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| {{merge to|Modified Newtonian dynamics|discuss=Talk:Modified Newtonian dynamics#Proposed merge with Gauge Vector-Tensor gravity|date=September 2013}}
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| {{technical|date=September 2013}}
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| {{ref improve|date=September 2013}}
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| '''Gauge Vector Tensor gravity'''<ref>{{cite journal|last=Exirifard|first=Qasem|title=GravitoMagnetic force in modified Newtonian dynamics|journal=Journal of Cosmology and Astroparticle Physics|date=27 August 2013|volume=2013|issue=08|pages=046–046|doi=10.1088/1475-7516/2013/08/046}}</ref> (GVT) is a relativistic generalization of [[Mordehai Milgrom]]'s [[Modified Newtonian dynamics|MOdified Newtonian Dynamics]] (MOND) paradigm
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| <ref>{{cite journal|last=Milgrom|first=M.|title=A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis|journal=The Astrophysical Journal|date=1 July 1983|volume=270|pages=365|doi=10.1086/161130}}</ref> where gauge fields cause the MOND behavior. The former covariant realizations of MOND such as the Bekenestein's [[Tensor–vector–scalar gravity]] and the Moffat's [[Scalar–tensor–vector gravity]] attribute MONDian behavior to some scalar fields. GVT is the first example wherein the MONDian behavior is mapped to the gauge vector fields.
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| The main features of GVT can be summarized as follows:
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| * As it is derived from the [[action principle]], GVT respects [[conservation laws]];
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| * In the [[weak-field approximation]] of the spherically symmetric, static solution, GVT reproduces the MOND acceleration formula;
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| * It can accommodate [[gravitational lens]]ing.
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| * It is in total agreement with the [[Einstein–Hilbert action]] in the strong and Newtonian gravities.
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| Its dynamical degrees of freedom are:
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| * Two [[gauge field]]s: <math>B_{\mu}, \tilde{B}_{\mu}</math>;
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| * A metric, <math>g_{\mu\nu}</math>.
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| ==Details== | |
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| The physical geometry, as seen by particles, represents the [[Finsler geometry]]-Randers type:
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| <math>
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| ds = \sqrt{-g_{\mu\nu} dx^\mu dx^\nu} + (B_\mu + \tilde{B}_\mu) dx^\mu
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| </math>
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| This implies that the orbit of a particle with mass <math>m</math> can be derived from the following effective action:
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| <math>
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| S= m \int d\tau (\frac{1}{2} \dot{x}^\mu \dot{x}^\nu g_{\mu\nu}+(B_\mu+\tilde{B}_\mu) \dot{x}^\mu )\,.
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| </math>
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| The geometrical quantities are Riemannian. GVT, thus, is a bi-geometric gravity.
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| ===Action===
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| The metric's action coincides to that of the Einstein-Hilbert gravity:
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| <math> | |
| S_{\text{Grav}} = \frac{1}{16 \pi G} \int d^4 x \, \sqrt{-g} R
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| </math>
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| where <math>R</math> is the Ricci scalar constructed out from the metric. The action of the gauge fields follow:
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| <math>
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| S_{B} = -\frac{1}{16 \pi G\kappa l^2} \int d^4x \sqrt{- g}\, {L}(\frac{l^2}{4} B_{\mu\nu} B^{\mu\nu})
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| </math>
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| and
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| <math>
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| S_{\tilde{B}} = -\frac{1}{16 \pi G\tilde{\kappa} \tilde{l}^2} \int d^4x \sqrt{- g}\, {L}(\frac{\tilde{l}^2}{4} \tilde{B}_{\mu\nu} \tilde{B}^{\mu\nu})
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| </math> | |
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| where L has the following [[MOND]] asymptotic behaviors
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| <math>
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| { L}(x) = \left\{
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| \begin{array}{ccc}
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| x & ,&\text{for}~ x \gg 1 \\
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| \frac{2}{3}|x|^{\frac{3}{2}} &,& \text{for}~ x \le 1
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| \end{array}
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| \right.~,
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| </math>
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| and <math>\kappa, \tilde{\kappa}</math> represent the coupling constants of the theory while <math> l, \tilde{l}</math> are the parameters of the theory and
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| <math>
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| l < \tilde{l}\,.
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| </math>
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| ===Coupling to the matter=== | |
| Metric couples to the energy-momentum tensor. The matter current is the source field of both gauge fields. The matter current is
| |
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| <math>
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| J^\mu = \rho u^\mu
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| </math> | |
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| where <math>\rho </math> is the density and <math>u^\mu</math> represents the four velocity.
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| ==Regimes of the GVT theory== | |
| GVT accommodates the Newtonian and MOND regime of gravity. But it also admits the post-MONDian regime.
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| ===Strong and Newtonian regimes===
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| The strong and Newtonian regime of the theory is defined to be where holds:
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| <math>
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| \begin{array}{ccc}
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| { L}(\frac{l^2}{4} B_{\mu\nu} B^{\mu\nu}) &=& \frac{l^2}{4} B_{\mu\nu} B^{\mu\nu} \,,\\
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| {L}(\frac{\tilde{l}^2}{4} \tilde{B}_{\mu\nu} \tilde{B}^{\mu\nu}) &=& \frac{\tilde{l}^2}{4} \tilde{B}_{\mu\nu} \tilde{B}^{\mu\nu} \,.
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| \end{array}
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| </math>
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| The consistency between the [[gravitoelectromagnetism]] approximation to the GVT theory and that predicted and measured by the [[Einstein-Hilbert gravity]] demands that
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| <math>
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| \kappa + \tilde{\kappa} =0
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| </math> | |
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| which results to
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| <math>
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| B_\mu+\tilde{B}_\mu = 0\,.
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| </math> | |
| So the theory coincides to the Einstein-Hilbert gravity in its Newtonian and strong regimes.
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| ===MOND regime===
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| The MOND regime of the theory is defined to be
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| <math> | |
| \begin{array}{ccc}
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| { L}(\frac{l^2}{4} B_{\mu\nu} B^{\mu\nu}) &=& \left|\frac{l^2}{4} B_{\mu\nu} B^{\mu\nu}\right|^\frac{3}{2} \,,\\
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| {L}(\frac{\tilde{l}^2}{4} \tilde{B}_{\mu\nu} \tilde{B}^{\mu\nu}) &=& \frac{\tilde{l}^2}{4} \tilde{B}_{\mu\nu} \tilde{B}^{\mu\nu} \,.
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| \end{array}
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| </math>
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| So the action for the <math>B_{\mu}</math> field becomes aquadratic. For the static mass distribution, the theory then converts to the AQUAL model of gravity <ref>{{cite journal|last=Bekenstein|first=J.|coauthors=Milgrom, M.|title=Does the missing mass problem signal the breakdown of Newtonian gravity?|journal=The Astrophysical Journal|date=1 November 1984|volume=286|pages=7|doi=10.1086/162570}}</ref> with the critical acceleration of
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| <math>
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| a_0 = \frac{4\sqrt{2}\kappa c^2}{l}
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| </math>
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| So the GVT theory is capable of reproducing the flat rotational velocity curves of galaxies. The current observations do not fix <math> \kappa </math> which is supposedly of order one.
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|
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| ===Post-MONDian regime===
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| The post-MONDian regime of the theory is defined where both of the actions of the <math> B_{\mu}, \tilde{B}_\mu </math> are aquadratic. The MOND type behavior is suppressed in this regime due to the contribution of the second gauge field.
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| ==See also==
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| * [[Modified Newtonian Dynamics]]
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| * [[Scalar–tensor–vector gravity]]
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| * [[General theory of relativity]]
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| * [[Law of universal gravitation]]
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| * [[Pioneer anomaly]]
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| * [[Nonsymmetric Gravitational Theory]]
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| * [[Dark matter]]
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| * [[Dark energy]]
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| * [[Dark fluid]]
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| * [[Tensor]]
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| * [[vector (geometric)|Vector]]
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| * [[Scalar (physics)|Scalar]] - [[scalar field]]
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| ==References==
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| {{reflist}}
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| {{theories of gravitation}}
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| {{DEFAULTSORT:Gauge Vector Tensor Gravity}}
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| [[Category:Theories of gravitation]]
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| [[Category:Theoretical physics]]
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| [[Category:Astrophysics]]
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