Lorentzian Einstein–Rosen Bridges and Two-Boundary States

The maximally extended eternal AdS black hole has two asymptotic boundaries joined on a spacelike slice by a nontraversable Einstein–Rosen bridge. In the standard semiclassical dictionary it is prepared by a thermofield-double state of two noninteracting boundary theories; entanglement supports cross-boundary correlations, but neither the bridge volume nor a local point behind the horizon is an invariant observable without a relational prescription.

Required background. Two-Sided Black Holes and Thermofield-Double States supplies the thermal state preparation. Relational, Boundary, and Asymptotic Observables supplies the observable standard.

Helpful background. Reflected Entropy and Canonical Purifications gives a purification comparison. Relational Bulk Observables and Dressing Choices explains why gravitational localization requires dressing.

The two-boundary geometry and state

In Lorentzian signature $(+---)$, a static asymptotically AdS$_{d+1}$ black hole may be written outside either horizon as

$$ds^2=f(r)dt^2-\frac{dr^2}{f(r)}-r^2d\Sigma_{d-1}^2,$$

with $f(r_h)=0$. Kruskal extension produces right and left exteriors. Their natural future-directed Killing times have opposite orientation with respect to a single global Killing vector. The bifurcation surface is minimal on the time-reflection-symmetric slice; causal curves from one boundary encounter the future singularity rather than reaching the other boundary.

Euclidean evolution through half the thermal circle prepares

$$\lvert\mathrm{TFD}\rangle =\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\beta E_n/2} \lvert E_n\rangle_L\lvert E_n\rangle_R .$$

Tracing either factor gives $\rho_R=e^{-\beta H_R}/Z(\beta)$. The two-sided correlator is obtained by analytic continuation with the thermal contour fixed:

$$\langle O_L(t_L)O_R(t_R)\rangle_{\mathrm{TFD}} =\frac{1}{Z(\beta)} \operatorname{Tr}\!\left( e^{-\beta H/2}O(t_L)e^{-\beta H/2}O(t_R) \right),$$

with the precise transpose or CPT map in $O_L$ determined by the chosen identification of the two Hilbert spaces. Maldacena’s two-sided AdS construction makes this state–geometry relation explicit at leading large $N$ Maldacena 2003.

Application: fixed marginals, different bridges

Apply an energy-diagonal unitary to the left:

$$\lvert\Psi_\theta\rangle =\frac{1}{\sqrt{Z}}\sum_n e^{-\beta E_n/2+i\theta_n} \lvert E_n\rangle_L\lvert E_n\rangle_R .$$

Both one-sided density matrices remain exactly thermal, but a cross-boundary correlator becomes

$$\langle\Psi_\theta|O_L O_R|\Psi_\theta\rangle =\frac{1}{Z}\sum_{m,n} e^{-\beta(E_m+E_n)/2} e^{i(\theta_n-\theta_m)} (O_L)_{mn}(O_R)_{mn}.$$

Generic rapidly varying phases suppress the coherent correlator even though no one-sided coarse observable changes. Therefore two thermal marginals do not determine the global state, the existence of a smooth semiclassical bridge, or its detailed interior.

The length or maximal volume of a bridge slice also depends on its anchoring times and extremization rule. Under the two-sided Hamiltonian convention, time-shifted TFD states can have identical reduced states while the maximal slice grows. Volume is consequently a useful geometrical diagnostic within a fixed prescription, not a diffeomorphism-invariant local observable by itself.

Causal and relational check

Choose boundary anchor times, a relational bulk construction, and a gravitational dressing before comparing interiors. Then hold both one-sided density matrices fixed while varying cross-boundary phases as above. Any claim that the one-sided thermal data uniquely determine a smooth bridge fails this check. Conversely, nonzero mutual information or a two-sided correlator diagnoses correlations, not traversability: without an interaction, boundary operations on $L$ cannot signal to $R$.

Limits and handoff

The geometric description requires a large-$N$, strongly coupled state with a controlled semiclassical saddle and an appropriate code sector. It does not identify every entangled state with a wormhole. Turning the nontraversable bridge into a channel requires a specified interaction and negative averaged null energy, developed in Traversable Wormholes, Couplings, and Energy Conditions.

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

• Maldacena, J. “Eternal Black Holes in Anti-de Sitter.” Journal of High Energy Physics 2003, 4 (2003): 021. DOI.
• Van Raamsdonk, M. “Building Up Spacetime with Quantum Entanglement.” General Relativity and Gravitation 42 (2010): 2323–2329. DOI.