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  General relativity 2 particle problem with negative mass

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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". 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: Tμν=muμuνdx0dτδ(xxparticle), where uμ is velocity and τ particle proper time and δ the Dirac delta. Further I make the first approximation, that since particles are far away, their total Tμν is sum of tensors of 2 particles: Tμνsum=Tμν1+Tμν2.

I wanted to solve the problem for trace reversed perturbation ¯hμν=hμν1/2ημνdet(hμν) (Lorenz gauge). In this Gauge I am solving the Einstein field equation in the following form:

hμν=16πTμν If I rewrite d'Almbertian and assume particles have same magnitude opposite sygn masses m1=M, m2=M, for M>0 one can get a differential equation:

ημνμν¯hμν=16πM(uμ1uν1dx01dτuμ2uν2dx02dτ). I want to solve this equation and determine hμν 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 Tμν 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
asked Jan 16, 2023 in Theoretical Physics by Vid (30 points) [ no revision ]
retagged Apr 2, 2024
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
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
Where have the deltas gone in the RHS of the last equation? Why would it not be 16πM(uμ1uν1dx01dτδ(xx1)uμ2uν2dx02dτδ(xx2))?

This post imported from StackExchange Physics at 2024-04-02 19:33 (UTC), posted by SE-user kricheli
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
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
@kricheli I made a mistake and forgot to lower indices, the right way is 16πM(u1μu1νdx01dτδ(xx1)u2μu2νdx02dτδ(xx2))?, right?

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

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