I'm looking at the answer Michael gave to my question. Does this imply that under parameterized post-Newtonian formalism, under the $β1$ factor for kinetic energy, the gravity of a mass doubles as it approaches $c$? Or is it some other factor?
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asked Apr 19, 2022 at 18:08
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$\begingroup$ The PPN formalism only makes sense for velocities much less than $c$. That's what the name means: "post-Newtonian" means the low velocity corrections to Newtonian gravity. $\endgroup$– knzhouCommented Apr 19, 2022 at 18:42
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$\begingroup$ What do you mean by gravity? The coordinate acceleration of a massive body with the near c relative velocity? $\endgroup$– g sCommented Apr 19, 2022 at 18:45
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1 Answer
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The kinetic energy of a relativistic object is given by $K = (\gamma - 1) m c^2$, where $\gamma = 1/\sqrt{ 1-v^2/c^2}$. Since $\gamma \to \infty$ as $v \to c$, the kinetic energy of an object diverges as $v \to c$, as does the ratio of its kinetic energy to its rest mass (which is what the PPN parameter $\beta_1$ is talking about.
That said, the PPN formalism is only really well-suited for sources moving at speeds much less than the speed of light (as has been pointed out in the comments). If you were to try to apply it to an object moving at relativistic speeds, there would be terms in the full metric that the PPN formalism ignores. Note that this also means that you probably can't use the PPN formalism to analyse a kugelblitz spacetime, as mentioned in your previous question.
answered Apr 19, 2022 at 18:47
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$\begingroup$ Thanks. Can you give me the formula for how gravity increases at non-relativistic velocities under PPNF? This is what I'm having trouble understanding. $\endgroup$ Commented Apr 19, 2022 at 18:55
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$\begingroup$ @foolishmuse: That's what the formula I gave you for $K$ tells you. The ratio of a particle's kinetic energy to its rest energy is $\gamma - 1$, and the $\beta$ parameters tells you (roughly) the ratio of the gravitational effect of kinetic energy relative to that of rest energy. $\endgroup$ Commented Apr 19, 2022 at 19:20