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NONLINEAR WAVE EQUATIONS FOR RELATIVITY

Graviational radiation is described by canonical Yang-Mills wave equations on the curved space-time manifold, together with evolution equations for the metric in the tangent bundle. The initial data problem is described in Yang-Mills scalar and vector potentials, resulting in Lie-constraints in addition to the familiar Gauss-Codacci relations.


Introduction and outline of the approach
The asymptotic gravitational wave structure resulting from the coalescence of astrophysical black holes or neutron stars is receiving wide attention in connection with the gravitational wave detectors currently under construction. A compact binary system produces strongly nonlinear gravitational waves of which a small residue escapes to infinity as radiation. The asymptotic wave structure at a distant observer is in quantitative relation to the system parameters of compact binaries, which makes gravitational radiation a new spectrum for astronomical observations. The prediction of gravitational wave forms is presently pursued by numerical simulation. The current approaches are based the 3+1 Hamiltonian formulation by Arnowitt, Deser and Misner [2], with the notable exception of strategies employing null-coordinates (e.g. [7]) or systems of conservation laws [5].
The significance of gravity waves in compact binaries, both in the process of coalescence and in radiation, suggests to focus on a description of relativity by nonlinear wave equations. Wave equations provide a proper frame-work for the physics of nonlinear wave motion, establish connection with electromagnet-ics (abelian and linear) and Yang-Mills theory (nonabelian and nonlinear; see, e.g., [1]) and may complement present approaches in numerical relativity by way of understanding or circumventing some of the computational difficulties. This paper establishes a first step in this direction. Generally, wave equations provide a setting appropriate for numerical implementation via cylindrical or spherical coordinate systems, including outgoing boundary conditions at grid boundaries, and offer a possible setting for numerical treatment of ingoing horizon boundary conditions.
Pirani [4] presented compelling arguments to concentrate on the Riemann tensor in describing gravitational radiation. One can proceed by proposing to establish a numerical algorithm based on the divergence of the Riemann tensor [11], R abcd , in which the tensor acts as a source term. A tetrad {(e µ ) b }-formalism takes the Riemannian formulation into equations of Yang-Mills type. With Lorentz gauge on the connection 1-forms, ω aµν , (∇ a denotes the covariant derivative associated with the metric, g ab ), fully nonlinear, canonical Yang-Mills wave equations are obtained Here,✷ is the non-abelian generalization of the Laplace-Beltrami wave operator. The wave equations are also derived for the scalar, Ricci rotation coefficients. Complemented by the equations of structure for the evolution of the tetrad legs, a complete system of evolution equations is obtained which is amenable to numerical implementation. This interwoveness of wave motion and causal structure distinguishes gravity from the other field theories. In the present description, this two-fold nature of gravity has thus been made explicit. Essentially, gravitational waves are now propagated by wave equations on the (curved) manifold, while the metric is evolved in the (flat) tangent bundle by the equations of structure. Of course, such two-fold description is only meaningful for wave motion with wave lengths above the Planck scale, below which the causal structure is not welldefined and wave motion can not be distinguished from quantum fluctuations.
Because the wave equations are second order in time, conditions on the initial data involve both ω aµν and their normal Lie derivates L n ω aµν . In addition to the Gauss-Codacci relations, ω aµν and L n ω aµν are related to the initial energy-momentum and τ bµν . The latter follow from Gauss-Riemann relations, as non-abelian generalizations of Gauss's law.

Equations for R abcd
We will work on a four-dimensional manifold, M , with hyperbolic metric g ab . In a given coordinate system {x b } the line-element is given by The natural volume element on M is ǫ abcd = √ −g∆ abcd , where g denotes the determinant of the metric in the given coordinate system, and ∆ abcd denotes the completely antisymmetric symbol. Covariant differentiation associated with g ab will be referred to by ∇ a . The geometry of M is contained in the Riemann tensor, R abcd , which satisfies the Bianchi identity Using the volume element ǫ abcd , the dual * R is defined as 1 2 ǫ ···ef ab R ef cd . The Bianchi identity then takes the form ∇ a * R abcd = 0.
Einstein proposed that the Ricci tensor R ab = R c acb and scalar curvature R = R ·c c are connected to energymomentum, T ab , through Geometry now becomes dynamical with (5) (see, e.g., [13]) In the presence of Einsteins equations (7), therefore, becomes The quantity on the right-hand side shall be referred to as τ bcd . In vacuo, E has been discussed by Klainerman [8], who refers to E (with τ bcd = 0) together with (5) as the spin-2 equations. The tensor τ bcd is divergence free: in consequence of the conservation laws ∇ a T ab = 0, and in agreement with ∇ b ∇ a R abcd = 0.

Yang-Mills equations
In the language of tetrads, additional invariance arises due to the liberty of choosing the tetrad position at each space-time point. This invariance is described by the Lorentz group, and introduces a vector gauge invariance in the form of the connection 1-forms, governed by Yang-Mills equations as outlined below. The formal arguments can be found in the theory of supersymmetry (see, e.g., [1,6,3]).
Neighboring tetrads introduce their connection 1forms, ω aµν , The connection 1-forms ω aµν serve as Yang-Mills connections in the gauge covariant derivativê satisfying∇ a (e µ ) b = 0. Here, the commutator is defined by its action on tensors φ a1···a k α1···α l as In particular, we have In what follows, Greek indices stand for contractions with tetrad elements: The Yang-Mills construction thus obtains for the Bianchi identity∇ and for the equivalent of E The Bianchi identity (16) introduces the representation (cf. [13]) which will be central in our discussion. The antisymmetries in the Riemann tensor introduce conditions on initial data on an initial hypersurface, Σ. If ν b denotes the normal to Σ, and we obtain the Gauss-Riemann relations where ρ cd = ν b τ bcd . These may be considered generalizations of Gauss's law in electromagnetism. Conditions (20) find their equivalents in the tetrad formulation. To this end, we write, analogous to (19), on Σ the derivative aŝ The Gauss-Riemann relations (20) thus become

Evolution equations for the tetrads
The definition of the connection 1-forms gives the equations of structure [13] become freely specifyable functions. The evolution equations for the tetrad legs thus become The tetrad lapse functions N µ are related to the familiar lapse and shift functions in the Hamiltonian formalism through We will now turn to evolution equations for the connection 1-forms.

Equations for ω aµν
We define a Lorentzian cross-section of the tangent bundle of the space-time manifold by [11] c µν := ∇ d ω dµν = 0.
The Lorentz gauge (27) provides 1 a complete, six-fold connection between neighboring tetrads. The six constraints c µν = 0 are incorporated in E ′ by application of the divergence technique [10,12]: Recall the transformation rule for the connection (see, e.g. [3]) In the present tetrad language, this gauge trans-

Existence of Lorentzian crosssection
To proceed, we consider in an open neighborhood N (Σ) of Σ with Gaussian normal coordinates {τ, It is convenient to employ language of scalar and vector potentials, Φ µν and A aµν , respectively, defined in Denoting the effect of (30) via (29) by a superscript (r), we have so that Using a geodesic extension of the normal ν b off Σ, constraints (27) become By choice of σ µν = c µν , the constraints (27) transform into Notice that bringing the tetrad in Lorentz gauge (37) by (30) is achieved by proper second time-derivative ∂ 2 τ (E µ ) b |τ =0 of its legs, leaving the tetrad position and its first time-derivative as invariants at τ = 0.
We shall now establish that E ′′ maintains the Lorentz gauge (27) in the future domain of dependence of Σ. The inhomogeneous Gauss-Riemann relation (20) is implied by antisymmetry of the Riemann tensor in its coordinate indices, and gives The inhomogeneous Gauss-Riemann relations are gauge covariant, so that we are at liberty to consider initial data satisfying both (38) and the gauge choice (37), whence In (28) c µν satisfies a homogeneous Yang-Mills wave equation (cf. [10,12]), It follows that the scalar fields c µν are solutions to an initial value problem for homogeneous wave equations (40) with trivial Cauchy data (39). Consequently, which establishes that solutions to E ′′ are solutions to E ′ . We now elaborate further on (28).

Canonical wave equations
In Lorentz gauge (27), the divergence equation E ′′ obtains wave equations for the connection 1-forms through the representation of the Riemann tensor (18). Indeed, by explicit calculation, we havê Here, we have used c µν = 0, so that∇ a c µν = ∇ a c µν .✷ is used to denote the Yang-Mills wave operator∇ c∇ c . The Ricci tensor R ab in (42) is understood in terms of T ab using Einstein's equations.
Similarly, we can obtain a system of scalar equations for the Ricci rotation coefficients, ω αµν = (e α ) a ω aµν . This involves a number of manipulations, the result of which iŝ The two-fold nature of gravity can now be expressed as The coupling of matter to the connections is accounted for by τ bcd . It is of interest to note that τ bcd contains vorticity; a detailed enumeration of the nature of τ bcd falls outside the scope of this dicsussion.
The Theorem suggests some computational approximations for weakly nonlinear gravity waves. Firstly, consider equations with uncoupled (prescribed or fixed) metric, with given Laplace-Beltrami wave operator, ✷ g . Thus, we have (in vacuo) with the metric g ab as a background field providing the underlying causal structure. The approximate radiation then follows from the fluctuations in the metric as would follow from the equations of structure. Secondly, small amplitude waves allow us to linearize, thereby obtaining the purely abelian wave equations where η refers to the Lorentz metric. Numerical experiments must show the validity of such approximations.

Initial value problem
Initial data for the wave equations are ω aµν and its Lie derivative L n ω aµν on Σ. These data must satisfy certain constraints on Σ, commensurate with the initial distribution of energy-momentum and the Gauss-Riemann equations (22). We shall express these equations in terms of scalar and vector potentials (31). The projection tensor, h ab , onto Σ is The covariant derivative induced by h ab shall be denoted byD a , i.e.,D a h cd = 0. In what follows, (ν c ∇ c )f is also denoted byḟ . The unit normal is extended geodesically off Σ, so that ν c ∇ c ν b = 0 and

Expressions for A aµν
Consider a normal tetrad, {(E µ ) b }, in which one of the legs is (initially) everywhere normal to the initial hypersurface: Extrinsic curvature, K ab , of Σ can be defined 'static' by projection of the unit normal following parallel transport over Σ, and 'dynamic' by the Lie derivative, 1 2 L n h ab , of the projection operator h ab with respect to a Gaussian normal coordinate, n. With a normal tetrad, its (extrinsic) helicity, i.e, the twist in a strip swept out by the integral curves of a leg (E ν ) b passing through an integral curve of leg (E µ ) b is For µ, ν = n, H µν describes twist in Σ when µ = ν, and bending when µ = ν. It follows that in view of (48): ν a (E α ) a = −δ ν α on Σ. The normal (µ = n) and tangent (µ = n) extrinsic helicities are now By (51), the symmetry of K ab and A µnν = −A µνn , we have for µ, ν = n the symmetry If µ = n, (E µ ) a = h ab (E µ ) b , and so If also ν = n,D a acts in (54) on tangent legs (E ν ) b , the result of which is determined by h ab . We shall writeĀ aµν = A aαβ h α µ h β ν for this intrinsic part of A aµν . In the normal tetrad, therefore, A aµν falls into two groups: (i) the extrinsic part A µνn = K µν , and the (ii) intrinsic partĀ aµν , with which we have

Expressions for R abµν
On Σ, R abµν contains intrinsic, three-dimensional, and extrinsic parts by substitution of (31): Using (55), the latter may be reduced to whereD a K µb = D a K µb + A γ aµ K γb . Notice that the intrinsic, three-curvature (3) R abcd of Σ associated with h ab satisfies It follows that This last equation generalizes a similar expression obtained in [9], where µ, ν = n is considered.

Energy-momentum relations
Constraints related to the initial distribution of energymomentum are obtained by consideration of the Ricci tensor. To this end, consider the two expressions (56) and (59) for R abµν of the previous section in Consequently, the elements of the Ricci tensor become The second expression (59) for the Riemann tensor also also gives resulting in the familiar identity By Einsteins equations, we have Combining the results above, we have the constraints on the Lie derivative of A aµν in together with the familiar Gauss-Codacci relations We shall refer to (65)-(66) as the energy-momentum constraints.
In going from a first order Hamiltonian description to a second order description, it is not surprising to encounter also constraints on the Lie derivative, reflecting the conditions that the Gauss-Codacci relations also must be satisfied on a future hypersurface in N (Σ).

Gauss-Riemann relations
The Gauss-Riemann relations (22) can similarly be obtained in terms of Φ µν and A aµν . The antisymmetry of the Riemann tensor in its coordinate indices implies By (56), it follows that Therefore, the first, inhomogeneous Gauss-Riemann constraint in (22) reads where the symmetry of the extrinsic curvature tensor was used in going from the third to the fourth equation.
These six constraints on L n A aµν arise in view of the functional relationship Because 1 2 L n h ab = K ab and K aµ = A aµn , six degrees of freedom in To summarize, the constraints on the initial data in terms of potentials are Gauss-Codacci: Lie-constraints: Lorentz gauge:
With this restricted Lorentz gauge, the Lie-constraints reduce toD We remark it does not appear to be feasible to ensure Φ = 0 throughout D + (Σ) by suitable choice of restricted Lorentz gauge on the initial data.