Negative mass
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 pseudonegative pressure density produced by the Casimir effect. Although general relativity well describes gravity and the laws of motion for both positive and negative energy particles, hence negative mass, it does not include the other fundamental forces. On the other hand, although the Standard Model well describes elementary particles and the other fundamental forces, it does not include gravity, even though gravity is intimately involved in the origin of mass and inertia. A model that explicitly includes gravity along with the other fundamental forces may be needed for a better understanding of the concept of negative mass.
Contents

In general relativity 1
 Inertial versus gravitational 1.1
 Runaway motion 1.2
 Arrow of time and space inversion 1.3
 In Gauss's law for gravity 1.4
 Gravitational interaction of antimatter 1.5

In quantum mechanics 2
 Schrödinger equation 2.1
 Negative bare mass of the electron 2.2
 See also 3
 References 4
 External links 5
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 to 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 Matt Visser.^{[1]}
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 his first prize essay for the 1951 Gravity Research Foundation competition, Joaquin Mazdak Luttinger considered the possibility of negative mass and how it would behave under gravitational and other forces.^{[2]}
In 1957, following Luttinger's idea, Hermann Bondi suggested in a paper in Reviews of Modern Physics that mass might be negative as well as positive.^{[3]} 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 counterintuitive 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.
There have been several other analyses of negative mass, for example R.H. Price,^{[4]} however none addressed the question of what kind of energy and momentum would be necessary to describe nonsingular negative mass. Indeed, the Schwarzschild solution for negative mass parameter has a naked singularity at a fixed spatial position. The question that immediately comes up is, would it not be possible to smooth out the singularity with some kind of negative mass density. The answer is yes, but not with energy and momentum that satisfies the dominant energy condition. This is because if the energy and momentum satisfies the dominant energy condition within a spacetime that is asymptotically flat, which would be the case of smoothing out the singular negative mass Schwarzschild solution, then it must satisfy the positive energy theorem, i.e. its ADM mass must be positive, which is of course not the case.^{[5]}^{[6]} However, it was noticed by Belletête and Paranjape that since the positive energy theorem does not apply to asymptotic de Sitter spacetime, it would actually be possible to smooth out, with energymomentum that does satisfy the dominant energy condition, the singularity of the corresponding exact solution of negative mass Schwarzschildde Sitter, which is the singular, exact solution of Einstein's equations with cosmological constant.^{[7]} In a subsequent article, Mbarek and Paranjape showed that it is in fact possible to obtain the required deformation through the introduction of the energymomentum of a perfect fluid.^{[8]}
Runaway motion
Although no particles are known to have negative mass, physicists (primarily Hermann Bondi in 1957,^{[3]} William B. Bonnor in 1989,^{[9]} then Robert L. Forward^{[10]}) 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, based on the Einstein field equations:
 Positive mass attracts both other positive masses and negative masses.
 Negative mass repels both other negative masses and positive masses.
For two positive masses, nothing changes and there is a gravitational pull on each other causing an attraction. Two negative masses would repel because of their negative inertial masses. For different signs however, there is a push that repels the positive mass from the negative mass, and a pull that attracts the negative mass towards the positive one at the same time.
Hence Bondi pointed out that two objects of equal and opposite mass would produce a constant acceleration of the system towards the positivemass object,^{[3]} an effect called "runaway motion" by Bonnor who disregarded its physical existence, stating:
“  I regard the runaway (or selfaccelerating) motion […] so preposterous that I prefer to rule it out by supposing that inertial mass is all positive or all negative.  ” 
— William B. Bonnor, in Negative mass in general relativity.^{[9]}

Such a couple of objects would accelerate without limit (except relativistic one); however, the total mass, momentum and energy of the system would remain 0.
This behavior is completely inconsistent with a commonsense 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:
 P_{sys} = mv + (m)v = [m+(m)]v = 0\times v = 0.
And equivalently for the kinetic energy:
 E_{k,sys} = {1 \over 2} mv^2 + {1 \over 2}(m)v^2 = {1 \over 2}[m+(m)]v^2 = {1 \over 2}(0)v^2 = 0
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,^{[11]} including noting that although negative mass particles would repel each other gravitationally, the electrostatic force would be attractive for likecharges and repulsive for opposite charges.
Forward used the properties of negativemass matter to create the concept of 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 "cancelout" or "nullify" each other's existence. An interaction between equal quantities of positive mass matter (hence of positive energy E = m c^2) and negative mass matter (of negative energy E = m c^2) 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. So once this runaway phenomenon has been revealed, the scientific community considered negative mass could not exist in the universe.
Arrow of time and space inversion
In 1970, JeanMarie Souriau demonstrated, through the complete Poincaré group of dynamic group theory, that reversing the energy of a particle (hence its mass, if the particle has one) is equal to reversing its arrow of time.^{[12]}^{[13]}
The universe according to general relativity is a Riemannian manifold associated to a metric tensor solution of Einstein’s field equations. In such a framework, the runaway motion prevents the existence of negative matter.^{[3]}^{[9]}
Some bimetric theories of the universe propose that two parallel universes instead of one may exist with an opposite arrow of time, linked together by the Big Bang and interacting only through gravitation.^{[14]}^{[15]}^{[16]} The universe is then described as a manifold associated to two Riemannian metrics (one with positive mass matter and the other with negative mass matter). According to group theory, the matter of the conjugated metric would appear to the matter of the other metric as having opposite mass and arrow of time (though its proper time would remain positive). The coupled metrics have their own geodesics and are solutions of two coupled field equations:^{[17]}^{[18]}
 R_{\mu \nu}^{(+)}  {1 \over 2}g_{\mu \nu}\,R^{(+)} g_{\mu \nu}^{(+)} = {8 \pi G \over c^4} [ T_{\mu \nu}^{(+)} + \varphi T_{\mu \nu}^{()} ]
 R_{\mu \nu}^{()}  {1 \over 2}g_{\mu \nu}\,R^{()} g_{\mu \nu}^{()} =  {8 \pi G \over c^4} [ \phi T_{\mu \nu}^{(+)} + T_{\mu \nu}^{()} ]
The Newtonian approximation then provides the following interaction laws:
 Positive mass attracts positive mass.
 Negative mass attracts negative mass.
 Positive mass and negative mass repel each other.
Those laws are different to the laws described by Bondi and Bonnor, and solve the runaway paradox. The negative matter of the coupled metric, interacting with the matter of the other metric via gravity, could be an alternative candidate for the explanation of dark matter, dark energy, cosmic inflation and accelerating universe.^{[17]}^{[18]}
In Gauss's law for gravity
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.
Gravitational interaction of antimatter
The overwhelming consensus among physicists is that antimatter has positive mass and should be affected by gravity just like normal matter. Direct experiments on neutral antihydrogen have not been sensitive enough to detect any difference between the gravitational interaction of antimatter, compared to normal matter.^{[19]}
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.
In quantum mechanics
In 1928, Paul Dirac's theory of elementary particles, now part of the Standard Model, already included negative solutions.^{[20]} The Standard Model is a generalization of quantum electrodynamics (QED) and negative mass is already built into the theory.
Morris, Thorne and Yurtsever^{[21]} pointed out that the quantum mechanics of the Casimir effect can be used to produce a locally massnegative 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 negativemass loops of cosmic string.^{[22]} 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;^{[23]} this proves, for example, that a finite Tipler cylinder cannot be used as a time machine.
Schrödinger equation
For energy eigenstates of the Schrödinger equation, the wavefunction is wavelike wherever the particle's energy is greater than the local potential, and exponentiallike (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 KleinGordon equation is required.
Negative bare mass of the electron
The mass contributed to the total mass of the electron by the cloud of virtual photons, by Einstein's second law, is positive, so the bare mass of the electron is necessarily less than its observed mass. Since the virtual photons have energies greater than twice the electron mass, so they can make the electronpositron pairs needed for charge renormalization, then the bare mass of the source electron must be negative.^{[24]}^{[25]}^{[26]}
See also
 Alcubierre drive
 Antimatter
 Dark energy
 Dark matter
 Exotic matter
 Mirror matter
 Warpfield experiments
 Woodward effect
References
 ^ Visser, M. (1995). Lorentzian Wormholes: from Einstein to Hawking. Woodbury NY: AIP Press.
 ^ Luttinger, J. M. (1951). "On "Negative" mass in the theory of gravitation" (PDF). Awards for Essays on Gravitation (Gravity Research Foundation).
 ^ ^{a} ^{b} ^{c} ^{d} Bondi, H. (July 1957). "Negative Mass in General Relativity". Rev. Mod. Phys. 29 (3): 423–428.
 ^ Price, R. M. (1993). "Negative mass can be positively amusing" (PDF). Am. J. Phys. 61: 216.
 ^ Shoen, R.; Yao, S.T. (1979). "On the proof of the positive mass conjecture in general relativity" (PDF). Commun. Math. Phys. 65: 45–76.
 ^ Witten, Edward (1981). "A new proof of the positive energy theorem". Comm. Math. Phys. 80: 381–402.
 ^ Belletête, Jonathan; Paranjape, Manu (2013). "On Negative Mass". Int.J.Mod.Phys. D22: 1341017.
 ^ Mbarek, Saoussen; Paranjape, Manu (2014). "Negative Mass Bubbles in De Sitter Spacetime". Phys. Rev. D (R) D90: 101502.
 ^ ^{a} ^{b} ^{c} Bonnor, W. B. (1989). "Negative mass in general relativity". General Relativity and Gravitation 21 (11): 1143.
 ^ Forward, R. L. (1990). "Negative matter propulsion". Journal of Propulsion and Power 6: 28.
 ^ Landis, G. (1991). "Comments on Negative Mass Propulsion". J. Propulsion and Power 7 (2): 304.
 ^ Souriau, J. M. (1970). Structure des Systèmes Dynamiques (in French). Paris: Dunod. p. 199.
 ^ Souriau, J. M. (1997). "A mechanistic description of elementary particles: Inversions of space and time". Structure of Dynamical Systems. Boston: Birkhäuser.
 ^ A.D. Sakharov: "Cosmological model of the Universe with a time vector inversion". ZhETF 79: 689–693 (1980); translation in JETP Lett. 52: 349–351 (1980)
 ^ Petit, J. P. (1995). "Twin universes cosmology" (PDF). Astrophysics and Space Science 226 (2): 273.
 ^ Barbour, J.; Koslowski, T.; Mercati, F. (2014). "Identification of a Gravitational Arrow of Time". Physical Review Letters 113 (18).
 ^ ^{a} ^{b} Petit, J. P.; d’Agostini, G. (2014). "Negative mass hypothesis in cosmology and the nature of dark energy" (PDF). Astrophysics and Space Science 354 (2).
 ^ ^{a} ^{b} Petit, J. P.; d'Agostini, G. (2014). "Cosmological bimetric model with interacting positive and negative masses and two different speeds of light, in agreement with the observed acceleration of the Universe" (PDF). Modern Physics Letters A 29 (34): 1450182.
 ^ Amole, C.; Charman, M. D.; Amole, M.; Ashkezari, W.; BaqueroRuiz, 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–.
 ^ Dirac, P. A. M. (1928). "The Quantum Theory of the Electron" (PDF). Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 117 (778): 610.
 ^ Morris, Michael; Thorne, Kip; Yurtsever, Ulvi (September 1988). "Wormholes, Time Machines, and the Weak Energy Condition". Physical Review 61 (13): 1446–1449.
 ^ 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.
 ^ Hawking, Stephen (2002). The Future of Spacetime. W. W. Norton. p. 96.
 ^ Woodward, J. F. (1993). "Maximal acceleration, Mach's principle, and the mass of the electron". Foundations of Physics Letters 6 (3): 233.
 ^ Woodward, J. F. (1994). "ADM electrons and the equivalence principle". Foundations of Physics Letters 7: 59.
 ^ Woodward, J. F. (2013). "Making Starships and Stargates".