The Spacetime Geometry of Fixed-Area States in Gravitational Systems

TL;DR

The paper investigates the Lorentzian spacetime geometry intrinsic to fixed-area states in gravitational systems, contrasting it with Euclidean saddles used to prepare such states. Using Schwinger-Keldysh constructions, it shows that the intrinsic fixed-area geometry is real at real times and largely free of conical singularities, with curvature divergences appearing only along null congruences from the fixed-area surface. Through explicit examples in JT gravity and AdS3 setups, it demonstrates that the Lorentzian spacetimes are smooth in the central regions, though power-law divergences can arise on lightcones when symmetry is broken. The authors argue that quantum fields in these states require smearing of the fixed-area surface to remain well-defined, and they discuss the implications for holography and potential extensions to higher-derivative theories.

Abstract

The concept of fixed-area states has proven useful for recent studies of quantum gravity, especially in connection with gravitational holography. We explore the Lorentz-signature spacetime geometry intrinsic to such fixed-area states in this paper. This contrasts with previous treatments which focused instead on Euclidean-signature saddles for path integrals that prepare such states. We analyze general features of fixed-area state geometries and construct explicit examples. The spacetime metrics are real at real times and have no conical singularities. With enough symmetry the classical metrics are in fact smooth, though more generally their curvatures feature power-law divergences along null congruences launched orthogonally from the fixed-area surface. While we argue that such divergences are not problematic at the classical level, quantum fields in fixed-area states feature stronger divergences. At the quantum level we thus expect fixed-area states to be well-defined only when the fixed-area surface is appropriately smeared.

Figures (7)

• Figure 1: Left: Starting from a TFD state with inverse temperature $\beta_0$, fixing the area of the HRT surface $\gamma_R$, corresponding to the time slice $\tau=0$, results in a Euclidean saddle with a conical singularity (red) with opening angle $2\pi m$, where $m=\frac{\beta_0}{\beta}$. Right: The same state prepared as a microcanonical TFD state by imposing asymptotic fixed energy boundary conditions. This results in a smooth Euclidean saddle with boundary length $\beta$. The $\mathbb{Z}_2$ symmetric Cauchy slice $\Sigma_{sym}$ (green and blue) on both saddles has identical data and thus, results in identical Lorentzian spacetimes upon time evolution.
• Figure 2: The saddle point manifold $\mathcal{M}$ for $\langle \psi| \Pi_R^2 | \psi \rangle$ can be split into two parts $\mathcal{M}_1$ and $\mathcal{M}_2$ along a slice $\Sigma_{cut}$ such that there are two conical defects (red) at $\gamma_R$, with one on the bra side of the cut and the other on the ket side. On the right, we have added a regulator $\epsilon$ that moves each of the resulting singularities away from the cut. The original $\mathcal{M}$ should be understood as the limit $\epsilon\rightarrow 0$ where the two $\gamma_R$ surfaces coincide.
• Figure 3: (a) The saddle point for $\langle \psi| \Pi_R (0) O(t_0) \Pi_R (0) | \psi \rangle$ has two conical defects (red) at $\gamma_R$ on the bra and ket side respectively. It has a $\mathbb{Z}_2$ symmetry that leaves invariant the Cauchy slice $\Sigma_{sym}$ (green). This slice splits the saddle into two parts $\mathcal{M}_1$ and $\mathcal{M}_2$. The region of the saddle point shaded light blue can be thought of as the spacetime inherent to the fixed-area state, while the yellow portion is involved in the preparation of the state. We may take the blue portion to lie at real Lorentzian times and the yellow portion to lie at real Euclidean times. In that sense, the two blue portions each involve the same interval of real Lorentzian times and should perhaps be drawn as being degenerate with each other, but we have separated the pieces for ease of visualization. (b) The corresponding Schwinger-Keldysh contour in the complex-time plane.
• Figure 4: The saddle can be extended in the real time direction by evolving using the data on Cauchy slice $\Sigma_{\text{sym}}$. As an example, we depict the Schwinger-Keldysh contour for forward evolution up to surface $\Sigma_2$ and backward evolution up to surface $\Sigma_1$.
• Figure 5: Left: Euclidean preparation of the state of a scalar field on a conical background of opening angle $2\pi m$. A source for the scalar field (pink) is present at the asymptotic boundary and which breaks the $U(1)$ symmetry of the geometry. This prepares initial data for Lorentzian evolution on the $\mathbb{Z}_2$ symmetric slice defined by the union of the blue half-line at $\theta=0$ line and the green half-line at $\theta=\pi$. Note that there is really only a single blue half-line; the marks on the diagram indicate that the two copies are to be identified. Right: The initial data prepared by the Euclidean solution is used to generate the Lorentzian solution in Minkowski space by solving the equation of motion as described in the main text. The solution at the small black dot in region II is obtained by propagating the data along left and right-moving light rays (for a free massless field).