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In
theoretical physics,
negative mass is matter whose
mass is of opposite sign to the mass of normal matter, e.g. −1 kg.
[1][2] Such matter would violate one or more
energy conditions
and show some strange properties, 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
wormholes and the
Alcubierre drive. Originally the closest known real representative of such exotic matter is a region of
negative pressure density produced by the
Casimir effect. In 2017, researchers at
Washington State University realized negative effective inertial mass experimentally by
cooling rubidium atoms with
lasers, although this is not negative mass in the fundamental sense.
[3]
General relativity describes
gravity and the
laws of motion for both positive and
negative energy particles, hence negative mass, but does not include the other
fundamental forces. On the other hand, the
Standard Model describes
elementary particles
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
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
Ever since
Newton first formulated his theory of
gravity, 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” gravitational mass – 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
equivalence principle postulates that inertial mass must equal passive gravitational mass. The law of
conservation of momentum
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
Gravity Research Foundation competition,
Joaquin Mazdak Luttinger considered the possibility of negative mass and how it would behave under gravitational and other forces.
[4]
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.
[5]
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,
[6]
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
ADM mass must be positive, which is of course not the case.
[7][8]
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.
[9] 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.
[10]
Runaway motion
Although no particles are known to have negative mass, physicists (primarily
Hermann Bondi in 1957,
[5] William B. Bonnor in 1989,
[11] then
Robert L. Forward[12])
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
Einstein field equations and the
Equivalence principle:
- 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,
[5] 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.[11] |
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
energy.
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:
And equivalently for the
kinetic energy:
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,
[13] 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, all such
interactions would leave a surplus of momentum, which is classically
forbidden. So once this runaway phenomenon has been revealed, the
scientific community considered negative mass could not exist in the universe.
Arrow of time and energy inversion
In 1970,
Jean-Marie Souriau demonstrated, through the complete
Poincaré group of dynamic
group theory, that reversing the energy of a particle (hence its mass, if the particle has one) is equal to reversing its
arrow of time.
[14][15]
The universe according to
general relativity is a
Riemannian manifold associated to a
metric tensor solution of Einstein’s field equations. In such a framework, the runaway motion prevents the existence of negative matter.
[5][11]
Some
bimetric theories of the universe propose that two
parallel universes instead of one may exist with an opposite arrow of time, linked together by the
Big Bang and interacting only through
gravitation.
[16][17][18]
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
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
geodesics and are solutions of two coupled field equations:
[19][20]
The
Newtonian approximation 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
dark matter,
dark energy,
cosmic inflation and
accelerating universe.
[19][20]
In Gauss's law of gravity
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.
Gravitational interaction of antimatter
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.
[21]
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.
Experimentation
Physicist Peter Engels and a team of colleagues at
Washington State University
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
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.
[22][23]
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.
[24] 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.
[25] [26]
In quantum mechanics
In 1928,
Paul Dirac's theory of
elementary particles, now part of the
Standard Model, already included negative solutions.
[27] The
Standard Model is a generalization of
quantum electrodynamics (QED) and negative mass is already built into the theory.
Morris,
Thorne and Yurtsever
[28] 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.
[29] Stephen Hawking has proved 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;
[30] this proves, for example, that a finite
Tipler cylinder cannot be used as a
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
photons),
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 special relativity
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
[31] such as
−c2.
[32] 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,
Under the circumstances,
and so,
When
v = 0,
Consequently,
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,
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
[33] because the force is defined as
F = dp/dt. Interestingly,
negative pressure exists in
dark energy too. Using these above equations, the
energy-momentum relation should be
Substituting the
Planck–Einstein relation E = ħω and
de Broglie's
p = ħk, we obtain the following
dispersion relation
when the wave consists of a stream of particles whose
energy-momentum relation is
(
wave–particle duality) and can be excited in a
negative index metamaterial. The velocity of such a particle is equal to
and range is from zero to infinity
Moreover, the
kinetic energy is also negative
In fact,
negative kinetic energy exists in some models
[34] 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.