Hexagonal lattice

Tensor–vector–scalar gravity (TeVeS), developed by Jacob Bekenstein, is a relativistic generalization of Mordehai Milgrom's Modified Newtonian dynamics (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;
• As it is a relativistic theory it can accommodate gravitational lensing.

The theory is based on the following ingredients:

• A unit vector field;
• A dynamical scalar field;
• A nondynamical scalar field;
• A matter Lagrangian constructed using an alternate metric;
• An arbitrary dimensionless function.

These components are combined into a relativistic Lagrangian density, which forms the basis of TeVeS theory.

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 $M$ 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

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

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

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

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

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 to a first, nonrelativistic generalization of MoND. This theory, called AQUAL (for A QUAdratic Lagrangian) is based on the Lagrangian

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

where $\mathrm{\Phi }$ is the Newtonian gravitational potential, $\rho$ 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 $a=-\mathrm{\nabla }\mathrm{\Phi }$ 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 $g_{\mu \nu }$, terms pertaining to a unit vector field $u^{\alpha }$ and two scalar fields $\sigma$ and $\varphi$, of which only $\varphi$ is dynamical. The TeVeS action, therefore, can be written as

$S_{\mathrm{T}\mathrm{e}\mathrm{V}\mathrm{e}\mathrm{S}}=\int ({\mathcal{ℒ}}_g+{\mathcal{ℒ}}_s+{\mathcal{ℒ}}_v)d^{4}x.$

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

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

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

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

where $B_{\alpha \beta }={\partial }_{\alpha }{u}_{\beta }-{\partial }_{\beta }{u}_{\alpha }$, while $K$ is a dimensionless parameter. $k$ and $K$ are respectively called the scalar and vector coupling constants of the theory. The consistency between the Gravitoelectromagnetism of the TeVeS theory and that predicted and measured by the general relativity leads to $K=\frac{k}{2\pi }$ .

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

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

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

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

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.

Problems and criticisms

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 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. A possible resolution may be in the form of massive (around 2eV) neutrinos.

A study in August 2006 reported an observation of a pair of colliding galaxy clusters, the Bullet Cluster, whose behavior, it was reported, was not compatible with any current modified gravity theories.

A quantity $E_G$ probing General Relativity (GR) on large scales (a hundred billion times the size of the solar system) for the first time has been measured with data from the Sloan Digital Sky Survey to be $E_G=0.392\pm 0.065$ (~16%) consistent with GR, GR plus Lambda CDM and the extended form of GR known as $f(R)$ theory, but ruling out a particular TeVeS model predicting $E_G=0.22$. This estimate should improve to ~1% with the next generation of sky surveys and may put tighter constraints on the parameter space of all modified gravity theories.

See also

• Modified Newtonian Dynamics
• Gauge Vector-Tensor gravity
• Scalar–tensor–vector gravity
• General theory of relativity
• Law of universal gravitation
• Pioneer anomaly
• Nonsymmetric Gravitational Theory
• Dark matter
• Dark energy
• Dark fluid
• Tensor
• Vector
• Scalar - scalar field

Further reading

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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• Dark Matter Observed (SLAC Today)
• Einstein's Theory 'Improved'? (PPARC)
• Einstein Was Right: General Relativity Confirmed ' TeVeS, however, made predictions that fell outside the observational error limits', (Space.com)