Negative Energy and Gravity

Why do many science communicators say that negative masses fall upwards? Is it the same in the physics literature that they say that negative masses fall upwards?

In general relativity, things don't fall down, it's the earth that accelerates upwards.This is from the Einstein Equivalence Principle. It states that standing on the ground is equivalent to standing in an accelerating rocket.

So according to general relativity, negative masses would also fall downwards not upwards. Why wouldn't the earth also accelerate upwards for a negative mass?

The only way negative mass would fall upwards is in Newtonian gravity. In Newtonian gravity, gravity is a force and since it's a vector, it has a direction. Thus according to Newtonian gravity, negative masses would accelerate upwards.

In general relativity, gravity is the curvature of spacetime. The force we experience from walking on earth is because the earth is accelerating upwards not because there's a force pushing us downwards.

Am I wrong on anything here? Is there any way in general relativity where a negative and positive mass accelerate from each other other than a Newtonian approximation?

• You can consider it a warning not to take the maths too seriously; not every solution is "physically" meaningful. If you want a headache, think about negative inertial mass - that is even more "fun"! Jan 6 at 12:09
• Hi, welcome to Physics SE. Could you quote or link to such a communicator's statement? What would be correct is that negative gravitational masses have the opposite effect on test masses.
– J.G.
Jan 6 at 15:39

Negative masses fall downwards, even in Newtonian gravity.

Negative mass has two separate implications. Negative gravitational mass means that the gravitational force (of a big positive mass planet) on negative masses points upwards. Negative inertial mass means that the force and the acceleration that results from it are in opposite directions. (The force is the rate of change of momentum, which is the mass times the velocity. Negative mass means momentum and velocity point in opposite directions. When we differentiate both, we find force is opposite to acceleration.) So the upwards gravitational repulsion on a negative mass particle accelerates it downwards.

Science communicators have recognised that the negative mass is repelled by the Earth with an upwards force, but haven't realised that this accelerates the mass downwards. (Assuming that gravitational mass and inertial mass are still the same, even for negative masses.)

Hermann Bondi, William Bonnor, and Robert Forward have studied some of the implications of negative mass. Bondi noted in 1957 that negative masses repel one another, positive masses attract one another, but a negative and positive mass both accelerate in the same direction, leading to the possibility of "runaway motion". William Bonnor thought this allowed perpetual motion and thus contradicted relativity, but Forward later showed that it contradicted no conservation laws. The combination of equal positive and negative masses has zero rest mass, zero momentum, and zero energy. (See Wikipedia for references.)

If we take a negative mass in Einstein's Theory of Relativity, the sum of the three stress components, on Einstein's stress-energy tensor is larger in magnitude than the mass density. This violates his positive energy condition, however, the positive energy condition is not a required condition for the mathematical consistency of the theory.

• In gravity, Newton's or Einstein's, acceleration is independent of the mass, negative or positive. Jan 6 at 14:56