Boundary of the future of a surface

We prove that the boundary of the future of a surface $K$ consists precisely of the points $p$ that lie on a null geodesic orthogonal to $K$ such that between $K$ and $p$ there are no points conjugate to $K$ nor intersections with another such geodesic. Our theorem has applications to holographic screens and their associated light sheets and in particular enters the proof that holographic screens satisfy an area law.

Figure 1

Possibilities for how a null geodesic orthogonal to a surface can exit the boundary of its future. In this example, a parabolic surface $K$ (blue line) lies in a particular spatial slice. A future-directed null geodesic (red line) is launched orthogonally from $p$. At $q$, it encounters a caustic, entering the interior of the future of $K$ (red dashed line). The point $q$ is conjugate to $K$. Other null geodesics orthogonal to $K$ (black lines) encounter nonlocal intersections with other such geodesics along the green line, where they exit the boundary of the future of $K$.

Figure 2

In this generic Penrose diagram, the codimension-two surface $K$ (black dot) splits a Cauchy surface $\mathrm{\Sigma }$ (dashed line) into two parts ${\mathrm{\Sigma }}_{\mathrm{in}}$, ${\mathrm{\Sigma }}_{\mathrm{out}}$. This induces a splitting of the spacetime $M$ into four parts: the past and future of $K$ (red, yellow) and the domains of dependence of ${\mathrm{\Sigma }}_{\mathrm{in}}$ and ${\mathrm{\Sigma }}_{\mathrm{out}}$ (green, blue) [8]. Theorem 1 guarantees that this splitting is fully characterized by the four orthogonal null congruences originating on $K$ (black diagonal lines).

Figure 3

An illustration of the exponential map exp, which takes a vector in $TM$ to a point in $M$, and the Jacobian of the exponential map, which takes a vector in the tangent space $TTM$ of $TM$ to a vector in $TM$.

Figure 4

An illustration of the surface-orthogonal exponential map ${\exp }_K$ evaluated at $p\in K$, which takes a vector in $T_p{K}^{\perp }$ to a point $c_{p,v}(1)$ in $M$. Here, as in text, the tangent space at $p$, $T_{p}M$, is broken up as a product $T_p{K}^{\perp }\times T_{p}K$. Also shown is the Jacobian ${\exp }_{K*}$ at $v\in T_p{K}^{\perp }$, which takes a vector $w=(w_1,w_2)\in T_{p,v}NK=T_{p}K\times T_v{T}_p{K}^{\perp }$ to a vector in $T_{c_{p,v}(1)}M$.

Figure 5

The two types of conjugate points defined in Defs. 17 and 19. The point $q_1$ is conjugate to the point $p_1$, with the Jacobi field illustrated by the red arrows. The point $q_2$ is conjugate to the surface $K$ (blue line), at the point $p_2$, with the Jacobi field illustrated by the green arrows. Geodesics orthogonal to $K$ are shown in black. If a general conjugate point lies along an orthogonal null geodesic, then by Lemma 16 there exists a Jacobi field such that the conjugate point is of the surface type. Hence, this type of conjugacy appears in Theorem 1.

Figure 6

Possibilities in the proof. The sequence of timelike geodesics ${\gamma }_n$ (black) connects $K$ with a sequence of points $q_n\in I^{+}(K)$ on the orthogonal null geodesic $\gamma$ (red) that joins $p\in K$ with $q$, after which $\gamma$ leaves ${\dot{I}}^{+}(K)$ (red dashed). In the case on the left, ${\gamma }^{\prime }$ (green) is distinct from $\gamma$, so condition (iii) fails. In the case on the right, ${\gamma }^{\prime }=\gamma$, which we prove corresponds to a failure of condition (ii).