Covariant Extremal Surfaces and HRT

In a time-dependent semiclassical Einstein bulk, the leading entropy is assigned to a spacelike codimension-two surface whose area is stationary under both independent normal deformations, anchored on $\partial A$, and homologous to $A$. Among admissible extremal surfaces, the least-area one is selected. This is the HRT prescription. It is not “minimize on whichever time slice is convenient”: that result depends on the slice and generally has nonzero timelike expansion. Here the bulk is Lorentzian $\mathrm{AdS}_3$, and the quench example uses an ingoing AdS–Vaidya patch with a thin null shell.

Required background. RT supplies anchoring, homology, and static saddle competition; time-dependent holographic geometries supplies the AdS–Vaidya quench.

Helpful background. Global hyperbolicity and Cauchy surfaces state the causal assumptions, and entropy-calculation verification supplies regulator and replica checks.

Lorentzian extremality

Let $X^\mu(\sigma^a)$ embed a spacelike codimension-two surface $\gamma$. With the site convention $(+,-,\ldots,-)$, its positive-definite spatial metric and area are

$$q_{ab}=-g_{\mu\nu}\partial_aX^\mu\partial_bX^\nu, \qquad \mathcal A[\gamma]=\int d^{d-1}\sigma\sqrt{\det q}.$$

There are two independent null normals $k^\mu$ and $\ell^\mu$. First variation gives

$$\delta\mathcal A =\int_\gamma\sqrt{\det q}\, \left(\theta_{(k)}\delta X^{(k)} +\theta_{(\ell)}\delta X^{(\ell)}\right) +\text{anchor terms}.$$

With $\partial\gamma=\partial A$ fixed, spacetime extremality is

$$\theta_{(k)}=\theta_{(\ell)}=0.$$

The HRT surface $\gamma_A^{\rm HRT}$ is the least-area member of the admissible extremal set satisfying the homology condition, and

$$S_A^{(0)}=\frac{\mathcal A(\gamma_A^{\rm HRT})}{4G_N}.$$

Extremality is a saddle condition in Lorentzian signature, not a minimum under timelike displacements. Causal placement and existence require additional hypotheses, supplied by maximin rather than by the variational equation alone. Hubeny, Rangamani, and Takayanagi introduced this covariant replacement of RT (Hubeny, Rangamani, and Takayanagi 2007, §§2–3).

Extremal geodesics through an AdS–Vaidya shell

Use ingoing coordinates

$$ds^2=\frac{L^2}{z^2} \left[f(v,z)dv^2+2\,dv\,dz-dx^2\right], \qquad f(v,z)=1-Mz^2\Theta(v).$$

The region $v<0$ is pure AdS; $v>0$ is planar BTZ. Anchor an interval of length $\ell$ at equal boundary time $t_b$, and exploit reflection symmetry about $x=0$. A geodesic $(v(x),z(x))$ has

$$\mathcal L =L\int dx\,\frac{1}{z} \sqrt{1-fv'^2-2v'z'}.$$

Because the integrand has no explicit $x$, a first integral is

$$\sqrt{1-fv'^2-2v'z'}=\frac{z_*}{z},$$

where $z_*$ is the turning-point depth. Away from the shell, the conserved momentum associated with the stationary coordinate of each region reduces the problem to one-dimensional quadratures for $\ell/2$, $t_b$, and the regulated length. Across $v=0$, variation gives refraction conditions. With

$$p_v=-\frac{L(fv'+z')}{z\sqrt{1-fv'^2-2v'z'}}, \qquad p_z=-\frac{Lv'}{z\sqrt{1-fv'^2-2v'z'}},$$

$p_z$ is continuous, while integrating the $v$ equation through the delta-function in $\partial_vf$ fixes the jump in $p_v$:

$$[p_v]_-^+ =-\int_{x_c-0}^{x_c+0}dx\, \frac{L(\partial_v f)v'^2} {2z\sqrt{1-fv'^2-2v'z'}}.$$

These equations determine the shell-crossing geodesic rather than allowing the two segments to be joined at an arbitrary angle.

Three regimes provide independent checks.

• Before the shell, the geodesic lies in pure AdS and $$\mathcal L_{\rm ren}=2L\log(\ell/\epsilon).$$
• At sufficiently late $t_b$, the geodesic lies entirely in BTZ and $$\mathcal L_{\rm ren} =2L\log\!\left[ \frac{\beta}{\pi\epsilon} \sinh\!\left(\frac{\pi\ell}{\beta}\right) \right].$$
• At intermediate time, the extremal geodesic crosses the shell, satisfies the refraction equation, and interpolates between these limits. If multiple crossing and noncrossing extrema coexist, their regulated lengths must be compared; the dominant branch can switch.

This construction is the requested quench application. It tracks both the geometric transition across the shell and the saddle transition. Numerical studies of AdS–Vaidya extremal surfaces implement precisely these matching and competition conditions (Abajo-Arrastia, Aparício, and López 2010, §§3–4).

Why minimizing on an arbitrary slice fails

The failure is visible even in pure Poincaré AdS. Anchor the interval at boundary time $t=0$, but minimize length on the spacelike slice

$$t=\alpha z, \qquad |\alpha|<1.$$

The positive spatial metric $q=-g|_{\rm slice}$ is

$$ds^2_{\rm slice}=\frac{L^2}{z^2} \left[(1-\alpha^2)dz^2+dx^2\right].$$

Writing $y=\sqrt{1-\alpha^2}\,z$, the slice minimum is a semicircle in $(x,y)$ and has

$$\mathcal L_\alpha =2L\sqrt{1-\alpha^2} \log\!\left[ \frac{\ell}{\sqrt{1-\alpha^2}\,\epsilon} \right]+O(\epsilon^2).$$

It depends on $\alpha$, even though the boundary state and interval do not. The curve is stationary against deformations tangent to that slice but not against the second, timelike normal deformation. Only $\alpha=0$, the reflection-symmetric slice, recovers the spacetime extremal geodesic. This direct falsifier rules out “minimum on any slice” as a covariant entropy rule.

Scope and handoff

HRT is a leading classical prescription. It requires a spacelike extremal surface, fixed anchoring, homology, global comparison among extrema, and the correct Lorentzian geometry. A thin shell is an idealization; smooth quenches replace refraction by continuous integration and should converge to the thin-shell answer. Null-energy violation, singular causal structure, or noncompact variational sets can invalidate standard existence and nesting arguments.

Maximin states the hypotheses that connect HRT to a slice optimization and prove causal properties. FLM/QES changes both the functional and, at next order, the surface.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Abajo-Arrastia, J., Aparício, J., and López, E. (2010). “Holographic evolution of entanglement entropy.” Journal of High Energy Physics 2010(11), 149. DOI.
• Hubeny, V. E., Rangamani, M., and Takayanagi, T. (2007). “A covariant holographic entanglement entropy proposal.” Journal of High Energy Physics 2007(7), 062. DOI.