A Generally Covariant Field Equation For Gravitation And Electromagnetism

Transcription

1 3 A Generally Covariant Field Equation For Gravitation And Electromagnetism Summary. A generally covariant field equation is developed for gravitation and electromagnetism by considering the metric vector q µ in curvilinear, non-euclidean space-time. The field equation is: R µ 1 2 Rqµ = kt µ (3.1) where T µ is the canonical energy-momentum four-vector, k the Einstein s constant, R µ is the curvature four-vector, and R the Riemann scalar curvature. It is shown that eqn. (3.1) can be written as: R µ = αq µ, (3.2) where α is a coefficient defined in terms of R, k, and the scale factors of the curvilinear coordinate system. Gravitation is described from eqn. (3.2) through the Einstein s field equation, which is recovered by multiplying both sides of eqn. (3.2) by q µ. Generally covariant electromagnetism is described from eqn. (3.2) by multiplying it on both sides by the wedge q ν. Therefore gravitation is described by the symmetric metric q µ q ν and electromagnetism by the anti-symmetric metric defined by the wedge product q µ q ν. Key words: Generally covariant field equation for gravitation and electromagnetism, O(3) electrodynamics, B (3) field. 3.1 Introduction The Principle of General Relativity states that every theory of physics should be generally covariant, i.e., retain its form under the general coordinate transformation in non-euclidean space-time defined by any well-defined set of curvilinear coordinates [1]. This is a well known and accepted principle, [2], so a unified field theory should also be generally covariant. At present however, only one out of the four known fields of nature: gravitational, electromagnetic, weak and strong, is described by a generally covariant field equation, the Einstein s field equation of gravitation:

2 54 3 Generally Covariant Field Equations: Gravitation & EM R µν(s) 1 2 Rqµν(S) = kt µν(s) (3.3) Here q µν(s) is the symmetric metric tensor, R µν(s) the symmetric Ricci tensor defined in Riemann geometry, R the scalar curvature, k the Einstein s constant and T µν(s) the symmetric canonical energy-momentum tensor. In this Letter a generally covariant field equation for gravitation and electromagnetism is inferred through fundamental geometry: in non-euclidean space-time the existence of a symmetric metric tensor q µν(s) implies the existence of an anti-symmetric metric tensor q µν(a). The former is defined by the line element ds 2 formed from the square of the arc length and the latter by the area element da. In differential geometry the one-form ds 2 is dual to the two-form da. The symmetric metric tensor is defined by the symmetric tensor product of two metric four- vectors: q µν(s) = q µ q ν (3.4) and the anti-symmetric metric tensor by the wedge product: where the metric four-vector is: q µν(a) = q µ q ν (3.5) q ν = (h 0, h 1, h 2, h 3 ) (3.6) Here h i are the scale factors of the generally covariant curvilinear coordinate system defining non-euclidean space-time. Therefore both types of metric tensor are defined by the metric vector q µ. From this result of geometry it is inferred that if gravitation be identified through q µν(s), through the well known eqn. (3.3), then electromagnetism is identified through q µν(a). This inference is developed in Section 3.3 into a generally covariant field equation of gravitation and electromagnetism, an equation written in terms of the metric four-vector q µ, which is at the root of both gravitation and electromagnetism. The following section defines the fundamental geometrical concepts needed for the field equation inferred in Section Fundamental Geometrical Concepts Restrict attention initially to three non-euclidean space dimensions. The set of curvilinear coordinates is defined as (u 1, u 2, u 3 ), where the functions are single valued and continuously differentiable, and where there is a one to one relation between (u 1, u 2, u 3 ) and the Cartesian coordinates. The position vector is r(u 1, u 2, u 3 ), and the arc length is the modulus of the infinitesimal displacement vector: ds = dr = r du 1 + r du 2 + r du 3. u 1 u 2 u 3 (3.7)

3 3.2 Fundamental Geometrical Concepts 55 The metric coefficients are r/ u i, and the scale factors are: h i = r u i. (3.8) The unit vectors are e i = 1 r (3.9) h i u i and form the O(3) symmetry cyclic relations: e 1 e 2 = e 3, e 2 e 3 = e 1, e 3 e 1 = e 2, (3.10) where O(3) is the rotation group of three dimensional space [3-8]. The curvilinear coordinates are orthogonal if: e 1 e 2 = 0, e 2 e 3 = 0, e 3 e 1 = 0. (3.11) The symmetric metric tensor is then defined through the line element, a one form of differential geometry: ω 1 = ds 2 = q ij(s) du i du j, (3.12) and the anti-symmetric metric tensor through the area element, a two form of differential geometry: ω 2 = da = 1 2 qij(a) du i du j. (3.13) These results generalize as follows to the four dimensions of any non-euclidean space-time: ω 1 = ds 2 = q µν(s) du µ du ν, (3.14) ω 2 = ω 1 = da = 1 2 qµν(a) du µ du ν. (3.15) In differential geometry the element du σ is dual to the wedge product du µ du ν. The symmetric metric tensor is: h 2 0 h 0 h 1 h 0 h 2 h 0 h 3 q µν(s) h = 1 h 0 h 2 1 h 1 h 2 h 1 h 3 h 2 h 0 h 2 h 1 h 2, (3.16) 2 h 2 h 3 h 3 h 0 h 3 h 1 h 3 h 2 h 2 3 and the anti-symmetric metric tensor is: 0 h 0 h 1 h 0 h 2 h 0 h 3 q µν(a) h = 1 h 0 0 h 1 h 2 h 1 h 3 h 2 h 0 h 2 h 1 0 h 2 h 3 h 3 h 0 h 3 h 1 h 3 h 2 0 (3.17)

4 56 3 Generally Covariant Field Equations: Gravitation & EM 3.3 The Generally Covariant Field Equation It has been shown that both the symmetric and anti-symmetric metric can be built up of individual metric four vectors q µ in any non-euclidean spacetime, including the Riemannian space-time used in eqn. (3.3). It can therefore be inferred that the Einstein s field equation (3.3) can be built up from the generally covariant field equation: R µ 1 2 Rqµ = kt µ (3.18) Eqn. (3.3) is recovered from eqn. (3.18) by multiplying both sides of the latter by the metric four vector q ν. We may therefore define the familiar symmetric tensors appearing in the Einstein s field equation of gravitation as follows: R µν(s) = R µ q ν, (3.19) q µν(s) = q µ q ν, (3.20) T µν(s) = T µ q ν, (3.21) in terms of the more fundamental four vectors R µ, q µ, and T µ. Eqn. (3.18) gives the generally covariant form of Newtons second law: f µ = T µ τ, (3.22) where f µ is a force four-vector and τ the proper time. Newtons third Law, and the Noethers Theorem (conservation of energy-momentum) is expressed through the invariant: T µ T µ = constant, (3.23) and Newtons Law of universal gravitation in generally covariant form is given from eqn. (3.1) by: f µ = 1 k G µ τ, Gµ := R µ 1 2 Rqµ. (3.24) The results of generally covariant gravitational theory are also given by equation (3.18) because it is the basis of Einsteins field equation (3.3). We have argued that the metric four vector q σ is dual to the wedge product q µ q ν. From this fundamental result in differential geometry [8] it follows that the Einstein s field equation is dual to the following equation between two forms: R µ q ν 1 2 Rqµ q ν = kt µ q ν, (3.25) an equation which is derived by multiplying both sides of eqn. (3.18) by the wedge q ν and in which appear the anti-symmetric Ricci tensor, antisymmetric metric, and anti-symmetric energy-momentum tensor, respectively defined as follows:

5 3.3 The Generally Covariant Field Equation 57 R µν(a) = R µ q ν, (3.26) q µν(a) = q µ q ν, (3.27) T µν(a) = T µ q ν. (3.28) Define the following anti-symmetric field tensor: G µν(a) = G (0) ( R µν(a) 1 2 Rqµν(A) ). (3.29) The anti-symmetry of the tensor implies the following Jacobi identity of non- Euclidean space-time [3-8]: D ρ G (A) µν + D µ G (A) νρ + D ν G (A) ρµ := 0, (3.30) where D ρ are generally covariant four derivatives. In Riemannian space-time they can be defined through Christoffel symbols which are anti-symmetric in their lower two indices. The Jacobi identity (3.30) can be rewritten as: where is the dual of G (A) ρσ, and orthogonal to G (A) D ρ Gµν(A) := 0, (3.31) G µν(a) = 1 2 εµνρσ G (A) ρσ (3.32) ρσ : G µν(a) G (A) µν = 0. (3.33) Taking covariant derivatives either side of eqn. (3.25) gives: Define the generally covariant four-vector: D µ G µν(a) = G (0) kd µ T µν(a). (3.34) j ν = µ 0 G (0) kd µ T µν(a) (3.35) where µ 0 is the vacuum permeability, a fundamental constant [3-8]. Then eqn. (3.34) can be written as: D µ G µν(a) = j ν /µ 0. (3.36) We define eqns. (3.31) and (3.36) as respectively the homogenous and inhomogeneous field equations of generally covariant electromagnetism. The generally covariant electric and magnetic fields are defined as: cb k = 1 2 εijk G ij(a), E k = G 0k(A), Ẽ k = 1 2 εijk Gij(A), c B k = G 0k(A), (3.37)

6 58 3 Generally Covariant Field Equations: Gravitation & EM and the generally covariant electromagnetic field tensors as: 0 E 1 E 2 E 3 0 cb 1 cb 2 cb 3 G µν(a) = E 1 0 cb 3 cb 2 E 2 cb 3 0 cb 1, G µν(a) = cb 1 0 E 3 E 2 cb 2 E 3 0 E 1. E 3 cb 2 cb 1 0 cb 3 E 2 E 1 0 (3.38) The Jacobi identity (3.31) becomes an identity of a gauge invariant Yang-Mills type field theory [3-8] of electromagnetism if we define the field tensor as a commutator of covariant derivatives: where G µν(a) = i g [Dµ, D ν ], (3.39) D µ = µ iga µ, (3.40) and where A µ is the vector potential. The field tensor G µν(a) is invariant under the gauge transformation: A µ = SA µ S 1 i g ( µs)s 1 (3.41) for any internal gauge field symmetry. The Jacobi identity (3.31) can be expressed as the identity [3-8]: [D ρ, [D µ, D ν ]] := 0 (3.42) cyclic for any gauge group symmetry. It has therefore been shown that a gauge invariant Yang Mills field theory for electromagnetism can be derived from the generally covariant field equation (3.18), which also produces the gauge invariant and generally covariant theory of gravitation first proposed contemporaneously by Einstein s and Hilbert in late Discussion The simplest expression of eqn. (3.18) can be obtained from the well known definition of the scalar curvature R in Riemann geometry: (S) R = g µν(s) R µν = (h 2 0 h 2 1 h 3 2 h 3 3)q µ R µ. (3.43) It can be seen that this equation can be obtained from the equation: R µ = 1 h 4 Rqµ, h 2 := h 2 0 h 2 1 h 3 2 h 2 3, (3.44)

7 3.4 Discussion 59 by multiplying eqn. (3.44) on both sides by h 2 q µ. Using eqn. (3.44) in eqn. (3.18) gives: T µ = αq µ, (3.45) where the proportionality coefficient is defined as: α := R ( 1 k h 4 1 ). (3.46) 2 On the unit hyper-sphere: the proportionality simplifies to: h 2 0 h 2 1 h 2 2 h 2 3 = 1, (3.47) α = R/2k. (3.48) Using the definition (3.19) for the symmetric Ricci tensor, and multiplying on both sides by q ν, we also obtain the following useful expression for the curvature four-vector as a contraction of the symmetric Ricci tensor in Riemann geometry: R µ = (1/h 2 )q ν R µν(s) (3.49) Eqn. (3.45) shows that for both gravitation and electromagnetism the generally covariant energy momentum four-vector T µ is proportional to the generally covariant metric four-vector q µ through the metric dependent proportionality coefficient α. It is likely that such a result is also true for the weak and strong fields, because it is known that the electromagnetic field can be unified with the weak field [3-8] and because both the weak and strong fields are gauge fields. It is likely therefore that eqn. (3.18) is a generally covariant field equation of classical grand unified field theory. This result is required by the Principle of General Relativity. In the special case where te covariant derivatives of the Yang Mills field theory have O(3) internal symmetry, with indices (1), (2) and (3), where ((1), (2) (3)) is the complex circular representation of space, eqns. (3.31) and (3.36) become the field equations of O(3) electrodynamics [3-7]. The latter has been extensively discussed in the literature and tested against experimental data from several sources, and can now be recognised as an example of eqn. (3.18), illustrating the advantages of eqn. (3.18) over the received view of electromagnetism. Therefore O(3) electrodynamics is a theory of general relativity. In the received opinion [3-7] electromagnetism is a theory of special relativity in Euclidean space-time in which the field is an entity superimposed on the frame of reference, a Yang Mills gauge field theory with U(1) internal gauge group symmetry. The several advantages of O(3) electrodynamics over the received opinion have been discussed in the literature [3-7] and it can now be seen that these advantages stem from the fact that eqn. (3.16) gives a theory of generally covariant electromagnetism and also the well known generally

8 60 3 Generally Covariant Field Equations: Gravitation & EM covariant theory of gravitation. Using eqn. (3.45) it can be seen that both gravitation and electromagnetism are defined by the metric vector q µ within the proportionality coefficient α: both fields being essentially the frame of reference itself. We may now conclude that the non-euclidean nature of space-time gives rise to both gravitation and electromagnetism through eqn. (3.18). A similar conclusion has been reached by Sachs [9] using Clifford algebra, but The important Sachs unification scheme is based on Clifford algebra and is considerably more complicated than eqn. (3.18), which is therefore preferred by Okhams Razor - choose the simpler of two theories. Acknowledgments Extensive discussions are gratefully acknowledged with Fellows and Emeriti of the Alpha Foundation.

9 References 1. E. G. Milewski (Chief Ed.), The Vector Analysis Problem Solver, (Research and Education Association, New York, 1987). 2. A. Einstein, The Meaning of Relativity (Princeton Univ. Press, 5th ed., 1955). 3. M. W. Evans, J.-P. Vigier et al., The Enigmatic Photon (Kluwer, Dordrecht, 1994 to 2002, hardback and paperback), vols M. W. Evans and L. B. Crowell.Classical and Quantum Electrodynamics and the B (3) Field (World Scientific, Singapore, 2001). 5. M. W. Evans and A. A. Hasanein, The Photomagneton in Quantum Field Theory (World Scientific, Singapore, 1994). 6. M. W. Evans and S. Kielich (eds.), Modern Non-Linear Optics, in I Prigogine and S. A. Rice (series eds.), Advances in Chemical Physics (Wiley Inter-science, New York, 1995 to 1997, hardback and paperback), vol. 85, first edition. 7. M. W. Evans, (ed.) second edition of ref. 6, vol. 119 (Wiley Inter-science, 2001). 8. L. H. Ryder, Quantum Field Theory (Cambridge Uniov. Press, papervack, 1987 and 1996, second edition). 9. M. Sachs in ref. 7, vol. 119(1), and references therein.