## On Generalized Einstein Field Equation Geodesic Equation Conservation Laws and Torsion Tensor in a Non-Symmetric Geometry

Delso J

Bachelor's Degree in Physics by Zaragoza University, Spain

Article
Article Info
Author Info
Figures & Data

## Abstract

In four dimensions, the Minkowski metric leads to the 16-dimensional Clifford algebra C(1,3), Dirac's equation [1] is using four of these 16 matrices that form a basis of this algebra, a new operator is defined using all of these matrices and also generalized for a curved space. This new multilevel operator generalizes the Dirac's equation, the value of the generalized Dirac's operator is calculated in the Schwarzschild's metric. The torsion tensor is calculated taking into account the non-symmetric part of the metric tensor in the vanishing of its covariant derivative and applied to Kerr's metric generalizing the Clifford algebra. Geodesic equation, conservation laws, torsion tensor and Einstein field equation are obtained in a non-symmetric geometry.

## Key Words

Gravitomagnetic Tensor; Gravitational Magnetic Field; Energymomentum1-Form; Clifford Algebra; Dirac Equation; Dirac Operator; Gravity and Quantum Mechanics Unification; Multilevel Operator; Schwarzschild's Metric; Torsion Tensor; Rearranged Kerr's Metric; Generalized Clifford Algebra; Generalized Einstein Field Equation; Generalized Geodesic Equation; Conservation Laws; Non-Symmetric Geometry

## Introduction

Dirac's equation is the relativistic wave equation derived by physicist Paul Dirac in 1928. The wave functions in the Dirac theory are vectors of four complex components (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrodinger equation which described wave functions of only one complex component.

Dirac's operator is just the tip of the iceberg, the tip of a generalized operator that is obtained by operating on all members of the Clifford algebra basis and not just on four of them.

The Schwarzschild's metric is named in honour of Karl Schwarzschild, who found the exact solution in 1915 and published it in January 1916, a little more than a month after the publication of Einstein's theory of general relativity. It was the first exact solution of the Einstein field equations other than the trivial at space solution. Schwarzschild died shortly after his paper was published, as a result of a disease he developed while serving in the German army during World War I. Johannes Droste in 1916 independently produced the same solution as Schwarzschild.

Schwarzschild's metric is an exact solution to the Einstein's field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant is all zero.

The new generalized Dirac's operator, the multilevel operator, is calculated in the Schwarzschild's metric, torsion tensor and new gravitomagnetic tensor appear in level 2, curvature tensor appears in levels 3 and 4.

The Kerr's metric is a generalization to a rotating body of the Schwarzschild's metric. The Einstein field equation relates the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form of a tensor equation which related the local spacetime curvature with the local energy, momentum and stress within that spacetime expressed by the stress-energy tensor.

## Multilevel operator

We are using Pauli matrices σ, electromagnetic four-potential and charge e with

1

In four dimensions, Minkowski's metric ημν = diag (+1,-1,-1,-1) leads to the Clifford algebra C(1,3)[2],  Dirac matrices

2

Multilevel operator Dn  acts on level n, n is the number of matrices in the product of the algebra members, for example, D3 acts on γ10, γ11, γ12, and γ13. Total multilevel operator = D0 +D1 +D2 +D3 +D4, the action of on the spinor function vanishes ψ=0                                      3

4

5

6

7

8

9

Multilevel operator can be generalized for a curved space with four-potential P, field charge q and covariant derivative [3] instead of derivative  in the definition of

10

11

12

13

with                                                                                                                                                                                     14

15

16

17

For gravity  is the new gravitomagnetic tensor.  is the torsion tensor [4].

18

with  the Riemann-Christoffel tensor [5].                                                                                                                                       19

20

21

22

23

## Gravitomagnetic Tensor Defined in Schwarzschild's Metric

We are using  with , this metric is defined by [6]

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

, from equations (16) and (17)                                                                                                                  40

41

42

43

44

45

46

47

Energy-momentum form is a 1-form [7]

48

dp is a 2-form

49

50

51

52

Comparing equations (41-43) and (50-52) we can infer

53

is related to the scalar 0 -form  is related to the Energy-momentum 1-form, is related to the Electromagnetic 2 -form, is related to
3-form [8].

54

is related to L 4-form [9].

55

is the proyector matrix, historically

56

## Torsion Tensor in a Rearranged Kerr's Metric

We are using  M is the black hole's mass and a is the angular momentum per unit mass with G = c = 1. The invariance of the length of vectors under parallel transport means that the connection is compatible with the metric, it is a metric connection, the requirement of the preservation of the length by parallel transport may be stated as [10].

57

58

, with                                                                                                              59

60

61

62

Solving these equations, we get the torsion applying its definition [11].

63

Expanding the line element in powers of  and examining the leading terms [12].

64

Rearranging the line elements

65

66

67

68

69

70

71

72

73

74

75

76

77

78

Generalizing Clifford algebra with , if  and  then 1, else 0

79

80

81

82

83

, from equations (16) and (17)                                                                                                              84

85

86

87

88

89

90

is the proyector matrix, historically , but

91

## Geodesic Equation and Torsion Tensor

A geodesic that is not a null geodesic has the property that , taken along a section of the track with the end points  and , is stationary if one makes a small variation of the track keeping the end points fixed. If  denotes an element along the track [13].

92

93

94

and                                                                                                                                                                                   95

96

By partial integration with  at end points  and Q, we get

97

The condition for this to vanish with arbitrary  is

98

99

, and with                                                                                             100

101

From equation (98) with

102

Thus the condition (102) becomes

103

104

105

Multiplying equation (103) by , we obtain the geodesic equation

106

107

108

are the Christoffel symbols of the symmetric part, so

109

110

We directly obtain the torsion tensor without solving equations (62) and (63)

111

## Einstein Field Equation and Conservation Laws

From equation (108) where  are the symbols of the symmetric part

112

113

114

Equation (112) becomes

115

116

117

The vector  has the covariant divergence

118

119

If the left-hand side of equation (119) equals zero then the right-hand side gives us the first conservation law.

For the antisymmetric tensor

120

121

122

If the left-hand side of equation (122) equals zero then the right-hand side gives us the second conservation law.

For the antisymmetric tensor

123

124

125

126

Adding equations (124), (125) and (126)

127

From the definition of the curvature tensor

128

is called the Ricci tensor

129

Now  is not symmetric,  is the antisymmetric part and  where  is the symmetric part in the Einstein's equation [14].

130

,

131

132

133

134

## Conclusions

Multilevel operator  has been generalized for a curved space with a general four-potential P. For gravity  is the new gravitomagnetic tensor and torsion tensor  appears in its definition.

In a at space  and operators vanish. In a curved space the curvature tensor  appears in levels 3 and 4.

The appearance of torsion tensor  and curvature tensor  in multilevel operator  means that this operator is a fundamental operator in Quantum Field Theory.

, have been calculated for Schwarzschild's metric, then , the gravitomagnetic tensor has been obtained.

Each, where n is the number of γ matrices in the product of the algebra members, is related to an n-form.

The invariance of the length of vectors under parallel transport requires the vanishing of the metric tensor covariant derivative, a new term appears in equation (59) with  measuring the non-symmetric part of the metric tensor, solving these equations we get the torsion tensor.

Rearranging Kerr's metric we obtained , the non-symmetric part of the metric tensor, gravitomagnetic tensor has also been calculated generalizing the Clifford algebra.

Taking into account the  in the geodesic equation we have obtained the torsion tensor, conservation laws and Einstein field equation in a non-symmetric geometry.

## References

1. Marian Fecko (2006) Differential Geometry and Lie Groups for Physicists. Cambridge University Press 637.
2. Marian Fecko (2006) Differential Geometry and Lie Groups for Physicists. Cambridge University Press 650652.
3. Marian Fecko (2006) Differential Geometry and Lie Groups for Physicists. Cambridge University Press 380.
4. Marian Fecko (2006) Differential Geometry and Lie Groups for Physicists. Cambridge University Press 384.
5. Marian Fecko (2006) Differential Geometry and Lie Groups for Physicists. Cambridge University Press 404.
6. P.A.M. Dirac (1996) General Theory of Relativity. Princeton University Press 30-32.
7. J.A. Wheeler., C. Misner., K.S. Thorne (2017) Gravitation. Princeton University Press 91.
8. Roger Penrose (2006) El camino a la realidad. Random House Mondadori, Barcelona 603.
9. J.A. Wheeler., C. Misner., K.S. Thorne (2017) Gravitation. Princeton University Press 119.
10. Marian Fecko (2006) Differential Geometry and Lie Groups for Physicists. Cambridge University Press 383.
11. Marian Fecko (2006) Differential Geometry and Lie Groups for Physicists. Cambridge University Press 389.
12. J.A. Wheeler., C. Misner., K.S. Thorne (2017) Gravitation. Princeton University Press 891.
13. P.A.M. Dirac (1996) General Theory of Relativity. Princeton University Press 16-17.
14. J.A. Wheeler., C. Misner., K.S. Thorne (2017) Gravitation. Princeton University Press 406.

Review Article

#### Publication history

Accepted date: April 29, 2023
Published date: May 10, 2023

©2023 Delso J. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

#### Citation

Delso J. (2023) On Generalized Einstein Field Equation Geodesic Equation Conservation Laws and Torsion Tensor in a Non-Symmetric Geometry. OSP Journal of Physics and Astronomy 4: JPA-4-147

#### Jesus Delso Lapuerta

Bachelor's Degree in Physics by Zaragoza University, Spain, jesus.delso@gmail.com

No Figures