Metric-affine gravitation theory explained

X . Metric-affine gravitation theory has been suggested as a natural generalization of Einstein–Cartan theory of gravity with torsion where a linear connection obeys the condition that a covariant derivative of a metric equals zero.^[1]

Metric-affine gravitation theory straightforwardly comes from gauge gravitation theory where a general linear connection plays the role of a gauge field.^[2] Let

TX be the tangent bundle over a manifold X provided with bundle coordinates (x ^{• \mu, x} ^{\mu)} . A general linear connection on TX is represented by a connection tangent-valued form:\Gamma = dx ^{λ ⊗ (\partial} _{λ} + \Gamma ^{• \nu \partial} _{\mu).} ^[3] FX of frames in the tangent spaces to X whose structure group is a general linear group GL( 4 , R) .^[4] Consequently, it can be treated as a gauge field. A pseudo-Riemannian metric g = g _{\mu\nu} dx ^{\mu ⊗} dx ^{\nu} on TX is defined as a global section of the quotient bundle FX/SO( 1 , 3 )\to X , where SO( 1 , 3 ) is the Lorentz group. Therefore, one can regard it as a classical Higgs field in gauge gravitation theory. Gauge symmetries of metric-affine gravitation theory are general covariant transformations.

It is essential that, given a pseudo-Riemannian metric

g , any linear connection \Gamma on TX admits a splitting\Gamma _{\mu\nu\alpha} = \{ _{\mu\nu\alpha} \} + (12)/(C) _{\mu\nu\alpha} + S _{\mu\nu\alpha}

in the Christoffel symbols

\{ _{\mu\nu\alpha} \} =- (12 (\partial)/(_{\mu}) g _{\nu\alpha} + \partial _{\alpha g} _{\nu\mu} - \partial _{\nu} g _{\mu\alpha} ),

a nonmetricity tensor

C _{\mu\nu\alpha} = C _{\mu\alpha\nu} = \nabla ^{\Gamma} _{\mu} g _{\nu\alpha} = \partial _{\mu} g _{\nu\alpha} + \Gamma _{\mu\nu\alpha} + \Gamma _{\mu\alpha\nu}

and a contorsion tensor

S _{\mu\nu\alpha} =- S _{\mu\alpha\nu} = (12 (T)/(_{\nu\mu\alpha}) + T _{\nu\alpha\mu} + T _{\mu\nu\alpha} + C _{\alpha\nu\mu} - C _{\nu\alpha\mu} ),

where

T _{\mu\nu\alpha} = (12 (\Gamma)/(_{\mu\nu\alpha}) - \Gamma _{\alpha\nu\mu} )

is the torsion tensor of

\Gamma .

Due to this splitting, metric-affine gravitation theory possesses a different collection of dynamic variables which are a pseudo-Riemannian metric, a non-metricity tensor and a torsion tensor. As a consequence, a Lagrangian of metric-affine gravitation theory can contain different terms expressed both in a curvature of a connection

\Gamma and its torsion and non-metricity tensors. In particular, a metric-affine f(R) gravity, whose Lagrangian is an arbitrary function of a scalar curvature R of \Gamma , is considered.

A linear connection

\Gamma is called the metric connection for apseudo-Riemannian metric g if g is its integral section, i.e.,the metricity condition\nabla ^{\Gamma} _{\mu} g _{\nu\alpha} =0

holds. A metric connection reads

\Gamma _{\mu\nu\alpha} = \{ _{\mu\nu\alpha} \} + (12 (T)/(_{\nu\mu\alpha}) + T _{\nu\alpha\mu} + T _{\mu\nu\alpha} ).

For instance, the Levi-Civita connection in General Relativity is a torsion-free metric connection.

F ^{gX} of the frame bundle FX corresponding to a section g of the quotient bundle FX/SO( 1 , 3 )\to X . Restricted to metric connections, metric-affine gravitation theory comes to the above-mentioned Einstein – Cartan gravitation theory.

At the same time, any linear connection

\Gamma defines a principal adapted connection \Gamma ^{g} on a Lorentz reduced subbundle F ^{gX} by its restriction to a Lorentz subalgebra of a Lie algebra of a general linear group GL( 4 , R) . For instance, the Dirac operator in metric-affine gravitation theory in the presence of a general linear connection \Gamma is well defined, and it depends just of the adapted connection \Gamma ^{g} . Therefore, Einstein–Cartan gravitation theory can be formulated as the metric-affine one, without appealing to the metricity constraint.

In metric-affine gravitation theory, in comparison with the Einstein – Cartan one, a question on a matter source of a non-metricity tensor arises. It is so called hypermomentum, e.g., a Noether current of a scaling symmetry.

See also

• Gauge gravitation theory
• Einstein–Cartan theory
• Affine gauge theory
• Classical unified field theories

References

• Hehl . F. . McCrea . J. . Ne'eman . Y. . Metric-affine gauge theory of gravity: field equations . Physics Reports . 258 . 1–2 . 1995 . 0370-1573 . 10.1016/0370-1573(94)00111-F . 1–171. gr-qc/9402012 .
• Vitagliano . V. . Sotiriou . T. . Liberati . S. . The dynamics of metric-affine gravity . Annals of Physics . 326 . 5 . 2011 . 10.1016/j.aop.2011.02.008 . 1259–1273. 1008.0171 .
• G. Sardanashvily, Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Phys. 8 (2011) 1869–1895;
• C. Karahan, A. Altas, D. Demir, Scalars, vectors and tensors from metric-affine gravity, General Relativity and Gravitation 45 (2013) 319–343;

Notes and References

1. Hehl . F. W. . McCrea . J. D. . Mielke . E. W. . Ne'eman . Y. . July 1995 . Metric-Affine Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invariance . Physics Reports . 258 . 1–2 . 1–171 . 10.1016/0370-1573(94)00111-F. gr-qc/9402012 .
2. Lord . Eric A. . February 1978 . The metric-affine gravitational theory as the gauge theory of the affine group . Physics Letters A . en . 65 . 1 . 1–4 . 10.1016/0375-9601(78)90113-5.
3. Gubser . S. S. . Klebanov . I. R. . Polyakov . A. M. . 1998-05-28 . Gauge theory correlators from non-critical string theory . Physics Letters B . 428 . 1 . 105–114 . 10.1016/S0370-2693(98)00377-3 . hep-th/9802109 . 0370-2693.
4. gr-qc/0201074 . Sardanashvily . G. . On the geometric foundation of classical gauge gravitation theory . 2002 .