Scalar tensor vector gravity theory

The scalar tensor vector theory of gravity ( STVG ) is a modified theory of gravity , which was developed by John Moffat at the Perimeter Institute for Theoretical Physics . The theory is often referred to as Modified Gravity ( MOG ).

overview

The scalar-tensor-vector-gravitation theory is based on an operating principle and postulates the existence of a vector field, it treats the contained cosmological constants as vector fields.

For weak fields, the theory produces a modification of the gravitational force that is similar to the Yukawa potential . This means that the gravitational force at great distances is stronger than is predicted by Newton's law of gravitation , while the gravitation at shorter distances counteracts a fifth force with a repulsive effect.

The scalar tensor vector gravitation theory wants the distribution of the rotational speeds of galaxies in accordance with the Tully-Fisher relation as well as the mass distributions of galaxy clusters, the gravitational lensing effect in the galaxy 1E 0657-558 , as well as further cosmological observations without the existence of darker ones Explain matter and without Einstein's cosmological constant . In smaller orders of magnitude, such as the solar system , no observable deviation from general relativity is predicted. In addition, the scalar tensor vector gravitation theory offers an explanation of the origin of the inertia effect .

Mathematical description

The STVG is formulated on the basis of the Hamiltonian principle . In the following the metric is used and the speed of light is set by using natural units . The Ricci tensor is defined by: ${\displaystyle [+,-,-,-]}$ ${\displaystyle c=1}$

${\displaystyle R_{\mu \nu }=\partial _{\alpha }\Gamma _{\mu \nu }^{\alpha }-\partial _{\nu }\Gamma _{\mu \alpha }^{\alpha }+\Gamma _{\mu \nu }^{\alpha }\Gamma _{\alpha \beta }^{\beta }-\Gamma _{\mu \beta }^{\alpha }\Gamma _{\alpha \nu }^{\beta }}$.

We start with the Einstein-Hilbert effect

${\displaystyle {\mathcal {L}}_{G}=-{\frac {1}{16\pi G}}\left(R+2\Lambda \right){\sqrt {-g}}}$,

where is the trace of the Ricci tensor, the gravitational constant , the determinant of the metric tensor and the cosmological constant . ${\displaystyle R}$ ${\displaystyle G}$ ${\displaystyle g}$ ${\displaystyle g_{\mu \nu }}$ ${\displaystyle \Lambda }$

Next, we introduce the Maxwell Proca - Lagrangian for the STVG- vector field a: ${\displaystyle \phi _{\mu }}$

${\displaystyle {\mathcal {L}}_{\phi }=-{\frac {1}{4\pi }}\omega \left[{\frac {1}{4}}B^{\mu \nu }B_{\mu \nu }-{\frac {1}{2}}\mu ^{2}\phi _{\mu }\phi ^{\mu }+V_{\phi }(\phi )\right]{\sqrt {-g}}}$,

where , the mass of the vector field is the strength of the coupling between the vector field of the fifth force with matter and the self-interaction potential. ${\displaystyle B_{\mu \nu }=\partial _{\mu }\phi _{\nu }-\partial _{\nu }\phi _{\mu }}$ ${\displaystyle \mu }$ ${\displaystyle \omega }$ ${\displaystyle V_{\phi }}$

The three constants of the theory , and are, by associated kinetic and potential terms introduced in the Lagrangian density to scalar fields: ${\displaystyle G}$ ${\displaystyle \mu }$ ${\displaystyle \omega }$

${\displaystyle {\mathcal {L}}_{S}=-{\frac {1}{G}}\left[{\frac {1}{2}}g^{\mu \nu }\left({\frac {\nabla _{\mu }G\nabla _{\nu }G}{G^{2}}}+{\frac {\nabla _{\mu }\mu \nabla _{\nu }\mu }{\mu ^{2}}}-\nabla _{\mu }\omega \nabla _{\nu }\omega \right)+{\frac {V_{G}(G)}{G^{2}}}+{\frac {V_{\mu }(\mu )}{\mu ^{2}}}+V_{\omega }(\omega )\right]{\sqrt {-g}}}$,

where the covariant derivative describes with reference to the metric , while , and represent the self-interaction potentials associated with the scalar fields. ${\displaystyle \nabla _{\mu }}$ ${\displaystyle g_{\mu \nu }}$ ${\displaystyle V_{G}}$ ${\displaystyle V_{\mu }}$ ${\displaystyle V_{\omega }}$

The STVG action integral has the form

${\displaystyle S=\int {({\mathcal {L}}_{G}+{\mathcal {L}}_{\phi }+{\mathcal {L}}_{S}+{\mathcal {L}}_{M})}~d^{4}x}$,

where is the Lagrangian density of ordinary matter. ${\displaystyle {\mathcal {L}}_{M}}$

Spherically symmetrical solution in a static vacuum

The field equations of the STVG can be derived from the action potential using the variation principle.

First, the Lagrange formalism is postulated for a test particle:

${\displaystyle {\mathcal {L}}_{\mathrm {TP} }=-m+\alpha \omega q_{5}\phi _{\mu }u^{\mu }}$,

where the mass of the particle, a factor for the nonlinearity of the theory, is the charge of the fifth force and the quadruple velocity of the fifth force. ${\displaystyle m}$ ${\displaystyle \alpha }$ ${\displaystyle q_{5}}$ ${\displaystyle u^{\mu }=dx^{\mu }/ds}$

Assuming that the charge of the fifth force is proportional to its mass , then the value of is determined by the following equation of motion in the spherical-symmetrical gravitational-static field of a point mass : ${\displaystyle q_{5}=\kappa m}$ ${\displaystyle \kappa ={\sqrt {G_{N}/\omega }}}$ ${\displaystyle M}$

${\displaystyle {\ddot {r}}=-{\frac {G_{N}M}{r^{2}}}\left[1+\alpha -\alpha (1+\mu r)e^{-\mu r}\right]}$,

where is Newton's constant of gravity . ${\displaystyle G_{N}}$

A further analysis of the field equations allows the determination of and for a point source of gravitational mass in the form ${\displaystyle \alpha }$ ${\displaystyle \mu }$ ${\displaystyle M}$

${\displaystyle \mu ={\frac {D}{\sqrt {M}}}}$,

${\displaystyle \alpha ={\frac {G_{\infty }-G_{N}}{G_{N}}}{\frac {M}{({\sqrt {M}}+E)^{2}}}}$,

where it is determined by cosmological observation, while the constants and the galaxy rotation curves give the following values: ${\displaystyle G_{\infty }\simeq 20\,G_{N}}$ ${\displaystyle D}$ ${\displaystyle E}$

${\displaystyle D\simeq 6250\,M_{\odot }^{1/2}\mathrm {kpc} ^{-1}}$,

${\displaystyle E\simeq 25000\,M_{\odot }^{1/2}}$,

where is the solar mass . These results provide the basis for calculations that can be used to test the theory based on astronomical observation. ${\displaystyle M_{\odot }}$

swell

1. ↑ ^{a b} J. W. Moffat: Scalar-Tensor-Vector Gravity Theory. In: arxiv. Cornell University Library, December 11, 2005, accessed June 8, 2017 (English, arxiv : gr-qc / 0506021 ).  
2. ^ Maggie McKee: Gravity theory dispenses with dark matter. In: New Scientist. January 25, 2006, accessed June 8, 2017 .  
3. ^ JR Brownstein, JW Moffat: Galaxy Rotation Curves Without Non-Baryonic Dark Matter. In: arxiv. Cornell University Library, September 22, 2005, accessed June 8, 2017 (English, arxiv : astro-ph / 0506370 ).  
4. ^ JR Brownstein, JW Moffat: Galaxy Cluster Masses Without Non-Baryonic Dark Matter. In: arxiv. Cornell University Library, July 8, 2005, accessed June 8, 2017 (English, arxiv : astro-ph / 0507222 ).  
5. ^ JR Brownstein, JW Moffat: The Bullet Cluster 1E0657-558 evidence shows Modified Gravity in the absence of Dark Matter. In: arxiv. Cornell University Library, September 13, 2007, accessed June 8, 2017 (English, arxiv : astro-ph / 0702146 ).  
6. ^ ^{A b} J. W. Moffat, VT Toth: Modified Gravity: Cosmology without dark matter or Einstein's cosmological constant. In: arxiv. Cornell University Library, January 4, 2012, accessed June 8, 2017 (English, arxiv : 0710.0364 ).  
7. ↑ JW Moffat, VT Toth: Testing modified gravity with globular cluster velocity dispersions. In: arxiv. Cornell University Library, February 29, 2008, accessed June 8, 2017 (English, arxiv : 0708.1935 ).  
8. ^ JW Moffat, VT Toth: Modified gravity and the origin of inertia. In: arxiv. Cornell University Library, April 10, 2009, accessed June 8, 2017 (English, arxiv : 0710.3415 ).