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I am looking at a problem of 2-particle system of which one has negative mass. I have situation described on Wiki under section "Runaway motion". Particulary, if we assume, that negative mass is possible, then:

  • Positive mass attracts both other positive masses and negative masses.
  • Negative mass repels both other negative masses and positive masses.

Or in practical terms, the negative mass will start chasing the positive mass (both will start accelerating). The problem is well studied in paper by H. Bondi, which uses the Weyl metric. Apparently the solution is also possible using Minkowski metric with help of linearized Einstein field equations, but I have some problems finding the solution for metric and solving the geodesic equation.

Stress tensor for a single particle is: $$ T^{\mu \nu}=m u^{\mu}u^{\nu}\frac{dx^0}{d \tau} \delta(\mathbf{x}-\mathbf{x_{particle}}), $$ where $u^{\mu}$ is velocity and $\tau$ particle proper time and $\delta$ the Dirac delta. Further I make the first approximation, that since particles are far away, their total $T^{\mu \nu}$ is sum of tensors of 2 particles: $T^{\mu \nu}_{sum}=T^{\mu \nu}_{1}+T^{\mu \nu}_{2}$.

I wanted to solve the problem for trace reversed perturbation $$\overline{h_{\mu \nu}}=h_{\mu \nu}-1/2 \eta_{\mu \nu} \det(h_{\mu \nu})$$ (Lorenz gauge). In this Gauge I am solving the Einstein field equation in the following form:

$$ \Box h_{\mu \nu}=-16 \pi T_{\mu \nu} $$ If I rewrite d'Almbertian and assume particles have same magnitude opposite sygn masses $m_1=M$, $m_2=-M$, for $M>0$ one can get a differential equation:

$$ \eta^{\mu \nu}\partial_{\mu }\partial_{\nu}\overline{h_{\mu \nu}}=-16\pi M \left( u_1^{\mu} u_1^{\nu} \frac{dx^0_1}{d \tau} -u_2^{\mu} u_2^{\nu} \frac{dx^0_2}{d \tau}\right). $$ I want to solve this equation and determine $h_{\mu \nu}$ and then using the geodesic equation calculate the trajectories of particles. I have 2 questions; first, was my treatment of Dirac deltas correct, when I wrote Einstein field equation only for area where the RHS of equation for $T^{\mu \nu}$ is non-zero? Secondly, how to approach last equation and find solutions?

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  • $\begingroup$ That link for the paper by H. Bondi is not working for me $\endgroup$
    – paul230_x
    Commented Jan 16, 2023 at 16:54
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    $\begingroup$ Where have the deltas gone in the RHS of the last equation? Why would it not be $-16\pi M \left( u_1^{\mu} u_1^{\nu} \frac{dx^0_1}{d \tau} \delta(x-x_1) -u_2^{\mu} u_2^{\nu} \frac{dx^0_2}{d \tau} \delta(x-x_2)\right)?$ $\endgroup$
    – kricheli
    Commented Jan 18, 2023 at 21:10
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    $\begingroup$ I think you could get some insight by breaking this into two easier subproblems. (1) Starting from purely Newtonian gravity, can you see how negative mass as you've described it would lead to a runaway instability? (2) Can you derive the Newtonian limit from Einstein's equations? (This part won't change much using negative mass). $\endgroup$
    – Andrew
    Commented Jan 18, 2023 at 22:33
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    $\begingroup$ @kricheli I made a mistake and forgot to lower indices, the right way is $-16\pi M \left( u_{1\mu} u_{1\nu} \frac{dx^0_1}{d \tau} \delta(x-x_1) -u_{2 \mu} u_{2\nu} \frac{dx^0_2}{d \tau} \delta(x-x_2)\right)$?, right? $\endgroup$
    – Vid
    Commented Jan 18, 2023 at 23:55
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    $\begingroup$ I can't comment due to low reputation, but pages 301-305 in Carroll GR book should answer your question on how to solve the equation. $\endgroup$
    – displayname17
    Commented Jan 20, 2023 at 8:58

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