---
title: Off-Shell Wormholes in Quantum Gravity
url: https://www.emergentmind.com/topics/off-shell-wormholes
type: topic
---

# Off-Shell Wormholes in Quantum Gravity

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 [1904.02187]. 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 [1904.02187].

Source: https://www.emergentmind.com/topics/off-shell-wormholes