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Tensor-vector-scalar gravity (TeVeS)[1] (not to be confused with Scalar-tensor-vector 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 106 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].

Details

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

$a = -\frac{GM}{r^2},$

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 = μ(a / a0)ma,

where μ(x) is an arbitrary function subject to the following conditions:

$\mu(x)=1~\mathrm{if}~|x|\gg 1,$

$\mu(x)=x~\mathrm{if}~|x|\ll 1.$

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

${\mathcal L}=-\frac{a_0^2}{8\pi G}f\left(\frac{|\nabla\Phi|^2}{a_0^2}\right)-\rho\Phi,$

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 $a=-\nabla\Phi$ and $\mu(\sqrt{y})=df(y)/dy$ 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 Einstein-Hilbert action for the metric field gμν, terms pertaining to a unit vector field uα and two scalar fields σ and φ, of which only φ is dynamical. The TeVeS action, therefore, can be written as

$S_\mathrm{TeVeS}=\int\left({\mathcal L}_g+{\mathcal L}_s+{\mathcal L}_v\right)d^4x.$

The terms in this action include the Einstein-Hilbert Lagrangian (using a metric signature [ + , − , − , − ] and setting the speed of light, c = 1):

${\mathcal L}_g=-\frac{1}{16\pi G}R\sqrt{-g},$

where R is the Ricci scalar and g is the determinant of the metric tensor.

The scalar field Lagrangian is

${\mathcal L}_s=-\frac{1}{2}\left[\sigma^2h^{\alpha\beta}\partial_\alpha\phi\partial_\beta\phi+\frac{1}{2}\frac{G}{l^2}\sigma^4F(kG\sigma^2)\right]\sqrt{-g},$

with hαβ = gαβuαuβ, l is a constant length, and F an unspecified dimensionless function; while the vector field Lagrangian is

${\mathcal L}_v=-\frac{K}{32\pi G}\left[g^{\alpha\beta}g^{\mu\nu}(B_{\alpha\mu}B_{\beta\nu})+2\frac{\lambda}{K}(g^{\mu\nu}u_\mu u_\nu-1)\right]\sqrt{-g}$

where $B_{\alpha\beta}=\partial_\alpha u_\beta-\partial_\beta u_\alpha$, while K is a dimensionless parameter.

In particular, ${\mathcal L}_v$ 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

${\hat g}^{\mu\nu}=e^{2\phi}g^{\mu\nu}-2u^\alpha u^\beta\sinh(2\phi).$

The action of ordinary matter is defined using the physical metric:

$S_m=\int{\mathcal L}({\hat g}_{\mu\nu},f^\alpha,f^\alpha_{|\mu},...)\sqrt{-{\hat g}}d^4x,$

where covariant derivatives with respect to ${\hat g}_{\mu\nu}$ 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.

References

1. ^ a b Jacob D. Bekenstein (2004), "Relativistic gravitation theory for the modified Newtonian dynamics paradigm", Phys. Rev. D 70: 083509, doi:10.1103/PhysRevD.70.083509
2. ^ a b M. Milgrom (1983), "A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis", Astrophys. J. 270: 365–370, doi:10.1086/161130
3. ^ Michael D. Seifert (2007), "Stability of spherically symmetric solutions in modified theories of gravity", Phys. Rev. D 76: 064002, doi:10.1103/PhysRevD.76.064002
4. ^ Mavromatos, Nick E.; Sakellariadou, Mairi; Yusaf, Muhammad Furqaan, "Can TeVeS avoid Dark Matter on galactic scales?", Phys. Rev. D 79: 081301, doi:10.1103/PhysRevD.79.081301
5. ^ Angus, G. W.; Shan, H. Y.; Zhao, H. S.; Famaey, B. (2007), "On the Proof of Dark Matter, the Law of Gravity, and the Mass of Neutrinos"], Astrophys. J. 654: L13-L16, doi:10.1086/510738

Tensor-vector-scalar gravity, TeVeS[1] (not to be confused with Scalar-tensor-vector 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:

• 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:

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].

Details

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

$a = -\frac\left\{GM\right\}\left\{r^2\right\},$

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

$F=\mu\left(a/a_0\right)ma,$

where $\mu\left(x\right)$ is an arbitrary function subject to the following conditions:

$\mu\left(x\right)=1~\mathrm\left\{if\right\}~|x|\gg 1,$

$\mu\left(x\right)=x~\mathrm\left\{if\right\}~|x|\ll 1.$

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

$\left\{\mathcal L\right\}=-\frac\left\{a_0^2\right\}\left\{8\pi G\right\}f\left\left(\frac\left\{|\nabla\Phi|^2\right\}\left\{a_0^2\right\}\right\right)-\rho\Phi,$

where $\Phi$ is the Newtonian gravitational potential, $\rho$ is the mass density, and $f\left(y\right)$ is a dimensionless function.

In the case of a spherically symmetric, static gravitational field, this Lagrangian reproduces the MOND acceleration law after the substitutions $a=-\nabla\Phi$ and $\mu\left(\sqrt\left\{y\right\}\right)=df\left(y\right)/dy$ 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 Einstein-Hilbert action for the metric field $g_\left\{\mu\nu\right\}$, terms pertaining to a unit vector field $u^\alpha$ and two scalar fields $\sigma$ and $\phi$, of which only $\phi$ is dynamical. The TeVeS action, therefore, can be written as

$S_\mathrm\left\{TeVeS\right\}=\int\left\left(\left\{\mathcal L\right\}_g+\left\{\mathcal L\right\}_s+\left\{\mathcal L\right\}_v\right\right)d^4x.$

The terms in this action include the Einstein-Hilbert Lagrangian (using a metric signature $\left[+,-,-,-\right]$ and setting the speed of light, $c=1$):

$\left\{\mathcal L\right\}_g=-\frac\left\{1\right\}\left\{16\pi G\right\}R\sqrt\left\{-g\right\},$

where $R$ is the Ricci scalar and $g$ is the determinant of the metric tensor.

The scalar field Lagrangian is

$\left\{\mathcal L\right\}_s=-\frac\left\{1\right\}\left\{2\right\}\left\left[\sigma^2h^\left\{\alpha\beta\right\}\partial_\alpha\phi\partial_\beta\phi+\frac\left\{1\right\}\left\{2\right\}\frac\left\{G\right\}\left\{l^2\right\}\sigma^4F\left(kG\sigma^2\right)\right\right]\sqrt\left\{-g\right\},$

with $h^\left\{\alpha\beta\right\}=g^\left\{\alpha\beta\right\}-u^\alpha u^\beta$, $l$ is a constant length, and $F$ an unspecified dimensionless function; while the vector field Lagrangian is

$\left\{\mathcal L\right\}_v=-\frac\left\{K\right\}\left\{32\pi G\right\}\left\left[g^\left\{\alpha\beta\right\}g^\left\{\mu\nu\right\}\left(B_\left\{\alpha\mu\right\}B_\left\{\beta\nu\right\}\right)+2\frac\left\{\lambda\right\}\left\{K\right\}\left(g^\left\{\mu\nu\right\}u_\mu u_\nu-1\right)\right\right]\sqrt\left\{-g\right\}$

where $B_\left\{\alpha\beta\right\}=\partial_\alpha u_\beta-\partial_\beta u_\alpha$, while $K$ is a dimensionless parameter.

In particular, $\left\{\mathcal L\right\}_v$ 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

$\left\{\hat g\right\}^\left\{\mu\nu\right\}=e^\left\{2\phi\right\}g^\left\{\mu\nu\right\}-2u^\alpha u^\beta\sinh\left(2\phi\right).$

The action of ordinary matter is defined using the physical metric:

$S_m=\int\left\{\mathcal L\right\}\left(\left\{\hat g\right\}_\left\{\mu\nu\right\},f^\alpha,f^\alpha_\left\{|\mu\right\},...\right)\sqrt\left\{-\left\{\hat g\right\}\right\}d^4x,$

where covariant derivatives with respect to $\left\{\hat g\right\}_\left\{\mu\nu\right\}$ 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.

References

1. 1.0 1.1 Jacob D. Bekenstein (2004), "Relativistic gravitation theory for the modified Newtonian dynamics paradigm", Phys. Rev. D 70: 083509, doi:10.1103/PhysRevD.70.083509
2. 2.0 2.1 M. Milgrom (1983), "", Astrophys. J. 270: 365-370, doi:[http://dx.doi.org/10.1086%2F161130 10.1086/161130
3. Michael D. Seifert (2007), "Stability of spherically symmetric solutions in modified theories of gravity", Phys. Rev. D 76: 064002, doi:10.1103/PhysRevD.76.064002
4. Mavromatos, Nick E.; Sakellariadou, Mairi; Yusaf, Muhammad Furqaan, "Can TeVeS avoid Dark Matter on galactic scales?", Phys. Rev. D 79: 081301, doi:10.1103/PhysRevD.79.081301
5. Angus, G. W.; Shan, H. Y.; Zhao, H. S.; Famaey, B. (2007), "On the Proof of Dark Matter, the Law of Gravity, and the Mass of Neutrinos", Astrophys. J. 654: L13-L16, doi:10.1086/510738