Gravitational action with null boundaries

We present a complete discussion of the boundary term in the action functional of general relativity when the boundary includes null segments in addition to the more usual timelike and spacelike segments. We confirm that ambiguities appear in the contribution from a null segment, because it depends on an arbitrary choice of parametrization for the generators. We also show that similar ambiguities appear in the contribution from a codimension-two surface at which a null segment is joined to another (spacelike, timelike, or null) segment. The parametrization ambiguity can be tamed by insisting that the null generators be affinely parametrized; this forces each null contribution to the boundary action to vanish, but leaves intact the fredom to rescale the affine parameter by a constant factor on each generator. Once a choice of parametrization is made, the ambiguity in the joint contributions can be eliminated by formulating well-motivated rules that ensure the additivity of the gravitational action. Enforcing these rules, we calculate the time rate of change of the action when it is evaluated for a so-called “Wheeler-DeWitt patch” of a black hole in asymptotically anti de Sitter space. We recover a number of results cited in the literature, obtained with a less complete analysis.

Figure 1

Wheeler-DeWitt patch of an eternal Schwarzschild-anti de Sitter black hole. The patch is defined by a future light cone originating inside the white-hole horizon and reaching the left boundary at time $t_{\mathrm{L}}$ and the right boundary at time $t_{\mathrm{R}}$. This light cone is joined to a past light cone converging to the future singularity.

Figure 2

A region $풱$ of spacetime with its broken boundary $\partial 풱$. The boundary consists of four spacelike segments ${\mathcal{S}}_1$, ${\mathcal{S}}_2$, ${\mathcal{S}}_3$, ${\mathcal{S}}_4$, and four null segments ${\mathcal{N}}_1$, ${\overline{\mathcal{N}}}_2$, ${\overline{\mathcal{N}}}_3$, ${\mathcal{N}}_4$. These are joined at codimension-two surfaces denoted ${\mathcal{B}}_{jk}$.

Figure 3

Domain $풱$ bounded by a closed hypersurface $\partial 풱$ consisting of a timelike segment $\mathcal{T}$ and two spacelike segments ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$. The intersection between $\mathcal{T}$ and ${\mathcal{S}}_j$ is the closed two-surface ${\mathcal{B}}_j$.

Figure 4

Joint terms in the boundary action. In (a) a past boundary is broken at $\mathcal{B}$ into two spacelike segments of normal $n_1^{\alpha }$ and $n_2^{\alpha }$; the contribution from $\mathcal{B}$ to the boundary action is $-2{\oint }_{\mathcal{B}}\eta dS$. In (b) a future boundary is broken into two spacelike segments; the contribution to the boundary action is $2{\oint }_{\mathcal{B}}\eta dS$. In (c), a timelike boundary is broken into two timelike segments of normal $s_1^{\alpha }$ and $s_2^{\alpha }$; the contribution to the boundary action is $-2{\oint }_{\mathcal{B}}\eta dS$. In (d), a timelike boundary of normal $s^{\alpha }$ is joined at $\mathcal{B}$ to a future, spacelike boundary of normal $n^{\alpha }$; the contribution to the boundary action is $2{\oint }_{\mathcal{B}}\eta dS$. In (e), a timelike boundary is joined to a past boundary, with contribution $-2{\oint }_{\mathcal{B}}\eta dS$. In panel $f$, two spacelike boundaries are joined, with contribution $2{\oint }_{\mathcal{B}}\eta dS$. Finally, two timelike boundaries are joined in (g), with contribution $2{\oint }_{\mathcal{B}}\eta dS$. In all panels the shaded region represents the interior of $풱$. The figure also shows the null vectors $k^{\alpha }$ and ${\overline{k}}^{\alpha }$, which are introduced in the main text.

Figure 5

A closed hypersurface $\partial 풱$ consisting of a past spacelike surface ${\mathcal{S}}_1$, a truncated past null cone $\mathcal{N}$, and a future spacelike surface ${\mathcal{S}}_2$.

Figure 6

A closed hypersurface $\partial 풱$ consisting of a past spacelike surface ${\mathcal{S}}_1$, a truncated future null cone $\overline{\mathcal{N}}$, and a future spacelike surface ${\mathcal{S}}_2$.

Figure 7

A closed hypersurface $\partial 풱$ consisting of a past spacelike surface ${\mathcal{S}}_1$, a truncated past null cone $\mathcal{N}$, a truncated future null cone $\overline{\mathcal{N}}$, and a future spacelike surface ${\mathcal{S}}_2$.

Figure 8

Adapted coordinates $(\lambda ,r,{\theta }^A)$ in $풱$.

Figure 9

Joint terms in the boundary action. (a) displays a null segment joined at $\mathcal{B}$ to another segment (not shown) which can be either spacelike, timelike, or null; because $\mathcal{B}$ is a future boundary to the null segment and the outward direction is a future direction, the contribution from $\mathcal{B}$ to the boundary action is $2{\oint }_{\mathcal{B}}adS$. In (b), the joint $\mathcal{B}$ is a past boundary, the outward direction continues to be a future direction, and the contribution to the boundary action is $-2{\oint }_{\mathcal{B}}adS$. In (c), the joint $\mathcal{B}$ is again a future boundary, but the outward direction is now a past direction; for these cases the contribution to the boundary action is $-2{\oint }_{\mathcal{B}}adS$. In (d), where the joint $\mathcal{B}$ is a past boundary, the outward direction is a past direction, and the contribution to the boundary action is $2{\oint }_{\mathcal{B}}adS$. In all panels the shaded region indicates the interior of $풱$.

Figure 10

Composition of boundary actions. In (a),(b), a spacelike/spacelike joint (i.e. the joint formed by the intersection of two spacelike surfaces) is obtained by taking the union of two null/spacelike joints. In (c), a timelike/timelike joint is the union of two null/timelike joints. In (d),(e), a spacelike/timelike joint is the union of a null/spacelike joint and a null/timelike joint.

Figure 11

Composition of boundary actions. In (a), a null/timelike joint is obtained by taking the union of a null/spacelike joint with a spacelike/timelike joint. In (b), a null/null joint is the union of two null/spacelike joints.

Figure 12

Wheeler-DeWitt patch of a Schwarzschild-anti de Sitter spacetime. On the left panel, the patch at coordinate time $t_0$ is shown in dark color, and the patch at time $t_0+\delta t$ is shown in light color. The difference between the two patches is shown on the right panel. In the left panel, the curved arrows indicate the flow of the Killing vector ${\partial }_t$ in each of the quadrants of the Penrose diagram. Further the red dashed curves indicate the cut-off surfaces at $r=r_{\max }$ near the asymptotic AdS boundaries.

Figure 13

Wheeler-DeWitt patch of a Reissner-Nordström anti–de Sitter spacetime. On the left, the patch at coordinate time $t_L=t_0$ is shown in dark color, and the patch at time $t_L=t_0+\delta t$ is shown in light color. The difference between the two patches is shown on the right. The red dashed curves on the left indicate the cutoff surfaces at $r=r_{\max }$ near the asymptotic AdS boundaries.