New Geometric Formalism for Gravity Equation in Empty Space Xin-Bing Huang Department of Physics, Peking University, arxiv:hep-th/0402139v2 23 Feb 2004 100871 Beijing, China Abstract In this paper, complex daor field which can be regarded as the square root of spacetime metric is proposed to represent gravity. Daor field equations in empty space are submitted, which are one-order differential equations and not conflict with Einstein s gravity theory. This paper also shows that complex affine connection constructs a bridge between gravity and U(1,3) gauge field. PACS numbers: 04.20.Cv, 04.20.Gz, 02.40.Sf huangxb@mail.phy.pku.edu.cn 1
In general relativity Einstein made the assumption that gravity equation in empty space is[1, 2] where R µν is Ricci tensor, which is symmetrical. R µν = 0, (1) In Minkowski space-time, Dirac equation is usually written as( h = c = 1)[3] ( iγ a ) x m a Where γ s are Dirac matrices, which satisfy ψ = 0. a = 0, 1, 2, 3. (2) γ a γ b + γ b γ a = 2η ab. (3) Where η ab is the metric tensor of Minkowski space-time, i.e. η 00 = +1, η 11 = η 22 = η 33 = 1, η ab = 0 for a b. (4) Eq.(2) describes a free massive spinor field. Physicist will find many advantages of Eq.(2) when he tries to realize the quantization of fields. Dirac equation is a one-order differential equation and spinor ψ is complex, which is consistent with the wave function in Schödinger equation of quantum mechanics. In this paper we will give a new geometric formalism for gravity equation in empty space which looks more like Dirac equation. At first we propose two basic principles: 1. Space-time is a 3 + 1 manifold, which looks like a Minkowski space-time around each point. 2. Intrinsic distance ds 2 = dx µ g µν dx ν (5) is invariant under any physical transformations. One kind of those local transformations is corresponding to one kind of interactions. Here empty means that there is no matter present and no physical fields except the gravitational field. x µ s(µ = 0, 1, 2, 3) are a system of curvilinear coordinates of space-time manifold. 2
In the literature, decomposing the metric into vierbeins or tetrads e a µ(x) has been used extensively[3, 4]. But vierbein decomposition g µν = η ab e a µ (x)eb ν (x) (6) just keeps ds 2 invariant under local SO(1, 3) group transformation, which is not the largest symmetry group transformation as we will show. Now we will discuss the larger symmetry group transformations which keep ds 2 invariant. To do so, we introduce complex vierbein field h a µ or H µ a, which satisfies g µν = h a µ η abh b ν, Gµν = H µ a ηab H ν b, g µν G νλ = G λν g νµ = δ λ µ. (7) where denotes complex conjugation, and we would like to stress that g µν is still real and symmetrical in Eq.(7). In the process of preparing this paper, we found that Ali H. Chamseddine had introduced complex vierbein[5] to construct gravity theory under noncommutative geometry. We suggest to call h a = h a µdx µ daor field, which is a 1-form. By defining the Hermitean conjugate of daor field h a µ or H µ a as follows (h a µ ) = h a µ, (H µ a ) = H µ a, (8) Eq.(7) can be rewritten in the following simple form g = h ηh, G = H ηh, G = g 1. (9) From Eq.(7), it is obvious that h and H satisfy H = h 1, H = (h ) 1. (10) In this paper, using Roman suffixes to refer to the bases of local Minkowski frame; using Greek suffixes to refer to the space-time coordinates. Dao is a basic and important concept in ancient Chinese philosophy. Because h a plays such a central role in physics as will be shown in the following paragraphs, we believe that daor field is a nice name for h a. The local complex vierbein is the component of daor field, which can also be called daor field for simplicity. 3
In differential geometry the inner product is given by[6] It is easy to extend the inner product to < x µ,dxν >= δ ν µ. (11) < H, h >= H ν ah a µ = δ ν µ. (12) So the inner product of the vector U = U µ x µ expressed as follows and the covector v = v ν dx ν can be < U, v >=< v, U >= U µ v µ = v µ U µ = U a v a = v a Ua. (13) Where U a, U a, v a and v a are given by U a = U µ h a µ, U a = h a µu µ, v a = v µ H µ a, v a = H µ a v µ. (14) For simplicity, we will don t distinguish between the component U a, v a and its complex conjugate because they all can be transferred into real component in curvilinear coordinates by corresponding form of daor field. In this complex vierbein geometry, the exterior derivative and exterior product have the same definitions and properties as in ordinary real vierbein geometry. We propose the daor field equations in empty space should be written as follows (δ a b d + ωa b )hb = 0. (15) ω ab = ± 1 2 ǫ abcd ω cd. (16) Where d and are exterior derivative and exterior product operators respectively, ǫ abcd is the totally antisymmetric tensor in 4-dimensions, and ω a b is the complex affine connection 1-form[6]. By defining the torsion operator 1-form ˆT δ a b d + ωa b, (17) Eq.(15) becomes ˆTh = 0. Multiplying both sides of Eq.(15) by torsion operator 1-form ˆT, we acquire ˆT ˆTh = (dω a b + ωa c ωc b ) hb ν = Ra b hb = 0. (18) 4
Where R a b = dω a b +ω a c ω c b is the curvature 2-form defined by Cartan. This equation shows that the operator ˆT can be regarded as the square root of the curvature 2-form. In Cartan s vierbein method, the Levi-Civita affine connection is obtained by requiring the complex affine connection ω ab satisfies the following conditions ω a bµ is then determined in terms of daor field no torsion : T a = dh a + ω a b hb = 0, (19) metricity : ω ab = ω ba. (20) ω a bµ = ha ν H ν b;µ = H ν b ha ν;µ = H ν b ( ha ν x µ Γλ νµ ha λ ). (21) Where Γ λ νµ is Levi-Civita affine connection, which is real and unrelated with daor field. It is obvious that Eq.(15) is torsion-free condition (19) and Eq.(16) implies metricity condition (20). Einstein s empty space equation (1) may be rewritten as[6] R a b h b = 0, (22) Where R ab is the dual of R ab, 2 R ab = ǫ abcd R cd. R ab is self-dual (or anti self-dual), namely, R ab = ±R ab, because Eq.(16) shows that ω ab is self-dual(+) (or anti self-dual(- )). So Eq.(22) can be deduced from Eq.(18). The self-duality or anti self-duality of R ab tells us that the daor field equations (15) and (16) are equivalent to Eq.(1). The daor field equations are covariant under two kinds of transformations: One is the general coordinate transformation x x (x); Another is the U(1,3) group transformation of the Roman suffixes. The covariance request of physical laws under the general coordinate transformation leads to Einstein s gravity theory. Covariance of the daor field equations under local U(1,3) group directly leads to the introduction of Yang-Mills gauge fields[7]. Consider a intrinsic rotation of daor field h a here S a b satisfies h a h a = S a bh b, (23) S c a η cds d b = η ab. (24) 5
So S is an element of U(1,3) group. According to the stand results of Yang and Mills, It is found that ω a b = S a cω c d(s 1 ) d b + S a c(ds 1 ) c b, (25) T a = S a bt b. (26) The transformation laws for the complex torsion 2-form and curvature 2-form are given by T a = (δ a b d + ω a b )h b, (27) R a b = dω a b + ω a c ω c b = Sa c Rc d (S 1 ) d b. (28) In empty space, the Levi-Civita affine connection shows that Cartan s spin connection ω ab is real. This demonstrates that there are no observable physical phenomena of U(1,3) gauge fields. But in more general cases, ω ab must be complex. Adding stressenergy tensor of gauge fields in Einstein s equation should give the couplings of daor field with gauge fields. More results on this problem and the coupling of daor field with spinor field will be given in forthcoming papers. It is stressed that only daor field can embody all the symmetries of space-time. Because complex spinor connection unifies gauge fields and real space-time Cartan s spinor connection, daor field reflects the gravitational effect of gauge fields also. Furthermore, daor field will be a powerful tool to realize the quantization of gravity. Conclusion: In this paper, the daor field which represent gravity is suggested. Oneorder differential equations of daor field in empty space are acquired, being proven to be consistent with Einstein s empty space equation. There are two kinds of symmetry transformations keeping ds 2 invariant. The general coordinate transformation is corresponding to gravity and U(1,3) group transformation to gauge fields. Acknowledgement: I would like to thank C. B. Guan, Y. Q. Li, J. F. Cheng and Y. P. Dai for their useful discussion. 6
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