Tensor-vector-scalar gravity, TeVeS (not to be confused with
Scalar-tensor-vector gravityScalar–tensor–vector gravity is a modified theory of gravity developed by John Moffat, a researcher at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario...
), developed by
Jacob BekensteinJacob David Bekenstein is a physicist who has contributed to the foundation of black hole thermodynamics and to other aspects of the connections between information and gravitation.-Biography:...
, is a relativistic generalization of
Mordehai MilgromMordehai Milgrom is an Israeli physicist and professor in the department of Condensed Matter Physics at the Weizmann Institute in Rehovot. He is most famous for his proposal of Modified Newtonian dynamics as an alternative to the dark matter and galaxy rotation curve problems, in 1981...
's
MOdified Newtonian DynamicsIn physics, Modified Newtonian dynamics is a theory that proposes a modification of Newton's law of gravity to explain the galaxy rotation problem. When the uniform velocity of rotation of galaxies was first observed, it was unexpected because Newtonian theory of gravity predicts that objects that...
(MOND) paradigm.
The main features of TeVeS can be summarized as follows:
- As it is derived from the action principle, TeVeS respects conservation laws;
- In the weak-field approximation of the spherically symmetric, static solution, TeVeS reproduces the MOND acceleration formula;
- TeVeS avoids the problems of earlier attempts to generalize MOND, such as superluminal propagation.
The theory is based on the following ingredients:
- A unit vector field
In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of...
;
- A dynamical scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity...
;
- A nondynamical scalar field;
- A matter Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics. In classical mechanics, the...
constructed using an alternate metricIn mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
;
- An arbitrary dimensionless function.
These components are combined into a relativistic Lagrangian density, which forms the basis of TeVeS theory.
In addition to its ability to account for the
flat rotation curvesThe rotation curve of a galaxy can be represented by a graph that plots the orbital velocity of the stars or gas in the galaxy on the y-axis against the distance from the center of the galaxy on the x-axis....
of galaxies (which is what MOND was originally designed to address), TeVeS is claimed to be consistent with a range of other phenomena, such as gravitational lensing and cosmological observations.
Tensor-vector-scalar gravity, TeVeS (not to be confused with
Scalar-tensor-vector gravityScalar–tensor–vector gravity is a modified theory of gravity developed by John Moffat, a researcher at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario...
), developed by
Jacob BekensteinJacob David Bekenstein is a physicist who has contributed to the foundation of black hole thermodynamics and to other aspects of the connections between information and gravitation.-Biography:...
, is a relativistic generalization of
Mordehai MilgromMordehai Milgrom is an Israeli physicist and professor in the department of Condensed Matter Physics at the Weizmann Institute in Rehovot. He is most famous for his proposal of Modified Newtonian dynamics as an alternative to the dark matter and galaxy rotation curve problems, in 1981...
's
MOdified Newtonian DynamicsIn physics, Modified Newtonian dynamics is a theory that proposes a modification of Newton's law of gravity to explain the galaxy rotation problem. When the uniform velocity of rotation of galaxies was first observed, it was unexpected because Newtonian theory of gravity predicts that objects that...
(MOND) paradigm.
The main features of TeVeS can be summarized as follows:
- As it is derived from the action principle, TeVeS respects conservation laws;
- In the weak-field approximation of the spherically symmetric, static solution, TeVeS reproduces the MOND acceleration formula;
- TeVeS avoids the problems of earlier attempts to generalize MOND, such as superluminal propagation.
The theory is based on the following ingredients:
- A unit vector field
In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of...
;
- A dynamical scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity...
;
- A nondynamical scalar field;
- A matter Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics. In classical mechanics, the...
constructed using an alternate metricIn mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
;
- An arbitrary dimensionless function.
These components are combined into a relativistic Lagrangian density, which forms the basis of TeVeS theory.
In addition to its ability to account for the
flat rotation curvesThe rotation curve of a galaxy can be represented by a graph that plots the orbital velocity of the stars or gas in the galaxy on the y-axis against the distance from the center of the galaxy on the x-axis....
of galaxies (which is what MOND was originally designed to address), TeVeS is claimed to be consistent with a range of other phenomena, such as gravitational lensing and cosmological observations. However, Seifert shows that with Bekenstein's proposed parameters, a TeVeS star is highly unstable, on the scale of approximately 10
6 seconds (two weeks). The ability of the theory to simultaneously account for galactic dynamics and lensing is also challenged. A possible resolution may be in the form of massive (around 2eV)
neutrinosNeutrinos are elementary particles that often travel close to the speed of light, lack an electric charge, are able to pass through ordinary matter almost undisturbed and are thus extremely difficult to detect. Neutrinos have a minuscule, but nonzero mass...
.
Details
MOND is a phenomenological modification of the Newtonian acceleration law. In Newtonian gravity theory, the gravitational acceleration in the spherically symmetric, static field of a point mass at distance from the source can be written as
where is Newton's constant of gravitation. The corresponding force acting on a test mass is
To account for the anomalous rotation curves of spiral galaxies, Milgrom proposed a modification of this force law in the form
where is an arbitrary function subject to the following conditions:
In this form, MOND is not a complete theory: for instance, it violates the law of momentum conservation.
However, such conservation laws are automatically satisfied for physical theories that are derived using an action principle. This led Bekenstein to a first, nonrelativistic generalization of MOND. This theory, called AQUAL (for AQUAdratic Lagrangian) is based on the Lagrangian
where is the Newtonian gravitational potential, is the mass density, and is a dimensionless function.
In the case of a spherically symmetric, static gravitational field, this Lagrangian reproduces the MOND acceleration law after the substitutions and are made.
Bekenstein further found that AQUAL can be obtained as the nonrelativistic limit of a relativistic field theory. This theory is written in terms of a Lagrangian that contains, in addition to the Einstein-Hilbert action for the metric field , terms pertaining to a unit vector field and two scalar fields and , of which only is dynamical. The TeVeS action, therefore, can be written as
The terms in this action include the Einstein-Hilbert Lagrangian (using a metric signature and setting the speed of light, ):
where is the Ricci scalar and is the determinant of the metric tensor.
The scalar field Lagrangian is
with , is a constant length, and an unspecified dimensionless function; while the vector field Lagrangian is
where , while is a dimensionless parameter.
In particular, incorporates a Lagrange multiplier term that guarantees that the vector field remains a unit vector field.
The function in TeVeS is unspecified.
TeVeS also introduces a "physical metric" in the form
The action of ordinary matter is defined using the physical metric:
where covariant derivatives with respect to are denoted by .
TeVeS solves problems associated with earlier attempts to generalize MOND, such as superluminal propagation. In his paper, Bekenstein also investigated the consequences of TeVeS in relation to gravitational lensing and cosmology.
See also
- Modified Newtonian Dynamics
In physics, Modified Newtonian dynamics is a theory that proposes a modification of Newton's law of gravity to explain the galaxy rotation problem. When the uniform velocity of rotation of galaxies was first observed, it was unexpected because Newtonian theory of gravity predicts that objects that...
- Scalar-tensor-vector gravity
Scalar–tensor–vector gravity is a modified theory of gravity developed by John Moffat, a researcher at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario...
- General theory of relativity
- Law of universal gravitation
- Pioneer anomaly
The Pioneer anomaly or Pioneer effect is the observed deviation from predicted trajectories and velocities of various unmanned spacecraft visiting the outer solar system, most notably Pioneer 10 and Pioneer 11....
- Nonsymmetric Gravitational Theory
In theoretical physics, the nonsymmetric gravitational theory of John Moffat is a classical theory of gravitation which tries to explain the mystery of dark matter....
- Dark matter
In astronomy and cosmology, dark matter is hypothetical matter that is undetectable by its emitted radiation, but whose presence can be inferred from gravitational effects on visible matter...
- Tensor
Tensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another...
- Vector
- Scalar
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . -Physical quantity:...
- scalar fieldIn mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity...
Further reading
- Dark Matter Observed (SLAC today)
- Einstein's Theory 'Improved'? (PPARC)
- J.D. Bekenstein and R.H. Sanders, "A Primer to Relativistic MOND Theory", arXiv:astro-ph/0509519
- H.S. Zhao and B. Famaey, "Refining MOND interpolating function and TeVeS Lagrangian", Astrophys. J. 638, L9-L12 (2006), arXiv:astro-ph/0512425
- Douglas Clowe, et al. "A direct empirical proof of the existence of dark matter", Astrophys. J. 648, L109-L113 (2006), arXiv:astro-ph/0608407