Timelike Boundary Geometry and Data

A finite timelike boundary carries an intrinsic Lorentzian metric, an embedding curvature, a foliation by spacelike cuts, and gauge information describing how the boundary is threaded. These fields cannot all be prescribed independently. The Gauss–Codazzi equations constrain the intrinsic and extrinsic geometry, and a hyperbolic reduction permits free data only for incoming characteristic fields. The geometric constraint count and the evolution count answer related but different questions.

Required background. The Gibbons–Hawking–York Boundary Term supplies the boundary metric and embedding variation, while PDEs and Green Functions for Gravitation supplies noncharacteristic boundaries, energy estimates, and incoming modes.

Helpful background. The Einstein Equation supplies the bulk equations; Hypersurface Normals and Tangential Projectors supplies induced geometry; Formulations of General Relativity orients the volume; and Causality and Global Structure supplies causal boundaries and domains of dependence.

Intrinsic and extrinsic boundary geometry

Let $\mathcal T$ be a smooth timelike hypersurface with outward unit spacelike normal $s^a$, so $s^as_a=1$. Its induced metric and projector are

$$h_{ab}=g_{ab}-s_as_b, \qquad h_a{}^b=\delta_a{}^b-s_as^b.$$

The metric $h_{ab}$ has signature $(-,+,+)$. To avoid confusing it with the spacelike-slice convention $K_{ij}=-\gamma_i{}^a\gamma_j{}^b\nabla_an_b$, define the timelike-boundary second fundamental form with the GHY sign

$$\mathcal K_{ab} =h_a{}^ch_b{}^d\nabla_cs_d, \qquad \mathcal K=h^{ab}\mathcal K_{ab}.$$

Reversing $s^a$ reverses $\mathcal K_{ab}$. The Brown–York-type momentum conjugate to $h_{ab}$ is proportional to $\mathcal K_{ab}-\mathcal K h_{ab}$, but a finite stress tensor also depends on the chosen action counterterms and reference prescription.

The hypersurface projections of Einstein’s equation give boundary constraints. With the stated conventions,

$${}^{(3)}R[h] +\mathcal K_{ab}\mathcal K^{ab}-\mathcal K^2 =2\Lambda-16\pi G\,T_{ss},$$ $$D_b\mathcal K^b{}_a-D_a\mathcal K =8\pi G\,T_{sa},$$

where $D_a$ is compatible with $h_{ab}$, $T_{ss}=T_{ab}s^as^b$, and $T_{sa}=h_a{}^cT_{cd}s^d$. The scalar equation and three tangential equations show directly that arbitrary $h_{ab}$ and $\mathcal K_{ab}$ do not define an Einstein boundary embedding.

Foliation by spacelike cuts

Let $u^a$ be a future unit timelike vector tangent to $\mathcal T$ and normal within $\mathcal T$ to two-dimensional cuts $S_t$. Then

$$\sigma_{ab}=h_{ab}+u_au_b =g_{ab}+u_au_b-s_as_b$$

is the cut metric. Decompose the boundary curvature into

$$\kappa_{AB}=\sigma_A{}^a\sigma_B{}^b\mathcal K_{ab}, \qquad j_A=-\sigma_A{}^a u^b\mathcal K_{ab}, \qquad \eta=u^au^b\mathcal K_{ab}.$$

These contain three, two, and one components respectively. They are not six unrestricted boundary functions: the Gauss–Codazzi equations couple their tangential and time derivatives to $h_{ab}$ and bulk matter. The choice of $u^a$, boundary coordinates, lapse, and tangential shift also contains gauge freedom rather than additional gravitational radiation.

Friedrich and Nagy formulate a smooth timelike boundary with an adapted timelike frame vector and explain why the mean extrinsic curvature and two curvature combinations, together with gauge source functions, supply their free boundary input in Friedrich and Nagy 1999, §§1–3, pp. 619–625. That is one formulation-specific parameterization, not a statement that every IBVP prescribes the same variables.

Geometric data versus characteristic data

For a first-order symmetric-hyperbolic reduction

$$A^0\partial_tU+A^s\partial_sU+A^A\partial_AU=F(U),$$

the boundary matrix $A^s$ splits fields into incoming, outgoing, and zero-speed subspaces. Only the incoming subspace accepts freely specified evolution data. Maximally dissipative conditions have the schematic form

$$U_{\mathrm{in}}=H U_{\mathrm{out}}+q, \qquad H^\dagger H\le I,$$

with no independent values assigned to zero-speed fields. Friedrich and Nagy derive this maximal nonpositive-subspace condition in Friedrich and Nagy 1999, §3, pp. 624–626.

The simple matrix bound assumes characteristic fields have been normalized by the boundary energy. Before that normalization, the outward quadratic flux is

$$\mathcal F =U_{\mathrm{out}}^\dagger\Lambda_{\mathrm{out}}U_{\mathrm{out}} -U_{\mathrm{in}}^\dagger\Lambda_{\mathrm{in}}U_{\mathrm{in}},$$

where $\Lambda_{\mathrm{out}}$ and $\Lambda_{\mathrm{in}}$ are positive matrices containing the characteristic-speed magnitudes in the symmetrizer metric. The correct condition is $H^\dagger\Lambda_{\mathrm{in}}H\le\Lambda_{\mathrm{out}}$. Absorbing the positive square roots into the fields produces the unit-weight formula above. An unweighted Euclidean bound on arbitrarily scaled components is not an energy estimate.

The number of incoming fields depends on the formulation, gauge, boundary motion, and characteristic convention. Here “incoming” means a characteristic whose future propagation enters the computational domain across the outward normal $s^a$. Ten metric Dirichlet conditions can have the correct count for ten reduced wave equations, but that count does not ensure compatibility with the Einstein constraints and gauge boundary hierarchy.

The boundary values must also agree with the initial data where $S_0$ meets $\mathcal T$; characteristic admissibility alone does not supply this corner compatibility.

Independent flat-cylinder check

In Minkowski spacetime, take the timelike cylinder $r=R$ with outward normal $s^a=(\partial_r)^a$. Its intrinsic metric is

$$h=-dt^2+R^2d\Omega_2^2,$$

and its nonzero embedding curvature lies on the sphere: $\kappa_{AB}=R^{-1}\sigma_{AB}$, while $\eta=j_A=0$. Hence $\mathcal K=2/R$, $\mathcal K_{ab}\mathcal K^{ab}=2/R^2$, and ${}^{(3)}R[h]=2/R^2$. The vacuum scalar constraint gives

$${}^{(3)}R+\mathcal K_{ab}\mathcal K^{ab}-\mathcal K^2 =\frac2{R^2}+\frac2{R^2}-\frac4{R^2}=0,$$

confirming the signs. Prescribing this same intrinsically curved cylinder $h=-dt^2+R^2d\Omega_2^2$ together with $\mathcal K_{ab}=0$ would fail the equation and cannot be realized as the stated Minkowski boundary. By contrast, a planar timelike boundary has flat $h_{ab}$ and $\mathcal K_{ab}=0$ and satisfies the vacuum constraints.

Common pitfalls

Boundary metric and boundary curvature are not independent Cauchy data. They obey four hypersurface constraints before evolution is considered.

Gauge threading is not radiation. A boundary time vector and coordinates are needed to state components, but their choice does not add physical modes.

Geometric component counting does not replace characteristic counting. Well-posedness is tested on the incoming fields of a named hyperbolic reduction.

Exercise: reverse the normal

Reverse the boundary normal $s^a\mapsto-s^a$. Determine how $h_{ab}$, $\mathcal K_{ab}$, and the two Gauss–Codazzi equations change.

Solution

The induced metric is unchanged because it depends quadratically on $s^a$. The second fundamental form and its trace reverse sign. The scalar constraint contains only quadratic combinations of $\mathcal K_{ab}$, so it is unchanged; $T_{ss}$ is also unchanged. Both sides of the momentum constraint reverse: $D_b\mathcal K^b{}_a-D_a\mathcal K$ is linear in $\mathcal K$, and $T_{sa}$ is linear in $s^a$. Thus the geometric statement is orientation consistent.

References

• Friedrich, Helmut, and Gabriel Nagy. “The Initial Boundary Value Problem for Einstein’s Vacuum Field Equation.” Communications in Mathematical Physics 201 (1999): 619–655. DOI.
• Kreiss, Heinz-Otto, Oscar Reula, Olivier Sarbach, and Jeffrey Winicour. “Well-Posed Initial-Boundary Value Problem for the Harmonic Einstein Equations Using Energy Estimates.” Classical and Quantum Gravity 24, no. 23 (2007): 5973–5984. DOI; Open PDF.