# Decoupling the momentum constraints in general relativity

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## Abstract

We present a \(2+1\) decomposition of the vacuum initial conditions in general relativity. For a constant mean curvature one of the momentum constraints decouples in quasi isotropic coordinates and it can be solved by quadrature. The remaining momentum constraints are written in the form of the tangential Cauchy–Riemann equation. Under additional assumptions its solutions can be written in terms of integrals of known functions. We show how to obtain initial data with a marginally outer trapped surface. A generalization of the Kerr data is presented.

## Keywords

Initial constraints Conformal method Marginally trapped surfaces## 1 Introduction

*S*. These data have to satisfy the constraint equations

*S*embedded in a 4-dimensional spacetime developing from these data in accordance with the Einstein equations.

*R*are, respectively, the covariant Laplace operator and the Ricci scalar of metric

*g*.

*g*,

*K*) of (5) and then consider Eq. (4) for \(\psi \). Equation (5) were solved analytically or reduced to a simpler system only for conformally flat initial metrics [4, 5, 6, 7] or symmetric data [8, 9, 10, 11, 12]. Exact solutions of (4) are known for the simplest class of data with \(K=0\) and conformally flat

*g*. In other cases, the best what one can do is to prove the existence of solutions [13, 14, 15, 16] or to find them numerically (see [17, 18] for a review).

The aim of this paper is to reduce Eq. (5) to a simpler system. We show that these equations decouple in coordinates closely related to quasi isotropic coordinates of Smarr [20]. One of the equations yields a component *W* of *K* as an integral of free data. Then the remaining two equations can be written as a single complex equation using the Cauchy–Riemann structure related to the metric *g*. Under additional assumptions solutions of this equation can be also represented as integrals of known functions.

In the last section we consider data admitting a marginally outer trapped surface (MOTS) which can be considered as an attribute of a black hole. Known constructions of such data are based on the puncture method of Brill and Lindquist [4], the conformal-imaging method of Misner [5] or the boundary condition method proposed by Thornburg [21]. In the spirit of Misner’s approach we define a class of maximal non-conformally flat data with a reflection symmetry which assures existence of MOTS. These data generalize the Kerr metric data but they don’t have to be axially symmetric.

## 2 The \(2+1\) decomposition of initial data

*S*into surfaces given by constant levels of a function \(\varphi \). In coordinates \(x^i=x^a,\varphi \), where \(i=1,2,3\) and \(a=1,2\), an arbitrary metric

*g*on

*S*can be written in the form

*g*to three. For instance, in this way one can obtain conditions satisfied by the quasi isotropic coordinates [20].

We are not able to describe all initial metrics which admit solutions of (8). Throughout the paper we will assume existence of coordinates \(x^a,\varphi \) such that *g* is given by (7). It is not very strong restriction compared to the assumption \(\tilde{H}=const\). We will work in a single coordinate system. Given a solution of (5) one can try to extend it to an acceptable initial manifold. Coordinates \(x^i\) may be analogs of the Cartesian coordinates in flat space but they may be also related to spherical-like coordinates with a radial distance defined by \(x^1\) and angles \(x^2=\theta \) and \(\varphi \). In the latter case initial data should be periodic in \(\varphi \) and satisfy appropriate conditions at \(\theta =0,\pi \) (see Sect. 3).

*U*,

*V*,

*W*as follows

*g*reads

**Proposition 2.1**

*W*and a complex function

*U*

*E*is given by (21) and

*F*is given up to

*W*by (22). Free data consist of metric

*g*, complex function

*V*, a real \(\varphi \)-independent function \(W_0\) and a complex function \(U_0\) satisfying

*Proof*

*W*up to a real function \(f(\xi ,\bar{\xi })\)

*f*is in one to one correspondence to function \(W_0=W|_{\varphi =\varphi _0}\). Substituting (23) into (22) yields

*F*up to

*f*. Then (15) becomes an equation for

*U*. Its solution, if it exists, is defined up to a solution \(U_0\) of the homogeneous part of (15).

Note that Eqs. (14)–(15) are invariant under transformation (3) with \(\tilde{H}=0\) since (5) is invariant. Using this freedom one can fix one of components of *g*. Another component can be fixed by transformation of coordinate \(\varphi \).

**Proposition 2.2**

*U*is given by

*C*and

*h*is an arbitrary function holomorphic in \(\xi \).

*Proof*

*F*provided that the r. h. s. of (26) still makes sense.

**Proposition 2.3**

*V*be analytic with respect to coordinate \(\varphi \). Then

*Proof*

*W*,

*F*and \(\hat{F}\) are also analytic in \(\varphi \). Thus, we can complexify \(\varphi \) and pass to coordinates \(\xi \), \(\bar{\xi }\) and \(\chi \). In these coordinates Eq. (24) reads

*F*is determined by metric

*g*and functions

*W*and \(\omega \).

*g*as in a known maximal (\(\tilde{H}=0\)) asymptotically flat solution of the full set of initial conditions. Then \(R\ge 0\) since the Hamiltonian constraint is satisfied by this solution. If we take as

*K*another solution of (5) with the same

*g*we can be sure that the Lichnerowicz equation can be solved but there is still problem to find new \(\psi \) numerically.

*U*,

*V*and

*W*are preserved by transformation (3) and \(\rho \rightarrow \psi ^2\rho \), \(\alpha \rightarrow \psi ^2\alpha \)). Equations (37) and (38) have to be considered simultaneously with (4) except the case \(\tilde{H}_{,\varphi }=0\) for which

*W*is given by (23). Existence of solutions of this system is much more difficult to prove [3, 19].

## 3 Horizons

In the theory of black holes it is important to construct initial data with one or more 2-dimensional surfaces which can represent horizons of black holes. This can be done within the conformal method by imposing an appropriate condition on the conformal factor \(\psi \) on an internal boundary of the initial manifold [15]. This boundary becomes MOTS for initial data \(({\tilde{g}},\tilde{K})\) given by (3). Existence of \(\psi \) for asymptotically flat data depends on the positivity of the Yamabe type invariant what is even more difficult to prove than in the case of an unbounded initial manifold (see [12] for partial results). Another problem is that a continuation of initial data through this internal boundary is not assured. This problem does not appear in the inversion symmetry approach of Misner [5] which is applicable to a restricted class of data. In this section we propose a construction of data which follows the latter method.

*t*is the Boyer–Lindquist time coordinate. The initial metric induced by the Kerr solution reads

*r*by

*K*are given by (34) with [12]

*g*and

*K*are invariant under the reflection

*g*has the form (13) and

*g*and

*K*

**Proposition 3.1**

*g*and

*K*satisfy (14)–(15), (46)–(47) and condition (trivial for \(\epsilon =-1\))

*Proof*

The part (i) of the proposition can be easily proved by considering induced transformations of functions *E* and *F* given by (21) and (22). The same refers to (34) and integral formulas for *W* and *U* in Sect. 2 if transformations (48) are taken into account.

*g*implies that the exterior curvature of surface \(x^1=0\) embedded in the initial manifold vanishes. For \(\epsilon =-1\) function \(K^{11}\) is antisymmetric with respect to \(x^1\), hence automatically

*g*,

*K*) and also to data \(({\tilde{g}},\tilde{K})\).

*E*and

*F*in Eqs. (14) and (15) are also invariant. Formula (23) defines periodic

*W*provided that

*U*Eq. (15) should be completed by the condition

*U*under suitable assumptions on functions \(\chi \) and

*h*.

*K*we assume that near \(\theta =0,\pi \) components \(K_{ij}\) of

*K*are proportional to the following powers of \(\sin {\theta }\)

*U*should behave as

*f*of order \(n\le k\) fall off as \(r^{l-n}\) when \(r\rightarrow \infty \).

According to Proposition 3.1 one can relatively easily construct initial data with MOTS assuming symmetry (45) with \(\epsilon =-1\). For instance, let *g* be the Kerr initial metric (39) in coordinates \(\tilde{r}\), \(\theta \), \(\varphi \) and only non-vanishing components of *K* are given by (34) with \(\omega \) which is an even function of \(x^1\). Since \(R\ge 0\) the Lichnerowicz equation with \(\tilde{H}=0\) admits a unique solution \(\psi \) which is reflection symmetric. Given \(\psi \) transformation (3) yields ultimate data which satisfies all initial constraints. Note that we can choose \(\omega \) which tends to (42) if \(x^1=\tilde{r}\rightarrow \infty \). Then the data approaches the Kerr data with the angular momentum *a* if \(x^1\rightarrow \infty \) and with the angular momentum \(-a\) if \(x^1\rightarrow -\infty \).

*W*and

*U*from Sect. 2. Function

*f*in (23) cannot solve this constraint because

*f*is independent of \(\varphi \). Function

*h*in integral formulas for

*U*can be used only for analytic fields. A radical way to avoid problem with (49) is to assume \(W=ReU=\beta =0\). Then Eq. (14) yields \(V=2i\omega _{,\xi }\) and (15) is equivalent to the following conjugate equations for

*ImU*

*f*is a \(\varphi \)-independent function. One can consider (60) as an equation for \(\omega \), in which \(\varphi \) is a parameter, or one can formally solve (60) to obtain

*f*are given. Then \(\omega \) has to satisfy the first order linear equation

## 4 Summary

We have been considering initial constraints for the vacuum Einstein equations in the framework of the conformal approach. We assumed that metric can be put into form (13). One can relate with coordinates \(\xi ,\bar{\xi },\varphi \) the Cauchy–Riemann operator \(\partial \). The momentum constraint with \(\tilde{H}=const\) splits into Eqs. (14) and (15) (Proposition 2.1). Equation (14) can be directly integrated with respect to *W* giving formula (23). In the case of fields analytic in coordinate \(\varphi \) or data with \(\beta =0\) solutions of equation (15) can be also written as integrals of known functions (Propositions 2.2 and 2.3). In order to complete the construction of initial data one has still to solve the Lichnerowicz equation (4) for the conformal factor \(\psi \). Its existence and uniqueness can be easily proved in some cases, e.g. if data (*g*, *K*) are asymptotically flat, \(\tilde{H}=0\) and the Ricci scalar of *g* is nonnegative.

In Sect. 3 we propose a construction of maximal data with a reflection symmetry which, together with (49), implies existence of a horizon in the form of a marginally outer trapped surface (Proposition 3.1). As an example we present data obtained by a modification of the Kerr initial data.

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