Off-Shell Wormholes in Quantum Gravity

• Off-shell wormholes are spacetime configurations arising from nonperturbative Euclidean instantons that nucleate a handle connecting two black hole horizons.
• Their formation involves cosmic string breaking and quantum tunneling, with detailed geometric analysis and matching conditions ensuring smooth transitions at the horizons.
• Quantum back-reaction induces negative null energy in the throat, rendering the wormhole traversable under appropriate semiclassical conditions.

An off-shell wormhole refers to a spacetime configuration arising from a nonperturbative Euclidean (off-shell) instanton in quantum gravity, nucleating a handle in space that connects two black hole horizons. The concept is grounded in the semiclassical description of processes by which a straight cosmic string breaks into two endpoint black holes whose horizons are identified, ultimately yielding a traversable wormhole after quantum back-reaction effects are taken into account (Horowitz et al., 2019). The process is fundamentally nonperturbative, with instanton configurations mediating tunneling transitions between distinct spacetime topologies. This approach leverages both geometric analysis and quantum field theoretic back-reaction to render such wormholes physically traversable.

1. Euclidean Background Geometry

The construction begins with a four-dimensional, static, spherically symmetric Lorentzian spacetime

$$ds^2_L = -f(r)\,dt^2 + \frac{dr^2}{g(r)} + r^2\,d\Omega_2^2,$$

where $d\Omega_2^2$ denotes the round metric on $S^2$, with $f(r)$ and $g(r)$ both positive outside any event horizon. The corresponding Euclidean section arises via $t\to -i\tau$:

$$ds^2_E = f(r)\,d\tau^2 + \frac{dr^2}{g(r)} + r^2\,d\Omega_2^2.$$

This geometry possesses a $\mathbb{Z}_2$ symmetry, $r\leftrightarrow -r$, in the $(\tau,r)$ subspace, and features totally geodesic two-surfaces at $\{\theta=0\}\cup\{\theta=\pi\}$. These structures ensure that test particles attached to the string evolve strictly in the $(t,r)$ plane.

Explicit instantiations include:

• The “AdS star” geometry, parameterized by

$$f(r) = g(r) = \begin{cases} 1+\frac{r^2}{L^2}-\frac{2M}{r}, & r>R, \ 1+\frac{r^2}{L^2}+A\,r^6-B\,r^2, & r<R, \end{cases}$$

with matching conditions for $f, f'$ at $r=R$ by appropriate $A, B$.

• Vacuum “boundary-deformed global AdS” solutions, where the conformal boundary is squashed by a spherical harmonic of order $\ell$ and the bulk metric is found perturbatively or with a numerical DeTurck construction.

2. World-Sheet Action and Instanton Trajectories

The dynamics are governed by a cosmic string of tension $\mu$ with two endpoint particles of mass $m$, whose Lorentzian action is

$S_L = -\mu \int_W dA - m\!\int_{\partial W} ds,$

where $W$ is the world-sheet of the (potentially broken) string and $\partial W$ corresponds to its endpoints. The action difference relative to the unbroken, infinite string is

$$\Delta S_L = 2\int_{-\infty}^{+\infty}dt\,\mathcal{L}(r,\dot r),$$

with

$$\mathcal{L}(r,\dot r) = \mu\,P(r) - m\sqrt{f(r)-\frac{\dot r^2}{g(r)}}, \qquad P(r) = \int_0^r\sqrt{\frac{f(\tilde r)}{g(\tilde r)}}\,d\tilde r.$$

The equations of motion are derived from energy conservation ($E=0$),

$$\dot r^2 + V(r) = 0, \qquad V(r) = f(r)g(r)\left\{\frac{m^2f(r)}{\left[\mu P(r)\right]^2} - 1\right\}.$$

Employing Wick rotation ($\tau = i t$), the Euclidean instanton trajectory $r(\tau)$ satisfies

$$\frac{dr}{d\tau} = \sqrt{f\,g}\,\sqrt{\frac{f}{(\mu/m)^2\,P^2}-1}, \qquad r(\tau_0)=0,~r(0)=r_1,$$

with $r_1$ determined by $V(r_1)=V'(r_1)=0$. In the limit of small acceleration (nearly degenerate minima), the instanton stretches in $\tau$, but yields finite action.

Upon replacing endpoint particles by small black holes of equal mass (and suitable charge or spin), the instantonic structure changes only within a tubular $S^1\times\mathbb{R}^3$ neighborhood about the world-line. Black hole temperature is matched to ensure proper Euclidean periodicity around the horizon.

3. Horizon Identification and Wormhole Mouth Formation

For $\tau=0$, the Euclidean configuration features two small black hole horizons at $r=r_1$. Identification of these horizons by Rindler-like gluing results in a single handle at the $\tau=0$ slice. Upon analytic continuation to Lorentzian signature, the solution yields two black hole mouths at rest, whose near-horizon geometries are identified—forming the two mouths of a wormhole.

Smooth extension at the horizons is enforced by matching the Euclidean period to $4\pi/\kappa$, where $\kappa$ is the horizon surface gravity; otherwise, a conical deficit proportional to the period mismatch is allowed. This construction avoids the problematic relative acceleration that would otherwise impair traversability via Unruh radiation.

4. Euclidean (Off-Shell) Action and Nucleation Probability

The (on-shell) Euclidean action relative to the original unbroken string is

$$\Delta S = 2m\!\int_{0}^{r_1}\frac{dr}{\sqrt{g(r)}\sqrt{1 - \frac{\mu^2}{m^2}\frac{P(r)^2}{f(r)}}},$$

and the semiclassical nucleation probability for the wormhole is given by

$P \simeq e^{-\Delta S}.$

For the “AdS star” geometry, this integral can be evaluated analytically. In the vacuum-deformed AdS case, to quadratic order in the boundary squashing parameter $\epsilon$ (for $\ell=2$),

$$\Delta S_{\ell=2} = \pi m L \left\{1 + \frac{3}{16}\epsilon\left[1+\frac{\pi}{2} + 2\log\left(\frac{3\pi\epsilon}{32}\right)\right]\right\} + \mathcal{O}(\epsilon^2 \log \epsilon).$$

5. Quantum Back-Reaction and Traversability

The Lorentzian evolution describes two extremal or near-extremal black holes whose horizons are identified. For vanishing acceleration, the Killing horizon's surface gravity approaches zero, permitting quantum fields in a Hartle–Hawking–like state to develop negative null energy in the throat:

$$\langle T_{kk}\rangle = -\frac{\hbar}{L^4} F(\text{geometry}, m, \mu) < 0,$$

for any null vector $k$ pointing into the throat. Inputting this quantum stress tensor into the semiclassical Einstein equations,

$G_{ab} = 8\pi G_N \langle T_{ab} \rangle,$

produces an $O(G_N\hbar)$ shift in the metric that opens the wormhole throat, enabling causal traversability between asymptotic regions. For pure AdS, a large field content is required to suppress fluctuations, but for small acceleration, an eternal traversable wormhole is supported by the negative averaged null energy in the throat.

6. Key Equations and Formal Summary

The fundamental equations organizing the off-shell wormhole nucleation process are as follows:

Quantity	Equation	Description
Background metric	$$ds^2_E=f(r)\,d\tau^2+\frac{dr^2}{g(r)}+r^2d\Omega_2^2$$	Euclidean background used in instanton construction
World-sheet action	$$S_E=\mu\int dA+m\int ds \rightarrow \Delta S =2m\int_0^{r_1} [g(r)(1-\mu^2 P(r)^2/(m^2 f(r)))]^{-1/2}dr$$	Instanton action to break string and nucleate wormhole
Static orbit condition	$V(r_1) = V'(r_1) = 0$, $V(r) = f(r)g(r) \{\frac{m^2 f}{(\mu P)^2}-1\}$	Determines position and stability of black hole mouths
Nucleation rate	$P \sim e^{-\Delta S}$	Probability for wormhole formation
Horizon identification	match Euclidean period $\beta = 4\pi/\kappa$ with period of $\tau$-circle	Ensures smoothness or introduces controlled conical defect at horizons
Back-reaction for traversability	$\langle T_{kk}\rangle<0$ in the throat $\Longrightarrow$ traversability	Negative null energy supports an open wormhole throat

The off-shell instanton thus mediates the quantum tunneling process creating a wormhole connecting two black holes at rest, with quantum effects ensuring traversability under appropriate boundary and field content conditions (Horowitz et al., 2019).