Some Structural Considerations on the Theory of Gravitational Field

Satoshi Ikeda

Abstract

When the gravitational field is regarded as the time-sequence of space-slices (i.e., the evolution space), the gravitational field itself is treated by means of the differential geometry of total space of the vector bundle whose base manifold is the one-dimensional time-axis and fibre at each time is the three-dimensional space. From this vector bundle-like standpoint, new field equations and new conservation laws are proposed.

Mathematics Subject Classification: 83C40 Key words: space-time, bundle-like structures, field equations, conservation laws

1 Introduction

In this paper, the gravitational field is regarded as the ensemble of space-time events and then, as the time-sequence of space-slices. That is to say, the four-dimensional gravitational field itself is decomposed into the one-dimensional time-part and the three-dimensional space-part and the three-dimensional space evolves along the timeaxis. This idea arises from the concept of evolution space in the theory of dynamical systems (cf. [1]). Therefore, from the vector bundle-like standpoint, the gravitational field can be adapted to the total space of the vector bundle whose base manifold is the one-dimensional time-axis (x^{0} = t:time) and the fibre at each time is the threedimensional space spanned by points {x^{i}} (i = 1, 2, 3).

Therefore, the gravitational field can be treated by means of the differential geometry of total space of the vector bundle [2], [3]. From this standpoint, the metrical and connection structures will be introduced and then, new field equations and new conservation laws will be proposed in the following.

2 On the vector bundle-like structures - I

Now, in the total space mentioned above, the so-called adapted frame is set as follows:

Balkan Journal of Geometry and Its Applications, Vol.4, No.2, 1999, pp. 63-68 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

(2.1)  \begin{matrix}dX^{A} ≡ (dx^{0}, δx^{i} = dx^{i} + N_{0}^{i}dx^{0}), \\ \frac{∂}{∂X^{A}} ≡ (\frac{δ}{δx^{0}} = \frac{∂}{∂x^{0}} − N_{0}^{i} \frac{∂}{∂x^{i}} , \frac{∂}{∂x^{i}} ) , \end{matrix}

where X^{A} = (x^{0}, x^{i})(A = (0, i); i = 1, 2, 3) and x^{0} = t (time). The quantity N_{0}^{i} denotes the nonlinear connection playing physically the role of velocity.

On the basis of (2.1), the connection structures is introduced by

(2.2)  \begin{matrix}∇\frac{∂}{∂X^{C}} \frac{∂}{∂X^{B}} = Γ_{BC}^{A} \frac{∂}{∂X^{A}}; \\ Γ_{BC}^{A} ≡ (L_{00}^{0}, L_{j0}^{i}, C_{0k}^{0}, C_{jk}^{i}), \end{matrix}

where

∇\frac{δ}{δx^{0}} \frac{δ}{δx^{0}} = L_{00}^{0} \frac{δ}{δx^{0}} , ∇\frac{∂}{∂x^{k}} \frac{∂}{∂x^{j}} = C_{jk}^{i} \frac{∂}{∂x^{i}} , etc.

Namely, the following four kinds of covariant derivatives can be defined, for an arbitrary vector V^{A} = (V^{0}, V^{i})

(2.3)  \{ \begin{matrix}V_{|0}^{0} = \frac{δV^{0}}{δx^{0}} + L_{00}^{0}V^{0}, \\ V^{0}|_{k} = \frac{∂V^{0}}{∂x^{k}} + C_{0k}^{0}V^{0}, \\ V_{|0}^{i} = \frac{δV^{i}}{δx^{0}} + L_{j0}^{i}V^{j}, \\ V^{i}|_{k} = \frac{∂V^{i}}{∂x^{k}} + C_{jk}^{i}V^{j}. \end{matrix}

On the other hand, the metrical structure is introduced by

(2.4)  G ≡ G_{AB}dX^{A}dX^{B} = g_{00}dx^{0} ⊗ dx^{0} + g_{ij}δx^{i} ⊗ δx^{j},

where g_{00} and g_{ij} are metric tensors depending on (x^{0}, x^{i}). The connection (2.2) can be made metrical by imposing the metrical conditions such as g_{00|0} = 0, g_{00}|_{k} = 0, g_{ij|0} = 0 and g_{ij}|_{k} = 0. In the metrical case, the canonical connection coefficients can be determined as follows [2], [3]

(2.5)  \{ \begin{matrix}L_{00}^{0} = \frac{1}{2}g^{00} \frac{δg_{00}}{δx^{0}} , \\ L_{j0}^{i} = \frac{1}{2}g^{il} \frac{δg_{lj}}{δx^{0}} , \\ C_{0k}^{0} = \frac{1}{2}g^{00} \frac{∂g_{00}}{∂x^{k}} , \\ C_{jk}^{i} = \frac{1}{2}g^{il} (\frac{∂g_{lj}}{∂x^{k}} + \frac{∂g_{kl}}{∂x^{j}} − \frac{∂g_{jk}}{∂x^{l}} ) , \end{matrix}

where g^{00} and g^{il} are the inverse of g_{00} and g_{li} respectively and the torsion tensor S_{jk}^{i}(= C_{jk}^{i} − C_{kj}^{i}) is assumed to vanish (see (3.5)). C_{jk}^{i} is nothing but the threedimensional Christoffel symbol formed with g_{ij}. And also, in (2.5), if g_{00} = constant, then L_{00}^{0} = 0 and C_{0k}^{0} = 0.

Some Structural Considerations on the Theory of Gravitational Field 65

3 On the vector bundle-like structures - II

Now, the torsion and curvature tensors are defined by [2], [3], respectively,

(3.1)  T_{BC}^{A} = Γ_{BC}^{A} − Γ_{CB}^{A} + ω_{BC}^{A},
(3.2)  R_{BCD}^{A} = \frac{∂}{∂X^{D}} Γ_{BC}^{A} − \frac{∂}{∂X^{C}} Γ_{BD}^{A} + Γ_{BC}^{E}Γ_{ED}^{A} − Γ_{BD}^{E}Γ_{EC}^{A} + Γ_{DE}^{A}ω_{CD}^{E},

where ω_{BC}^{A} means the non-holonomic object defined by

(3.3)  [\frac{∂}{∂X^{B}} , \frac{∂}{∂X^{C}} ] = ω_{BC}^{A} \frac{∂}{∂X^{A}} .

In our case, only one component ω_{0j}^{i} = −ω_{j0}^{i} = \frac{∂N_{0}^{i}}{∂x^{j}} appears (because ω_{00}^{i} = R_{00}^{i} = 0, see below).

Five components of the torsion tensor

T_{BC}^{A} ≡ (T_{00}^{0} = 0, R_{00}^{i} = 0, C_{0i}^{0}, P_{0j}^{i}, S_{jk}^{i})

and six components of the curvature tensor

R_{BCD}^{A} ≡ (R_{000}^{0} = 0, R_{j00}^{i} = 0, P_{k0j}^{i}, P_{00i}^{0}, S_{0ij}^{0}, S_{jkl}^{i})

appear explicitly in the following Ricci-identities:

(3.4)  \begin{matrix}\{ & \begin{matrix}V^{0}_{|0|0} − V^{0}_{|0|0} = R_{000}^{0}V^{0} − T_{00}^{0}V^{0}_{|0} − R_{00}^{i}V^{0}|_{i}(= 0), \\ V^{0}_{|0}|_{i} − V^{0}|_{i|0} = P_{00i}^{0}V^{0} − C_{0i}^{0}V_{|0}^{0} − P_{0i}^{j}V^{0}|_{j}, \\ V^{0}|_{i}|_{j} − V^{0}|_{j}|_{i} = S_{0ij}^{0}V^{0} − S_{ij}^{k}V^{0}|_{k}, \end{matrix} \\ \{ & \begin{matrix}V^{i}_{|0|0} − V^{i}_{|0|0} = R_{j00}^{i}V^{j} − T_{00}^{0}V^{i}_{|0} − R_{00}^{j}V^{i}|_{j}(= 0), \\ V^{i}_{|0}|_{j} − V^{i}|_{j}_{|0} = P_{k0j}^{i}V^{k} − C_{0j}^{0}V^{i}_{|0} − P_{0j}^{k}V^{i}|_{k}, \\ V^{i}|_{j}|_{k} − V^{i}|_{k}|_{j} = S_{ljk}^{i}V^{l} − S_{jk}^{l}V^{i}|_{l}. \end{matrix} \end{matrix}

The concrete definitions of those components are given as follows

(3.5)  \{ \begin{matrix}T_{00}^{0} = L_{00}^{0} − L_{00}^{0} = 0, \\ R_{00}^{i} = \frac{δN_{0}^{i}}{δx^{0}} − \frac{δN_{0}^{i}}{δx^{0}} = 0, \\ P_{0j}^{i} = \frac{∂N_{0}^{i}}{∂x^{j}} − L_{j0}^{i}, \\ S_{jk}^{i} = C_{jk}^{i} − C_{kj}^{i}. \end{matrix}

(3.6)  \{ \begin{matrix}R_{000}^{0} = \frac{δL_{00}^{0}}{δx^{0}} − \frac{δL_{00}^{0}}{δx^{0}} + L_{00}^{0}L_{00}^{0} − L_{00}^{0}L_{00}^{0} + C_{0i}^{0}R_{00}^{i} = 0, \\ R_{j00}^{i} = \frac{δL_{j0}^{i}}{δx^{0}} − \frac{δL_{j0}^{i}}{δx^{0}} + L_{j0}^{l}L_{l0}^{i} − L_{j0}^{l}L_{l0}^{i} + C_{jl}^{i}R_{00}^{l} = 0, \\ P_{j0k}^{i} = \frac{∂L_{j0}^{i}}{∂x^{k}} − C_{jk|0}^{i} + C_{jl}^{i}P_{0k}^{l}, \\ P_{00k}^{0} = \frac{∂L_{00}^{0}}{∂x^{k}} − C_{0k|0}^{0} + C_{0l}^{0}P_{0k}^{l}, \\ S_{0kh}^{0} = \frac{∂C_{0k}^{0}}{∂x^{h}} − \frac{∂C_{0h}^{0}}{δx^{k}} + C_{0k}^{0}C_{0h}^{0} − C_{0h}^{0}C_{0k}^{0}, \\ S_{jkh}^{i} = \frac{∂C_{jk}^{i}}{δx^{h}} − \frac{∂C_{jh}^{i}}{δx^{k}} + C_{jk}^{l}C_{lh}^{i} − C_{jh}^{l}C_{lk}^{i}. \end{matrix}

In the canonical case of (2.5), the tensor S_{jkh}^{i} is just the three-dimensional Riemann-Christoffel curvature tensor and also, if g_{00} = constant, then P_{00k}^{0} = 0 and S_{0kh}^{0} = 0 in (3.6).

From R_{BCD}^{A}, the Ricci-tensor is given by (3.7)

R_{BC} ≡ R_{BCA}^{A} ≡ (R_{00} ≡ R_{000}^{0} = 0, P^{1}_{j0}≡ P_{j0i}^{i}, − P^{2}_{0k}≡ −P_{0k0}^{0} = P_{00k}^{0}, S_{ij} ≡ S_{ijk}^{k}).

Namely, three non-vanishing components appear. And the total scalar is given by

(3.8)  R ≡ R_{AB}G^{AB} = R_{00}g^{00} + S_{ij}g^{ij} = S_{ij}g^{ij} ≡ S.

(3.7) and (3.8) will be used in the next Section.

4 On the field equations and the conservation laws

As the field equation for the total space (i.e., the gravitational field), we shall put it in the form [2], [3], with use of the Ricci-tensor R_{AB} (3.7) and the total scalar R (3.8),

(4.1)  R_{AB} − \frac{1}{2}RG_{AB} = τ_{AB},

where τ_{AB} represents the energy-momentum tensor with four components τ_{AB} ≡ (τ_{00}, τ_{j0}, τ_{0i}, τ_{ij}). Then, by use of the components of (3.7) and (3.8), we can obtain the following four kinds of field equations

(4.2)  \{ \begin{matrix}−\frac{1}{2}Sg_{00} = τ_{00}, \\ P^{1}_{j0}= τ_{j0}, \\ P^{2}_{0i}= −τ_{0i}, \\ S_{ij} − \frac{1}{2}Sg_{ij} = τ_{ij}. \end{matrix}

These are new equations, different from those obtained in [2], [3]. In the case of the canonical connection (2.5), the last equation is just the (three-dimensional) Einstein’s field equation.

As to the conservative law, we can formulate it in the form [2], [3] (i.e., the divergence-zero of (4.1))

(4.3)  \frac{∇_{∂}}{∂X^{A}} (R_{B}^{A} − \frac{1}{2}Rδ_{B}^{A}) = 0,

where

(4.4)  \begin{aligned}R_{B}^{A} & ≡ & R_{BC}G^{CA} ≡ (R_{0}^{0} ≡ R_{00}g^{00} = 0, P_{0}^{i}^{1}≡P_{j0}^{1} g^{ji}, \\ & − & P_{j}^{0}^{2}≡ − P^{2}_{0j} g^{00}, S_{j}^{i} ≡ S_{jl}g^{li}). \end{aligned}

Therefore, we can obtain the following two kinds of conservation laws

(4.5)  \{ \begin{matrix}P_{0}^{i}^{1} |_{i} − \frac{1}{2}S_{|0} = 0, \\ (S_{j}^{i} − \frac{1}{2}Sδ_{j}^{i})|_{i}− P_{j|0}^{0}^{2}= 0. \end{matrix}

These are new conservation laws, different from those obtained in [2], [3]. In some

special cases where P_{j}^{0}^{2}= 0 or P_{j}^{0}^{2} _{|0} = 0, we can obtain the pure conservation law

(S_{j}^{i} − \frac{1}{2}Sδ_{j}^{i}) |_{i} = 0,

which is the same as the Einstein’s one. In those special cases, the conditions such as

(g_{00} = g_{00}(x^{0})&g_{ij} = g_{ij}(x^{i})) or (g_{00} = g_{00}(x^{i})&g_{ij} = g_{ij}(x^{i})),

etc. must be taken into account.

5 Conclusion

Thus, we can treat the gravitational field by means of the differential geometry of total space of the vector bundle whose base manifold is the time-axis and fibre at each time is the space-slice. And we can propose new field equations (4.2) and new conservation laws (4.5). (Some physical aspects of these subjects are referred to [4]).

References

[1] J.M. Souriau, Structure de systemes dynamiques, Dunos, Paris, 1970.

[2] R.Miron and M.Anastasiei, The geometry of Lagrange spaces: Theory and Applications, Kluwer Academic Publishers, Dordrecht, 1994.

[3] R.Miron and M.Anastasiei, Vector bundles and Lagrange spaces with applications to relativity, Geometry Balkan Press, Bucharest, 1997.

[4] S.Ikeda, Some physical aspects underlying the Lagrangian theory of relativity, Chap. 12 of [3], 183-198.

Department of Mechanical Enginerring Faculty of Science and Technology Science University of Tokyo, Noda Chiba 278-8510, Japan