Negative mass

In December 2018, astrophysicist Jamie Farnes from the University of Oxford proposed a "dark fluid" theory,[4][5][6][7][8][9][10][11][12][13][14] 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.[15][16]

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 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.

Runaway motion

Although no particles are known to have negative mass, physicists (primarily Hermann Bondi in 1957,[18] William B. Bonnor in 1964 and 1989,[24][25] then Robert L. Forward[26]) 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 Newtonian approximation of the Einstein field equations. The interaction laws are then:

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,[18] 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.[25]

Such a couple of objects would accelerate without limit (except 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 as a circular motion:

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.

— Thomas Gold, in Negative mass in general relativity.[27]

But Forward showed that the phenomenon is mathematically consistent and introduces no violation of conservation laws.[26] 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:

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

And equivalently for the kinetic energy:

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

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 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,[28] including noting that although negative mass particles would repel each other gravitationally, the electrostatic force would be attractive for like 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 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 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, such interactions would leave a surplus of momentum.

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.

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.[36]

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.

Physicist Peter Engels and a team of colleagues at Washington State University reported the observation of 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 absolute zero, generating a Bose–Einstein condensate. 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.[37][38] 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.[39] However, neither case is negative mass for the purposes of the stress–energy tensor.

Some recent work with metamaterials 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.[40][41]

In 1928, Paul Dirac's theory of elementary particles, now part of the Standard Model, already included negative solutions.[42] The Standard Model is a generalization of quantum electrodynamics (QED) and negative mass is already built into the theory.

Morris, Thorne and Yurtsever[43] pointed out that the quantum mechanics of the Casimir effect 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 wormhole. Cramer et al. argue that such wormholes might have been created in the early universe, stabilized by negative-mass loops of cosmic string.[44] 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;[45] this implies, for example, that a finite Tipler cylinder cannot be used as a time machine.

One can achieve a negative mass independent of negative energy. According to mass–energy equivalence, 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[46] such as −c².[47] In this case, it is unnecessary to introduce a negative energy because the mass can be negative although the energy is positive. That is to say,

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

Under the circumstances,

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

and so,

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

When v = 0,

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

Consequently,

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

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

From the above relation,

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

The negative momentum is applied to explain negative refraction, the inverse Doppler effect and the reverse Cherenkov effect observed in a negative index metamaterial. The radiation pressure in the metamaterial is also negative[48] because the force is defined as F = dp/dt. Negative pressure exists in dark energy too. Using these above equations, the energy–momentum relation should be

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

Substituting the Planck–Einstein relation E = ħω and de Broglie's p = ħk, we obtain the following dispersion relation

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

when the wave consists of a stream of particles whose energy–momentum relation is ${\displaystyle E^{2}=-p^{2}c^{2}+m_{0}^{2}c^{4}}$ (wave–particle duality) and can be excited in a negative index metamaterial. The velocity of such a particle is equal to

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

and range is from zero to infinity

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

Moreover, the kinetic energy is also negative

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

In fact, negative kinetic energy exists in some models[49] to describe dark energy (phantom energy) whose pressure is negative. In this way, the negative mass of exotic matter is now associated with negative momentum, negative pressure, negative kinetic energy and faster-than-light phenomena.

Figure 1. A core with mass ${\displaystyle m_{2}}$ is connected internally through the spring with ${\displaystyle k_{2}}$ to a shell with mass ${\displaystyle m_{1}}$. The system is subjected to the sinusoidal force F(t).

The mechanical model giving rise to the negative effective mass effect is depicted in Figure 1. A core with mass ${\displaystyle m_{2}}$is connected internally through the spring with constant ${\displaystyle k_{2}}$ to a shell with mass ${\displaystyle m_{1}}$. The system is subjected to the external sinusoidal force ${\displaystyle F(t)={\widehat {F}}sin\omega t}$. If we solve the equations of motion for the masses ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ and replace the entire system with a single effective mass ${\displaystyle m_{eff}}$ we obtain:[50][51][52][53]

${\displaystyle m_{eff}=m_{1}+{(m_{2}\omega _{0}^{2}) \over (\omega _{0}^{2}-\omega ^{2})}}$,

where ${\displaystyle \omega _{0}={\sqrt {k_{2} \over m_{2}}}}$.

Figure 2. Free electrons gas ${\displaystyle m_{2}}$ is embedded into the ionic lattice ${\displaystyle m_{1}}$; ${\displaystyle \omega _{p}}$  is the plasma frequency (the left sketch). The equivalent mechanical scheme of the system (right sketch).

When the frequency ${\displaystyle \omega }$ approaches ${\displaystyle \omega _{0}}$ from above the effective mass ${\displaystyle m_{eff}}$ will be negative.[50][51][52][53]

The negative effective mass (density) becomes also possible based on the electro-mechanical coupling exploiting plasma oscillations of a free electron gas (see Figure 2).[54][55] The negative mass appears as a result of vibration of a metallic particle with a frequency of ${\displaystyle \omega }$ which is close the frequency of the plasma oscillations of the electron gas ${\displaystyle m_{2}}$ relatively to the ionic lattice ${\displaystyle m_{1}}$. The plasma oscillations are represented with the elastic spring ${\displaystyle k_{2}=\omega _{p}^{2}m_{2}}$, where ${\displaystyle \omega _{p}}$ is the plasma frequency.[54][55] Thus, the metallic particle vibrated with the external frequency ω is described by the effective mass

${\displaystyle m_{eff}=m_{1}+{(m_{2}\omega _{p}^{2}) \over (\omega _{p}^{2}-\omega ^{2})}}$,

which is negative when the frequency  approaches  from above. Metamaterials exploiting the effect of the negative mass in the vicinity of the plasma frequency were reported.[54][55]