Negative mass is a (theoretical/hypothetical) concept in which the
mass of a particle can take up negative mass.
There is no empirical proof for the existence of negative mass, and
there are good reasons why particles with negative mass can not
exist, and are considered non-physical. Mainstream physics theory therefore does not contemplate the concept of negative mass.

Negative mass (if it could exist physically) would have some peculiar properties based on physical laws. For example in Newton's law:

(1) **F** = **m** x **a**

Where **F** is the force, **m** the mass and **a** the acceleration, when the
mass is taken negative, such a mass would accelerate in the opposite
direction of the force applied to it.

For the law of gravity (Newton's law of gravitation) in the case of one of the masses taken negative this would yield:

(2)** F** = -**G** x **m** x **M** / **R**^2

Where **F** is the gravitational force, **G** the gravitational constant, **m** and **M** the masses and **R** the distance between the mass-centers of those
masses.

If one of these masses is taken as negative, the force between them, this would yield a negative force.

Contrary to intuition though, the negative mass would not be
repelled by the positive mass, since (based on eq. (1)), the acceleration
would be negative too, yielding an acceleration TOWARDS the positive
mass (negative x negative = positive).

Positive masses attract all masses (both negative and positive).
Negative masses repel all masses (both positive and negative). A
negative mass and a positive mass have the peculiarity that the negative
mass is accelerated towards the positive mass, while the positive mass
is accelerated in the opposite direction.

Two equal masses but with opposite signs would have the strange
property of both accelerating continuously in the same direction in the
direction from the negative mass to the positive mass. Note however that this does not violate the conservation of mass/energy principle since the accelerating system has a zero nett-mass since the contribution of both the two rest masses and their kinetic energies cancel.

### Notes

- Negative mass != Anti-matter

Negative masses are not the same as anti-matter, since the latter has
normal positive mass, just that the electric charge is opposite to
normal matter (for example a positron is the anti-particle of the electron with opposite electric charge, but with the same mass as the electron.
- Negative energy

Although the negative mass concept is considered hypothetical and not physical reality, the concept of negative energy (which can be associated to mass through the equation E = m c^{2}) as for example in the case of gravitational fields, is a physical concept, and thus, this concept of negative mass might not be that hypothetical at all. See the reference below for an argument in which the force/energy that speeds up the expansion of the universe can be conceived as a form of negative mass.
- Since negative masses repel each other (that is: they still have an attracting force, but due to the negative mass value, this would result in an opposite acceleration), there could not be any large material bodies of negative mass (as far as negative massed objects/particles have any physical reality at all).

- In the above write-up we have assumed that (like ordinary, positive mass) negative mass has equal inertial and gravitational mass (i.e. both negative and same absolute magnitude). The described cased of attraction resp. repulsion between positive/negative masses would of course be different if for negative masses the equality between inertial and gravitational mass would not hold, but would be opposite. In that case negative masses would not repulse each other, but attract, and a positive and negative mass would both repel each other.
- Tachyons (another hypothetical type of particle) have yet another property, namely that they have speeds strictly above light speed, and hence, based on the relativistic equations, posses a mass value which is imaginary (which means that the mass squared is negative).

### References