Negative Mass

I’ve been trying to lose weight, so I’ve been studying the physics of negative mass.

Basically it doesn’t exist. But physicists are have run into a serious problem. They think they can use astronomical measurements to put upper bounds on the total mass of all 3 kinds of neutrinos. But when they do this, the upper bound tends to come out negative!

This doesn’t really make sense. In fact, if you look at the details, you’ll see it’s the result of extrapolating an approximate formula beyond its realm of validity:

In more exact formula, equation (1), only squares of neutrino masses show up. Indeed in particle physics we never see masses, only squares of masses—so the sign of the mass never matters. But when the neutrino masses are positive and big enough, you can approximate the stuff in equation (1) by something involving the sum of the neutrino masses, getting equation (2).

So far so good. But then people consider what happens when the sum of the neutrino masses is negative. This is crazy, but the people doing it know this—or at least the smart ones do. If this theory fits the data best when the sum of neutrino masses is negative, it’s a sign that something has gone wrong somewhere.

The equations above are from here:

• Willem Elbers, Carlos S. Frenk, Adrian Jenkins, Baojiu Li, Silvia Pascoli Negative neutrino masses as a mirage of dark energy, Physical Review D 111 (2025), 063534.

but also read this:

• Daniel Green and Joel Meyers, The cosmological preference for negative neutrino mass.

This is a very interesting story, but let’s turn to something simpler: negative mass in Newtonian mechanics.

Suppose you have a positive mass star of mass m and a negative mass star of mass -m. Thanks to Newton’s laws of gravity they will each feel a force pointing outward. The positive mass star will accelerate away thanks to Newton’s other law, F = ma. But by the same rule, the negative mass star will accelerate toward the positive mass star, by the same amount!

So, the negative mass star will wind up chasing the positive mass star, faster and faster, never catching it.

That’s in Newtonian gravity… at least if we assume the equivalence principle, saying gravitational and inertial mass are equal.

Puzzle. What happens in Newtonian gravity if you have an ordinary positive mass star and a star with negative gravitational mass but positive inertial mass?

Puzzle. What happens in Newtonian gravity if you have an ordinary positive mass star and a star with negative inertial mass but positive gravitational mass?

What about general relativity? The famous physicist Bondi studied this question in 1957!

• Hermann Bondi, Negative mass in general relativity, Reviews of Modern Physics 29 (1957), 423–428.

He argued that essentially the same thing happens as in Newtonian mechanics: the two stars start accelerating, with the negative mass star chasing the positive mass star. One difference is that both stars approach but never exceed the speed of light.

But he seems to consider two infinitesimal masses, not thinking about the deeper question of how a finite-sized negative point mass would behave in general relativity.

What would a negative mass black hole be like? The only reasonable answer to this silly question is to take the Schwarzschild solution describing a black hole of mass m, and stick in a negative number for m.

You’ll get a solution of the equations of general relativity. I haven’t studied it carefully, but here’s what I know so far.

First, a negative mass black hole not a white hole.

A white hole is a time-reversed black hole. It still attracts you gravitationally! To see this, take a movie of someone throwing a rock up into the air. Then play this movie backwards. The rock still goes up and falls back down! The time reverse of gravitational attraction is not gravitational repulsion. It’s still attraction. In fact, if you’re near a white hole, it’s hard to tell it’s not a black hole…. until you see someone fall out.

A negative mass black hole is different. You’ll be pushed away from it, and this repulsive ‘force’ will become infinite as you approach the central singularity. But unlike an ordinary black hole, this is a naked singularity: it’s not hidden behind an event horizon, an imaginary surface such that if you cross it, you can never get out.

If you shine light at a negative mass black hole, it will be repelled and scatter away. Thus, calling it a ‘black hole’ is inaccurate. It would look roughly like some sort of strange mirrored sphere. Someone should simulate this thing and animate it.

II’ve had some good conversations about negative mass black holes on Mastodon, but I want to understand them better. So far I just know one paper on them, pointed out by Robert Low:

• Servando V. Serdio, Hernando Quevedo, Singularity theorems in Schwarzschild spacetimes, The European Physical Journal Plus 135 (2020), 1–21.

This paper is very mathematical: for example, it points out that negative mass black hole spacetimes are not globally hyperbolic. But I would like to see a paper written by someone like John Wheeler or Kip Thorne.

For more, check out this:

• Wikipedia, Negative mass.

And for even more negativity, try this:

• John Baez, Negative probabilities, Azimuth, 19 July 2013.