負質量

負質量（negative mass）是理论物理学的一個概念，指一種具有特殊質量的物筫：其質量的正負值與正常物質相反，例如：−1 kg。照現時我們生存的世界為參照，負物質擁有的質量是負數。由於根據相對論質量和能量可互相轉換，負質量在某種意義上又可以等同負能量。這種存在會違反現實世界至少一個能量條件，令其出現會顯現某些奇怪的物理特性， 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 traversable 虫洞s and the 阿庫別瑞引擎. Initially, the closest known real representative of such exotic matter is a region of 压强 density produced by the 卡西米爾效應. In 2017, researchers at 華盛頓州立大學 realized negative effective inertial mass experimentally by 雷射冷卻 铷 atoms with 激光, although this is not negative mass in the fundamental sense.

注意這裏的「負物質」與「反物质」（antimatter）是完全不同的概念，負物質擁有負質量/負能量，而反物質具有正質量/正能量。反物質與普通物質一樣會被重力場吸引，但另一方面負物質不會受重力場吸引，反而會受其排斥。

廣義相對論所描述的引力及牛顿运动定律同樣適用於具有正能量值或負能量值的粒子，也就是說亦適用於具有負質量的物質，但並不包括其他基本相互作用力。On the other hand, the 标准模型 describes 基本粒子s and the other fundamental forces, but it does not include gravity. A unified theory that explicitly includes gravity along with the other fundamental forces may be needed for a better understanding of the concept of negative mass.

In general relativity

Negative mass is 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 應力-能量張量 is larger in magnitude than the mass density. All of these are violations of one or another variant of the positive 能量條件 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

Ever since 艾萨克·牛顿 first formulated his theory of 引力, there have been at least three conceptually distinct quantities called mass:

• inertial mass – the mass m that appears in Newtons second law of motion, F = m a
• “active” 质量 – the mass that produces a gravitational field that other masses respond to
• “passive” gravitational mass – the mass that responds to an external gravitational field by accelerating.

Einstein’s 等效原理 postulates that inertial mass must equal passive gravitational mass. The law of 动量 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 4th-prize essay for the 1951 competition, considered the possibility of negative mass and how it would behave under gravitational and other forces.

In 1957, following Luttinger's idea, 赫爾曼·邦迪 suggested in a paper in 现代物理评论 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 (non-gravitationally).

There have been several other analyses of negative mass, such as the studies conducted by R. M. Price, however 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 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 must be positive, which is of course not the case. 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. 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.

Runaway motion

Although no particles are known to have negative mass, physicists (primarily 赫爾曼·邦迪 in 1957, in 1989, then ) 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 爱因斯坦场方程 and the 等效原理:

• 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, an effect called "runaway motion" by Bonnor who disregarded its physical existence, stating:

“ 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. ” ——William B. Bonnor，in Negative mass in general relativity.

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 common-sense approach and the expected behaviour of 'normal' matter; but is completely mathematically consistent and introduces no violation of conservation of momentum or 能量守恒定律. 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:

$p_{\mathrm {sys} }=mv+(-m)v={\big (}m+(-m){\big )}v=0\times v=0.$

And equivalently for the 动能:

$E_{\mathrm {k,sys} }={\tfrac {1}{2}}mv^{2}+{\tfrac {1}{2}}(-m)v^{2}={\tfrac {1}{2}}{\big (}m+(-m){\big )}v^{2}={\tfrac {1}{2}}(0)v^{2}=0$

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 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 温度 without bound. However, the negative matter portion gains negative temperature at the same rate, again balancing out. 杰弗里·兰迪斯 pointed out other implications of Forward's analysis, including noting that although negative mass particles would repel each other gravitationally, the 库仑定律 would be attractive for like 荷 (物理) and repulsive for opposite charges.

Forward used the properties of negative-mass matter to create the concept of diametric drive, a design for 航天器推进 using negative mass that requires no energy input and no 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 E = mc²) and negative mass matter (of negative energy −E = −mc²) 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 科學界 considered negative mass could not exist in the universe.

Arrow of time and energy inversion

In 1970, demonstrated, through the complete 龐加萊群 of dynamic 群论, that reversing the energy of a particle (hence its mass, if the particle has one) is equal to reversing its 时间箭头.

The universe according to 廣義相對論 is a 黎曼流形 associated to a 度量张量 solution of Einstein’s field equations. In such a framework, the runaway motion prevents the existence of negative matter.

Some of the universe propose that two 多重宇宙論 instead of one may exist with an opposite arrow of time, linked together by the 大爆炸 and interacting only through 引力. 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 metric would appear to the matter of the other metric as having opposite mass and arrow of time (though its 原時 would remain positive). The coupled metrics have their own 测地线s and are solutions of two coupled field equations:

$R_{\mu \nu }^{(+)}-{\tfrac {1}{2}}\,R^{(+)}g_{\mu \nu }^{(+)}=\chi \left(T_{\mu \nu }^{(+)}+{\sqrt {\frac {g^{(-)}}{g^{(+)}}}}T_{\mu \nu }^{(-)}\right)$

$R_{\mu \nu }^{(-)}-{\tfrac {1}{2}}\,R^{(-)}g_{\mu \nu }^{(-)}=-\chi \left({\sqrt {\frac {g^{(+)}}{g^{(-)}}}}T_{\mu \nu }^{(+)}+T_{\mu \nu }^{(-)}\right)$

The 后牛顿力学近似方法 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 暗物质, 暗能量, 宇宙暴脹 and 宇宙加速膨脹.

In Gauss's law of gravity

In 电磁学 one can derive the energy density of a field from 高斯定律, assuming the curl of the field is 0. Performing the same calculation using 高斯重力定律 produces a negative energy density for a gravitational field.

Gravitational interaction of antimatter

主条目：Gravitational interaction of antimatter

The overwhelming consensus among physicists is that 反物质 has positive mass and should be affected by gravity just like normal matter. Direct experiments on neutral 反氫 have not been sensitive enough to detect any difference between the gravitational interaction of antimatter, compared to normal matter.

氣泡室 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 螺旋 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.

Experimentation

Physicist Peter Engels and a team of colleagues at 華盛頓州立大學 claimed to have observed negative mass behavior in rubidium atoms. On 10 April 2017 Engels team created negative "effective" mass by reducing the temperature of rubidium atoms to near 绝对零度, generating a 玻色–爱因斯坦凝聚. By using a laser-trap, the team were able to reverse the spin of some of the rubidium atoms in this state, and observed that once released from the trap, the atoms expanded and displayed properties of negative mass, in particular accelerating towards a pushing force instead of away from it. This kind of negative effective mass is analogous to the well-known apparent negative effective mass of electrons in the upper part of the dispersion bands in solids. However, neither case is negative mass for the purposes of the 應力-能量張量.

Some recent work with 超材料s suggests that some as-yet-undiscovered composite of superconductors, metamaterials and normal matter could exhibit signs of negative effective mass in much the same way as low temperature alloys melt at below the melting point of their components or some semiconductors have negative differential resistance.

In quantum mechanics

In 1928, 保罗·狄拉克's theory of 基本粒子s, now part of the 标准模型, already included negative solutions. The 标准模型 is a generalization of 量子電動力學 (QED) and negative mass is already built into the theory.

, 基普·索恩 and Yurtsever pointed out that the quantum mechanics of the 卡西米爾效應 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 虫洞. Cramer et al. argue that such wormholes might have been created in the early universe, stabilized by negative-mass loops of . 史蒂芬·霍金 has proved that is a necessary condition for the creation of a 封閉類時曲線 by manipulation of gravitational fields within a finite region of space; this proves, for example, that a finite cannot be used as a 时间旅行.

Schrödinger equation

For energy eigenstates of the 薛定谔方程, 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 量子力学, 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 光子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 克莱因-戈尔登方程 is required.

In special relativity

参见：Exotic matter和Faster-than-light

One can achieve a negative mass independent of . According to 質能等價, mass m is in proportion to energy E and the coefficient of proportionality is c². Actually, m is still equivalent to E although the coefficient is another constant such as −c². In this case, it is unnecessary to introduce a because the mass can be negative although the energy is positive. That is to say,

${\begin{aligned}E&=-mc^{2}>0\\m&=-{\frac {E}{c^{2}}}<0\end{aligned}}$

Under the circumstances,

$dE=F\,ds={\frac {dp}{dt}}\,ds={\frac {ds}{dt}}\,dp=v\,dp=v\,d(mv)$

and so,

${\begin{aligned}-c^{2}\,dm&=v\,d(mv)\\-c^{2}(2m)\,dm&=2mv\,d(mv)\\-c^{2}\,d(m^{2})&=d(m^{2}v^{2})\\-m^{2}c^{2}&=m^{2}v^{2}+C\end{aligned}}$

When v = 0,

$C=-m_{0}^{2}c^{2}$

Consequently,

${\begin{aligned}-m^{2}c^{2}&=m^{2}v^{2}-m_{0}^{2}c^{2}\\m&={\frac {m_{0}}{\sqrt {1+{\frac {v^{2}}{c^{2}}}}}}\end{aligned}}$

where m₀ < 0 is 不变质量 and invariant energy equals E₀ = −m₀c² > 0. The squared mass is still positive and the particle can be stable.

From the above relation,

$p=mv={\frac {m_{0}v}{\sqrt {1+{\frac {v^{2}}{c^{2}}}}}}<0$

The negative momentum is applied to explain 负折射, the 多普勒效应 and the 契忍可夫輻射 observed in a 负折射率超材料. The 輻射壓 in the 超材料 is also negative because the force is defined as F = dp/dt. 压强 exists in 暗能量 too. Using these above equations, the should be

$E^{2}=-p^{2}c^{2}+m_{0}^{2}c^{4}$

Substituting the 普朗克-愛因斯坦關係式 E = ħω and 路易·德布罗意's p = ħk, we obtain the following 色散关系

$\omega ^{2}=-k^{2}c^{2}+\omega _{\mathrm {p} }^{2}\,,\quad \left(E_{0}=\hbar \omega _{\mathrm {p} }=-m_{0}c^{2}>0\right)$

when the wave consists of a stream of particles whose is $E^{2}=-p^{2}c^{2}+m_{0}^{2}c^{4}$ (波粒二象性) and can be excited in a 负折射率超材料. The velocity of such a particle is equal to

$v=c{\sqrt {{\frac {E_{0}^{2}}{E^{2}}}-1}}=c{\sqrt {{\frac {\omega _{\mathrm {p} }^{2}}{\omega ^{2}}}-1}}$

and range is from zero to infinity

${\begin{aligned}{\frac {\omega _{\mathrm {p} }^{2}}{\omega ^{2}}}&<2\,,\quad {\mbox{when }}v<c\\{\frac {\omega _{\mathrm {p} }^{2}}{\omega ^{2}}}&>2\,,\quad {\mbox{when }}v>c\end{aligned}}$

Moreover, the 动能 is also negative

${\begin{aligned}E_{\mathrm {k} }&=E-E_{0}\\&=-mc^{2}-\left(-m_{0}c^{2}\right)\\&=-{\frac {m_{0}c^{2}}{\sqrt {1+{\frac {v^{2}}{c^{2}}}}}}+m_{0}c^{2}\\&=m_{0}c^{2}\left(1-{\frac {1}{\sqrt {1+{\frac {v^{2}}{c^{2}}}}}}\right)<0\,,\quad \left({\mbox{where }}m_{0}<0\right)\end{aligned}}$

In fact, negative kinetic energy exists in some models to describe 暗能量 (幻能量) whose pressure is negative. In this way, the negative mass of exotic matter is now associated with negative momentum, 压强, negative kinetic energy and 超光速 phenomena.

See also

• 阿庫別瑞引擎
• 反物质
• 暗能量
• 暗物质
• 异常物质
• 质量

外部連結

• YouTube上的Physics Lecture: Negative Mass
• YouTube上的Negative mass, Dark matter, Dark Energy, Bullet Cluster, Antigravity-1