Applications of Theorems of Jean Leray to the Einstein-Scalar Field

Applications of theorems of Jean Leray to the Einstein-scalar field equations Yvonne Choquet-Bruhat James Isenberg∗ University of Paris University of Oregon Daniel Pollack† University of Washington November 1, 2006 Dedicated to the memory of Jean Leray Abstract The Einstein-scalar field theory can be used to model gravitational physics with scalar field matter sources. We discuss the initial value formulation of this field theory, and show that the ideas of Leray can be used to show that the Einstein-scalar field system of partial differential equations is well-posed as an evolutionary system. We also show that one can generate solutions of the Einstein-scalar field constraint equations using conformal methods. 1 Introduction In Newtonian theory, one models gravitational physics by studying a lin- ear elliptic Poisson equation for the Newtonian potential on a fixed absolute background space and time, with the motion of material bodies governed by the Newtonian force equation on this fixed background. By contrast, in general relativity the gravitational field is modeled using Lorentzian space- times whose curvature reflects the material and field content of the spacetime. ∗Partially supported by the NSF under Grant PHY-0354659 †Partially supported by the NSF under Grant DMS-0305048 1 Mathematically, a Lorentzian spacetime is a pair (M n+1, g) where M n+1 is a smooth manifold of dimension n+1 (in everyday physics n = 3, but higher dimensions are sometimes considered for modelling electromagnetism and other interactions with gravity), and g is a pseudo-Riemannian metric of signature (−, +,..., +). The metric distinguishes timelike directions g(X,X) < 0 for a tangent vector X (along the possible path for a massive physical object), null directions g(X,X) = 0 (along the possible path for a massless physical particle), and spacelike directions g(X,X) < 0. The physical time between a pair of events, as would be marked by a proper clock, corresponds to the g-length of the timelike trajectory followed by that clock. The path followed by a test particle which is free of non gravitational forces corresponds to a timelike geodesic in the spacetime. It is well known that theoretical studies of general relativity have pre- dicted such strange and interesting phenomena as the expansion of the universe, black holes, gravitational lenses, and gravitational waves. These phenomena, all of which have now been confirmed either by direct or indirect observation, were originally discovered via studies of solutions of the Ein- stein field equations. It is also well known, at least among mathematicians, that Einstein’s equations present a number of very challenging mathematical problems, such as the cosmic censorship conjectures and the question of the nonlinear stability of black holes. In this brief review, we show that in two important ways, the ideas of Leray have played an important role in the study of Einstein’s equations. In previous studies these ideas were applied in the absence of a scalar field. Here, since scalar fields are now viewed as possibly important for understanding the apparent acceleration of the expansion of the universe, and since including them results in some additional interesting features in the analysis, we work with the Einstein-scalar field system. We introduce this system in section 2 and discuss its Cauchy formulation in section 3. We then show in section 4 how Leray’s ideas play a role in understanding the evolutionary aspect of the Einstein-scalar field system. In section 5 we prove the existence of solutions to the Einstein-scalar constraint equations via the conformal methods, where seminal ideas of Leray regarding solutions of nonlinear elliptic equations have played a crucial role. We end this section with a note about notation. Wherever there may arise a possible confusion, we use bold faced symbols for spacetime variables and tensors. For initial data sets, we reserve the notation of over-barred symbols for physical variables, which satisfy the relevant constraint equations, and 2 may or may not be “time-dependent” depending on the context. Unadorned symbols are used for free conformal data as described in section 5. Finally the occasional appearance of a “tilded”-symbol (as in (5.4) below) appears when we need to introduce an intermediate quantity which is neither a free variable, nor a physical one. Acknowledgement: The authors would like to thank the Isaac Newton Institute of Mathematical Sciences in Cambridge, England for providing an excellent research environment during the program on Global Problems in Mathematical Relativity during Autumn, 2005, and again in October 2006, where some of this research was carried out. 2 Einstein - scalar field equations. For general source fields, the Einstein gravitational field equations take the tensorial form G(g)= T(Φ, g), (2.1) where G(g) is the Einstein tensor, which is a second order differential oper- 1 ator on the metric defined by G(g) := Ric(g) − 2 R g with Ric(g) denoting the Ricci tensor of g and R denoting the scalar curvature of g, and where T(Φ, g) is the stress-energy or energy-momentum tensor1, a specified func- tional of the matter source fields Φ and the metric. The specific form that the stress-energy tensor takes depends upon the matter source fields presumed to be present in the physical system being modeled. Here, we presume that Φ is a scalar field, which we label Ψ, with potential function V (Ψ), and we set 1 T = ∂Ψ ⊗ ∂Ψ − |∂Ψ|2 + V (Ψ) g. (2.2) 2 g In addition to the equation (2.1), the Einstein-scalar field theory includes a field equation for Ψ, which reads dV ∇α∂ Ψ= , (2.3) α dΨ where ∇ denotes the covariant derivative compatible with g. While this extra equation may simply be added to the theory by hand, it also follows directly 1Note that we have chosen units so that 8π times the Newtonian gravitational constant is set equal to one. 3 as a necessary consequence of equation (2.1) together with the geometric properties of the Einstein tensor: One readily verifies that the Bianchi iden- tities for the curvature imply that the Einstein tensor satisfies the identity divgG(g)=0 (2.4) where divg denotes the divergence operator for the metric g. This condition together with (2.1) implies the conservation law divgT =0. (2.5) We readily verify that (2.5) applied to the scalar field stress-energy tensor (2.2) results in the field equation (2.3). How does one choose the scalar field potential function V (Ψ)? While m 2 there are many possibilities, we note that V (Ψ) = 2 Ψ corresponds to the massive Klein-Gordon field, while setting V (Ψ) = Λ for a non zero constant Λ and requiring that Ψ = 0 produces the vacuum Einstein theory with non zero cosmological constant Λ. 3 The Cauchy problem: constraints and evolution. The Einstein-scalar field system of partial differential equations on a 3+1 spacetime consists of eleven equations (2.1) and (2.3) for the eleven field vari- 1 ables gµν and Ψ (for n+1 dimensions, there are 2 (n+1)(n+2)+1 equations for the same number of field variables). One of its most characteristic features, however, is that it is both an under-determined and an over-determined system, in the sense that if one formulates the Einstein-scalar-field system as a Cauchy problem, there are constraint equations which must be satisfied by any candidate set of initial data, and as the data evolves there are certain of the field variables whose evolution is entirely at one’s discretion. Both of these features reflect the spacetime covariance of the theory (i.e., the theory has the spacetime diffeomorphism group as its gauge group). To see these features explicitly, we now sketch out an n + 1 decomposi- tion of the Einstein-scalar field variables and equations. Given a spacetime (M n+1, g), we start by choosing an n + 1 foliation of the spacetime manifold n n+1 n Ft : Σ → M (t ∈ R), for which each of the leaves Ft(Σ ) of the foliation 4 is presumed spacelike. We also choose a threading of the spacetime by a con- n+1 n gruence of timelike observer paths Tp : R → M (p ∈ Σ ). The choice of a foliation and a threading, together with a choice of coordinate patches for Σn, automatically determines local coordinates (x0 = t, x1,...,xn) and local ∂ ∂ ∂ n+1 coordinate bases ( ∂t , ∂x1 ..., ∂xn ) covering M . We may then, without loss of generality, write the metric (locally) in the form 2 t t i j g = −N θ ⊗ θ +¯γijθ ⊗ θ where (θt = dt, θj = dxj + βjdt) is the one-form basis dual to the surface- 1 ∂ j ∂ ∂ ∂ ⊥ ⊥ compatible tangent vector basis (e = N ( ∂t − β ∂xj ), ∂x1 ..., ∂xn ) with e n the normal vector field to Ft(Σ ). Here N is the positive definite “lapse j function”, β are the components of the spacelike “shift vector”, andγ ¯ij are the components of the spatial metric tensor. We note that for each choice i j of t,γ ¯(t)=¯γij(t)dx ⊗ dx is the induced Riemannian metric on the leaf n i j Ft(Σ ). We use the notation K¯ (t) = K¯ij(t)dx ⊗ dx to denote the second fundamental form defined by the foliation. For the scalar field Ψ, there is no need to do any space + time decom- position. However, we shall use the notation ψ¯ to denote the restriction of Ψ to one of the leaves of the chosen foliation, and we shall use the definition 1 ∂ ¯ m ∂ ¯ π¯ := N ( ∂t ψ − β ∂xm ψ) for convenience in working with the time derivative of Ψ.

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