In theoretical physics, negative mass is a hypothetical concept of matter whose mass is of opposite sign to the mass of normal matter, e.g. −2 kg. Such matter would violate one or more energy conditions and show some strange properties, stemming from the ambiguity as to whether attraction should refer to force or the oppositely oriented acceleration for negative mass. It is used in certain speculative theories, such as on the construction of wormholes. The closest known real representative of such exotic matter is a region of pseudo-negative pressure density produced by the Casimir effect.
Inertial versus gravitational
The earliest references to negative weight are due to the observation that metals gain weight when oxidizing in the study of phlogiston theory in the early 1700s.
Ever since Newton first formulated his theory of gravity, there have been at least three conceptually distinct quantities called mass: inertial mass, "active" gravitational mass (that is, the source of the gravitational field), and "passive" gravitational mass (that is, the mass that is evident from the force produced in a gravitational field). The Einstein equivalence principle postulates that inertial mass must equal passive gravitational mass. The law of conservation of momentum requires that active and passive gravitational mass be identical. All experimental evidence to date has found these are, indeed, always the same. In considering negative mass, it is important to consider which of these concepts of mass are negative. In most analyses of negative mass, it is assumed that the equivalence principle and conservation of momentum continue to apply, and therefore all three forms of mass are still the same.
In 1957, Hermann Bondi suggested in a paper in Reviews of Modern Physics that mass might be negative as well as positive. He pointed out that this does not entail a logical contradiction, as long as all three forms of mass are negative, but that the assumption of negative mass involves some counter-intuitive form of motion. For example, an object with negative inertial mass would be expected to accelerate in the opposite direction to that in which it was pushed.
Although no particles are known to have negative mass, physicists (primarily Hermann Bondi and Robert L. Forward) have been able to describe some of the anticipated properties such particles may have. Assuming that all three concepts of mass are equivalent the gravitational interactions between masses of arbitrary sign can be explored.
For two positive masses, nothing changes and there is a pull on each other causing an attraction. Two negative masses would produce a pull on one another, but would repel because of their negative inertial masses. For different signs there is a push that repels the positive mass but attracts the negative mass.
Bondi pointed out that two objects of equal and opposite mass would produce a constant acceleration of the system towards the positive-mass object. However, the total mass, momentum and energy of the system would remain 0.
This behavior is completely inconsistent with a common-sense approach and the expected behaviour of 'normal' matter; but is completely mathematically consistent and introduces no violation of conservation of momentum or energy. If the masses are equal in magnitude but opposite in sign, then the momentum of the system remains zero if they both travel together and accelerate together, no matter what their speed:
And equivalently for the kinetic energy :
Forward extended Bondi's analysis to additional cases, and showed that even if the two masses m(-) and m(+) are not the same, the conservation laws remain unbroken. This is true even when relativistic effects are considered, so long as inertial mass, not rest mass, is equal to gravitational mass.
This behaviour can produce bizarre results: for instance, a gas containing a mixture of positive and negative matter particles will have the positive matter portion increase in temperature without bound. However, the negative matter portion gains negative temperature at the same rate, again balancing out. Geoffrey A. Landis pointed out other implications of Forward's analysis, including noting that although negative mass particles would repel each other gravitationally, the electrostatic force would be attractive for like-charges and repulsive for opposite charges.
Forward used the properties of negative-mass matter to create the diametric drive, a design for spacecraft propulsion using negative mass that requires no energy input and no reaction mass to achieve arbitrarily high acceleration.
Forward also coined a term, "nullification" to describe what happens when ordinary matter and negative matter meet: they are expected to be able to "cancel-out" or "nullify" each other's existence. An interaction between equal quantities of positive and negative mass matter would release no energy, but because the only configuration of such particles that has zero momentum (both particles moving with the same velocity in the same direction) does not produce a collision, all such interactions would leave a surplus of momentum, which is classically forbidden.
Classical gravitational field theory
In electromagnetism one can derive the energy density of a field from Gauss's law, assuming the curl of the field is 0. Performing the same calculation using Gauss's law for gravity produces a negative energy density for a gravitational field.
For energy eigenstates of the Schrödinger equation, the wavefunction is wavelike wherever the particle's energy is greater than the local potential, and exponential-like (evanescent) wherever it is less. Naively, this would imply kinetic energy is negative in evanescent regions (to cancel the local potential). However, kinetic energy is an operator in quantum mechanics, and its expectation value is always positive, summing with the expectation value of the potential energy to yield the energy eigenvalue.
For wavefunctions of particles with zero rest mass (such as photons), this means that any evanescent portions of the wavefunction would be associated with a local negative mass–energy. However, the Schrödinger equation does not apply to massless particles; instead the Klein-Gordon equation is required.
In general relativity
In general relativity, negative mass is generalized to refer to any region of space in which for some observers the mass density is measured to be negative. This could occur due a region of space in which the stress component of the Einstein stress–energy tensor is larger in magnitude than the mass density. All of these are violations of one or another variant of the positive energy condition of Einstein's general theory of relativity; however, the positive energy condition is not a required condition for the mathematical consistency of the theory. (Various versions of the positive energy condition, weak energy condition, dominant energy condition, etc., are discussed in mathematical detail by Visser.)
Morris, Thorne and Yurtsever pointed out that the quantum mechanics of the Casimir effect can be used to produce a locally mass-negative region of space–time. In this article, and subsequent work by others, they showed that negative matter could be used to stabilize a wormhole. Cramer et al. argue that such wormholes might have been created in the early universe, stabilized by negative-mass loops of cosmic string. Stephen Hawking has proved that negative energy is a necessary condition for the creation of a closed timelike curve by manipulation of gravitational fields within a finite region of space; this proves, for example, that a finite Tipler cylinder cannot be used as a time machine.
Gravitational interaction of antimatter
Antimatter has positive mass and should be affected by gravity just like normal matter. Direct experiments on neutral antihydrogen have not detected any difference between the gravitational interaction of antimatter, compared to normal matter.
Bubble chamber experiments provide further evidence that antiparticles have the same inertial mass as their normal counterparts. In these experiments, the chamber is subjected to a constant magnetic field that causes charged particles to travel in helical paths, the radius and direction of which correspond to the ratio of electric charge to inertial mass. Particle–antiparticle pairs are seen to travel in helices with opposite directions but identical radii, implying that the ratios differ only in sign; but this does not indicate whether it is the charge or the inertial mass that is inverted. However, particle–antiparticle pairs are observed to electrically attract one another. This behavior implies that both have positive inertial mass and opposite charges; if the reverse were true, then the particle with positive inertial mass would be repelled from its antiparticle partner.
- Bondi, H. (July 1957). "Negative Mass in General Relativity". Rev. Mod. Phys. 29 (3): 423. Bibcode:1957RvMP...29..423B. doi:10.1103/RevModPhys.29.423.
- Landis, G. (1991). "Comments on Negative Mass Propulsion". J. Propulsion and Power 7 (2): 304. doi:10.2514/3.23327.
- Visser, M. (1995). Lorentzian Wormholes: from Einstein to Hawking. Woodbury NY: AIP Press. ISBN 1-56396-394-9.
- Morris, Michael; Thorne, Kip; Yurtsever, Ulvi (September 1988). "Wormholes, Time Machines, and the Weak Energy Condition". Physical Review 61 (13): 1446–1449. Bibcode:1988PhRvL..61.1446M. doi:10.1103/PhysRevLett.61.1446. PMID 10038800.
- Cramer, John; Forward, Robert; Morris, Michael; Visser, Matt; Benford, Gregory; Landis, Geoffrey (1995). "Natural Wormholes as Gravitational Lenses". Phys. Rev. D 51 (6): 3117–3120. arXiv:astro-ph/9409051. Bibcode:1995PhRvD..51.3117C. doi:10.1103/PhysRevD.51.3117.
- Hawking, Stephen (2002). The Future of Spacetime. W. W. Norton. p. 96. ISBN 0-393-02022-3.
- Amole, C.; Charman, M. D.; Amole, M.; Ashkezari, W.; Baquero-Ruiz, E.; Bertsche, A.; Butler, C. L.; Capra, M.; Cesar, S.; Charlton, J.; Eriksson, T.; Fajans, M. C.; Friesen, D. R.; Fujiwara, A.; Gill, J. S.; Gutierrez, W. N.; Hangst, M. E.; Hardy, C. A.; Hayden, S.; Isaac, L.; Jonsell, A.; Kurchaninov, N.; Little, J. T. K.; Madsen, S.; McKenna, S. C.; Menary, P.; Napoli, A.; Nolan, P.; Olin, C. Ø.; Pusa, F. (2013). "Description and first application of a new technique to measure the gravitational mass of antihydrogen". Nature Communications 4: 1785–. doi:10.1038/ncomms2787. PMC 3644108. PMID 23653197.