does its influence on gravity increase, and if so, does it increase proportionally?

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possible duplicate of reletivistic mass as seen by different observers –  John Rennie Oct 22 '13 at 16:53
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Also see physics.stackexchange.com/q/3436 –  John Rennie Oct 22 '13 at 16:58
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marked as duplicate by John Rennie, Qmechanic Oct 22 '13 at 18:18

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The mass (the true mass) is an intrinsic property of a body, i.e, it does not depends on the observer's frame of reference. The mass used when calculating the gravitational force, via Newton's law, does not change. So in answer to your question: "No, the gravitational force doesn't change in that situation".

I strongly suggest to read this popular article by Lev Okun where he calls the "relativistic mass" a pedagogical virus.

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The concept of a so-called relativistic mass which changes with velocity is no longer considered the best interpretation of mass in special relativity. Instead, the invariant/rest mass is used. See here for a discussion. The increase in energy (through momentum) of energy associated with speeding up, however, does increase the extent to which an object contributes to the stress-energy tensor of the system under investigation. Hence, it is a stronger source of gravity.

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Note that, for a single particle, the trace of the stress energy tensor is invariant under a boost, and that this is the one number you typically associate with the "source of the gravitational field," though of course, you should generically use the whole stress-energy tensor. –  Jerry Schirmer Oct 22 '13 at 16:45
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