Massive gravity
Encyclopedia
In theoretical physics
, massive gravity is a particular generalization of general relativity
studied by Hendrik van Dam, Martinus J. G. Veltman
(1)
, and Vladimir E. Zakharov
.(2)
One assumes that physics takes place in Minkowski space
and gravity is caused by a massive spin
-2 field h that couples to matter like the graviton
, namely by the term
where T is the stress-energy tensor
.
The adjective "massive" means that there are also mass terms in the Lagrangian proportional to
.
At distances shorter than the corresponding Compton wavelength
, one recovers the Newton's gravitational law. In the same limit, however, the bending of light is only three quarters of the result Albert Einstein
obtained in general relativity. This is known as the vDVZ discontinuity.
More technically stated by E. Babichev, et al (3), "The (quadratic) Pauli-Fierz theory is known to suffer from the van Dam-Veltman-Zakharov (vDVZ) discontinuity,i.e. the fact that when one lets the mass m of the graviton vanish, one does not recover predictions of General
Relativity. E.g., if one adjusts the parameters (namely the Planck scale) such that the Newton constant agrees with the one measured by some type of Cavendish experiment, then the light bending as predicted by Pauli-Fierz theory (and for a vanishingly small graviton mass) will be 3/4 of the one obtained by linearizing GR.
The fact that it is smaller is easy to understand: the essential difference between Pauli-Fierz theory and linearized GR comes from an extra propagating scalar mode present in the massive theory. This mode exerts an extra attraction in the massive case compared to the
massless case. Hence, if one wants measurements of the force exerted between non relativistic masses to agree, the coupling constant of the massive theory should be smaller than that of the massless theory. But light bending is blind to the scalar sector - because the light energy momentum tensor is traceless. Hence, provided the two theories agree on the force between non relativistic probes, the massive theory would predict a smaller light bending than the massless one.
It has been recently argued that although this discontinuity survives in many particular realizations of the situation, it may disappear if the theory becomes fully covariant. More precisely, this discontinuity states that in the limit as the mass goes to zero, we get a spin-2 graviton and a scalar boson
which couples to the stress-energy tensor. This scalar boson has not been observed experimentally.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, massive gravity is a particular generalization of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
studied by Hendrik van Dam, Martinus J. G. Veltman
Martinus J. G. Veltman
Martinus Justinus Godefriedus Veltman is a Dutch theoretical physicist. He shared the 1999 Nobel Prize in physics with his former student Gerardus 't Hooft for their work on particle theory.-Biography:...
(1)
, and Vladimir E. Zakharov
Vladimir E. Zakharov
Vladimir Evgen'evich Zakharov is a Soviet and Russian mathematician and physicist. He is currently Regents' Professor of mathematics at The University of Arizona and director of the Mathematical Physics Sector at the Lebedev Physical Institute...
.(2)
One assumes that physics takes place in Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
and gravity is caused by a massive spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
-2 field h that couples to matter like the graviton
Graviton
In physics, the graviton is a hypothetical elementary particle that mediates the force of gravitation in the framework of quantum field theory. If it exists, the graviton must be massless and must have a spin of 2...
, namely by the term
where T is the stress-energy tensor
Stress-energy tensor
The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
.
The adjective "massive" means that there are also mass terms in the Lagrangian proportional to
.At distances shorter than the corresponding Compton wavelength
Compton wavelength
The Compton wavelength is a quantum mechanical property of a particle. It was introduced by Arthur Compton in his explanation of the scattering of photons by electrons...
, one recovers the Newton's gravitational law. In the same limit, however, the bending of light is only three quarters of the result Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
obtained in general relativity. This is known as the vDVZ discontinuity.
More technically stated by E. Babichev, et al (3), "The (quadratic) Pauli-Fierz theory is known to suffer from the van Dam-Veltman-Zakharov (vDVZ) discontinuity,i.e. the fact that when one lets the mass m of the graviton vanish, one does not recover predictions of General
Relativity. E.g., if one adjusts the parameters (namely the Planck scale) such that the Newton constant agrees with the one measured by some type of Cavendish experiment, then the light bending as predicted by Pauli-Fierz theory (and for a vanishingly small graviton mass) will be 3/4 of the one obtained by linearizing GR.
The fact that it is smaller is easy to understand: the essential difference between Pauli-Fierz theory and linearized GR comes from an extra propagating scalar mode present in the massive theory. This mode exerts an extra attraction in the massive case compared to the
massless case. Hence, if one wants measurements of the force exerted between non relativistic masses to agree, the coupling constant of the massive theory should be smaller than that of the massless theory. But light bending is blind to the scalar sector - because the light energy momentum tensor is traceless. Hence, provided the two theories agree on the force between non relativistic probes, the massive theory would predict a smaller light bending than the massless one.
It has been recently argued that although this discontinuity survives in many particular realizations of the situation, it may disappear if the theory becomes fully covariant. More precisely, this discontinuity states that in the limit as the mass goes to zero, we get a spin-2 graviton and a scalar boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....
which couples to the stress-energy tensor. This scalar boson has not been observed experimentally.