Corners, Joints, and Intersecting Boundaries

When two smooth boundary segments meet, their induced two-geometry can agree while their normals jump. The missing datum is an oriented angle or Lorentzian boost in the two-dimensional normal plane of the joint. It controls geometric compatibility and the localized term required for a differentiable gravitational action. The additivity argument below concerns spacelike joints; timelike joints require additional care and are not covered by the same cancellation rule.

Required background. We use Hypersurface Normals and Tangential Projectors, The Einstein Equation, and Flat-Background Hilbert Stress-Energy from Metric Variation.

Helpful background. The Gibbons-Hawking-York Boundary Term derives the smooth-segment term whose corner variation is completed here.

Joint geometry and oriented normal frames

Let two nonnull boundary segments $\mathcal B_1$ and $\mathcal B_2$ meet on a spacelike codimension-two surface $J$. Its induced metric $\sigma_{AB}$ is positive definite and its normal plane is Lorentzian. Each segment has an outward unit normal $r_i^\mu$, induced metric $h^{(i)}_{ab}$, and extrinsic curvature

$$K^{(i)}_{ab} =-h^{(i)\mu}{}_a h^{(i)\nu}{}_b\nabla_\mu r^{(i)}_\nu.$$

This is the site’s negative projected-derivative convention. The pullbacks of both segment metrics to $J$ must give the same two-metric $\sigma_{AB}$. But $\sigma_{AB}$ does not determine how the two boundary segments sit in spacetime. That information is the Lorentz transformation or rotation relating their oriented normal frames.

For example, when a spacelike and a timelike segment meet, choose at $J$ a future-directed unit timelike vector $u^\mu$ and an outward unit spacelike vector $s^\mu$ spanning the normal plane. A signed rapidity $\eta$ can be fixed by

$$\sinh\eta=-u\cdot s,$$

together with the declared ordering and outward orientations. Here $u$ and $s$ are normals to the two different segments, so they need not be mutually orthogonal. If $J$ is instead timelike, its normal plane is Euclidean and the analogous datum is a rotation angle; the integration measure is then $\sqrt{|\det\sigma|}\,d^2z$. That case is outside the boost and additivity formulas below. Absolute values such as $|u\cdot s|$ lose the orientation needed by the boost term.

Boundary-segment labels $1,2$ are not junction side labels. If $J$ also lies on an interface between $M_-$ and $M_+$, first pull both side data to one abstract surface, use one normal directed from minus to plus, and only then define

$$[Q]\equiv Q_+-Q_-.$$

The two outward normals of glued regions are opposite and must never be inserted directly into this jump. The same separation of conventions underlies Hayward’s smoothing construction for nonsmooth boundaries Hayward 1993, pp. 3276–3279.

Segment and joint terms in the action

The existing GHY page uses $\widehat K_{ab}=+h_a{}^\mu h_b{}^\nu\nabla_\mu r_\nu$. Since $\widehat K_{ab}=-K_{ab}$ here, the smooth nonnull boundary contribution written in the present convention is

$$S_{\mathrm{seg}} =-\frac{1}{8\pi G} \sum_i\epsilon_i \int_{\mathcal B_i}\sqrt{|h_i|}\,K_i\,d^3y, \qquad \epsilon_i=r_i^\mu r_{i\mu}.$$

Changing the definition of $K$ does not change the physical action; it changes the explicit sign multiplying its symbol. A universal unsigned corner formula is likewise unsafe. With an orientation factor $s_J$ fixed by smoothing the ordered outward normal from $\mathcal B_1$ to $\mathcal B_2$, write

$$S_J =\frac{s_J}{8\pi G} \int_J\sqrt{\sigma}\,\eta\,d^2z.$$

Under fixed intrinsic two-geometry, variation of the bulk-plus-segment action leaves

$$\delta(S_{\mathrm{EH}}+S_{\mathrm{seg}})\big|_J =-\frac{s_J}{8\pi G} \int_J\sqrt{\sigma}\,\delta\eta\,d^2z.$$

The joint term cancels it. The segment metrics must restrict consistently to $\sigma_{AB}$, but the boost angle need not be fixed independently: allowing $\delta\eta\ne0$ is exactly why the joint term is needed. The sign and angle reference must nevertheless be declared to specify the action’s value. Lehner and collaborators give the orientation prescription in Lehner et al. 2016, §§II D–II E, pp. 14–19, PDF.

Null joints and additivity

If one segment is null, its normal is a tangent generator $k^\mu$ and no unit-normal rapidity exists. Representative joint data become logarithms such as

$$a=\ln|k\cdot r|$$

for a null–nonnull joint, with an additional sign determined by whether the joint is the past or future endpoint and which side contains the region. Rescaling $k^\mu\mapsto c\,k^\mu$ shifts $a$ by $\ln c$. Thus generator normalization is part of the joint convention; the complete cases and their signs are tabulated in Lehner et al. 2016, Appendix C, pp. 43–44, Eqs. (C1)–(C5), PDF.

Additivity is a useful check within the spacelike-joint class. If an artificial internal boundary divides a region, its outward normals in the two subregions are opposite. Their segment terms cancel after normal reversal; the prescribed joint terms must combine to reproduce the original joints, or cancel if the original boundary was smooth. For null segments the generator normalization and the additive constants in the joint terms must also agree. The corresponding rules are established in Lehner et al. 2016, §II H, pp. 25–27, PDF.

This is not a general theorem for timelike joints. Their usual smoothing prescription can leave a nonadditive area contribution; changing the constant reference angle can restore additivity but changes that prescription. The source explicitly excludes timelike joints from its additivity argument Lehner et al. 2016, p. 5, footnote 7, and p. 38, footnote 23, PDF. Differentiability under fixed intrinsic data and additivity under subdivision are distinct properties.

Common failures are to infer the action’s joint value from $\sigma_{AB}$ alone, to use an unsigned boost angle, or to import the positive-$K$ GHY formula without translating its symbol. At a null joint, fixing an affine parameter does not by itself fix its constant normalization, so it does not remove the logarithmic ambiguity.

Exercises

A smooth region $\mathcal M$ is divided into $\mathcal M_1$ and $\mathcal M_2$ by an artificial nonnull hypersurface $\mathcal H$. Assume the intersection $J=\mathcal H\cap\partial\mathcal M$ is spacelike and use the same oriented boost prescription on both subregions. Show why the artificial segment and joint contributions cancel. Explain why the hypothesis on $J$ matters.

Solution

Let $r^\mu$ be the outward normal to $\mathcal H$ as a boundary of $\mathcal M_1$. It is $-r^\mu$ as a boundary of $\mathcal M_2$. Since the site curvature is linear in the normal,

$$K_{ab}[-r]=-K_{ab}[r].$$

The induced metric and $\epsilon=r^2$ are the same, so the two copies of the segment integral are equal and opposite.

At $J$, the ordered rotation from the external-boundary normal to $r^\mu$ in the first subregion is reversed in the second subregion, where the internal normal is $-r^\mu$ and the boundary orientation is opposite. The oriented joint data therefore enter as $s_J\eta$ and $-s_J\eta$. Their integrals cancel. Only the original external boundary and its genuine joints remain, so

$$S[\mathcal M]=S[\mathcal M_1]+S[\mathcal M_2].$$

Using an absolute angle would make both artificial joint terms have the same sign and would fail this check.

The assumption that $J$ is spacelike permits the additive boost prescription. A timelike joint instead uses an angular reference with the separate nonadditivity issue described above; reversing a segment normal alone does not prove cancellation for that class.

References

• Hayward, Geoff. “Gravitational Action for Spacetimes with Nonsmooth Boundaries.” Physical Review D 47, no. 8 (1993): 3275–3280. doi:10.1103/PhysRevD.47.3275.
• Lehner, Luis, Robert C. Myers, Eric Poisson, and Rafael D. Sorkin. “Gravitational Action with Null Boundaries.” Physical Review D 94, no. 8 (2016): 084046. arXiv:1609.00207. doi:10.1103/PhysRevD.94.084046.