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Linear approximation in Möller gravity. E. Rakhmetov , S. Keyzerov SINP MSU, Moscow. QFTHEP 2011, 24-30 September, Luchezarny , Russia. Outline. Motivations Brief introduction in Möller ( vielbien ) gravity Equations for the small variations over an arbitrary background
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Linear approximation in Möller gravity E. Rakhmetov, S. Keyzerov SINP MSU, Moscow QFTHEP 2011, 24-30 September, Luchezarny, Russia
Outline • Motivations • Brief introduction in Möller (vielbien) gravity • Equations for the small variations over an arbitrary background • Minkovskyspace as a background • Schwarzschild solution as a background • Self-consistent solutions for Kaluza-Klein theories • Conclusions
Motivations • Searching for such generalization of General Relativity, which might be useful in understanding what “Dark Matter” and “Dark Energy” might be. • Obtaining some important exact solutions in Möller Gravity, and making a comparison with GR. • Trying to use linear approximation for interpretation of some results for this solutions. • Make an attempt to find Schwarzschild solution in Möller Gravity with an arbitraryconstants in Lagrangian • Checkingwhetherself-consistentsolutionappeared in this theory in 7 dimensional space-time can be treated as spontaneous compactification of Freund-Rubin type or not. • Finally, we demonstrate, that the small variations over backgraund, can be considered as an antisymmetric second rank tensor field.
Möller (vielbein ) Gravity Introdution • Was suggested by C. Möller in 1978. • Metric theory with metric tensor • Constructed from four vielbiene (frame) vectors • Which areorthonormal: • Here indexes in brackets are frame indexes, from 1 to 4, and summation is considered over repeated indexes • Stress tensor for this vector fields is • Coordinate indexes can be turned into frame indexes:
Vielbein (Möller) Gravity Introdution • Curvature tensor can be obtaned from stress tensor : • We can produce from stress tensor 3 different scalars: • Then the simplest action
Vielbein (Möller) Gravity Introdution • Finally action: • Motion equations: Symmetric part: Asymmetric part: • If k’1= k’2 = 0, asymmetric part vanish, and symmetric part gives us General Relativity
Linear approximation One can divide variation on 2 parts: -metric variation, - some remnant, then . , - antisymmetric tensor. After integration over obtain for variation - is a pure rotations, because of - generators of the rotation Thus, in Moller Gravity, we can consider frame rotations as dynamical variables
Equations for the small variations over an arbitrary background If and are close solutions of motion equations, then is background and are small variations over this background. If we consider pure rotations , then from asymmetric part of motion equations we obtain equation for small deviations: Where
Minkovsky space Because of for Minkovsky space we can take frames with If 2k1+k2+k3=0, we have the eq. for masslessantisymmetric second rank tensor field with spin 1: that have, as as usual, two different transverse polarizations. In common case we have longitudinal polarizations also, but later we will consider only last case, because of spherically symmetric Shwarzshield-like solutions appears, only if 2k1+k2+k3=0
Schwarzschild solution in Möller Gravity • Metric: • Metric tensor: • Ansatzfor the vielbein: • Where
Schwarzschild solution in Möller Gravity • Finally obtain 3 equations for two parameters • Where • System has solution only if K=0 or (this is the same) 2k1+k2+k3=0 and this is pure GR Schwarzschild solution: where
Equations for the small variations over Schwarzschild background For Shwarzschield reference frame we have two non trivial expressions for Thus the equation for small variations is where As we can see, this eq. describe the waves, which have some additional interaction with background and some effective mass, that depends from radial component.
Self-consistent Solutions for Kaluza-Klein theories • Here large latin letters are changed from 0 to 7, except 4, small latin letters -from 5 to 7, small greece letters - from 0 to 3 • Manifold is M4XSn . Ansatz for the vielbein • Then nontrivial equations: • If constants are we have a searching solution.
Conclusions • Schwarzschild solution appears not only in case, when the constants of the Möller theory are small, as was shown by Möller, but in case of arbitrary constants too, when some relations is valid. There are wide area of theory parameters, in wich we have not find spherically symmetric Shwarzshield-like solutions. Linear approximation does not help us whether such solutions exist or not. • Small variations over backgraund, can be considered as an antisymmetric second rank tensor field with spin 1, which “feels” reference frame structure, and in the arbitrary case have an effective mass, depending from coordinates. • With large spectra of theory parameters, 4-dimantional dynamics allows solutions like plane Minkovsky space, and the other dimensions are spontaneously compactifed in to n-dimentional sphere. This is very similar on compactification of Freund-Rubin type, but our antisymmetric second rank tensor field can not play role of Freund-Rubin field.
