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The Global Nonlinear Stability of the Minkowski Space
By Demetrios Christodoulou, Sergiu Klainerman PRINCETON UNIVERSITY PRESS
Copyright © 1993 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08777-1
CHAPTER 1
Introduction
The aim of this book is to provide a proof of the nonlinear gravitational stability of the Minkowski space-time. More precisely, our work accomplishes the following goals:
1. It provides a constructive proof of global, smooth, nontrivial solutions to the Einstein-Vacuum equations, which look, in the large, like the Minkowski space-time. In particular, these solutions are free of black holes and singularities.
2. It provides a detailed description of the sense in which these solutions are close to the Minkowski space-time in all directions and gives a rigorous derivation of the laws of gravitational radiation proposed by Bondi. It also describes our new results concerning the behavior of the gravitational field at null infinity.
3. It obtains these solutions as dynamic developments of all initial data sets, which are close, in a precise manner, to the initial data set of the Minkowski space-time, and thus it establishes the global dynamic stability of the latter.
4. Though our results are established only for developments of initial data sets which are uniformly close to the trivial one, they are in fact valid in the complement of the domain of influence of a sufficiently large compact subset of the initial manifold of any "strongly asymptotically flat" initial data set.
According to Einstein, the underlying geometry of space-time is that given by a pair (M, g) where M is a 3+1 -dimensional manifold and g is an Einstein metric on M, that is, a nondegenerate, 2-covariant tensorfield with the property that at each point one can choose 3+1 vectors e0, e1, e2, e3 such that g(eα, eβ) = ηαβ; α, β = 0, 1, 2, 3 where η is the diagonal matrix with entries -1, 1, 1, 1. The Einstein metric divides the nonzero vectors X in the tangent space at each point into timelike, null, or spacelike vectors according to whether the quadratic form = gαβXαXβ is, respectively, negative, zero, or positive.
The set of null vectors forms a double cone, called the null cone of the corresponding point. The set of timelike vectors forms the interior of this cone. It has two connected components whose boundaries are the corresponding components of the null cone. The set of spacelike vectors is the exterior of the null cone, a connected open set. Any physically meaningful space-time should be time orientable, that is, one can choose in a continuous fashion a future-directed component of the set of timelike vectors. This allows us to specify the causal future and past of any point in space-time. More generally, the causal future of a set S [subset] M, denoted by J+(S), is defined as the set of points q that can be reached by a future-directed causal curve that initiates at S. Similarly, J-(S) consists of the set of all points q that can be reached from S by a past-directed causal curve.
The boundaries of past and future sets of points in M are null geodesic cones, often called light cones. Their specification defines the causal structure of the space-time, which, up to a conformal factor, uniquely determines the metric.
A hypersurface M in M is said to be spacelike if its normal direction is timelike at every point on M. We denote by g the Riemannian metric induced by g on M The covariant differentiation on the space-time M will be denoted by D, while that on M will be written with the symbols D or [nabla]. Similarly, we denote by R, respectively R, the Riemann curvature tensors of M, respectively M. Recall that for any given vector fields X, Y, Z on (M, g),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
or, in components, relative to an arbitrary frame eα, α= 0, 1,2, 3,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The extrinsic curvature, or second fundamental form, of M in M will be denoted by k. Recall that if T denotes the future-directed unit normal to M, we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
with ei, i = 1, 2, 3, an arbitrary frame on M.
We will use the notation [member of]αβγδ to express the components of the volume element d]µITLM relative to an arbitrary frame. Similarly, if ei, i = 1, 2, 3 is an arbitrary frame on M, then [member of]ijk = [member of]oijk are the components of dµM, the volume element of M, with respect to the frame e0 = T, e1, e2, e3.
The Riemann curvature tensor R of the space-time satisfies the following:
Biarichi Identities
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The traceless part of the curvature tensor is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where the 2-tensor Rαβ and scalar R are respectively the Ricci tensor and the scalar curvature of the space-time. We call this the conformal curvature tensor of the space-time. We notice that the Riemann curvature tensor has 20 independent components while the conformal curvature and Ricci tensors have 10 components each.
The conformal curvature tensor is a particular example of a Weyl tensor. These refer to an arbitrary 4-tensors W that satisfy all the symmetry properties of the curvature tensor and in addition are traceless. We say that such W's satisfy the Bianchi equation if, with respect to the covariant differentiation on M,
Bianchi Equation
D[εWαβ]γδ = 0.
For a Weyl tensorfield W the following definitions of left and right Hodge duals are equivalent:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [member of]αβγδ are the components of the volume element in M. One can easily check that *W = W* is also a Weyl tensorfield and *(*W) = -W. Given an arbitrary vectorfield X, we can define the electric-magnetic decomposition of W to be the pair of 2-tensors formed by contracting W with X according to the formulas
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
These new tensors are symmetric, traceless, and orthogonal to X. Moreover, they completely determine W, provided that X is not null (see [CH-K]).
Given a vectorfield X and a Weyl field W, Lx W is not, in general, a Weyl field, since it fails to be traceless. To compensate for this, we define its modified Lie derivative
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where π is the deformation tensor of X, that is
π = Lxg.
One can associate [Be-Ro] to the conformal curvature tensor or, more generally, to any Weyl tensorfield W, a 4-tensor that is quadratic in W and plays precisely the same role for solutions of the Bianchi equations as the energy-momentum tensor of an electromagnetic field plays for solutions of the Maxwell equations.
Bel-Robinson Tensor
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Q is fully symmetric and traceless; moreover, it satisfies the positive energy condition, namely, Q(X, Y, Z, I) is positive whenever X, Y, Z, I are future-directed timelike vectors (see [Ch-K]) for a proof of these properties of Q). Moreover,
DδQαβγδ = 0
whenever W satisfies the Bianchi equations. This remarkable property of the Bianchi equations is intimately connected with their conformal properties. Indeed, they are covariant under conformal isometries. That is, if φ : M -> M is a conformal isometry of the space-time, that is, φ*g = Ω2g for some scalar Ω and W is a solution, then so is Ω-1φ*W.
It is well known that the causal structure of an arbitrary Einstein space-time can have undesirable pathologies. All these can be avoided by postulating the existence of a Cauchy hypersurface in M—a hypersurface Σ with the property that any causal curve intersects it at precisely one point. Einstein space-times with this property are called globally hyperbolic. Such space-times are, in particular, stable causal, that is, they allow the existence of a globally defined differentiate function t whose gradient Dt is timelike everywhere. We call t a time function, and the foliation given by its level surfaces a t-foliation. We denote by T the future-directed unit normal to the foliation.
Topologically, a space-time foliated by the level surfaces of a time function is diffeomorphic to a product manifold R×Σ where Σ is a 3-dimensional manifold. Indeed, the space-time can be parametrized by points on the slice t = 0 by following the integral curves of Dt. Moreover, relative to this parametrization, the space-time metric takes the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.1)
where x = (x1, x2, x3) are arbitrary coordinates on the slice t = 0. The function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called the lapse function of the foliation; gij is its first fundamental form. We refer to 1.0.1 as the canonical form of the space-time metric with respect to the foliation.
The foliation is said to be normalized at infinity if
Normal Foliation Condition
φ -> 1 as x -> ∞ on each leaf Σt
The second fundamental form of the foliation, the extrinsic curvature of the leaves Σt, is given by
kij = -(2φ)-1[partial derivative]tgij. (1.0.2)
We denote by [nabla] the induced covariant derivative on the leaves Σt, and by Rij the corresponding Ricci curvature tensor. Relative to an orthonormal frame e1, e2, e3 tangent to the leaves of the foliation, we have the formulas
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [bar.D]Tei denotes the projection of [DTei to the tangent space of the foliation. It is convenient to calculate relative to a frame for which [bar.D]Tei = 0.
Since Σt is three dimensional, we recall that the Ricci curvature Rij completely determines the induced Riemann curvature tensor Rijkl according to the formula
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where R is the scalar curvature gij Rij. The second fundamental form k, the lapse function φ, and the Ricci curvature tensor Rij of the foliation are connected to the space-time curvature tensor Rαβγδ according to the following equations:
The Structure Equations of the Foliation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.3a)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.3b)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.3c)
where [partial derivative]t denotes the partial derivative with respect to t, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the components R([partial derivative]1, T [partial derivative]j, T) and R ([partial derivative]m, T, [partial derivative]i, [partial derivative]j), respectively, of the space-time curvature relative to arbitrary coordinates on Σ. Equation 1.0.3a is the second-variation formula, while 1.0.3b and 1.0.3c are, respectively, the classical Gauss-Codazzi and Gauss equations of the foliation.
In view of 1.0.3c, the equation 1.0.3a becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.3d)
Taking the trace of the equations 1.0.3c, 1.0.3b, and 1.0.3a, respectively, we derive
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.4a)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.4b)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.0.4c)
where |k|2 = kijkij.
In contrast to Riemannian geometry, where the basic covariant equations one encounters are of elliptic type, in Einstein geometry the basic equations are hyperbolic. The causal structure of the space-time is tied to the evolutions feature of the corresponding equations. This is particularly true for the Einstein field equations, where the space-time itself is the dynamic variable.
The Einstein field equations were proposed by Einstein as a unified theory of space-time and gravitation. The space-time (M, g) is the unknown; one has to find an Einstein metric g such that
Einstein Field Equations
Gµv = 8πTµv
where Gµv is the tensor Rµv - ½gµvR, with Rµv the Ricci curvature of the metric, R its scalar curvature, and Tµv the energy momentum tensor of a matter field (e.g., the Maxwell equations). Contracting twice the Bianchi identities D[εRαβ]γβ, we derive
Contracted Bianchi Identities
DvGµv = 0,
which are equivalent to the divergence equations of the matter field
DvTµv = 0.
In the simplest situation of the physical vacuum, T = 0, the Einstein equations take the form
Einstein-Vacuum Equations
Rµv = 0.
In view of the four contracted Bianchi identities mentioned previously, the Einstein-Vacuum equations, or E-V for short, can be viewed as a system of 10–4 = 6 equations for the 10 components of the metric tensor g. The remaining 4 degrees of freedom correspond to the general covariance of the equations. Indeed, if Φ : M -> M is a diffeomorphism, then the pairs (M, g) and (M, Φ*g) represent the same solution of the field equations.
(Continues...)
Excerpted from The Global Nonlinear Stability of the Minkowski Space by Demetrios Christodoulou, Sergiu Klainerman. Copyright © 1993 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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