Tensorvectorscalar gravity (TeVeS)^{[1]} (not to be confused with Scalartensorvector gravity), developed by Jacob Bekenstein, is a relativistic generalization of Mordehai Milgrom's MOdified Newtonian Dynamics (MOND) paradigm^{[2]}.
The main features of TeVeS can be summarized as follows:
The theory is based on the following ingredients:
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 curves 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^{[3]} 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^{[4]}. A possible resolution may be in the form of massive (around 2eV) neutrinos^{[5]}.
Contents 
MOND^{[2]} 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 M at distance r from the source can be written as
where G is Newton's constant of gravitation. The corresponding force acting on a test mass m is
F = ma.
To account for the anomalous rotation curves of spiral galaxies, Milgrom proposed a modification of this force law in the form
F = μ(a / a_{0})ma,
where μ(x) 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^{[1]} to a first, nonrelativistic generalization of MOND. This theory, called AQUAL (for AQUAdratic Lagrangian) is based on the Lagrangian
where Φ is the Newtonian gravitational potential, ρ is the mass density, and f(y) 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 EinsteinHilbert action for the metric field g_{μν}, terms pertaining to a unit vector field u^{α} and two scalar fields σ and φ, of which only φ is dynamical. The TeVeS action, therefore, can be written as
The terms in this action include the EinsteinHilbert Lagrangian (using a metric signature [ + , − , − , − ] and setting the speed of light, c = 1):
where R is the Ricci scalar and g is the determinant of the metric tensor.
The scalar field Lagrangian is
with h^{αβ} = g^{αβ} − u^{α}u^{β}, l is a constant length, and F an unspecified dimensionless function; while the vector field Lagrangian is
where , while K is a dimensionless parameter.
In particular, incorporates a Lagrange multiplier term that guarantees that the vector field remains a unit vector field.
The function F 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.

Tensorvectorscalar gravity, TeVeS^{[1]} (not to be confused with Scalartensorvector gravity), developed by Jacob Bekenstein, is a relativistic generalization of Mordechai Milgrom's MOdified Newtonian Dynamics (MOND) paradigm^{[2]}.
The main features of TeVeS can be summarized as follows:
The theory is based on the following ingredients:
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 curves 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^{[3]} 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^{[4]}. A possible resolution may be in the form of massive (around 2eV) neutrinos^{[5]}.
Contents 
MOND^{[2]} 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 $M$ at distance $r$ from the source can be written as
$a\; =\; \backslash frac\{GM\}\{r^2\},$
where $G$ is Newton's constant of gravitation. The corresponding force acting on a test mass $m$ is
$F=ma.$
To account for the anomalous rotation curves of spiral galaxies, Milgrom proposed a modification of this force law in the form
$F=\backslash mu(a/a\_0)ma,$
where $\backslash mu(x)$ is an arbitrary function subject to the following conditions:
$\backslash mu(x)=1~\backslash mathrm\{if\}~x\backslash gg\; 1,$
$\backslash mu(x)=x~\backslash mathrm\{if\}~x\backslash ll\; 1.$
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^{[1]} to a first, nonrelativistic generalization of MOND. This theory, called AQUAL (for AQUAdratic Lagrangian) is based on the Lagrangian
$\{\backslash mathcal\; L\}=\backslash frac\{a\_0^2\}\{8\backslash pi\; G\}f\backslash left(\backslash frac\{\backslash nabla\backslash Phi^2\}\{a\_0^2\}\backslash right)\backslash rho\backslash Phi,$
where $\backslash Phi$ is the Newtonian gravitational potential, $\backslash rho$ is the mass density, and $f(y)$ is a dimensionless function.
In the case of a spherically symmetric, static gravitational field, this Lagrangian reproduces the MOND acceleration law after the substitutions $a=\backslash nabla\backslash Phi$ and $\backslash mu(\backslash sqrt\{y\})=df(y)/dy$ 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 EinsteinHilbert action for the metric field $g\_\{\backslash mu\backslash nu\}$, terms pertaining to a unit vector field $u^\backslash alpha$ and two scalar fields $\backslash sigma$ and $\backslash phi$, of which only $\backslash phi$ is dynamical. The TeVeS action, therefore, can be written as
$S\_\backslash mathrm\{TeVeS\}=\backslash int\backslash left(\{\backslash mathcal\; L\}\_g+\{\backslash mathcal\; L\}\_s+\{\backslash mathcal\; L\}\_v\backslash right)d^4x.$
The terms in this action include the EinsteinHilbert Lagrangian (using a metric signature $[+,,,]$ and setting the speed of light, $c=1$):
$\{\backslash mathcal\; L\}\_g=\backslash frac\{1\}\{16\backslash pi\; G\}R\backslash sqrt\{g\},$
where $R$ is the Ricci scalar and $g$ is the determinant of the metric tensor.
The scalar field Lagrangian is
$\{\backslash mathcal\; L\}\_s=\backslash frac\{1\}\{2\}\backslash left[\backslash sigma^2h^\{\backslash alpha\backslash beta\}\backslash partial\_\backslash alpha\backslash phi\backslash partial\_\backslash beta\backslash phi+\backslash frac\{1\}\{2\}\backslash frac\{G\}\{l^2\}\backslash sigma^4F(kG\backslash sigma^2)\backslash right]\backslash sqrt\{g\},$
with $h^\{\backslash alpha\backslash beta\}=g^\{\backslash alpha\backslash beta\}u^\backslash alpha\; u^\backslash beta$, $l$ is a constant length, and $F$ an unspecified dimensionless function; while the vector field Lagrangian is
$\{\backslash mathcal\; L\}\_v=\backslash frac\{K\}\{32\backslash pi\; G\}\backslash left[g^\{\backslash alpha\backslash beta\}g^\{\backslash mu\backslash nu\}(B\_\{\backslash alpha\backslash mu\}B\_\{\backslash beta\backslash nu\})+2\backslash frac\{\backslash lambda\}\{K\}(g^\{\backslash mu\backslash nu\}u\_\backslash mu\; u\_\backslash nu1)\backslash right]\backslash sqrt\{g\}$
where $B\_\{\backslash alpha\backslash beta\}=\backslash partial\_\backslash alpha\; u\_\backslash beta\backslash partial\_\backslash beta\; u\_\backslash alpha$, while $K$ is a dimensionless parameter.
In particular, $\{\backslash mathcal\; L\}\_v$ incorporates a Lagrange multiplier term that guarantees that the vector field remains a unit vector field.
The function $F$ in TeVeS is unspecified.
TeVeS also introduces a "physical metric" in the form
$\{\backslash hat\; g\}^\{\backslash mu\backslash nu\}=e^\{2\backslash phi\}g^\{\backslash mu\backslash nu\}2u^\backslash alpha\; u^\backslash beta\backslash sinh(2\backslash phi).$
The action of ordinary matter is defined using the physical metric:
$S\_m=\backslash int\{\backslash mathcal\; L\}(\{\backslash hat\; g\}\_\{\backslash mu\backslash nu\},f^\backslash alpha,f^\backslash alpha\_\{\backslash mu\},...)\backslash sqrt\{\{\backslash hat\; g\}\}d^4x,$
where covariant derivatives with respect to $\{\backslash hat\; g\}\_\{\backslash mu\backslash nu\}$ 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.

