## [GR-QC] Negative Mass Bu-...Wait, What?

Level: Possibly public, supplemented by undergraduate material

So I will begin by acknowledging that I am stepping outside my proposed subfield for this blog a little bit, here, but that this is something that I feel I should address.

This preprint [as of yet unpublished!] has been making the rounds in a few internet news sites like Hacker New and Slashdot. To begin with, while the arxiv is a wonderful source of information, you do have to take any unpublished results with a proverbial grain of salt, or a truckload, take your pick.

For some garbage on the arxiv, you can start with Luboš Motl's examples here. For some questionably named content, see Sabine Hossenfelder's Stupid Title List here. While Sabine's list is not necessarily bad science...it gives you some examples of the silliness that you might see. Example of awesome science with a silly title? Escape from the Menace of the Giant Wormholes by Coleman and Lee, Phys. Lett. B 221:242, 1989 :)

Abstract [taken verbatim]:
We study the possibility of the existence of negative mass bubbles within a de Sitter space-time background with matter content corresponding to a perfect fluid. It is shown that there exist configurations of the perfect fluid, that everywhere satisfy the dominant energy condition, the Einstein equations and the equations of hydrostatic equilibrium, however asymptotically approach the exact solution of Schwarzschid-de Sitter space-time with a negative mass. [Typo on Schwarzschild theirs']

Basically they showed that you can form a distribution of stuff with negative mass such that it approaches the classic point distribution solution when you shrink it down, but remains finite otherwise. This is cool, but not actually a particularly challenging calculation nor particularly surprising, if you ask me.

One of my favorite lines from the paper: "Thus the negative mass Schwarzschild solution remains an unphysical solution and the question of its meaning is still unanswered."
Not to be a jerk, but I'm pretty sure the first half of that sentence answers the second half. While tuning parameters into regimes they would not traditionally be allowed into has given some beautiful results in the past (cross section calculations and analytic continuation being my favorite example), having a solution that's mathematically consistent does not imply that it's physically meaningful! There are examples of this everywhere from projectile motion through quantum gravity. The paragraph proceeding that statement effectively lays out an argument showing that a negative mass Schwarzschild solution cannot obey the conditions they desire. As such, they consider a background which is de Sitter instead of Minkowski.

One of the implications that may merit further investigation is that a process which allowed for the production of positive/negative mass pairs in the early universe could allow for a phase of negative mass that might screen gravitational waves. This is something that I will have to think about more. They provided no calculations on the impact of these screening effects; so while there might be a simple mechanism allowing for this, it might be completely washed out by other things.

Some background for the uninitiated:
1. Negative mass in this context is at least naively what you might expect. Their motivation for this study comes from taking the mass parameter, $M$, of the Schwarzschild metric (general relativity's description of a black hole) and extending its domain to the negative numbers.
$$ds^2 = - \left( 1 - \frac{2 GM}{r c^2} \right)dt^2 + - \left( 1 - \frac{2 GM}{r c^2} \right)^{-1}dr^2 + r^2 d \Omega$$

2. A de Sitter space-time is, roughly, the geometry we believe the universe conforms to on large scales - $\sim 10^9$ parsecs is probably safe if you want a number. It's a lovely space with a constant positive curvature that's simply connected. When you solve Einstein's field equations for a maximally symmetric vacuum with a cosmological constant, $\Lambda$, this is the class of $\Lambda >0$ solutions. A cute fact about de Sitter space is that it was simultaneously and independently discovered by Tullio Levi-Civita of $\epsilon_{ij \ldots n}$ fame. Another cute fact, for the group-theoretically minded, is that the isometry group of de Sitter is $O(1,n)$ for an $n+1$ dimensional space.

3. A perfect fluid is completely characterized by its density, $\rho$, and pressure, $p$, in its rest frame. The stress-energy tensor, the object which tells you the densities of the energy and momentum for a system, for such a fluid is given by:
$$T^{\mu \nu} = \left( \rho + \frac{p}{c^2}\right) U^\mu U^\nu + p \eta^{\mu \nu}$$
where $\eta_{mu \nu}$ is the Minkowski metric and and $U$ is the velocity vector field of the fluid. It's worth noting that they're devoid of viscosity and heat conduction.

4. The dominant energy condition is a generalization of the weak energy condition (matter density is always non-negative for time-like vectors) that prevents mass-energy from being observed to be flowing faster than light.
Weak energy condition:
$$\rho = T_{ab} X^a X^b \geq 0$$
Dominant energy condition: $- T^a_b Y^b$ must be future-pointing for every timelike or null vector field $Y$.
NB: Each of these is an ansatz about a system. Einstein's equations are agnostic to a lot of the properties of the matter that you could feed them. While these conditions are things people would consider reasonable they're by no means necessary nor fundamental.

5. Hydrostatic equilibrium is basically the requirement that the mechanics of pressures and energy densities are consistent in a system. It's pretty straightforward to arrive at the equations from simply summing forces in terms of the densities and pressures present such that they sum to zero. For example, a simple fluid with a pressure $p$ with density $\rho$, in a local gravitational acceleration $g$ in the $z$ direction gives:
$$\frac{\partial p}{\partial z} + \rho g = 0$$

6. Finally, the de Sitter-Schwarzschild metric described a black hole sitting in a de Sitter space as described above. It's roughly what the space of a black hole in our universe is expected to look like. The metric is what you might expect, if you're familiar with these things:
$$ds^2 = - f(r) dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega$$
where $f(r) = 1 - \frac{2a}{r} - b r^2$ where $a$ and $b$ are related to the mass of the black hole $M$ and the cosmological constant $\Lambda$.
These are particularly interesting since they have a maximum size called the Nariai spacetime AND they're the simplest objects that have both an event horizon and cosmological horizon.

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