View source for Negative mass

{{Short description|Concept in physical models}}
In [[theoretical physics]], '''negative mass''' is a hypothetical type of [[exotic matter]] whose [[mass]] is of [[Negative number|opposite sign]] to the mass of [[Matter|normal matter]], e.g. −1&nbsp;kg.<ref>{{cite web|url=https://www.independent.co.uk/news/science/negative-mass-object-washington-state-physics-isaac-newton-upside-down-a7693701.html |archive-url=https://ghostarchive.org/archive/20220618/https://www.independent.co.uk/news/science/negative-mass-object-washington-state-physics-isaac-newton-upside-down-a7693701.html |archive-date=18 June 2022 |url-access=subscription |url-status=live |title=Scientists observe liquid with 'negative mass', which turns physics completely backwards |first=Andrew |last=Griffin |website=[[The Independent]] |date=2017-04-20 |access-date=2020-12-11}}</ref><ref>{{cite web|url=http://www.cbc.ca/news/technology/fluid-negative-mass-1.4073937 |title=Scientists create fluid that seems to defy physics:'Negative mass' reacts opposite to any known physical property we know |website=[[CBC News]] |date=2017-04-20 |access-date=2020-12-11 |first=Nicole |last=Mortillaro }}</ref> Such matter would violate one or more [[energy condition]]s and exhibit strange properties such as the oppositely oriented [[acceleration]] for an applied force orientation. It is used in certain [[wikt:speculative|speculative]] [[hypotheses|hypothetical]] technologies such as [[time travel]] to the past and future,<ref name="urlTime travel is possible – but only if you have an object with infinite mass">{{cite web |url=https://theconversation.com/time-travel-is-possible-but-only-if-you-have-an-object-with-infinite-mass-107063 |title=Time travel is possible – but only if you have an object with infinite mass |website=[[The Conversation (website)|The Conversation]] |date=2019-01-28 |access-date=2020-12-11 |first=Gaurav |last=Khanna }}</ref> construction of traversable [[artificial wormhole]]s, which may also allow for time travel, [[Krasnikov tube]]s, the [[Alcubierre drive]], and potentially other types of [[faster-than-light]] [[warp drive]]s. Currently, the closest known real representative of such exotic matter is a region of [[Pressure#Negative pressures|negative pressure]] density produced by the [[Casimir effect]].

==In cosmology==
In December 2018, astrophysicist [[Jamie Farnes]] from the [[University of Oxford]] proposed a "[[dark fluid]]" theory, related, in part, to notions of gravitationally repulsive negative masses, presented earlier by [[Albert Einstein]], that may help better understand, in a testable manner, the considerable amounts of unknown [[dark matter]] and [[dark energy]] in the [[cosmos]].<ref name="EA-20181205">{{cite web |author=University of Oxford |title=Bringing balance to the universe: New theory could explain missing 95 percent of the cosmos |url=https://www.eurekalert.org/pub_releases/2018-12/uoo-bbt120318.php |date=5 December 2018 |work=[[EurekAlert!]] |access-date=6 December 2018 |author-link=University of Oxford }}</ref><ref name="ARX-2018">{{cite journal |last=Farnes |first=J.S. |title=A Unifying Theory of Dark Energy and Dark Matter: Negative Masses and Matter Creation within a Modified ΛCDM Framework |journal=Astronomy & Astrophysics |volume=620 |pages=A92 |year=2018 |doi=10.1051/0004-6361/201832898 |bibcode=2018A&A...620A..92F |s2cid=53600834 | arxiv = 1712.07962 }}</ref>

==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.

==Arrow of time and energy inversion==

In [[general relativity]], the universe is described as a [[Riemannian manifold]] associated to a [[metric tensor]] solution of Einstein's field equations. In such a framework, the runaway motion forbids the existence of negative matter.<ref name="Bondi 1957" /><ref name="Bonnor 1989" />

Some [[bimetric theory|bimetric theories]] of the universe propose that two [[multiverse|parallel universes]] with an opposite arrow of time may exist instead of one, linked together by the [[Big Bang]] and interacting only through [[gravitation]].<ref name="Sakharov 1980">{{cite journal |first=A.D. |last=Sakharov |title=&nbsp; |trans-title=Cosmological model of the Universe with a time vector inversion |journal=ZhETF |volume=79 |pages=689–693 |year=1980 |language=ru}}{{full citation needed|reason=original Russian title, full name of journal, DOI or link URL|date=January 2020}} <br/>translation in {{cite journal |title=Cosmological model of the Universe with a time vector inversion |journal=JETP Lett. |volume=52 |pages=349–351 |year=1980 |language=en}}{{full citation needed|reason=translator name, full name of journal, DOI or link URL|date=January 2020}}
</ref><ref name="Barbour 2014">{{cite journal|last1=Barbour|first1=Julian|author-link=Julian Barbour|last2=Koslowski|first2=Tim|last3=Mercati|first3=Flavio|year=2014|title=Identification of a Gravitational Arrow of Time|journal=Physical Review Letters|volume=113|issue=18|pages=181101|arxiv=1409.0917|bibcode=2014PhRvL.113r1101B|doi=10.1103/PhysRevLett.113.181101|pmid=25396357|s2cid=25038135}}</ref> 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 [[Topological conjugacy|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 [[geodesic]]s and are solutions of two coupled field equations.<ref name="Hossenfelder 2008">{{cite journal|last1=Hossenfelder|first1=S.|author-link=Sabine Hossenfelder|date=15 August 2008|title=A Bi-Metric Theory with Exchange Symmetry|journal=Physical Review D|volume=78|issue=4|page=044015|arxiv=0807.2838|bibcode=2008PhRvD..78d4015H|doi=10.1103/PhysRevD.78.044015|s2cid=119152509}}</ref>

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]], [[Inflation (cosmology)|cosmic inflation]] and an [[accelerating universe]].<ref name="Hossenfelder 2008"/>

==Gravitational interaction of antimatter==
{{Main|Gravitational interaction of antimatter}}

The gravitational interaction of antimatter with [[matter]] has been observed by physicists.<ref name="AnderssonEtAl2023">{{cite journal |last1=Andersson |first1=E. K. |last2=Baker |first2=C. J. |last3=Bertsche |first3=W. |last4=Bhatt |first4=N. M. |date=September 27, 2023 |title=Observation of the effect of gravity on the motion of antimatter. |journal=Nature |volume=621 |issue=7980 |pages=716–722 |doi=10.1038/s41586-023-06527-1 |pmid=37758891 |pmc=10533407 |bibcode=2023Natur.621..716A }}</ref> As was the consensus among physicists previously, it was experimentally confirmed that [[gravity]] attracts both matter and antimatter at the same rate within experimental error.

[[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 [[Helix|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 particle]]s, now part of the [[Standard Model]], already included negative solutions.<ref name="Dirac 1928">{{cite journal |doi=10.1098/rspa.1928.0023 |title=The Quantum Theory of the Electron |journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |volume=117 |issue=778 |pages=610–624 |year=1928 |last1=Dirac |first1=P. A. M. |bibcode=1928RSPSA.117..610D |doi-access=free }}</ref> The Standard Model is a generalization of [[quantum electrodynamics]] (QED) and negative mass is already built into the theory.

[[Mike Morris (physicist)|Morris]], [[Kip Thorne|Thorne]] and [[Ulvi Yurtsever|Yurtsever]]<ref>{{cite journal |doi=10.1103/PhysRevLett.61.1446 |pmid=10038800 |title=Wormholes, Time Machines, and the Weak Energy Condition |journal=Physical Review Letters |volume=61 |issue=13 |pages=1446–1449 |year=1988 |last1=Morris |first1=Michael S. |last2=Thorne |first2=Kip S. |last3=Yurtsever |first3=Ulvi |bibcode=1988PhRvL..61.1446M |url=https://authors.library.caltech.edu/9262/1/MORprl88.pdf }}</ref> pointed out that the quantum mechanics of the [[Casimir effect]] can be used to produce a locally energy-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]]. [[John G. Cramer|Cramer]] ''et al.'' argue that such wormholes might have been created in the early universe, stabilized by negative-mass loops of [[cosmic string]].<ref>{{cite journal |arxiv=astro-ph/9409051 |doi=10.1103/PhysRevD.51.3117 |title=Natural wormholes as gravitational lenses |journal=Physical Review D |volume=51 |issue=6 |pages=3117–3120 |year=1995 |last1=Cramer |first1=John G. |last2=Forward |first2=Robert L. |last3=Morris |first3=Michael S. |last4=Visser |first4=Matt |last5=Benford |first5=Gregory |last6=Landis |first6=Geoffrey A. |bibcode=1995PhRvD..51.3117C |pmid=10018782 |s2cid=42837620 }}</ref> [[Stephen Hawking]] has argued 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;<ref name="futureofspacetime">{{Cite book | last = Hawking | first = Stephen | title = The Future of Spacetime | publisher = W. W. Norton | year = 2002 | pages = [https://archive.org/details/futureofspacetim0000unse/page/96 96] | isbn = 978-0-393-02022-9 | url-access = registration | url = https://archive.org/details/futureofspacetim0000unse/page/96 }}</ref> this implies, for example, that a finite [[Tipler cylinder]] cannot be used as a [[Time travel|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 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 [[photon]]s), 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 theory of vibrations and metamaterials ==
[[File:A mechanical model giving rise to the negative effective mass effect..jpg|alt=A core with mass m1 is connected internally through the spring with k2 to a shell with mass m1. The system is subjected to the sinusoidal force.|thumb|'''Figure 1'''. A core with mass <math>m_2</math> is connected internally through the spring with <math>k_2</math> to a shell with mass <math>m_1</math>. The system is subjected to the sinusoidal force ''F''(''t'').|left]]
The mechanical model giving rise to the negative effective mass effect is depicted in '''Figure 1'''. A core with mass <math>m_2</math> is connected internally through the spring with constant <math>k_2</math> to a shell with mass <math>m_1</math>. The system is subjected to the external sinusoidal force <math>F(t)=F_{0} \sin\omega t</math>. If we solve the equations of motion for the masses <math>m_1</math> and <math>m_2</math> and replace the entire system with a single effective mass <math>m_{\text{eff}}</math> we obtain:<ref name=":0">{{Cite journal|last1=Milton|first1=Graeme W|last2=Willis|first2=John R| date=2007-03-08| title=On modifications of Newton's second law and linear continuum elastodynamics| url=https://royalsocietypublishing.org/doi/10.1098/rspa.2006.1795| journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences| volume=463|issue=2079|pages=855–880|doi=10.1098/rspa.2006.1795|bibcode=2007RSPSA.463..855M|s2cid=122990527|url-access=subscription}}</ref><ref name=":1">{{Cite journal| last1=Chan|first1=C. T.|last2=Li|first2=Jensen|last3=Fung|first3=K. H.|date=2006-01-01|title=On extending the concept of double negativity to acoustic waves|url=https://doi.org/10.1631/jzus.2006.A0024|journal= Journal of Zhejiang University Science A|language=en|volume=7|issue=1|pages=24–28| doi=10.1631/jzus.2006.A0024|bibcode=2006JZUSA...7...24C |s2cid=120899746|issn=1862-1775|url-access=subscription}}</ref><ref name=":2">{{Cite journal|last1=Huang|first1=H. H.|last2=Sun|first2=C. T.| last3=Huang|first3=G. L.|date=2009-04-01|title=On the negative effective mass density in acoustic metamaterials| url=http://www.sciencedirect.com/science/article/pii/S0020722508002279|journal=International Journal of Engineering Science| language=en| volume=47| issue=4|pages=610–617|doi=10.1016/j.ijengsci.2008.12.007|issn=0020-7225|url-access=subscription}}</ref><ref name=":3">{{Cite journal| last1=Yao|first1=Shanshan| last2=Zhou|first2=Xiaoming|last3=Hu|first3=Gengkai|date=2008-04-14|title=Experimental study on negative effective mass in a 1D mass–spring system|journal=New Journal of Physics|volume=10|issue=4|pages=043020|doi=10.1088/1367-2630/10/4/043020|bibcode=2008NJPh...10d3020Y|issn=1367-2630| doi-access=free}}</ref>
<math display="block">m_{\text{eff}}=m_{1}+{{m_2\omega_0^2}\over{\omega_0^2-\omega^2}},</math>
where <math>\omega_0=\sqrt{k_2 \over m_2}</math>.
[[File:Equivalent mechanical scheme of electron gas in ionic lattice..jpg|alt=Free electrons gas is embedded into the ionic lattice (the left sketch). The equivalent mechanical scheme of the system (right sketch).|thumb|'''Figure 2.''' Free electrons gas <math>m_2</math> is embedded into the ionic lattice <math>m_1</math>; <math>\omega_p</math>  is the plasma frequency (the left sketch). The equivalent mechanical scheme of the system (right sketch).]]
When the frequency <math>\omega</math> approaches <math>\omega_0</math> from above the effective mass <math>m_{\text{eff}}</math> will be negative.<ref name=":0" /><ref name=":1" /><ref name=":2" /><ref name=":3" />

The negative effective mass (density) becomes also possible based on the electro-mechanical coupling exploiting [[plasma oscillation|plasma oscillations]] of a [[Fermi gas|free electron gas]] (see '''Figure 2''').<ref name=":4">{{Cite journal|last1=Bormashenko|first1=Edward|last2=Legchenkova|first2=Irina|date=April 2020|title=Negative Effective Mass in Plasmonic Systems|journal=Materials|language=en|volume=13|issue=8|pages=1890|doi=10.3390/ma13081890|pmc=7215794|pmid=32316640|bibcode=2020Mate...13.1890B|doi-access=free}}</ref><ref name=":5">{{Cite journal|last1=Bormashenko|first1=Edward|last2=Legchenkova|first2=Irina|last3=Frenkel|first3=Mark|date=August 2020|title=Negative Effective Mass in Plasmonic Systems II: Elucidating the Optical and Acoustical Branches of Vibrations and the Possibility of Anti-Resonance Propagation|journal=Materials|language=en|volume=13|issue=16|pages=3512|doi=10.3390/ma13163512|pmc=7476018|pmid=32784869|bibcode=2020Mate...13.3512B|doi-access=free}}</ref> The negative mass appears as a result of vibration of a metallic particle with a frequency of <math>\omega</math> which is close the frequency of the plasma oscillations of the electron gas <math>m_2</math> relatively to the ionic lattice <math>m_1</math>. The plasma oscillations are represented with the elastic spring <math>k_2=\omega_p^2m_2</math>, where <math>\omega_p</math> is the plasma frequency.<ref name=":4" /><ref name=":5" /> Thus, the metallic particle vibrated with the external frequency ''ω'' is described by the effective mass

:<math>m_{\text{eff}}=m_1+{{m_2\omega_p^2}\over{\omega_p^2-\omega^2}}</math>,

which is negative when the frequency ''ω'' approaches <math>\omega_p</math> from above. Metamaterials exploiting the effect of the negative mass in the vicinity of the plasma frequency were reported.<ref name=":4" /><ref name=":5" />

==See also==
{{Portal|Physics}}
* [[Dark fluid]]
* [[Imaginary mass]]
* [[Mirror matter]]
{{clear}}

==References==
{{Reflist}}

==External links==
*{{Commons category-inline|Negative mass}}

{{Use dmy dates|date=September 2019}}

{{DEFAULTSORT:Negative Mass}}
[[Category:Mass]]
[[Category:Gravity]]
[[Category:Wormhole theory]]
[[Category:Warp drive theory]]
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