Unification of Gravitational and Electromagnetic Fields in Curved Space-Time Using Gauge Symmetry of Bianchi Identities

This paper deals with the generalization of the linear theory of the unification of gravitational and electromagnetic fields using 4-dimensional gauge symmetry in order to solve the contradictions from the Kaluza-Klein theory’s unification of the gravitational and electromagnetic fields. The unification of gravitational and electromagnetic fields in curved space-time starts from the Bianchi identity, which is well known as a mathematical generalization of the gravitational equation, and by using the existing gauge symmetry condition, equations for the gravitational and electromagnetic fields can be obtained. In particular, the homogeneous Maxwell’s equation can be obtained from the first Bianchi identity, and the inhomogeneous Maxwell’s equation can be obtained from the second Bianchi identity by using Killing’s equation condition of the curved space-time. This paper demonstrates that gravitational and electromagnetic fields can be derived from one equation without contradiction even in curved space-time, thus proving that the 4-dimensional metric tensor using the gauge used for this unification is more complete. In addition, geodesic equations can also be derived in the form of coordinate transformation, showing that they are consistent with the existing equations, and as a result, they are consistent with the existing physical equations.

• Bianchi Identity
• Killing’s Equation
• Maxwell’s Equation
• Gravity
• Kaluza-Klein Theory
• Unification
• Gauge Symmetry
• Relativity

1. Introduction

After Einstein explained gravity as a geometry based on Newton’s equivalence principle through the theory of general relativity in 1915, many scientists including Einstein himself have studied the unified field theory, a new theory encompassing electromagnetic and gravitational phenomena. In 1918 Hermann Weyl, a mathematician, first published the unification theory of general relativity and electromagnetic phenomena [1]. In his unified field theory, Weyl identified electrons as substances continuously distributed in space and tried to unify gravity and electromagnetic forces by expanding them into multidimensional spaces (such as Riemann geometry or four-dimensional space) using a method currently called gauge transformation. However, Einstein rejected Weyl’s result and studied the unified field theory in his own way, but did not solve the problem. Kaluza and Klein also came up with the unification theory of gravity and electromagnetic force in 1921 and 1926, which is now leading to string theory [2] [3]. Even though it made cornerstone for the later multi-dimension theories and gave much inspiration to physicists for the dream of unification theory, it has not been established as a complete theory. The main reason for this is that the Kaluza-Klein theory uses a 5-dimensional metric tensor, resulting in various contradictions. Recently, a paper saying that gravitational and electromagnetic field can be unified even in the fourth dimension has been published [4]. It is consistent with existing physical phenomena and there is no mathematical contradiction, but it still has a limitation of the linear approximated space-time. This paper is expanded to show that it can be unified even in a 4-dimensional curved space-time. Starting from the Bianchi identities, Maxwell’s equation is obtained by using the conditions of the gauge symmetry. Particularly, since the 2^nd Bianchi identity is a general form of the gravitational equation, it tells that the gravitational and electromagnetic fields are naturally unified.

Table 1 and Table 2 below introduce Bianchi identities and Maxwell’s equations. Both equations can be categorized into 2 types, and we can see some formal similarities by comparing their types. The 2nd Bianchi identity and Maxwell’s equations in the curved space-time have the notations of covariant derivatives [5]. We are going to make relations with Bianchi identities and Maxwell’s equations.

We also introduce the gauge of metric tensor in a curved space-time by using previously calculated general gauge transformation [6] [7]. The result is (1-1).

g ′ j k ( x ) ≅ g j k ( x ) − ε ( ∇ k ξ j + ∇ j ξ k ) with ε → 0 (1-1)

ξ j means the infinitesimal coordinate transformation

x ′ j = x j + ε ξ j with ε → 0 (1-2)

Under Killing’s equation, the coordinate transformation of the metric tensor is invariant, thus Equation (1-3) can be used. The solution to this equation, ξ j is called the Killing vector [6] [7] [8].

∇ k ξ j + ∇ j ξ k = 0 (1-3)

By using Killing’s equation and the gauge symmetry condition representing the conservation property, we would like to show the deriving process of Maxwell’s equation in curved space-time in detail.

Table 1

Types of Bianchi identity

Types	Bianchi identity
The first identity	R i j k l + R i k l j + R i l j k = 0
The second identity	∇ l R m i j k + ∇ j R m i k l + ∇ k R m i l j = 0 Or ∇ j ( R j k − 1 2 g j k R ) = 0 Or R j k − 1 2 g j k R = k T j k , ∇ j T j k = 0

(Where ∇ k : covariant derivative, ∂ k : ordinary derivative).

Table 2

Types of Maxwell’s equations

Types	Linear approximated space-time	Curved space-time
Homogeneous	∂ k F l i + ∂ i F k l + ∂ l F i k = 0	∇ k F l i + ∇ i F k l + ∇ l F i k = 0
Inhomogeneous	J i = ∂ j F i j	J i = ∇ j F i j

2. Derivation of Homogeneous Maxwell’s Equation from the First Bianchi Identity

The first Bianchi identity is as follows [6] [9].

R i j k l + R i k l j + R i l j k = 0 (2-1)

where

R i j k l = ∂ k Γ i j l − ∂ l Γ i j k + Γ j l p Γ i p k − Γ j k p Γ i p l (2-2)

where

Γ i j l = 1 2 ( ∂ j g i l + ∂ l g i j − ∂ i g j l ) (2-3)

When Equation (2-3) is changed as gauge-transform, the metric tensor g j k is replaced with g ′ j k and Equation (1-1) can be used in Equation (2-3). If we intend to check the gauge symmetry, only gauge term ε ( ∇ j ξ k + ∇ k ξ j ) can be substituted into Equation (2-3) and we have Equation (2-4).

Γ i j l = 1 2 ε ( ∂ j ( ∇ l ξ i + ∇ i ξ l ) + ∂ l ( ∇ j ξ i + ∇ i ξ j ) − ∂ i ( ∇ j ξ l + ∇ l ξ j ) ) (2-4)

When Equation (2-4) is put into the Riemann tensor Equation (2-2), the 3rd and the 4th terms of right side can be ignored because they are O ( ε 2 ) and ε is infinitesimal. For convenience, 1/2 is omitted and the 1st and 2nd terms of right side of Equation (2-2) are expanded like Equation (2-5).

R i j k l = ∂ k Γ i j l − ∂ l Γ i j k = ∂ k ( ∂ j g i l + ∂ l g i j − ∂ i g j l ) − ∂ l ( ∂ j g i k + ∂ k g i j − ∂ i g j k ) (2-5)

When the indexes are rotated, we have Equation (2-6) and Equation (2-7).

R i k l j = ∂ l Γ i k j − ∂ j Γ i k l = ∂ l ( ∂ j g i k + ∂ k g i j − ∂ i g j k ) − ∂ j ( ∂ k g i l + ∂ l g i k − ∂ i g k l ) (2-6)

R i l j k = ∂ j Γ i l k − ∂ k Γ i l j = ∂ j ( ∂ l g i k + ∂ k g i l − ∂ i g l k ) − ∂ k ( ∂ j g i l + ∂ l g i j − ∂ i g l j ) (2-7)

Therefore, Equation (2-5) is

R i j k l = ∂ k Γ i j l − ∂ l Γ i j k =

∂ k ( ∂ j ( ∇ l ξ i + ∇ i ξ l ) + ∂ l ( ∇ j ξ i + ∇ i ξ j ) − ∂ i ( ∇ j ξ l + ∇ l ξ j ) ) (2-8)

− ∂ l ( ∂ j ( ∇ k ξ i + ∇ i ξ k ) + ∂ k ( ∇ i ξ j + ∇ j ξ i ) − ∂ i ( ∇ i ξ j + ∇ j ξ k ) )

Equation (2-6) is

R i k l j = ∂ l Γ i k j − ∂ j Γ i k l =

∂ l ( ∂ j ( ∇ k ξ i + ∇ i ξ k ) + ∂ k ( ∇ i ξ j + ∇ j ξ i ) − ∂ i ( ∇ i ξ j + ∇ j ξ k ) ) (2-9)

− ∂ j ( ∂ k ( ∇ l ξ i + ∇ i ξ l ) + ∂ l ( ∇ k ξ i + ∇ i ξ k ) − ∂ i ( ∇ k ξ l + ∇ k ξ k ) )

Equation (2-7) is

R i l j k = ∂ j Γ i l k − ∂ k Γ i l j =

∂ j ( ∂ k ( ∇ l ξ i + ∇ i ξ l ) + ∂ l ( ∇ k ξ i + ∇ i ξ k ) − ∂ i ( ∇ k ξ l + ∇ k ξ k ) ) (2-10)

− ∂ k ( ∂ j ( ∇ l ξ i + ∇ i ξ l ) + ∂ l ( ∇ j ξ i + ∇ i ξ j ) − ∂ i ( ∇ j ξ l + ∇ l ξ j ) )

According to Equation (2-1), when all the above three equations from Equation (2-8) to Equation (2-10) are added, we can see that the underlined terms of the same types disappear by each other and the final result becomes 0. Since only the gauge terms are used in the Bianchi identity, it can be said that the 1^st Bianchi identity equation is gauge symmetric.

Next, we will show that there are identities between the gauge terms as well. To calculate it, we put all the terms into Table 3 and organize them.

Each element in Table 3 is assigned to two numbers for each term. We search and rearrange pairs in the form like ∂ j ∂ i ∇ k ξ l − ∂ l ∂ j ∇ k ξ i with only indexes change. Table 4 is not the unique combination in finding pairs game, but just one of examples.

As previously shown, addition of all values in Table 5 results in 0, thus

∂ j ( ∇ k F i l + ∇ i F l k + ∇ k F l i + ∇ i F k l + ∇ l F i k + ∇ l F k i )   + ∂ l ( ∇ i F j k + ∇ j F k i + ∇ k F j i + ∇ k F i j + ∇ i F k j + ∇ j F i k )   + ∂ k ( ∇ j F l i + ∇ i F l j + ∇ i F j l + ∇ l F i j + ∇ l F j i + ∇ j F i l ) = 0 (2-11)

Table 3

Set of gauge terms for Equation (2-8), Equation (2-9) and Equation (2-10)

∂ k ∂ j ( ∇ l ξ i + ∇ i ξ l ) (1, 2)	∂ k ∂ l ( ∇ j ξ i + ∇ i ξ j ) (3, 4)	− ∂ k ∂ i ( ∇ j ξ l + ∇ l ξ j ) (5, 6)	− ∂ l ∂ j ( ∇ k ξ i + ∇ i ξ k ) (7, 8)	− ∂ l ∂ k ( ∇ i ξ j + ∇ j ξ i ) (9, 10)	∂ l ∂ i ( ∇ i ξ j + ∇ j ξ k ) (11, 12)
∂ l ∂ j ( ∇ k ξ i + ∇ i ξ k ) (13, 14)	∂ l ∂ k ( ∇ i ξ j + ∇ j ξ i ) (15, 16)	− ∂ l ∂ i ( ∇ i ξ j + ∇ j ξ k ) (17, 18)	− ∂ j ∂ k ( ∇ l ξ i + ∇ i ξ l ) (19, 20)	− ∂ j ∂ l ( ∇ k ξ i + ∇ i ξ k ) (21, 22)	∂ j ∂ i ( ∇ k ξ l + ∇ k ξ k ) (23, 24)
∂ j ∂ k ( ∇ l ξ i + ∇ i ξ l ) (25, 26)	∂ j ∂ l ( ∇ k ξ i + ∇ i ξ k ) (27, 28)	− ∂ j ∂ i ( ∇ k ξ l + ∇ k ξ k ) (29, 30)	− ∂ k ∂ j ( ∇ l ξ i + ∇ i ξ l ) (31, 32)	− ∂ k ∂ l ( ∇ j ξ i + ∇ i ξ j ) (33, 34)	∂ k ∂ i ( ∇ j ξ l + ∇ l ξ j ) (35, 36)

Table 4

Matched gauge terms and position number

∂ j ∂ i ∇ k ξ l − ∂ l ∂ j ∇ k ξ i (23, 7)	∂ j ∂ l ∇ i ξ k − ∂ l ∂ k ∇ i ξ j (28, 9)	∂ l ∂ k ∇ j ξ i − ∂ k ∂ i ∇ j ξ l (16, 5)
∂ l ∂ j ∇ i ξ k − ∂ k ∂ j ∇ i ξ l (14, 32)	∂ k ∂ l ∇ j ξ i − ∂ l ∂ i ∇ j ξ k (3, 18)	∂ k ∂ l ∇ i ξ j − ∂ j ∂ k ∇ i ξ l (4, 20)
∂ l ∂ j ∇ k ξ i − ∂ j ∂ i ∇ k ξ l (13, 29)	∂ j ∂ l ∇ k ξ i − ∂ l ∂ i ∇ k ξ j (27, 17)	∂ j ∂ k ∇ i ξ l − ∂ k ∂ l ∇ i ξ j (26, 34)
∂ k ∂ j ∇ i ξ l − ∂ l ∂ j ∇ i ξ k (2, 8)	∂ l ∂ i ∇ k ξ j − ∂ j ∂ l ∇ k ξ i (11, 21)	∂ k ∂ i ∇ l ξ j − ∂ j ∂ k ∇ l ξ i (36, 19)
∂ j ∂ i ∇ l ξ k − ∂ k ∂ j ∇ l ξ i (24, 31)	∂ l ∂ k ∇ i ξ j − ∂ j ∂ l ∇ i ξ k (15, 22)	∂ j ∂ k ∇ l ξ i − ∂ k ∂ i ∇ l ξ j (25, 6)
∂ k ∂ j ∇ l ξ i − ∂ j ∂ i ∇ l ξ k (1, 30)	∂ l ∂ i ∇ j ξ k − ∂ k ∂ l ∇ j ξ i (12, 33)	∂ k ∂ i ∇ j ξ l − ∂ l ∂ k ∇ j ξ i (35, 10)
→	→	→
∂ j ∇ k F i l	∂ l ∇ i F j k	∂ k ∇ j F l i
∂ j ∇ i F l k	∂ l ∇ j F k i	∂ k ∇ i F l j
∂ j ∇ k F l i	∂ l ∇ k F j i	∂ k ∇ i F j l
∂ j ∇ i F k l	∂ l ∇ k F i j	∂ k ∇ l F i j
∂ j ∇ l F i k	∂ l ∇ i F k j	∂ k ∇ l F j i
∂ j ∇ l F k i	∂ l ∇ j F i k	∂ k ∇ j F i l

Here, F i l = ∂ i ξ l − ∂ l ξ i .

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