Einstein-Hilbert actions with torsion

Nenad O. Vesi\'c; Dragoljub D Dimitrijevi\'c

arXiv:1906.05672·math.DG·June 14, 2019

Einstein-Hilbert actions with torsion

Nenad O. Vesi\'c, Dragoljub D Dimitrijevi\'c

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TL;DR

This paper explores Einstein-Hilbert actions incorporating torsion and non-symmetric metrics, deriving formulas for pressure, density, and energy-momentum tensor, and generalizing cosmological models to include torsion effects.

Contribution

It introduces general formulas for pressure, density, and energy-momentum tensor in the context of non-symmetric metrics with torsion, extending cosmological models.

Findings

01

Derived formulas for pressure and density based on non-symmetric metrics.

02

Obtained the energy-momentum tensor expression for non-symmetric metrics.

03

Generalized Bianchi type-I cosmological model to include torsion effects.

Abstract

In this paper, we studied the full Einstein-Hilbert actions with respect to non-symmetric metrics and the corresponding torsion. The first concrete result in this paper are the general formulae for pressure and density with respect to the Madsen's article (the equation (3.1), in [10]). Based on these results, we obtained the expression of energy-momentum tensor with respect to non-symmetric metrics. We started the generalization of the Bianchi type-I model of cosmology with respect to the corresponding non-symmetric metrics.

Equations214

\displaystyle g_{\underline{ij}}=\frac{1}{2}\big{(}g_{ij}+g_{ji}\big{)}

\displaystyle g_{\underline{ij}}=\frac{1}{2}\big{(}g_{ij}+g_{ji}\big{)}

\Gamma_{i.jk}=\frac{1}{2}\big{(}\partial_{k}g_{j\alpha}-\partial_{\alpha}g_{jk}+\partial_{j}g_{\alpha k}\big{)}.

\Gamma_{i.jk}=\frac{1}{2}\big{(}\partial_{k}g_{j\alpha}-\partial_{\alpha}g_{jk}+\partial_{j}g_{\alpha k}\big{)}.

\Gamma^{i}_{jk}=g^{\underline{i\alpha}}\Gamma_{\alpha.jk}=\frac{1}{2}g^{\underline{i\alpha}}\big{(}\partial_{k}g_{j\alpha}-\partial_{\alpha}g_{jk}+\partial_{j}g_{\alpha k}\big{)},

\Gamma^{i}_{jk}=g^{\underline{i\alpha}}\Gamma_{\alpha.jk}=\frac{1}{2}g^{\underline{i\alpha}}\big{(}\partial_{k}g_{j\alpha}-\partial_{\alpha}g_{jk}+\partial_{j}g_{\alpha k}\big{)},

\displaystyle\Gamma^{i}_{\underline{jk}}=\frac{1}{2}\big{(}\Gamma^{i}_{jk}+\Gamma^{i}_{kj}\big{)}

\displaystyle\Gamma^{i}_{\underline{jk}}=\frac{1}{2}\big{(}\Gamma^{i}_{jk}+\Gamma^{i}_{kj}\big{)}

\hat{T} (\hat{X}, \hat{Y}) = \nabla_{\hat{Y}} \hat{X} - \hat{\nabla}_{\hat{X}} \hat{Y} + [\hat{X}, \hat{Y}],

\hat{T} (\hat{X}, \hat{Y}) = \nabla_{\hat{Y}} \hat{X} - \hat{\nabla}_{\hat{X}} \hat{Y} + [\hat{X}, \hat{Y}],

\displaystyle\Gamma^{i}_{\underline{jk}}=\frac{1}{2}g^{\underline{i\alpha}}\big{(}\partial_{k}g_{\underline{j\alpha}}-\partial_{\alpha}g_{\underline{jk}}+\partial_{j}g_{\underline{\alpha k}}\big{)},

\displaystyle\Gamma^{i}_{\underline{jk}}=\frac{1}{2}g^{\underline{i\alpha}}\big{(}\partial_{k}g_{\underline{j\alpha}}-\partial_{\alpha}g_{\underline{jk}}+\partial_{j}g_{\underline{\alpha k}}\big{)},

\displaystyle\Gamma^{i}_{\underset{\vee}{jk}}=\frac{1}{2}g^{\underline{i\alpha}}\big{(}\partial_{k}g_{\underset{\vee}{j\alpha}}-\partial_{\alpha}g_{\underset{\vee}{jk}}+\partial_{j}g_{\underset{\vee}{\alpha k}}\big{)}.

\Gamma_{i.\underset{\vee}{jk}}=g_{\underline{i\alpha}}\Gamma^{\alpha}_{\underset{\vee}{jk}}=\frac{1}{2}\big{(}\partial_{k}g_{\underset{\vee}{ji}}-\partial_{i}g_{\underset{\vee}{jk}}+\partial_{j}g_{\underset{\vee}{ik}}\big{)}.

\Gamma_{i.\underset{\vee}{jk}}=g_{\underline{i\alpha}}\Gamma^{\alpha}_{\underset{\vee}{jk}}=\frac{1}{2}\big{(}\partial_{k}g_{\underset{\vee}{ji}}-\partial_{i}g_{\underset{\vee}{jk}}+\partial_{j}g_{\underset{\vee}{ik}}\big{)}.

\displaystyle\Gamma^{\alpha}_{\underline{i\alpha}}=\frac{1}{2{g}}\partial_{i}\big{|}{g}\big{|}

\displaystyle\Gamma^{\alpha}_{\underline{i\alpha}}=\frac{1}{2{g}}\partial_{i}\big{|}{g}\big{|}

\nabla g_{k} X_{j}^{i}

\nabla g_{k} X_{j}^{i}

R g_{j mn}^{i} = \partial_{n} Γ_{\underline{j m}}^{i} - \partial_{m} Γ_{\underline{j n}}^{i} + Γ_{\underline{j m}}^{α} Γ_{\underline{α n}}^{i} - Γ_{\underline{j n}}^{α} Γ_{\underline{α m}}^{i} .

R g_{j mn}^{i} = \partial_{n} Γ_{\underline{j m}}^{i} - \partial_{m} Γ_{\underline{j n}}^{i} + Γ_{\underline{j m}}^{α} Γ_{\underline{α n}}^{i} - Γ_{\underline{j n}}^{α} Γ_{\underline{α m}}^{i} .

R g_{ij} = R g_{ij α}^{α} = \partial_{α} Γ_{\underline{ij}}^{α} - \partial_{j} Γ_{\underline{i α}}^{α} + Γ_{\underline{ij}}^{β} Γ_{\underline{β α}}^{α} - Γ_{\underline{i α}}^{β} Γ_{\underline{j β}}^{α}

R g_{ij} = R g_{ij α}^{α} = \partial_{α} Γ_{\underline{ij}}^{α} - \partial_{j} Γ_{\underline{i α}}^{α} + Γ_{\underline{ij}}^{β} Γ_{\underline{β α}}^{α} - Γ_{\underline{i α}}^{β} Γ_{\underline{j β}}^{α}

0 \nabla g_{k} X_{j}^{i} = \partial_{k} X_{j}^{i} + Γ_{α k}^{i} X_{j}^{α} - Γ_{j k}^{α} X_{α}^{i},

0 \nabla g_{k} X_{j}^{i} = \partial_{k} X_{j}^{i} + Γ_{α k}^{i} X_{j}^{α} - Γ_{j k}^{α} X_{α}^{i},

1 \nabla g_{k} X_{j}^{i} = \partial_{k} X_{j} + Γ_{k α}^{i} X_{j}^{α} - Γ_{k j}^{α} X_{α}^{i},

2 \nabla g_{k} X_{j}^{i} = \partial_{k} X_{j}^{i} + Γ_{α k}^{i} X_{j}^{α} - Γ_{k j}^{α} X_{α}^{i},

3 \nabla g_{k} X_{j}^{i} = \partial_{k} X_{j}^{i} + Γ_{k α}^{i} X_{j}^{α} - Γ_{j k}^{α} X_{α}^{i} .

R g_{j mn}^{i} = R g_{j mn}^{i} + u \nabla g_{n} T g_{j m}^{i} + u^{'} \nabla g_{m} T g_{j n}^{i} + v T g_{j m}^{α} T g_{α n}^{i} + v^{'} T g_{j n}^{α} T g_{α m}^{i} + w T g_{mn}^{α} T g_{α j}^{i},

R g_{j mn}^{i} = R g_{j mn}^{i} + u \nabla g_{n} T g_{j m}^{i} + u^{'} \nabla g_{m} T g_{j n}^{i} + v T g_{j m}^{α} T g_{α n}^{i} + v^{'} T g_{j n}^{α} T g_{α m}^{i} + w T g_{mn}^{α} T g_{α j}^{i},

R g_{ij} = R g_{ij} + u \nabla g_{α} T g_{ij}^{α} - (v^{'} + w) T g_{i β}^{α} T g_{j α}^{β}

R g_{ij} = R g_{ij} + u \nabla g_{α} T g_{ij}^{α} - (v^{'} + w) T g_{i β}^{α} T g_{j α}^{β}

0 R g_{j mn}^{i} = R g_{j mn}^{i} + \frac{1}{2} \nabla g_{n} T g_{j m}^{i} - \frac{1}{2} \nabla g_{m} T g_{j n}^{i} + \frac{1}{4} T g_{j m}^{α} T g_{α n}^{i} - \frac{1}{4} T g_{j n}^{α} T g_{α m}^{i},

0 R g_{j mn}^{i} = R g_{j mn}^{i} + \frac{1}{2} \nabla g_{n} T g_{j m}^{i} - \frac{1}{2} \nabla g_{m} T g_{j n}^{i} + \frac{1}{4} T g_{j m}^{α} T g_{α n}^{i} - \frac{1}{4} T g_{j n}^{α} T g_{α m}^{i},

1 R g_{j mn}^{i} = R g_{j mn}^{i} - \frac{1}{2} \nabla g_{n} T g_{j m}^{i} + \frac{1}{2} \nabla g_{m} T g_{j n}^{i} + \frac{1}{4} T g_{j m}^{α} T g_{α n}^{i} - \frac{1}{4} T g_{j n}^{α} T g_{α m}^{i},

2 R g_{j mn}^{i} = R g_{j mn}^{i} + \frac{1}{2} \nabla g_{n} T g_{j m}^{i} + \frac{1}{2} \nabla g_{m} T g_{j n}^{i} - \frac{1}{4} T g_{j m}^{α} T g_{α n}^{i} + \frac{1}{4} T g_{j n}^{α} T g_{α m}^{i} - \frac{1}{2} T g_{mn}^{α} T g_{α j}^{i},

3 R g_{j mn}^{i} = R g_{j mn}^{i} + \frac{1}{2} \nabla g_{n} T g_{j m}^{i} + \frac{1}{2} \nabla g_{m} T g_{j n}^{i} - \frac{1}{4} T g_{j m}^{α} T g_{α n}^{i} + \frac{1}{4} T g_{j n}^{α} T g_{α m}^{i} + \frac{1}{2} T g_{mn}^{α} T g_{α j}^{i},

4 R g_{j mn}^{i} = R g_{j mn}^{i} - \frac{1}{4} T g_{j m}^{α} T g_{α n}^{i} + \frac{1}{4} T g_{j n}^{α} T g_{α m}^{i},

5 R g_{j mn}^{i} = R g_{j mn}^{i} + \frac{1}{4} T g_{j m}^{α} T g_{α n}^{i} + \frac{1}{4} T g_{j n}^{α} T g_{α m}^{i} .

\begin{array}[]{ll}\underset{0}{\overset{g}{\widetilde{R}}}{}_{ij}=\overset{g}{R}{}_{ij}+\frac{1}{2}\overset{g}{\nabla}{}_{\alpha}\overset{g}{T}{}^{\alpha}_{ij}+\frac{1}{4}\overset{g}{T}{}^{\beta}_{i\alpha}\overset{g}{T}{}^{\alpha}_{j\beta}&\underset{1}{\overset{g}{\widetilde{R}}}{}_{ij}=\overset{g}{R}{}_{ij}-\frac{1}{2}\overset{g}{\nabla}{}_{\alpha}\overset{g}{T}{}^{\alpha}_{ij}+\frac{1}{4}\overset{g}{T}{}^{\beta}_{i\alpha}\overset{g}{T}{}^{\alpha}_{j\beta},\\ \underset{2}{\overset{g}{\widetilde{R}}}{}_{ij}=\overset{g}{R}{}_{ij}+\frac{1}{2}\overset{g}{\nabla}{}_{\alpha}\overset{g}{T}{}^{\alpha}_{ij}+\frac{1}{4}\overset{g}{T}{}^{\beta}_{i\alpha}\overset{g}{T}{}^{\alpha}_{j\beta},&\underset{3}{\overset{g}{\widetilde{R}}}{}_{ij}=\overset{g}{R}{}_{ij}+\frac{1}{2}\overset{g}{\nabla}{}_{\alpha}\overset{g}{T}{}^{\alpha}_{ij}-\frac{3}{4}\overset{g}{T}{}^{\beta}_{i\alpha}\overset{g}{T}{}^{\alpha}_{j\beta},\\ \underset{4}{\overset{g}{\widetilde{R}}}{}_{ij}=\overset{g}{R}{}_{ij}-\frac{1}{4}\overset{g}{T}{}^{\beta}_{i\alpha}\overset{g}{T}{}^{\alpha}_{j\beta},&\underset{5}{\overset{g}{\widetilde{R}}}{}_{ij}=\overset{g}{R}{}_{ij}.\end{array}

\begin{array}[]{ll}\underset{0}{\overset{g}{\widetilde{R}}}{}_{ij}=\overset{g}{R}{}_{ij}+\frac{1}{2}\overset{g}{\nabla}{}_{\alpha}\overset{g}{T}{}^{\alpha}_{ij}+\frac{1}{4}\overset{g}{T}{}^{\beta}_{i\alpha}\overset{g}{T}{}^{\alpha}_{j\beta}&\underset{1}{\overset{g}{\widetilde{R}}}{}_{ij}=\overset{g}{R}{}_{ij}-\frac{1}{2}\overset{g}{\nabla}{}_{\alpha}\overset{g}{T}{}^{\alpha}_{ij}+\frac{1}{4}\overset{g}{T}{}^{\beta}_{i\alpha}\overset{g}{T}{}^{\alpha}_{j\beta},\\ \underset{2}{\overset{g}{\widetilde{R}}}{}_{ij}=\overset{g}{R}{}_{ij}+\frac{1}{2}\overset{g}{\nabla}{}_{\alpha}\overset{g}{T}{}^{\alpha}_{ij}+\frac{1}{4}\overset{g}{T}{}^{\beta}_{i\alpha}\overset{g}{T}{}^{\alpha}_{j\beta},&\underset{3}{\overset{g}{\widetilde{R}}}{}_{ij}=\overset{g}{R}{}_{ij}+\frac{1}{2}\overset{g}{\nabla}{}_{\alpha}\overset{g}{T}{}^{\alpha}_{ij}-\frac{3}{4}\overset{g}{T}{}^{\beta}_{i\alpha}\overset{g}{T}{}^{\alpha}_{j\beta},\\ \underset{4}{\overset{g}{\widetilde{R}}}{}_{ij}=\overset{g}{R}{}_{ij}-\frac{1}{4}\overset{g}{T}{}^{\beta}_{i\alpha}\overset{g}{T}{}^{\alpha}_{j\beta},&\underset{5}{\overset{g}{\widetilde{R}}}{}_{ij}=\overset{g}{R}{}_{ij}.\end{array}

\begin{array}[]{ll}\underset{0}{\overset{g}{\widetilde{R}}}=\overset{g}{R}+\frac{1}{4}g^{\underline{\gamma\delta}}\overset{g}{T}{}^{\beta}_{\gamma\alpha}\overset{g}{T}{}^{\alpha}_{\delta\beta}&\underset{1}{\overset{g}{\widetilde{R}}}=\overset{g}{R}+\frac{1}{4}g^{\underline{\gamma\delta}}\overset{g}{T}{}^{\beta}_{\gamma\alpha}\overset{g}{T}{}^{\alpha}_{\delta\beta},\\ \underset{2}{\overset{g}{\widetilde{R}}}=\overset{g}{R}+\frac{1}{4}g^{\underline{\gamma\delta}}\overset{g}{T}{}^{\beta}_{\gamma\alpha}\overset{g}{T}{}^{\alpha}_{\delta\beta},&\underset{3}{\overset{g}{\widetilde{R}}}=\overset{g}{R}-\frac{3}{4}g^{\underline{\gamma\delta}}\overset{g}{T}{}^{\beta}_{\gamma\alpha}\overset{g}{T}{}^{\alpha}_{\delta\beta},\\ \underset{4}{\overset{g}{\widetilde{R}}}=\overset{g}{R}-\frac{1}{4}g^{\underline{\gamma\delta}}\overset{g}{T}{}^{\beta}_{\gamma\alpha}\overset{g}{T}{}^{\alpha}_{\delta\beta},&\underset{5}{\overset{g}{\widetilde{R}}}=\overset{g}{R}.\end{array}

\begin{array}[]{ll}\underset{0}{\overset{g}{\widetilde{R}}}=\overset{g}{R}+\frac{1}{4}g^{\underline{\gamma\delta}}\overset{g}{T}{}^{\beta}_{\gamma\alpha}\overset{g}{T}{}^{\alpha}_{\delta\beta}&\underset{1}{\overset{g}{\widetilde{R}}}=\overset{g}{R}+\frac{1}{4}g^{\underline{\gamma\delta}}\overset{g}{T}{}^{\beta}_{\gamma\alpha}\overset{g}{T}{}^{\alpha}_{\delta\beta},\\ \underset{2}{\overset{g}{\widetilde{R}}}=\overset{g}{R}+\frac{1}{4}g^{\underline{\gamma\delta}}\overset{g}{T}{}^{\beta}_{\gamma\alpha}\overset{g}{T}{}^{\alpha}_{\delta\beta},&\underset{3}{\overset{g}{\widetilde{R}}}=\overset{g}{R}-\frac{3}{4}g^{\underline{\gamma\delta}}\overset{g}{T}{}^{\beta}_{\gamma\alpha}\overset{g}{T}{}^{\alpha}_{\delta\beta},\\ \underset{4}{\overset{g}{\widetilde{R}}}=\overset{g}{R}-\frac{1}{4}g^{\underline{\gamma\delta}}\overset{g}{T}{}^{\beta}_{\gamma\alpha}\overset{g}{T}{}^{\alpha}_{\delta\beta},&\underset{5}{\overset{g}{\widetilde{R}}}=\overset{g}{R}.\end{array}

\hat{g}=\left[\begin{array}[]{rrrc}s_{0}(t)&n_{0}(t)&n_{1}(t)&n_{2}(t)\\ -n_{0}(t)&s_{1}(t)&n_{3}(t)&n_{4}(t)\\ -n_{1}(t)&-n_{3}(t)&s_{2}(t)&n_{5}(t)\\ -n_{2}(t)&-n_{4}(t)&-n_{5}(t)&s_{3}(t)\end{array}\right],

\hat{g}=\left[\begin{array}[]{rrrc}s_{0}(t)&n_{0}(t)&n_{1}(t)&n_{2}(t)\\ -n_{0}(t)&s_{1}(t)&n_{3}(t)&n_{4}(t)\\ -n_{1}(t)&-n_{3}(t)&s_{2}(t)&n_{5}(t)\\ -n_{2}(t)&-n_{4}(t)&-n_{5}(t)&s_{3}(t)\end{array}\right],

\overline{x}^{i} = x^{i} + ε z^{i} (x^{j}), i, j = 0, \dots, N - 1,

\overline{x}^{i} = x^{i} + ε z^{i} (x^{j}), i, j = 0, \dots, N - 1,

\overline{A} (s^{'}, x) = A (s, \overline{x}),

\overline{A} (s^{'}, x) = A (s, \overline{x}),

δ A = ε \to 0 lim \frac{A ( s , x ) - A ( s , x )}{ε}

δ A = ε \to 0 lim \frac{A ( s , x ) - A ( s , x )}{ε}

\mathcal{I}[f]=\int{F\big{[}t,f(t),f^{\prime}(t)\big{]}dt},

\mathcal{I}[f]=\int{F\big{[}t,f(t),f^{\prime}(t)\big{]}dt},

δ I [f] = \int (\frac{\partial F}{\partial f} - \frac{d}{d t} \frac{\partial F}{\partial f ^{'}}) δ q d t .

δ I [f] = \int (\frac{\partial F}{\partial f} - \frac{d}{d t} \frac{\partial F}{\partial f ^{'}}) δ q d t .

\frac{δ F}{δ f ( t )} = \frac{\partial F}{\partial f} - \frac{d}{d t} \frac{\partial F}{\partial f ^{'}} .

\frac{δ F}{δ f ( t )} = \frac{\partial F}{\partial f} - \frac{d}{d t} \frac{\partial F}{\partial f ^{'}} .

X_{j_{0} \dots j_{q}}^{i_{0} \dots i_{p}} = u = 0 \sum p d_{(u)} \cdot X^{(u)} + v = 0 \sum q \partial_{(v)} X_{(v)},

X_{j_{0} \dots j_{q}}^{i_{0} \dots i_{p}} = u = 0 \sum p d_{(u)} \cdot X^{(u)} + v = 0 \sum q \partial_{(v)} X_{(v)},

F\left[\varepsilon,Y^{m_{0}\ldots m_{u}}_{n_{0}\ldots n_{v}}(\varepsilon),d\big{(}Y^{m_{0}\ldots m_{u}}_{n_{0}\ldots n_{v}}(\varepsilon)\big{)}/d\varepsilon\right]=F\left[\varepsilon,Y^{m_{0}\ldots m_{u}}_{n_{0}\ldots n_{v}}(\varepsilon),\delta Y^{m_{0}\ldots m_{u}}_{n_{0}\ldots n_{v}}\right].

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Taxonomy

TopicsCosmology and Gravitation Theories · Advanced Differential Geometry Research · Black Holes and Theoretical Physics

Full text

EINSTEIN-HILBERT ACTIONS WITH TORSION

Nenad O. Vesić a and Dragoljub D. Dimitrijević b

Abstract

In this paper, we studied the full Einstein-Hilbert actions with respect to non-symmetric metrics and the corresponding torsion. The first concrete result in this paper are the general formulae for pressure and density with respect to the Madsen’s article (the equation (3.1), in [10]). Based on these results, we obtained the expression of energy-momentum tensor with respect to non-symmetric metrics. We started the generalization of the Bianchi type-I model of cosmology with respect to the corresponding non-symmetric metrics.

Key words: Einstein-Hilbert action, energy-momentum tensor, non-symmetric metric tensor, torsion, cosmological model

$2010$ ** Math. Subj. Classification:** 53B50, 58Z05, 46G05

††a,bFaculty of Science and Mathematics, University of Niš††aDepartment of Mathematics, Serbian Ministry of Education, Science and Technological Development, Grant No. 174012††bDepartment of Physics, Serbian Ministry of Education, Science and Technological Development, Grant No. 174020, ICTP – SEENET-MTP NT-03 project ”Cosmology-Classical and Quantum Challenges”

1 Introduction

Cosmology aims to explain the origin and evolution of the Universe, the underlying physical processes, and to obtain a deeper understanding of the laws of physics [2, 1]. We have only one universe to study, and we cannot make experiments with it, only observations.

Cosmology is based on the Einstein’s theory of general relativity, i.e. theoretical studies take place in the context of gravitational theories based on Einstein’s theory of general relativity. Spacetime and hence the evolution of the Universe is determined by the matter present via the Einstein’s equations for gravitational field.

There are three ideas underlying Einstein’s theory of general relativity. The first is that spacetime may be described as a curved, four-dimensional pseudo-Riemannian manifold. The laws of physics must be expressed in a form that is valid independently of any coordinate system used to label points in spacetime. The second essential idea underlying the general relativity is that at every spacetime point there exist locally inertial reference frames, corresponding to locally flat coordinates (carried by freely falling observers), in which the physics of general relativity is locally indistinguishable from that of special relativity. This is Einstein’s famous strong equivalence principle and it makes general relativity an extension of special relativity to a curved spacetime. The third key idea is that mass curves spacetime in a manner described by the tensor field equations of Einstein.

Einstein’s theory of general relativity defines equations for gravity. It is a system of non-linear partial differential equations of up to second order for the components $g_{ij}$ of the spacetime metric tensor $\hat{g}$ (see [3]). They determine the structure of spacetime in a covariant and coordinate-independent way. This statement agrees with the first idea that the laws of physics must be expressed in a form that is valid independently of any coordinate system.

Very important fact is that the use of geometry provides important additional insights by which much information can be gained from Einstein’s equations in a systematic way. Spacetime itself is equipped with a pseudo-Riemannian structure and encodes gravity (gravitational field) in a geometrical way. Consequently, geometry (or more precisely differential geometry) provides means to understand the structure of the spacetime itself.

1.1 Motivation

Note that Einstein’s gravity is (appropriately) described by the pseudo-Riemann geometry which is torsion-free. The spacetime metric represents the gravitational field. The connections are given by the Christoffel symbol compatible with the metric structure.

Allowing spacetime to have non-zero torsion, which arise naturally in generalized gauge theories of gravity and in string theory, we can analyze cosmological models with a such geometry. We will consider an action of the generalized Einstein-Hilbert-like form constructed for non-zero torsion case.

In this paper we investigate cosmological aspects of spacetime with torsion and discuss modified expressions obtained involving torsion in general relativity. We will not discuss extended Einstein’s general relativity which includes spin, i.e. Einstein-Cartan-Kibble-Sciama theory of gravity [4, 5]. We will discuss a model with (dominant) cosmological perfect fluid described in the usual way. This perfect cosmological fluid naturally arises as a consequence of non-zero torsion.

This paper is composed of the introduction and the following sections:

In the second section, we will recall the necessary results from differential geometry. 2. -

In the third section, we will generalize the energy-momentum tensor. This part will start with the expression of the energy-momentum tensor from the Madsen’s article [10]. After that, we will consider $4$ -dimensional spacetimes equipped with different non-symmetric metrics. 3. -

In the fourth section, we will apply the obtained general results to study the Friedmann spacetime the Bianchi type-I spacetime with torsion. Precisely, we will consider the energy-momentum tensor with respect to special non-symmetric metrics.

2 Necessary observations from differential

geometry

Different Riemannian and generalized Riemannian spaces have been studied by a lot of authors. Some of them are L. P. Eisenhart [8, 9], M. Blau [6], M. S. Madsen [10], V. N. Ponomarev, A. O. Barvinsky, Y. N. Obukhov [15], N. S. Sinyukov [17], J. Mikeš and his research team [11], Lj. S. Velimirović, S. M. Minčić, M. S. Stanković [19, 18] and many others. S. M. Minčić [14, 13, 12] obtained curvature tensors for non-symmetric affine connection spaces (the affine connection spaces with torsion). Because the generalized Riemannian spaces are special non-symmetric affine connection spaces, these results will be useful in this article.

2.1 Riemannian and generalized Riemannian spaces

Let us present definitions and observations necessary for further research in this paper.

Definition 2.1.

[8, 9]* An $N$ -dimensional manifold $\mathcal{M}_{N}$ equipped with the non-symmetric metric tensor $\hat{g}$ with the components $g_{ij}=g_{ij}(x^{0},\ldots,x^{N-1})$ is the generalized Riemannian space $\mathbb{G}\overset{g}{\mathbb{R}}_{N}$ .*

Because the tensor $\hat{g}$ is non-symmetric, the symmetric and anti-symmetric part of the components $g_{ij}$ are

[TABLE]

It evidently holds the equality $g_{ij}=g_{\underline{ij}}+g_{\underset{\vee}{ij}}$ .

We guess that the matrix $\big{(}g_{\underline{ij}}\big{)}_{N\times N}$ is non-singular, i.e. $g=\det\big{(}g_{\underline{ij}}\big{)}_{N\times N}\neq 0$ . For this reason, the contravariant metric is the inverse matrix $g^{\underline{ij}}=\big{(}g_{\underline{ij}}\big{)}_{N\times N}^{-1}$ .

The generalized Christoffel symbols of the first kind of the space $\mathbb{G}\overset{g}{\mathbb{R}}_{N}$ are

[TABLE]

The affine connection coefficients of the space $\mathbb{G}\overset{g}{R}_{N}$ are the generalized Christoffel symbols of the second kind

[TABLE]

One may easily check that it holds $\Gamma^{i}_{jk}\neq\Gamma^{i}_{kj}$ . For this reason, the symmetric and anti-symmetric parts of the affine connection coefficient $\Gamma^{i}_{jk}$ are

[TABLE]

The differences $\overset{g}{T}{}^{i}_{jk}=\Gamma^{i}_{jk}-\Gamma^{i}_{kj}$ are called the components of the torsion tensor of the space $\mathbb{G}\overset{g}{\mathbb{R}}_{N}$ .

Remark 2.1.

The affine connection $\nabla$ of an affine connection space is the bilinear transformation of the set of differentiable vector spaces on a manifold $\mathcal{M}$ . With respect to the affine connections with or without torsion, the corresponding covariant derivatives are defined.

In the case of the geometrical objects $X^{i_{1}\ldots i_{p}}_{j_{1}\ldots j_{q}}$ and $Y^{i_{1}\ldots i_{p}}_{j_{1}\ldots j_{q}}$ , the commutator $[X^{i_{1}\ldots i_{p}}_{j_{1}\ldots j_{q}},Y^{i_{1}\ldots i_{p}}_{j_{1}\ldots j_{q}}]$ vanishes.

In [15], the anti-symmetric part $\Gamma^{i}_{\underset{\vee}{jk}}$ is called the torsion tensor. Generally, if $\nabla$ is the affine connection of the space $\mathbb{G}\overset{g}{\mathbb{R}}_{N}$ the torsion tensor is defined as

[TABLE]

for the commutator $[\hat{X},\hat{Y}]=\hat{X}\hat{Y}-\hat{Y}\hat{X}$ .

Coordinately, it has the form $\overset{g}{T}{}^{i}_{jk}=\Gamma^{i}_{jk}-\Gamma^{i}_{kj}$ and we will use this definition for components of torsion.

It is easy to prove that it is satisfied the following equations

[TABLE]

Moreover, the following expressions also hold

[TABLE]

Similarly as in the case of the symmetric and anti-symmetric part of metric tensor, we get $\Gamma^{i}_{jk}=\Gamma^{i}_{\underline{jk}}+\Gamma^{i}_{\underset{\vee}{jk}}$ . Moreover, with respect to the definition of the generalized Riemannian space $\mathbb{G}\overset{g}{\mathbb{R}}_{N}$ (see [9, 8]), it holds

[TABLE]

The affine connection space equipped with the affine connection which affine connection coefficients are $\Gamma^{i}_{\underline{jk}}$ is the Riemannian space $\overset{g}{\mathbb{R}}{}_{N}$ and it is called the associated space of the space $\mathbb{G}\overset{g}{\mathbb{R}}{}_{N}$ .

With respect to the affine connection of the associated space $\overset{g}{\mathbb{R}}_{N}$ , one kind of covariant derivative is defined as (see [8, 9, 17, 11])

[TABLE]

for a geometrical object $X^{i}_{j}$ of the type $(1,1)$ .

There is one identity of Ricci type with regard to the covariant derivative $\overset{g}{\nabla}$ . The components of the corresponding curvature tensor are

[TABLE]

The components of the Ricci-curvature tensor and scalar curvature for the space $\overset{g}{\mathbb{R}}{}_{N}$ are

[TABLE]

With respect to the affine connection $\overset{g}{\widetilde{\nabla}}$ of the generalized Riemannian space $\mathbb{G}\overset{g}{\mathbb{R}}_{N}$ , four kinds of the covariant derivative are defined [14, 13, 12]

[TABLE]

Based on these covariant derivatives, four curvature tensors, eight derived curvature tensors and fifteen curvature pseudotensors are obtained [14, 13, 12].

In this paper, we will deal with the lagrangian obtained with respect to the curvature and derived curvature tensors. The components of the curvature tensors and the derived curvature tensors for the space $\mathbb{G}\overset{g}{\mathbb{R}}{}_{N}$ are elements of the family [20]

[TABLE]

for real coefficients $u,u^{\prime},v,v^{\prime},w$ and the components $\overset{g}{T}{}^{i}_{jk}$ of the torsion tensor $\hat{\overset{g}{T}}$ .

With respect to the equations (2.7, 2.15), one obtains that the families of Ricci-curvature tensors and the scalar curvature of the space $\mathbb{G}\overset{g}{\mathbb{R}}_{N}$ are

[TABLE]

Six of the curvature tensors in the family (2.15) are linearly independent, for example

[TABLE]

The components of the corresponding Ricci-curvature tensors of the space $\mathbb{G}\overset{g}{\mathbb{R}}_{N}$ are

[TABLE]

Three of them, for instance $\underset{0}{\overset{g}{\widetilde{R}}}{}_{ij},\underset{1}{\overset{g}{\widetilde{R}}}{}_{ij},\underset{5}{\overset{g}{\widetilde{R}}}{}_{ij}$ , are linearly independent.

The corresponding scalar curvatures of the space $\mathbb{G}\overset{g}{\mathbb{R}}{}_{N}$ are

[TABLE]

Two of the scalar curvatures, e.g. $\underset{0}{\overset{g}{\widetilde{R}}},\underset{5}{\overset{g}{\widetilde{R}}}=\overset{g}{R}$ , are linearly independent.

In our research about physics, we will stay focused on the generalized Riemannian space $\mathbb{G}\overset{g}{\mathbb{R}}_{4}$ equipped with the metric tensor

[TABLE]

for differentiable functions $s_{1}(t),s_{2}(t),s_{3}(t),n_{0}(t),\ldots,n_{5}(t)$ , depending on the time coordinate $t$ . The variables in the space $\mathbb{GR}_{4}$ will be $x^{0}=t$ and the variables $x^{1},x^{2},x^{3}$ are the space variables.

2.2 Variations

We will recall the infinitesimal deformations as in [19, 18] in this part of the paper. A necessary rule will be obtained in here.

A transformation $f:\mathbb{G}\overset{g}{\mathbb{R}}_{N}\to\mathbb{G}\overset{g}{\overline{R}}_{N}$ defined as $x\equiv(x^{i})\to\overline{x}\equiv(\overline{x}^{i})$ , for

[TABLE]

where $\varepsilon$ is an infinitesimal, is the infinitesimal deformation of the space $\mathbb{G}\overset{g}{\mathbb{R}}_{N}$ determined by the vector field $z=z(x^{i})$ .

A local coordinate system in which the point $x$ is endowed with coordinates $x^{i}$ and the point $\overline{x}$ with the coordinates $\overline{x}^{i}$ will be denoted by $(s)$ . In another coordinate system $(s^{\prime})$ , corresponding to the point $x=(x^{i})$ new coordinates $x^{i^{\prime}}=\overline{x}^{i}$ , i.e. as new coordinates $x^{i^{\prime}}$ of the point $x=(x^{i})$ we choose old coordinates of the point $\overline{x}=(\overline{x}^{i})$ . In other words, the equalities $x=(x^{i^{\prime}})=(\overline{x}^{i})$ are satisfied at the system $(s^{\prime})$ .

A considered geometric object $\mathcal{A}$ with respect to the system $(s)$ at the point $x=x^{i}$ will be denoted as $\mathcal{A}(s,x)$ .

The point $\overline{x}$ is said to be deformed point of the point $x$ , if the equation (2.26) holds. Geometrical object $\overline{\mathcal{A}}(s,x)$ is the deformed object $\mathcal{A}(s,x)$ with respect to the deformation (2.26), if it holds

[TABLE]

for the coordinate systems $s$ and $s^{\prime}$ .

The limit

[TABLE]

is the (first) variation of the geometric object $\mathcal{A}(s,x)$ . We may notice $\overline{\mathcal{A}}(s,x)=\mathcal{A}(s,x)+\varepsilon\delta\mathcal{A}$ .

Let

[TABLE]

be a functional for $f$ , where $f^{\prime}(t)=df/dt$ . If $f$ is varied by adding to it a function $\delta f$ , and the integrand $F(x,f+\delta f,f^{\prime}+\delta f^{\prime})$ is expanded in powers of $\delta f$ , then the change in the value of $\mathcal{I}$ to first order in $\delta f$ is

[TABLE]

The function $\delta F/\delta f(t)$ is the functional derivative of $F$ with respect to $f$ at the point $t$ . This functional derivative may be computed as

[TABLE]

As we may conclude, if values of a function $F$ do not change when we change values of a function $f(t)$ , with respect to $t$ , then the functional derivative of $F$ with respect to $f$ vanishes. In other words, if the function $F$ is not expressed as a composition of the function $f(t)$ and some other function $g$ , then the functional derivative $\delta F/\delta f(x)$ is equal [math].

With respect to the equation (2.31), we may define the functional derivative of a tensor $\hat{X}$ with components $X^{i_{0}\ldots i_{p}}_{j_{0}\ldots j_{q}}$ with respect to a tensor $\hat{Y}$ with components $Y^{m_{0}\ldots m_{u}}_{n_{0}\ldots n_{v}}$ .

First of all, notice that the component $X^{i_{0}\ldots i_{p}}_{j_{0}\ldots j_{q}}$ of the tensor $\hat{X}$ is expressed as

[TABLE]

for the orthogonal base $(d_{u})\equiv(d/dx{}_{u}),u=0,\ldots,p$ , of the space of functionals and the base $(\partial_{v})=(\partial/\partial x^{v}),v=0,\ldots,q$ , of the space of positions.

If we treat the components $X^{i_{0}\ldots i_{p}}_{j_{0}\ldots j_{q}}$ and $Y^{m_{0}\ldots m_{u}}_{n_{0}\ldots n_{v}}$ as functions $X^{i_{0}\ldots i_{p}}_{j_{0}\ldots j_{q}}(\varepsilon)$ and $Y^{m_{0}\ldots m_{u}}_{n_{0}\ldots n_{v}}(\varepsilon)$ of an infinitesimal deformation, we may conclude that the function $F$ from the equation (2.29) is

[TABLE]

After comparing the equations (2.31), (2.33), we will conclude that if the components $X^{i_{0}\ldots i_{p}}_{j_{0}\ldots j_{q}}(\varepsilon)$ and $Y^{m_{0}\ldots m_{u}}_{n_{0}\ldots n_{v}}(\varepsilon)$ as well as the components $X^{i_{0}\ldots i_{p}}_{j_{0}\ldots j_{q}}(\varepsilon)$ and $\delta Y^{m_{0}\ldots m_{u}}_{n_{0}\ldots n_{v}}$ are functionally independent, the variational derivative of the component $X^{i_{0}\ldots i_{p}}_{j_{0}\ldots j_{q}}$ with respect to the component $Y^{m_{0}\ldots m_{u}}_{n_{0}\ldots n_{v}}$ is equal zero. Moreover, if the tensor $\hat{X}$ is not expressed as a composition of the tensor $\hat{Y}$ and arbitrary tensor $\hat{Z}$ , the functional derivative of the tensor $\hat{X}$ with respect to the tensor $\hat{Y}$ is equal zero.

3 Theoretical considerations

In this section, we will theoretically consider the Einstein-Hilbert action with torsion and the corresponding energy-momentum tensor.

3.1 Four-dimensional spacetime

Let $\mathbb{G}\overset{g}{\mathbb{R}}_{4}$ be the generalized Riemannian space equipped with the metric tensor (2.25). The symmetric and anti-symmetric parts of this tensor are

[TABLE]

The contravariant metric tensor $\hat{\underline{g}}{}^{-1}$ (with the components $g^{\underline{ij}}$ ) is

[TABLE]

The generalized Christoffel symbols of the first and the second kind of the space $\mathbb{G}\overset{g}{\mathbb{R}}_{4}$ are

[TABLE]

The covariant anti-symmetric Christoffel symbols are

[TABLE]

and $\Gamma_{i.\underset{\vee}{jk}}=0$ in all other cases.

The corresponding anti-symmetric parts of the generalized Christoffel symbols $\Gamma^{i}_{jk}$ are

[TABLE]

and $\Gamma^{i}_{\underset{\vee}{jk}}=0$ in all other cases.

Because $\Gamma^{\gamma}_{\underset{\vee}{i\delta}}\Gamma^{\delta}_{\underset{\vee}{j\gamma}}=\Gamma^{\gamma}_{\underset{\vee}{j\delta}}\Gamma^{\delta}_{\underset{\vee}{i\gamma}}$ , it is enough to obtain the following geometrical objects for further calculations:

[TABLE]

and $\Gamma^{\gamma}_{\underset{\vee}{i\delta}}\Gamma^{\delta}_{\underset{\vee}{j\gamma}}=0$ in all other cases for $i\leq j$ .

With respect to the tensor $\hat{\underline{g}}{}^{-1}$ , the above mentioned linearly independent scalars $\underset{0}{\overset{g}{\widetilde{R}}},\overset{g}{R}$ and the metric determinant $g=s_{0}(t)s_{1}(t)s_{2}(t)s_{3}(t)$ , we obtain that the scalar curvature of the space $\mathbb{G}\overset{g}{R}{}_{4}$ with torsion is (see the equations (2.24) and (2.25))

[TABLE]

The scalar curvature (3.23) can be used to define the full Einstein-Hilbert action.

Note that the Lagrangian density in [6] is $\sqrt{|g|}\overset{g}{R}$ . Here, we examine the Lagrangian density in the case of the space with torsion to be $\sqrt{|g|}\underset{0}{\overset{g}{\widetilde{R}}}$ .

We are aimed to express a term $\mathcal{L}_{M}$ describing a dominant cosmological fluid appearing in the model as the function of torsion. The full Lagrangian density of the model takes the form

[TABLE]

Remark 3.1.

The same effect would be achieved if we get the metric

[TABLE]

and the Lagrangian $\mathcal{L}=R+\frac{1}{4}g^{\underline{\gamma\delta}}\overset{g}{T}{}^{\alpha}_{\gamma\beta}\overset{g}{T}{}^{\beta}_{\delta\gamma}$ , for $\mathcal{L}_{M}=\frac{1}{4}g^{\underline{\gamma\delta}}\overset{g}{T}{}^{\alpha}_{\gamma\beta}\overset{g}{T}{}^{\beta}_{\delta\gamma}$ .

Precisely, we guess that the symmetric and anti-symmetric part of the metric $g$ will satisfy the equality $g_{ij}=g_{\underline{ij}}+2\kappa g_{\underset{\vee}{ij}}$ in our paper. It is done for the simplifying of the expression through the computation process.

Remark 3.2.

We could easily put $2\kappa=1$ . In this way, some researchers may find wrong conclusions. To avoid that, we will analyze the general case with the constant $\kappa$ .

Lemma 3.1.

In the space $\mathbb{GR}_{4}$ equipped with the metrics

[TABLE]

the tensor $\overset{g}{\hat{T}}$ whose components are $\overset{g}{T}{}_{i.jk}=\frac{1}{2}\big{(}\partial_{k}g_{\underset{\vee}{ji}}-\partial_{i}g_{\underset{\vee}{jk}}+\partial_{j}g_{\underset{\vee}{ik}}\big{)}$ is not a function of neither $g_{\underline{pq}}$ nor $\partial_{r}g_{\underline{pq}}$ .

Proof.

The following equalities are satisfied:

[TABLE]

That means the torsion tensor $\overset{g}{T}{}_{i.jk}$ may be expressed as

[TABLE]

for a geometrical object $t_{ijk}$ anti-symmetric in any pair of indices $i,j,k$ and $\tau_{ijk}=\tau_{kji}$ . For this reason, we get

[TABLE]

In this way, we proved that the torsion tensor $\overset{g}{T}{}_{i.jk}$ is totally anti-symmetric.

Assume that the covariant torsion tensor $\hat{T}$ with the components $\overset{g}{T}{}_{i.jk}$ is a function of some $\partial_{r}g_{\underline{pq}}$ . Because $\overset{g}{T}{}_{i.jk}$ is anti-symmetric in any pair of the indices $i,j,k$ but the geometrical object $\partial_{r}g_{\underline{pq}}$ is symmetric in the indices $p$ and $q$ , these indices should be mute (dummy) in the functional correspondence between the objects $\partial_{r}g_{\underline{pq}}$ and $\overset{g}{T}{}_{i.jk}$ .

That means that it is satisfied the equation

[TABLE]

for some geometrical object $\theta_{i.jk}$ which is not a function of $\partial_{j}g_{\underline{ik}}$ . Because the symmetric metric $\underline{\hat{g}}$ in this lemma is diagonal, the first summand in the last equation is not equal zero if and only if $i=k$ . Because the component $\overset{g}{T}{}_{i.ji}=0$ , i.e. it is the trivial function, its functional derivative by any function is equal [math]. ∎

3.2 Variations of the action

The variation of the action with the Lagrangian density (3.24) is

[TABLE]

Because $\delta\sqrt{|g|}=-\frac{1}{2}\sqrt{|g|}g_{\underline{\alpha\beta}}\delta g^{\underline{\alpha\beta}}$ , $\int{d^{4}x\sqrt{|g|}g^{\underline{\alpha\beta}}\delta\overset{g}{R}_{\alpha\beta}}=0$ (from standard calculations) and $\delta\overset{g}{R}=\overset{g}{R}_{\alpha\beta}\delta g^{\underline{\alpha\beta}}$ , the equation (3.26) transforms to

[TABLE]

i.e.

[TABLE]

Based on the equations (2.6, 2.6’), we conclude that the anti-symmetric part $\Gamma_{i.\underset{\vee}{jk}}$ is anti-symmetric by indices $i$ and $j$ as well as $j$ and $k$ , but it is symmetric by indices $i$ and $k$ .

Let us define the scalar object $F(g^{\underline{ij}})=g^{\underline{\epsilon\zeta}}g^{\underline{\rho\gamma}}g^{\underline{\sigma\delta}}\overset{g}{T}{}_{\rho.\epsilon\delta}\overset{g}{T}{}_{\sigma.\zeta\gamma}$ . Because $\overset{g}{T}{}_{i.jk}=-\overset{g}{T}{}_{i.kj}=-\overset{g}{T}{}_{j.ik}=\overset{g}{T}{}_{j.ki}=-\overset{g}{T}{}_{k.ji}=\overset{g}{T}{}_{k.ij}$ , $g_{\underline{\alpha\beta}}\delta g^{\underline{\alpha\beta}}=-g^{\underline{\alpha\beta}}\delta g_{\underline{\alpha\beta}},g^{\underline{\epsilon\zeta}}=g^{\underline{\zeta\epsilon}},\delta\overset{g}{T}{}_{i.jk}/\delta g_{\underline{pq}}=0$ and $g_{\underline{\epsilon\zeta}}=g_{\underline{\zeta\epsilon}}$ we obtain

[TABLE]

Hence, the equation (3.27’) is equivalent to

[TABLE]

Variation of the action is zero, $\delta S=0$ , if and only if

[TABLE]

In this way we obtained the set of generalized Einstein equations.

Because there are two linearly independent scalar curvatures of the space $\mathbb{G}\overset{g}{\mathbb{R}}_{4}$ , for example $\underset{0}{\overset{g}{\widetilde{R}}}$ and $\underset{5}{\overset{g}{\widetilde{R}}}=\overset{g}{R}$ , the generalized Einstein equations (3.29) reduce to the Einstein’s equations

[TABLE]

in the case of the lagrangian $\mathcal{L}=\sqrt{|g|}\underset{5}{\overset{g}{\widetilde{R}}}\equiv\sqrt{|g|}\overset{g}{R}$ ( $\mathcal{L}_{M}$ ).

The equations (3.29, 3.30) are the linearly independent dynamical equations for metric field with torsion. Any other equation of motion for models with torsion may be expressed as the corresponding linear combination of the equations (3.29, 3.30).

Components of the corresponding energy-momentum tensor for the cosmological fluid are written on the right side of (3.29)

[TABLE]

With respect to the expressions (3.16—3.22), we get

[TABLE]

but $\overset{g}{T}{}^{\alpha}_{0\beta}\overset{g}{T}{}^{\beta}_{i\alpha}=\overset{g}{T}{}^{\alpha}_{i\beta}\overset{g}{T}{}^{\beta}_{0\alpha}=0$ for $i\in\{1,2,3\}$ .

Because the matrix $\hat{\underline{g}}{}^{-1}$ is diagonal, the following equation holds

[TABLE]

3.3 General formulae

The components of the energy-momentum tensor for a non-ideal (cosmological) fluid are [10]

[TABLE]

for the components $u_{i}$ of a velocity $\hat{u}$ , the pressure $\overset{g}{p}$ and the density $\overset{g}{\rho}$ , as well as the tensor $\hat{\Delta}$ whose components $\Delta_{ij}$ are symmetric in the indices $i$ and $j$ . In the case of an ideal fluid, (3.43) reduces to

[TABLE]

From the M. S. Madsen’s article [10], we may read that the energy-momentum tensor for non-ideal fluid is expressed more explicitly as

[TABLE]

for the velocity components $u_{i}$ such that $u_{\alpha}u^{\alpha}=1$ , the energy density $\overset{g}{\rho}=\overset{g}{T}{}_{\alpha\beta}u^{\alpha}u^{\beta}$ , the geometrical objects $h_{ij}=g_{\underline{ij}}-u_{i}u_{j}$ , $\overset{g}{q}{}_{i}=\overset{g}{T}{}_{\alpha\beta}u^{\alpha}h^{\beta}_{i}$ , $\Pi_{ij}=\overset{g}{p}h_{ij}+\pi_{ij}=-\overset{g}{T}{}_{\alpha\beta}h^{\alpha}_{i}h^{\beta}_{j}$ , and the pressure $\overset{g}{p}=\frac{1}{3}\Pi^{\alpha}_{\alpha}$ .

From the expression of the objects $\Pi_{ij}$ and $h_{ij}$ , we get

[TABLE]

which leads to

[TABLE]

If we multiply the last equation with $g^{\underline{ij}}$ , we will obtain the following results

[TABLE]

i.e.

[TABLE]

Hence, the pressure $\overset{g}{p}$ of the fluid is

[TABLE]

In the comoving reference frame $u_{i}=\delta_{i0}$ , the last equation reduces to

[TABLE]

The following lemma holds.

Lemma 3.2.

The pressure $\overset{g}{p}$ , the density $\overset{g}{\rho}$ and the components of the energy-momentum tensor for a non-ideal fluid satisfy the equation (3.48). In the comoving reference frame, the pressure reduces to (3.48’).

The energy density $\overset{g}{\rho}$ is expressed as

[TABLE]

In the comoving reference frame the energy density reduces to

[TABLE]

Having in mind that the usual equation of state for a fluid is given by

[TABLE]

where $\overset{g}{\omega}$ is the state parameter, then $\overset{g}{\omega}$ can be expressed as

[TABLE]

In the comoving reference frame the last expression reduces to

[TABLE]

∎

With respect to the previous computations and the last Lemma, we obtain that the next theorem holds:

Theorem 3.1.

The components $\overset{g}{T}{}_{ij}$ of the energy-momentum tensor and the components $\overset{g}{T}{}^{i}_{jk}$ of the torsion tensor satisfy the equation

[TABLE]

The trace of the energy-momentum tensor is

[TABLE]

The energy-density is

[TABLE]

The pressure is

[TABLE]

The components of the $1$ -form $\hat{\overset{g}{q}}$ are

[TABLE]

The state parameter $\overset{g}{\omega}$ is

[TABLE]

The components of the energy-momentum tensor, the torsion-tensor, and the metric tensor satisfy the equation

[TABLE]

for the above obtained $\overset{g}{T}{}^{\alpha}_{i\beta}\overset{g}{T}{}^{\beta}_{j\alpha}$ . ∎

Corollary 3.1.

In the comoving reference frame, $u_{i}=\delta_{i0}$ , the expressions for $\overset{g}{\rho},\overset{g}{p},\overset{g}{q}{}_{i}$ and $\overset{g}{\omega}$ , reduce to

[TABLE]

∎

4 Some special cases

In this section, we examine Friedmann-like and Bianchi type-I-like models with torsion.

4.1 Friedmann spacetime with torsion

Let be $s_{0}(t)\equiv-1,s_{1}(t)=s_{2}(t)=s_{3}(t)=s(t),n_{3}(t)=n_{4}(t)=n_{5}(t)=n(t)$ . In this case, it is satisfied the following equations

[TABLE]

and $\overset{g}{T}{}_{i0}=\overset{g}{T}{}_{0i}=0$ for $i\in\{1,2,3\}$ .

In the comoving reference frame, we get

[TABLE]

Let be $s_{0}(t)\equiv-1,s_{1}(t)=s_{2}(t)=s_{3}(t)=s(t),n_{3}(t)=n_{4}(t)=n(t),n_{5}(t)=0$ . In this case, we get:

[TABLE]

and $\overset{g}{T}{}_{ij}=\overset{g}{T}{}_{ji}=0$ in all other cases for $i,j\in\{0,1,2,3\}$ .

In the comoving reference frame, we also obtain

[TABLE]

4.2 Bianchi type-I spacetime with torsion

The Bianchi type-I cosmological model is the main subject for our observations in this subsection. This model is characterized by the first square form

[TABLE]

All above obtained results may be treated as generalizations of the Bianchi type-I spatetime model with torsion. In this section, we will generalize the Bianchi type-I spatetime model with a bidiagonal metrics.

As we may see from the results above, the components $n_{0}(t),n_{1}(t),n_{2}(t)$ do not affect the action. For this reason, we will pay attention to the following metric

[TABLE]

The symmetric and anti-symmetric part of $\hat{b}$ are

[TABLE]

The determinant of the matrix $\underline{\hat{b}}$ is $f=-s_{1}(t)s_{2}(t)s_{3}(t)$ .

We obtain the following values of energy-momentum tensor components in this case:

[TABLE]

and $\overset{b}{T}{}_{ij}=0$ in all other cases.

In the comoving reference frame, we get the following values

[TABLE]

4.2.1 The Friedmann-Lemaitre-Robertson-Walker model

In this part of the paper, we are aimed to consider the Friedmann-Lemaitre-Robertson-Walker cosmological model as a special case of the Bianchi type-I model.

The metric (4.13) reduces to the metric for the FLRW model in the case of $s_{1}(t)=s_{2}(t)=s_{3}(t)=s(t)$ . Because our previous results for pressure and density are obtained for general non-symmetric metric, we conclude that the results for FLRW model are the special cases of the results obtained for the Bianchi type-I model. In this case, i.e. in the case of the metric

[TABLE]

we have

[TABLE]

5 Conclusion

After recalled the necessaries from differential geometry (Section 2), we geometrically studied and generalized the concept of the energy-momentum tensor.

In the Section 3 of this paper, we obtained the general formulae of the energy-momentum tensor, the pressure and the density with respect to the non-symmetric metrics and the torsion-tensor. After that, we analyzed the energy-momentum tensors obtained with respect to the special non-symmetric metrics.

In the Section 4, we applied the general formulae to generalize the Bianchi type-I spacetime model. We studied the special case of the non-symmetric metrics with non-zero components placed on both diagonals of the non-symmetric metric matrices.

It is worth mentioning that although the state parameters for Friedmann and Bianchi type-I spacetime model with torsion have the same numerical value, it is obvious that those two cosmological fluids have different dynamics, i.e. their pressures and densities are different.

Based on the results presented in this paper, in the future we will examine how many generalized Riemannian spaces in the Eisenhart’s sense (see [8, 9]) does a lagrangian $\mathcal{L}_{M}\neq 0$ generate. We will try to give the physical interpretations of these results. Moreover, we will use different concepts of the generalized Riemannian space (the Eisenhart’s model used in this paper is one of them) to expand the Shapiro’s model of the cosmology with torsion (presented in [16]).

Bibliography20

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] V. Mukhanov , Physical Foundations of Cosmology , Cambridge University Press, Cambridge, UK (2005).
2[2] S. Weinberg , Cosmology , Oxford University Press, Oxford, UK (2008).
3[3] M. Bojowald , Canonical Gravity and Applications: Cosmology, Black Holes, and Quantum Gravity , Cambridge University Press, Cambridge (2011).
4[4] T. W. B. Kibble , Lorentz Invariance and the Gravitational Field , J. Math. Phys. 2, 212 (1961), https://doi.org/10.1063/1.1703702.
5[5] D. W. Sciama , The Physical Structure of General Relativity , Rev. Mod. Phys. 36, 463 (1964), 463–469.
6[6] M. Blau , Lecture Notes on General Relativity .
7[7] L. P. Eisenhart , Non-Riemannian Geometry , New York, 1927.
8[8] L. P. Eisenhart , Generalized Riemannian Spaces I , Proc. Nat. Acad. Sci. USA, 37 (1951), 311–315.

TL;DR

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EINSTEIN-HILBERT ACTIONS WITH TORSION

Abstract

1 Introduction

1.1 Motivation

2 Necessary observations from differential

2.1 Riemannian and generalized Riemannian spaces

Definition 2.1**.**

Remark 2.1**.**

2.2 Variations

3 Theoretical considerations

3.1 Four-dimensional spacetime

Remark 3.1**.**

Remark 3.2**.**

Lemma 3.1**.**

Proof.

3.2 Variations of the action

3.3 General formulae

Lemma 3.2**.**

Theorem 3.1**.**

Corollary 3.1**.**

4 Some special cases

4.1 Friedmann spacetime with torsion

4.2 Bianchi type-I spacetime with torsion

4.2.1 The Friedmann-Lemaitre-Robertson-Walker model

5 Conclusion

Definition 2.1.

Remark 2.1.

Remark 3.1.

Remark 3.2.

Lemma 3.1.

Lemma 3.2.

Theorem 3.1.

Corollary 3.1.