General relativity 2 particle problem with negative mass

Question

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". Prticulary, 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 (booth will start accelerating). The problem is well studied in paper by H. Bondi, which uses 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; fisrt, was my treatment of Dirac delatas 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?

This post imported from StackExchange Physics at 2024-04-02 19:33 (UTC), posted by SE-user Vid

Comments

That link for the paper by H. Bondi is not working for me

This post imported from StackExchange Physics at 2024-04-02 19:33 (UTC), posted by SE-user KP99

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ayuba.fr/pdf/bondi1957.pdf Its reference [7] on the cited Wikipedia page. H. Bondi (1957), Negative Mass in General Relativity, Reviews of Modern Physics

This post imported from StackExchange Physics at 2024-04-02 19:33 (UTC), posted by SE-user Vid

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Where have the deltas gone in the RHS of the last equation? Why would it not be $-16\pi M \left( u_1^{\mu} u_1^{\nu} \frac{dx^0_1}{d \tau} \delta(x-x_1) -u_2^{\mu} u_2^{\nu} \frac{dx^0_2}{d \tau} \delta(x-x_2)\right)?$

This post imported from StackExchange Physics at 2024-04-02 19:33 (UTC), posted by SE-user kricheli

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My idea was to solve equation only for the particle trayectory (where delta=1), but thinking again this seems bad idea. But my main concern is, how to treat deltas in equation..

This post imported from StackExchange Physics at 2024-04-02 19:33 (UTC), posted by SE-user Vid

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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).

This post imported from StackExchange Physics at 2024-04-02 19:33 (UTC), posted by SE-user Andrew

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@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?

This post imported from StackExchange Physics at 2024-04-02 19:33 (UTC), posted by SE-user Vid

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Btw. are you aware how you would solve the last equation if the RHS was not dependent on $h$ via the geodesic equation and instead just a given function? I.e. are you familiar with the Green function technique or d'Alembert's formula?

This post imported from StackExchange Physics at 2024-04-02 19:33 (UTC), posted by SE-user kricheli

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Andrew's suggestion is quite good, but I would add as a word of caution that this should only work for low particle velocity (compared to the speed of light) because Newtonian gravity is incompatible with special relativity.

This post imported from StackExchange Physics at 2024-04-02 19:33 (UTC), posted by SE-user kricheli

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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.

This post imported from StackExchange Physics at 2024-04-02 19:33 (UTC), posted by SE-user displayname17