If someone has a list of counterintuitive things, then somewhere at the top of it there should be negative mass. Dealing with it is like riding a reverse steering bicycle, which instead of turning left goes right, and vice versa.
But wait, what's the negative mass? Does it exist? Has anyone seen a weight of minus one kilogram? Of course no, negative masses have not been registered (yet). But are they forbidden by the laws of physics? Let's figure it out.
Mass is a measure of the inertia of a body, as well as a gravitational "charge." Sometimes it makes sense to talk separately about the inert and gravitational mass, but the equivalence principle underlying General Relativity requires these masses to be equal. If we do not want to contradict GR, we will adhere to this principle for negative masses as well.
The body of negative mass will have negative inertia. That is, when you try to slow it down, it will instead accelerate, and when you try to speed it up, it will decelerate. In other words, the acceleration created by the external force will be in the opposite direction to the force. This is evident from Newton's second law: a = F/m. If the mass is negative, then the force and acceleration will have different signs (see the picture). You cannot just take a negative brick and carry with you. When you pull it towards you, it will go away from you as if you've pushed it.
For a similar reason, the direction of the velocity and momentum of a negative body are also opposite to each other (p = mv). Despite the fact that the brick is going away from you, its momentum is directed towards you.
And gravity should work in a similar way, right? If positive bodies fall to Earth, then negative ones should fly up? Not at all! The negative gravitational mass is compensated by the negative inert mass. Minus times minus is plus. Negative bodies will fall on Earth in the same way as positive ones. In full accordance with the equivalence principle.
See what happens. If, according to Newton's law (F = GMm/r2), we calculate the force that will act on a negative body in the gravitational field of Earth, we will find that this is a repulsive force. It will be directed upwards. But we already know that the body will receive acceleration in the direction opposite to the force, i. e. it will accelerate downwards like a usual positive mass. Alas, the problem of antigravity is not so easy to solve.
In general, a positive mass attracts all the matter, both positive and negative. A negative mass, on the contrary, pushes away everything from itself, both positive and negative. The forces of attraction arise between two negative bodies the same way as between two positive ones, but these forces lead to repulsive acceleration due to the negative inert masses (see the lower picture).
A funny effect is obtained if we take two masses equal in magnitude but different in sign (see the middle picture). The negative mass will "fall" on the positive one, but the positive one will fly away from it. And with acceleration. That is, the system will constantly accelerate to the left, in the direction of the positive mass. Without any external source of energy. It is better than antigravity, it is a perpetual motion machine!
Despite all the unnaturalness of such runaway motion, the conservation laws are not violated here. The negative particle has negative energy (E = mc2), which means that the total energy of the system is zero, no matter how fast it moves. And the same is true for the momentum. The momentum of the negative particle is directed oppositely to its velocity, which means that the total momentum of the system is also always zero. It can be shown that even if the masses are not equal in magnitude, the energy and momentum of the system is still preserved.
But something must be violated! After all, perpetual motion is forbidden, right? Almost. Runaway motion violates the second law of thermodynamics. The system does not converge to the equilibrium state but goes "crazy", accelerating almost to the speed of light. And this is supported by the constant transfer of energy from the "cold" to the "hot" - from the negative body to the positive one. Thermodynamics says that such processes cannot occur spontaneously, without external energy.
However. The second law of thermodynamics has a completely different status than the conservation laws. This is only an empirical statistical law and not an absolute strict prohibition. It is based on statistical analysis and can be violated temporarily, in spontaneous fluctuations. In this case, it is not a fluctuation, but such motion does not contradict any statistics. So, who knows, maybe someday we will be traveling on such zero-mass dipoles.
Also, a recent paper suggests that the effects of dark energy and dark matter can be both explained by negative mass particles dominating the Universe. In my next post I'll try to cover these ideas in more details.
In addition, it must be said that the intrinsic time of negative particles is directed to the past and that gas of negative particles is likely to have a negative (Kelvin) temperature and negative pressure.
P. S. Do not confuse negative mass with antimatter (which has positive mass) and with hypothetical superluminal particles (tachyons), whose square mass is negative, i. e. the mass is imaginary.
Bondi, H. (1957). Negative mass in general relativity. Reviews of Modern Physics, 29(3), 423.
Terletsky, Y. P. (1969). Paradoxes in the theory of relativity (New York, NY, 1968); W. English translation of the 1966 Moscow edition.
Forward, R. L. (1990). Negative matter propulsion. Journal of Propulsion and Power, 6(1), 28-37.
Pollard, D., & Dunning-Davies, J. (1995). A consideration of the possibility of negative mass. Il Nuovo Cimento B (1971-1996), 110(7), 857-864.
Farnes, J. S. (2018). A unifying theory of dark energy and dark matter: Negative masses and matter creation within a modified ΛCDM framework. Astronomy & Astrophysics, 620, A92.
This post is a translation of my original post in Russian. The picture is mine, although it was inspired by a very similar picture in the latter paper.
