Null Boundaries, Corners, and Action Additivity

Null boundaries are not obtained by replacing a unit normal in the Gibbons–Hawking–York term. A null normal is tangent to its own hypersurface and cannot be normalized to $\pm1$; the induced metric is degenerate. A well-defined action for a piecewise smooth region must instead specify null generators and their parameterization, a screen-space area element, segment terms, and logarithmic joint terms whose signs follow the orientation of the region.

Required background. We use variational boundary potentials, curvature, and null and non-null hypersurface geometry.

Helpful background. Special-relativistic null vectors, tensor fields, and classical action principles clarify the normalization and gluing issues.

Intrinsic data on a null segment

Let a null boundary segment $\mathcal N$ be generated by a future-directed tangent

$$k^a=\left(\frac{\partial}{\partial\lambda}\right)^a, \qquad k^b\nabla_bk^a=\kappa k^a.$$

The nonaffinity $\kappa$ vanishes for an affine parameter but changes under a reparameterization $\lambda\mapsto\bar\lambda$. Cross-sections carry a nondegenerate two-metric $\gamma_{AB}$ and area density $\sqrt\gamma$. Their expansion is

$$\Theta=\frac{1}{\sqrt\gamma}\frac{d\sqrt\gamma}{d\lambda}.$$

Following the explicit orientation scheme of Lehner and collaborators, let $\sigma_{\mathcal N}=+1$ when the region lies to the future of the future-directed null segment and $-1$ when it lies to its past. In the normalization where the bulk coefficient is $1/(16\pi G)$, the null-segment contribution is

$$S_{\mathcal N} =+\frac{\sigma_{\mathcal N}}{8\pi G} \int_{\mathcal N}d\lambda\,d^2\theta\,\sqrt\gamma\,\kappa.$$

Lehner et al. derive this term and distinguish it from the non-null GHY integral Lehner et al. 2016, § II C, journal pp. 084046-7–084046-10, Eqs. (2.34)–(2.73). The sign shown here follows the published action manual, Eq. (C1). This detail is version-sensitive: arXiv v1 printed a minus sign in front of the general null-segment sum, while the final journal article corrected it to a plus sign. Affine parameterization makes the segment integral zero, but it does not remove joint contributions or the normalization freedom of $k^a$.

Joints and logarithmic normalization

Where two boundary segments meet on a codimension-two surface $J$, discontinuity of their normals produces a joint term

$$S_J=\frac{\sigma_J}{8\pi G} \int_Jd^2\theta\,\sqrt\gamma\,a.$$

For a joint containing a null segment, representative logarithms are

$$a=\begin{cases} \ln|k\cdot n|,&\text{null–spacelike},\\ \ln|k\cdot s|,&\text{null–timelike},\\ \ln|k\cdot\bar k/2|,&\text{null–null}, \end{cases}$$

where $n^a$, $s^a$, and $\bar k^a$ follow the declared orientations. The sign $\sigma_J$ depends on whether the joint is the past or future end of the null segment and on which side contains the region. Memorizing one unsigned logarithm is not sufficient; the complete published prescription is summarized in Lehner et al. 2016, Appendix C, journal pp. 084046-34–084046-35, Eqs. (C1)–(C5).

Here is the executable sign rule for the future-directed $k^a$ already chosen. Set $\sigma_J=+1$ when either the region lies to the future of the null segment and $J$ is its past end, or the region lies to the past and $J$ is its future end; set $\sigma_J=-1$ in the other two cases. For the logarithms above, $n^a$ is the future-directed unit normal to a spacelike segment, $s^a$ is the outward unit normal to a timelike segment, and $\bar k^a$ is future-directed. We adopt Lehner et al.’s additive representatives: the variation-fixed constant is zero at null–non-null joints, while the corresponding null–null choice is encoded by the factor $1/2$ inside $\ln|k\cdot\bar k/2|$. These rules require a common normalization for $k^a$ wherever actions will be compared or glued Lehner et al. 2016, § II H, pp. 25–27, and Appendix C, p. 44, Eq. (C5), PDF.

The sign chain is best checked rather than memorized. Put $C=1/(8\pi G)$ and rescale $k^a\mapsto e^\beta k^a$, so

$$d\bar\lambda=e^{-\beta}d\lambda, \qquad \bar\kappa\,d\bar\lambda=(\kappa+\partial_\lambda\beta)d\lambda, \qquad \bar a=a+\beta.$$

The rescaling $\beta(\lambda,\theta^A)$ may differ between generators. Define its area-weighted endpoint integrals by

$$\mathcal B_{\mathrm p}=\int_{J_{\mathrm p}}\sqrt\gamma\,\beta_{\mathrm p}\,d^2\theta, \qquad \mathcal B_{\mathrm f}=\int_{J_{\mathrm f}}\sqrt\gamma\,\beta_{\mathrm f}\,d^2\theta.$$

Integration by parts along each generator gives

$$\begin{aligned} \Delta S_{\mathcal N} &=\sigma_{\mathcal N}C \left(\mathcal B_{\mathrm f}-\mathcal B_{\mathrm p}\right) -\sigma_{\mathcal N}C \int_{\mathcal N}\sqrt\gamma\,\Theta\beta\,d\lambda\,d^2\theta,\\ \Delta S_J &=-\sigma_{\mathcal N}C \left(\mathcal B_{\mathrm f}-\mathcal B_{\mathrm p}\right). \end{aligned}$$

The endpoint pieces cancel, leaving

$$\Delta(S_{\mathcal N}+S_J) =-\sigma_{\mathcal N}C \int_{\mathcal N}\sqrt\gamma\,\Theta\beta\,d\lambda\,d^2\theta.$$

For one constant $\beta$ on the whole segment, $\mathcal B_{\mathrm p,f}=\beta A_{\mathrm p,f}$, where $A_{\mathrm p}$ and $A_{\mathrm f}$ are the endpoint areas. The segment change vanishes and the joint change is $-\sigma_{\mathcal N}C\beta(A_{\mathrm f}-A_{\mathrm p})$; for a nontrivial rescaling it vanishes only when the endpoint areas agree. For generator-dependent $\beta$, the endpoint integrals cannot be replaced by one rescaling factor times each total area, but their cancellation remains exact.

Optional reparameterization counterterm

If one wants the on-shell boundary action to be invariant under arbitrary reparameterizations of each generator, an intrinsic counterterm can be added:

$$S_{\mathrm{ct}} =+\frac{\sigma_{\mathcal N}}{8\pi G} \int_{\mathcal N}d\lambda\,d^2\theta\, \sqrt\gamma\,\Theta\ln\!\left(\ell_{\mathrm{ct}}|\Theta|\right).$$

Here $\ell_{\mathrm{ct}}$ is an arbitrary reference length needed to make the logarithm dimensionless. Since $\bar\Theta=e^\beta\Theta$, the measure $d\lambda\,\sqrt\gamma\,\Theta$ is invariant while the logarithm gains $\beta$. Therefore

$$\Delta S_{\mathrm{ct}} =+\sigma_{\mathcal N}C \int_{\mathcal N}\sqrt\gamma\,\Theta\beta\,d\lambda\,d^2\theta,$$

which cancels the residual above exactly. The construction and its arbitrary constant are given in Lehner et al. 2016, Appendix B, journal pp. 084046-33–084046-34, Eqs. (B1)–(B8).

This counterterm is optional: it does not change the Dirichlet variational principle under the prescribed intrinsic data, but it changes the numerical action by an intrinsic boundary functional. Any physical use of the on-shell action must state whether it is included and what reference scale is chosen.

Gluing, null-joint additivity, and its limit

Suppose two spacetime regions share a boundary segment with opposite orientations. Bulk integrals add, and properly oriented hypersurface terms cancel on the shared internal boundary. For the spacelike-joint gluing configurations analyzed by Lehner et al., including joints involving null segments, the logarithmic representatives above remove the variation-fixed ambiguity and make the joint contributions compose correctly Lehner et al. 2016, § II H, pp. 25–27, Eqs. (2.127)–(2.129), PDF.

This does not make the standard gravitational action ordinarily additive for every piecewise-smooth composition. With the joint term obtained by smoothing a boundary, a timelike joint formed by two timelike boundary hypersurfaces can leave a nonzero residue. Lehner et al. explicitly retain this obstruction rather than claiming universal additivity Lehner et al. 2016, § II H, pp. 25–27, PDF. Brill and Hayward formulate a broader alternative: associate a finite action $I[S]$ with the identification surface so that the composition law is

$$I[\widetilde{\mathcal M}]=I[\mathcal M]-I[S].$$

They call this generalized additivity; it records rather than erases the residue Brill and Hayward 1994, §§ 1 and 5, pp. 1–14, Eqs. (14)–(16), Open PDF.

Null normalization is part of the narrower gluing check. If the generator is independently normalized on the two sides, the joint logarithms need not cancel. One must match the null tangent across a common null boundary or retain the compensating terms. The optional expansion counterterm removes reparameterization dependence, but it does not remove the separate timelike-joint obstruction.

The discussion concerns the classical action and its on-shell composition. It does not by itself define a Lorentzian gravitational path integral or its measure.

Exercises

Take a null segment for which the region lies to its future, with endpoint areas $A_{\mathrm p}=4A_0$ and $A_{\mathrm f}=7A_0$. Under a constant rescaling $k^a\mapsto c\,k^a$ with $c>0$, find the changes in the segment, the two joints together, and the counterterm. What goes wrong if the counterterm is assigned the opposite overall sign while the segment and joint rules are kept fixed?

Solution

Here $\sigma_{\mathcal N}=+1$, the past-end joint has $\sigma_J=+1$, and the future-end joint has $\sigma_J=-1$. Since $a=\ln|k\cdot s|$,

$$a\mapsto\ln|c\,k\cdot s|=a+\ln c.$$

Because $\beta=\ln c$ is constant, $\kappa\,d\lambda$ is unchanged and $\Delta S_{\mathcal N}=0$. The two joints give

$$\Delta S_J =\frac{\ln c}{8\pi G}(A_{\mathrm p}-A_{\mathrm f}) =-\frac{3A_0\ln c}{8\pi G}.$$

The counterterm changes by the positive of $3A_0\ln c/(8\pi G)$ because $\int_{\mathcal N}\sqrt\gamma\,\Theta\,d\lambda\,d^2\theta=A_{\mathrm f}-A_{\mathrm p}=3A_0$. Thus the complete change is zero. Giving the counterterm the opposite sign makes its change equal to the joint change; the two then add to $-6A_0\ln c/(8\pi G)$ instead of canceling. This unequal-area example exposes the sign error that an equal-area test would miss.

References

• Brill, Dieter R., and Geoffrey Hayward. “Is the Gravitational Action Additive?” Physical Review D 50 (1994): 4914–4919. DOI; arXiv.
• Lehner, Luis, Robert C. Myers, Eric Poisson, and Rafael D. Sorkin. “Gravitational Action with Null Boundaries.” Physical Review D 94 (2016): 084046. DOI; arXiv.