Editing Negative mass (section)

===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>