Black-Hole Initial Data

Black-hole initial data solve the Einstein constraints on a slice whose topology or inner boundary represents one or more holes. The familiar choices— isometry throats, punctures, excision, conformal flatness, and analytic spin/boost tensors—answer different mathematical questions. No single choice determines a unique astrophysical binary. Nonphysical initial radiation arises whenever the chosen free data fail to match the geometry and time derivatives of the intended quasiequilibrium spacetime.

Required background. Hypersurface Normals and Tangential Projectors supplies spatial and horizon geometry, while PDEs and Green Functions for Gravitation supplies elliptic domains, inner boundary conditions, and singular-source methods.

Helpful background. The Einstein Equation supplies the constraints; Formulations of General Relativity locates the construction within this volume; and Presymplectic Potentials and Currents prepares the distinction between prescribed canonical data and physical charges.

Three ways to represent a hole

For time-symmetric conformally flat Schwarzschild data,

$$\gamma_{ij}=\psi^4\delta_{ij}, \qquad \psi=1+\frac{GM}{2r}, \qquad K_{ij}=0.$$

Here $M\gt0$ is the physical mass, $r\gt0$ is an isotropic coordinate radius, and the cosmological constant vanishes. The convention is $K_{ij}=-\gamma_i{}^\mu\gamma_j{}^\nu\nabla_\mu n_\nu$ for the future unit normal $n^\mu$. Time symmetry means that the entire $K_{ij}$ vanishes; maximal slicing below requires only its trace $K=0$.

The sphere $r=GM/2$ is a minimal surface and an inversion isometry exchanges the two asymptotically flat ends. This leads to three related but operationally different descriptions:

Representation	Elliptic domain	Inner prescription	Main tradeoff
Isometry throat	One exterior copied across a minimal surface	Inversion or throat condition	Geometrically transparent but restrictive
Puncture	Euclidean space with isolated points removed	Analytic singular part plus regular correction	No inner numerical boundary, but another end is compressed to a point
Excision	Exterior of one or more compact regions	Horizon or quasiequilibrium boundary data	Flexible geometry, but boundary conditions must be mutually compatible

Gourgoulhon derives the minimal-surface condition and the two-ended Schwarzschild bridge in Gourgoulhon 2007, §8.2.5, eqs. (8.52)–(8.66), pp. 132–136, PDF. For moving holes, the distinction is no longer merely a choice of coordinates: different free data and inner conditions generally produce different physical initial slices.

Why the puncture is a second infinity

The area of an isotropic coordinate sphere is $4\pi R^2$, where

$$R=r\psi^2=r+GM+\frac{(GM)^2}{4r}.$$

Thus decreasing $r$ does not indefinitely decrease physical area. The derivative $dR/dr=1-(GM)^2/(4r^2)$ vanishes at $r=GM/2$, where $R=2GM$. The same areal radius grows without bound both as $r\to\infty$ and as $r\to0$. With $x=\ln(2r/(GM))$, this becomes the useful symmetric expression

$$\frac{R}{GM}=1+\cosh x.$$

The radius plot makes the two ends visible: follow the curve toward either large positive or large negative $x$, and compare each limit with the minimum at the throat.

For the time-symmetric Schwarzschild slice, $m=GM$ and $R/m=1+\cosh x$, where $x=\ln(2r/m)$. The minimum $R=2m$ is the throat; $r\to0$ and $r\to\infty$ are separate asymptotically flat ends. This is a quantitative radius plot, not an embedding diagram.

The inversion $r'=(GM)^2/(4r)$ sends $x\mapsto-x$ and leaves the complete three-metric unchanged. In the $r'$ chart the limit $r\to0$ is another ordinary asymptotically flat region $r'\to\infty$. It also lies at infinite proper distance: along a radial line, $d\ell=\psi^2dr\sim (GM)^2dr/(4r^2)$. The divergence of $\psi$ at the puncture is therefore a coordinate description of an end, not a material delta source on the physical three-manifold.

The throat condition can be checked directly rather than inferred from a coordinate picture. Its outward spatial unit normal is $s=\psi^{-2}\partial_r$, and its mean curvature within the initial slice is

$$D_i s^i =\frac{1}{\psi^6r^2}\partial_r(\psi^4r^2) =\frac{2}{\psi^2r}\left(1+2r\frac{\partial_r\psi}{\psi}\right).$$

For time-symmetric data the outgoing null expansion equals this mean curvature, so a minimal sphere is marginally outer trapped. Setting the last factor to zero gives $\partial_r\psi=-\psi/(2r)$ at the throat: this is the Robin boundary condition used if only one exterior is retained. In this exact Schwarzschild example the sphere is the bifurcation sphere; identifying an event horizon in general initial data additionally requires the spacetime development.

Bowen–York momentum and spin

On a flat conformal background with maximal slicing $K=0$, the Bowen–York trace-free tensor solves the momentum constraint analytically. Keep $G$ explicit in the interpretation of its parameters: the $P^i$ and $S^i$ used below are geometrized, so the corresponding physical ADM charges are $P^i/G$ and $S^i/G$. The barred tensor is conformal data, related to the physical extrinsic curvature by $K^{ij}=\psi^{-10}\bar A^{ij}$ in this $K=0$ sector, as derived in Conformal Decomposition of the Einstein Constraints. With $n^i=x^i/r$ the Euclidean radial unit vector (distinct from the spacetime normal), the linear-momentum part is

$$\bar A_P^{ij} =\frac{3}{2r^2} \left[ P^in^j+P^jn^i -(\delta^{ij}-n^in^j)P_kn^k \right],$$

and its spin part is

$$\bar A_S^{ij} =\frac{3}{r^3} \left( \epsilon^{ikl}S_kn_l n^j +\epsilon^{jkl}S_kn_l n^i \right).$$

Both are trace free and divergence free away from the puncture. Surface integrals recover the geometrized parameters $P^i$ and $S^i$; restoring physical units therefore introduces the factors of $1/G$ stated above. Bowen and York give the original construction and its surface charges in Bowen and York 1980, §§II–III, pp. 2049–2053. The Hamiltonian constraint remains nonlinear:

$$\Delta\psi +\frac18\bar A_{ij}\bar A^{ij}\psi^{-7}=0, \qquad \psi\longrightarrow1 \quad(r\longrightarrow\infty).$$

For puncture data one separates the known singular behavior, for example $\psi=1+m/(2r)+u$, where the geometrized mass parameter $m$ has dimensions of length (and equals $GM$ for a single Schwarzschild puncture in these units), and solves for a regular correction $u$. Gourgoulhon gives the Bowen–York solution and verifies its charges in Gourgoulhon 2007, §8.2.6, eqs. (8.67)–(8.84), pp. 136–139, PDF.

Where nonphysical radiation enters

Conformal flatness is exact for the standard time-symmetric Schwarzschild slice, but not for the analogous stationary rotating Kerr geometry. A spinning Bowen–York hole therefore cannot be exactly a stationary Kerr slice under the usual smoothness and symmetry assumptions. Its evolution must rearrange both the hole and the exterior gravitational field, producing a burst commonly called junk radiation.

For a binary, additional mismatches arise from superposing single-hole fields, choosing approximate orbital parameters, setting the wrong radial velocity, or imposing time symmetry where an inspiral has nonzero time derivatives. These effects should be separated conceptually:

• constraint violation is a failure to solve the elliptic system;
• junk radiation can remain even when the constraints are solved accurately;
• eccentricity reflects an orbital mismatch and need not coincide with the radiative mismatch;
• gauge transients can alter coordinate diagnostics without changing the physical radiation content.

The conformal transverse-traceless construction prescribes a TT curvature seed. Conformal thin-sandwich and extended thin-sandwich constructions instead prescribe conformal metric velocity and lapse-related data, making approximate quasiequilibrium conditions more direct. This advantage is not a theorem of uniqueness: the extended system can possess multiple solution branches.

Excision and horizon data

An excision surface intended to represent a quasiequilibrium black hole is often required to have vanishing outgoing null expansion. Conditions on its shear, tangential shift, lapse, or normal motion then encode additional physical choices. The expansion condition alone does not fix the entire elliptic boundary-value problem, and an apparent horizon is slice dependent.

Puncture data avoid an explicit inner boundary, but they do not remove the need to choose topology and regularity. Conversely, excision does not remove the interior from the spacetime solution; it restricts the initial-data domain to the exterior and supplies boundary information there. Neither representation is intrinsically more physical without reference to the free data and target spacetime.

Independent checks

The Bowen–York expressions pass three immediate tests. Contracting either with $\delta_{ij}$ gives zero. The momentum part scales as $r^{-2}$ and can contribute to ADM linear momentum, whereas the spin part scales as $r^{-3}$ and combines with a rotational generator of order $r$ to give finite angular momentum. Setting $P^i=S^i=0$ returns the time-symmetric Schwarzschild puncture equation.

Common pitfalls

A puncture is not a curvature singularity on the initial slice. It usually represents a compactified asymptotic end; regularity is judged in a chart adapted to that end.

Constraint satisfaction does not imply astrophysical fidelity. The elliptic equations constrain the data but do not select the correct inspiral state.

Conformal flatness is a physical restriction. It is not merely a harmless coordinate gauge for spinning black holes.

Exercise: charge-sensitive falloff

Explain from radial scaling why the $P^i$ and $S^i$ terms can encode different ADM charges even though both solve the same momentum constraint.

Solution

The linear-momentum surface integral contracts $A_{ij}$ with an asymptotic translation, which is $O(1)$, and an area element $O(r^2)$. Thus the $O(r^{-2})$ momentum term has a finite nonzero limit. The $O(r^{-3})$ spin term vanishes in that integral. Angular momentum instead uses a rotational generator $O(r)$, so the spin contribution scales as $r^{-3}r r^2=O(1)$. Its tensor parity makes the corresponding momentum-term contribution cancel under the standard asymptotic conditions. The distinction is encoded jointly by falloff, angular structure, and the asymptotic generator.

References

• Bowen, Jeffrey M., and James W. York Jr. “Time-Asymmetric Initial Data for Black Holes and Black-Hole Collisions.” Physical Review D 21, no. 8 (1980): 2047–2056. DOI.
• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, arXiv:gr-qc/0703035v1 (2007). arXiv.