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