Polynomial form of the Hilbert Einstein action

Transcription

1 1 Polynomial form of the Hilbert Einstein action M. O. Katanaev Steklov Mathematical Institute, Gubkin St. 8, Moscow, , Russia arxiv:gr-qc/ v1 7 Jul July 2005 Abstract Configuration space of general relativity is extended by inclusion of the determinant of the metric as a new independent variable. As the consequence the Hilbert Einstein action takes a polynomial form. Let us consider space-time M of arbitrary dimension dim M = n 3. We do not deliberately restrict ourselves to the most interesting case n = 4 because gravity models in lower and higher number of dimensions attract much interest last years. Local coordinates of the space-time are denoted by x α, α = 0, 1,..., n 1. Equations of motion for the metric g αβ (x) in general relativity in the absence of matter fields follow from the variational principle for the Hilbert Einstein action [1, 2] with a cosmological constant Λ S he = d n x g [R (n 2)Λ], g = det g αβ, (1) R = R(g) is the scalar curvature for the metric g αβ. We use the following definitions of the curvature and Ricci tensors and the scalar curvature R αβγ δ = α Γ βγ δ β Γ αγ δ Γ αγ ǫ Γ βǫ δ + Γ βγ ǫ Γ αǫ δ, R αβ = R αγβ γ, R = g αβ R αβ, Γ αβ γ = 1 2 gγδ ( α g βδ + β g αδ δ g αβ ) are Christoffel s symbols. The coefficient in front of the cosmological constant is chosen in such a way that the Einstein equations take the form R αβ = Λg αβ. (2) The expression for the scalar curvature contains the inverse metric g αβ whose components are not polynomial in g αβ. Therefore the Hilbert Einstein action in a perturbation theory is represented by a very complicated infinite series, which is the main difficulty in the analysis of the equations of motion and quantization. katanaev@mi.ras.ru

2 2 Coordinates of the configuration space M in general relativity are components of the metric g αβ (x). The dimensionality of this space is equal to dim M = n(n + 1) 2 (n 1), the symbolic factor (n 1) corresponds to points of space in a space-time M. Let us consider different configuration space N with coordinates (x), k αβ (x) assuming that > 0 and the matrix k αβ is symmetric and nondegenerate in every point of a spacetime: k αβ = k βα, det k αβ 0. We assume also that matrixes g αβ and k αβ have the same Lorentzian signature. The dimensionality of the new configuration space is ( ) n(n + 1) dim N = + 1 (n 1). 2 We define the subspace M in N using the constraint { 1, det g αβ > 0, det k αβ = 1, det g αβ < 0. Then we can define the one-to-one correspondence between points of the subspace M N and the original configuration space M The inverse transformation is (3) g αβ = 2k αβ. (4) = detg αβ 1/2n, k αβ = 2 g αβ. Hence we identify M = M. The representation for the inverse metric g αβ = 2 k αβ k αβ k βγ = δ α γ follows from Eq.(4). Let us make two important notes. Firstly, the components of the inverse matrix k αβ are polynomial in k αβ. In a general case they are polynomials of order n 1 with respect to components k αβ as the consequence of Eq.(3). Secondly, the components k αβ are components of a second rank tensor density but not a tensor. Indeed, we require Eq.(4) to be fulfilled in an arbitrary coordinate system. Since the determinant of the metric g is a scalar density of weight deg g = 2, then the matrix k αβ is the symmetrical tensor density of second rank and weight deg k αβ = 2/n, and the field is the scalar density of weight deg = 1/n. This means the following transformation under coordinate changes x α x α (x) k α β = α xα β x β k αβ J 2/n, = J 1/n. J = det α x α is the Jacobian of the coordinate transformation. It means that the constraint (3) is invariant under coordinate transformations. Hence we call the matrix valued field k αβ (x) the metric density. Now we rewrite the Hilbert Einstein action in terms of the new variables, k αβ. Differentiations of Eq.(3) yield the identities needed for the calculations k γδ α k γδ = 0, k γδ 2 αβ k γδ k γδ k ǫζ α k γǫ β k δζ = 0. The expression for the scalar curvature is now easily found R = 4 [ 2R (k) + 2(n 1) α (k αβ β ) + (n 1)(n 4) 2], (5)

3 3 we introduced the shorthand notation 2 = k αβ α β. The scalar curvature R (k) for the metric density k αβ takes the surprisingly simple form R (k) = αβ k αβ kαβ α k γδ γ k βδ 1 4 kαβ α k γδ β k γδ. (6) Note that this expression is polynomial in metric density k αβ as well as in its inverse k αβ. Similar to general relativity, the second derivatives 2 αβ kαβ and α (k αβ β ) can be excluded from the action (1) by substraction of the boundary term α ( n 2 β k αβ + 2(n 1) n 3 k αβ β ). The resulting action takes the form S he div = d n xl, (7) L = n 4 [ 1 2 2k αβ α k γδ γ k βδ 1 4 kαβ α k γδ β k γδ (n 2) α k αβ β (n 1)(n 2) 2 (n 2)Λ 4 ]. (8) For n 4 this Lagrangian is polynomial in the fields, k αβ. The action is invariant under coordinate transformations up to boundary terms by construction. One has only to remember that the fields and k αβ are not tensors but tensor densities. The Lagrangian (8) is similar to the Lagrangian of the dilaton gravity the determinant of the metric plays the role of the dilaton. It differs from the usual dilaton models by the presence of the cross derivative term α k αβ β. Besides, it contains less number of independent fields because the constraint (3) is imposed on the metric density. The extra constraint on the metric density (3) can be taken into account by addition of the constraint to the Lagrangian L L + λ( det k αβ ± 1), λ is a Lagrangian multiplier. This procedure is not necessary. Variations of the metric density are restricted due to Eq.(3) δ det k αβ = ±k αβ δk αβ = δk αβ k αβ = 0. The action (7) can be varied with respect to k αβ or k αβ considering all components as independent and afterwards taking the traceless part of the resulting equations. The Euler Lagrange equations for the action (7) are equivalent to the Einstein equations (2). In new variables they are 2 ( R (k) αβ 1 n k αβr (k) ) + 2(n 1) + (n 1)(n 4) 2 + R (k) 2 = nλ 4, (9) + (n 2)( α β α β ) n 2 n k αβ( 2) = 0, (10)

4 4 α = α, = α (k αβ β ), α β = αβ 2 Γ(k) γ αβ γ, (11) and Christoffel s symbols Γ (k) αβ γ are computed for the metric density k αβ. Note that covariant derivatives for tensors and tensor densities coincide as the consequence of Eq.(3) because Γ (k) αβ β = 0. The Ricci tensor in Eq.(10) has the form R (k) αβ = 1 2 kγδ ( 2 γδk αβ 2 αγk βδ 2 βγk αδ ) 1 2 γk γδ ( α k βδ + β k αδ δ k αβ ) 1 4 αk γδ β k γδ 1 2 kγδ k ǫζ ( γ k αǫ δ k βζ γ k αǫ ζ k βδ ). Equations (9) and (10) are the trace and traceless parts of Einstein s equations (2), respectively. Hence for n 4 the action (7), the Euler Lagrange equations (9), (10) and the constraint (3) are polynomial in the fields, k αβ. This important to our mind simplification is achieved by extending the configuration space with the introduction of additional field variable. If the constraint (3) is solved with respect to one of the metric density components k αβ and the solution is substituted back into the action (7), then polynomiality will be lost. Note that the introduction of new field and constraint is not an extraordinary trick: the original metric g αβ already contains nonphysical degrees of freedom to be excluded from the theory by means of solution of the gauge conditions and constraints present in general relativity. We increase simply the number of field variables and constraints leaving the physical degrees of freedom untouched. In the Hamiltonian framework the above procedure means the following. The phase space corresponding to the variables, k αβ is also extended. There arises the additional constraint on the momenta (the trace of the momenta corresponding to k αβ must be zero) which together with the constraint (3) form a pair of second class constraints. However the total phase space will be no longer symplectic but only Poisson manifold because the Poisson structure will be degenerate. This question will be discussed else [3]. Addition of a scalar ϕ(x) and electromagnetic A α (x) fields preserves polynomiality of the action and the Euler Lagrange equations. For the minimal coupling we have L s = g[g αβ α ϕ β ϕ V (ϕ)] = n 2 [k αβ α ϕ β ϕ 2V (ϕ)], L em = gg αβ g γδ F αγ F βδ = n 4 k αβ k γδ F αγ F βδ, V (ϕ) is a potential for a scalar field including the mass term, and F αβ = α A β β A α is the electromagnetic field strength. In four-dimensional space-time Eq.(9) is simplified R(k) = 2 3 Λ 3. (12) Let us drop for a moment the constraint (3) on the metric density and consider k αβ and as a metric and a scalar field. Then Eq.(12) is covariant with respect to conformal transformations k αβ = Ω 2 k αβ, = Ω 1, (13) Ω(x) > 0 is a two times differentiable function. It was considered in [4] for Λ = 0 and in [5] for Λ 0. The present approach is essentially different since the metric

5 5 density k αβ is subjected to the constraint (3) which clearly breaks conformal invariance. However the appearance of the factor 1/6 is not accidental because the parameterization (4) coincides with the conformal transformation (13) in its form. Gravity models for a metric with unit determinant were considered in physics repeatedly. In generally covariant theories like in general relativity we have an arbitrariness in choosing a coordinate system which can be utilized. It is not difficult to prove that there is a coordinate system in a neighborhood of an arbitrary point of space-time the modulus of the determinant of a metric equals unity detg αβ = 1. Such coordinate systems are defined up to coordinate transformations with unit Jacobian. The condition det g αβ = 1 simplifies essentially many formula, in particular, the expression for the scalar curvature. This fact was used by Einstein inventing general relativity [2]. The gravity model based on the metric with unit determinant and not invariant under general coordinate transformations was considered in [6]. There Eq.(10) for = 1 was taken as the equation of motion for the metric. More involved transformation of the phase space leading to polynomial constraints was considered in [7]. In the present case the polynomial action (7) is invariant under general coordinate transformations up to boundary terms and is equivalent to the Hilbert Einstein action. It is very likely that the proposed form of the action will be helpful in the construction of quantum gravity. The author is sincerely grateful to I. V. Volovich for the discussion of the article. This work is supported by RFBR (grant ) and the program of support for leading academic schools (grant AS ). References [1] D. Hilbert. Nachrichten K. Gesellschaft Wiss. Göttingen, Math.-phys., Heft 3:395, Klasse [2] A. Einstein. Ann. d. Phys., 49(1916)769. [3] M. O. Katanaev. In preparation. [4] R. Penrose. In Relativité, Groupes et Topology / Les Houches Lectures, 1963, Summer School of Theor. Phys., Univ. Grenoble, Gordon & Breach, New York, 1964, pp ; N. A. Chernikov and E. A. Tagirov. Ann. Inst. Henri Poincaré, A9(1968)109; N. H. Ibragimov. In Dynamics of continuous media. V.1, Novosibirsk, 1969, pp [in Russian]. [5] C. G. Callen, S. Coleman, and R. Jackiw. Ann. Phys., 59(1970)42. [6] W. G. Unruh. Phys. Rev., D40(1989)1048. [7] A. Ashtekar. Phys. Rev., D36(1987)1587.