Editing Negative mass (section)

==In general relativity==
Negative mass is any region of [[space]] in which for some observers the mass density is measured to be negative. This may occur due to a region of space in which the sum of the three normal stress components (pressure on each of three axes) 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.

===Inertial versus gravitational mass===
In considering negative mass, it is important to consider which of these concepts of mass are negative. Ever since [[Isaac Newton|Newton]] first formulated his theory of [[gravity]], there have been at least three conceptually distinct quantities called [[mass]]:
* [[Mass#Inertial mass|inertial mass]] &ndash; the mass ''m'' that appears in Newton's second law of motion, '''F'''&nbsp;=&nbsp;''m''&thinsp;'''a'''
* "active" [[gravitational mass]] &ndash; the mass that produces a gravitational field that other masses respond to 
* "passive" gravitational mass &ndash; the mass that responds to an external gravitational field by accelerating.
The law of [[Momentum|conservation of momentum]] requires that active and passive gravitational mass be identical. Einstein's [[equivalence principle]] postulates that inertial mass must equal passive gravitational mass, and all experimental evidence to date has found these are, indeed, always the same.

In most analyses of negative mass, it is assumed that the equivalence principle and conservation of momentum continue to apply without using any matter in the process, and therefore all three forms of mass are still the same, leading to the study of "negative mass". But the equivalence principle is simply an observational fact, and is not necessarily valid. If such a distinction is made, a "negative mass" can be of three kinds: whether the inertial mass is negative, the gravitational mass, or both.

In his 4th-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.<ref name="Luttinger 1951">{{cite web|last=Luttinger |first=J. M. |year=1951 |title=On "Negative" mass in the theory of gravitation |url=https://static1.squarespacewebcom/static/5852e579be659442a01f27b8/t/5873dc04d1758eea4b41c720/1483987972731/luttinger.pdf |publisher=Gravity Research Foundation}}</ref>

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.<ref name="Bondi 1957">{{cite journal |doi=10.1103/RevModPhys.29.423 |title=Negative Mass in General Relativity |journal=Reviews of Modern Physics |volume=29 |issue=3 |pages=423–428 |year=1957 |last1=Bondi |first1=H. |bibcode=1957RvMP...29..423B |url=http://ayuba.fr/pdf/bondi1957.pdf }}</ref> 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 (non-gravitationally).

There have been several other analyses of negative mass, such as the studies conducted by R. M. Price,<ref>{{cite journal|last1=Price|first1=R. M.|title=Negative mass can be positively amusing|journal=Am. J. Phys.|date=1993|volume=61|issue=3|page=216|doi=10.1119/1.17293 |url=http://people.westminstercollege.edu/faculty/ccline/courses/resources/wp/pdf/AJP000216.pdf|bibcode = 1993AmJPh..61..216P }}</ref> though none addressed the question of what kind of energy and momentum would be necessary to describe non-singular 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 [[Energy condition#Dominant energy condition|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 formalism|ADM mass]] must be positive, which is of course not the case.<ref>{{cite journal|last1=Shoen|first1=R.|last2=Yao|first2=S.-T.|title=On the proof of the positive mass conjecture in general relativity|journal=Commun. Math. Phys.|date=1979|volume=65|issue=1|pages=45–76|url=http://www.doctoryau.com/papers/PositiveMassConjecture.pdf|bibcode=1979CMaPh..65...45S|doi=10.1007/BF01940959|s2cid=54217085|access-date=20 December 2014|archive-url=https://web.archive.org/web/20170516232805/http://www.doctoryau.com/papers/PositiveMassConjecture.pdf|archive-date=16 May 2017|url-status=usurped}}</ref><ref>{{cite journal|last1=Witten|first1=Edward|title=A new proof of the positive energy theorem|journal=Comm. Math. Phys.|date=1981|volume=80|issue=3|pages=381–402 |url=http://projecteuclid.org/euclid.cmp/1103919981|doi=10.1007/bf01208277|bibcode = 1981CMaPh..80..381W |s2cid=1035111}}</ref> 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 energy–momentum that does satisfy the dominant energy condition, the singularity of the corresponding exact solution of negative mass Schwarzschild–de Sitter, which is the singular, exact solution of Einstein's equations with cosmological constant.<ref>{{cite journal|last1=Belletête|first1=Jonathan|last2=Paranjape|first2=Manu|title=On Negative Mass|journal=Int. J. Mod. Phys. D|date=2013|volume=22|issue=12|page=1341017|doi=10.1142/S0218271813410174 |arxiv=1304.1566|bibcode = 2013IJMPD..2241017B |s2cid=119258256}}</ref> In a subsequent article, Mbarek and Paranjape showed that it is in fact possible to obtain the required deformation through the introduction of the energy–momentum of a perfect fluid.<ref>{{cite journal|last1=Mbarek|first1=Saoussen|last2=Paranjape|first2=Manu|title=Negative Mass Bubbles in De Sitter Spacetime|journal=Phys. Rev. D|date=2014|volume=90|issue=10|page=101502|doi=10.1103/PhysRevD.90.101502 |arxiv=1407.1457|bibcode = 2014PhRvD..90j1502M |s2cid=119167780}}</ref>

===Runaway motion===
Although no particles are known to have negative mass, physicists (primarily [[Hermann Bondi]] in 1957,<ref name="Bondi 1957" /> [[William B. Bonnor]] in 1964 and 1989,<ref name="Bonnor 1964">{{cite journal |last1=Bonnor |first1=W. B. |last2=Swaminarayan |first2=N. S. |date=June 1964 |title=An exact solution for uniformly accelerated particles in general relativity |journal=Zeitschrift für Physik |volume=177 |issue=3 |pages=240–256 |doi=10.1007/BF01375497|bibcode=1964ZPhy..177..240B |s2cid=122830231 }}</ref><ref name="Bonnor 1989">{{Cite journal | doi = 10.1007/BF00763458| title = Negative mass in general relativity| journal = General Relativity and Gravitation| volume = 21| issue = 11| pages = 1143–1157| year = 1989| last1 = Bonnor | first1 = W. B.|bibcode = 1989GReGr..21.1143B | s2cid = 121243887}}</ref> then [[Robert L. Forward]]<ref name="Forward 1990">{{Cite journal | doi = 10.2514/3.23219| title = Negative matter propulsion| journal = [[Journal of Propulsion and Power]]| volume = 6| pages = 28–37| year = 1990| last1 = Forward | first1 = R. L. }}</ref>) have been able to describe some of the anticipated properties such particles may have. Assuming that all three concepts of mass are equivalent according to the [[equivalence principle]], the gravitational interactions between masses of arbitrary sign can be explored, based on the [[Post-Newtonian expansion|Newtonian approximation]] of the [[Einstein field equations]]. The interaction laws are then:[[File:Runaway motion.svg|thumb|200px|right|In yellow, the "preposterous" ''runaway motion'' of positive and negative masses described by Bondi and Bonnor.]]

* 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 positive-mass object,<ref name="Bondi 1957" /> an effect called "runaway motion" by Bonnor who disregarded its physical existence, stating: 
{{Centered pull quote
 |I regard the runaway (or self-accelerating) motion […] so preposterous that I prefer to rule it out by supposing that inertial mass is all positive or all negative.
|author= William B. Bonnor
|source=in ''Negative mass in general relativity''.<ref name="Bonnor 1989" />}}

Such a couple of objects would accelerate without limit (except a relativistic one); however, the total mass, momentum and energy of the system would remain zero. This behavior is completely inconsistent with a common-sense approach and the expected behavior of "normal" matter. [[Thomas Gold]] even hinted that the runaway linear motion could be used in a [[perpetual motion]] machine if converted to circular motion:
{{Centered pull quote
 |What happens if one attaches a negative and positive mass pair to the rim of a wheel? This is incompatible with general relativity, for the device gets more massive.
|author= Thomas Gold
|source=in ''Negative mass in general relativity''.<ref name="Gold">{{cite book
 |last1=Bondi |first1=H.
 |last2=Bergmann |first2=P.
 |last3=Gold |first3=T.
 |last4=Pirani |first4=F.
 |date=January 1957
 |chapter=Negative mass in general relativity
 |title=The Role of Gravitation in Physics: Report from the 1957 Chapel Hill Conference
 |publisher=Open Access Epubli 2011
 |editor1-last=M. DeWitt |editor1-first=Cécile
 |editor1-link=Cécile DeWitt-Morette
 |editor2-last=Rickles |editor2-first=Dean
 |isbn=978-3869319636
 |chapter-url=http://www.edition-open-sources.org/sources/5/24/index.html
 |access-date=21 December 2018
}}</ref>}}

But Forward showed that the phenomenon is mathematically consistent and introduces no violation of [[Conservation law (physics)|conservation law]]s.<ref name="Forward 1990" /> 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:

:<math>p_\mathrm{sys} = mv + (-m)v = \big(m+(-m)\big)v = 0\times v = 0.</math>

And equivalently for the [[kinetic energy]]:

:<math>E_\mathrm{k,sys} = \tfrac12 mv^2 + \tfrac12(-m)v^2 = \tfrac12\big(m+(-m)\big)v^2 = \tfrac12(0)v^2 = 0</math>

However, this is perhaps not exactly valid if the energy in the gravitational field is taken into account.

Forward extended Bondi's analysis to additional cases, and showed that even if the two masses {{math|''m''<sup>(−)</sup>}} and {{math|''m''<sup>(+)</sup>}} 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.{{Citation needed|date=October 2019}} 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,<ref>{{Cite journal|first=G.|last= Landis|title=Comments on Negative Mass Propulsion|journal=J. Propulsion and Power|volume= 7|issue= 2|pages= 304 |year=1991|doi=10.2514/3.23327}}</ref> including noting that although negative mass particles would repel each other gravitationally, the [[electrostatic force]] would be attractive for like [[charge (physics)|charges]] and repulsive for opposite charges.

Forward used the properties of negative-mass matter to create the concept of diametric drive, a design for [[spacecraft propulsion]] using negative mass that requires no energy input and no [[Working mass|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 mass matter (hence of positive energy {{math|''E'' {{=}} ''mc''<sup>2</sup>}}) and negative mass matter (of negative energy {{math|−''E'' {{=}} −''mc''<sup>2</sup>}}) 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, such interactions would leave a surplus of momentum.