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3. Morris-Thorne wormhole

Morris-Thorne wormhole

The Morris-Thorne wormhole is a class of static spherically symmetric traversible wormhole parametrized by two functions, the redshift function $\Phi$ and the shape function $b$.

1. History

The Morris-Thorne wormhole was created by a request from Carl Sagan to Michael Morris and Kip Thorne in 1985 to help him conceive a more realistic way of interstellar travel for his novel "Contact". This lead to the publication in 1987 of the paper "Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity".

2. Topology

The Morris-Thorne wormhole spacetime has topology $\mathbb{R}^2 \times S^2$.

3. Metrics and coordinates

3.1. Schwarzschild coordinates

The Schwarzschild coordinates are put in a form reminescent of the Schwarzschild metric, with coordinates $(t, r, \theta, \varphi)$, defined on $t \in \mathbb R$, $r \in \mathbb R$, $\theta = (0, \pi]$ and $\varphi = (0, 2\pi]$. The metric in those coordinates is

$$ds^2 = -e^{2\Phi(r)} dt^2 + \frac{dr^2}{1 - \frac{b(r)}{r}} + r^2(d\theta^2 + \sin^2 \theta d\varphi^2)$$

$\Phi(r)$ the redshift function and $b(r)$ the shape function

3.2. Proper time coordinates

$$ds^2 = -e^{2\Phi(l)} dt^2 + di^2 + r^2(l) (d\theta^2 + \sin^2 \theta d\varphi^2)$$

3.3. Isotropic coordinates

$$ds^2 = -e^{2\Phi(l)} dt^2 + e^{-2\psi(r)} \left[ dr^2 + r^2 (d\theta^2 + \sin^2 \theta d\varphi^2)\right]$$

4. Tensor quantities

4.1. In Schwarzschild coordinates

4.1.1. Christoffel symbols

$$\begin{eqnarray} {\Gamma^t}_{rt} &=& \Phi'(r)\\ {\Gamma^r}_{\theta\theta} &=& -r + b\\ {\Gamma^\theta}_{\varphi\varphi} &=& -\sin\theta\cos\theta \end{eqnarray}$$

$$\begin{eqnarray} {\Gamma^r}_{tt} &=& (1 - \frac{b}{r}) \Phi'(r) e^{2\Phi(r)}\\ {\Gamma^r}_{\varphi\varphi} &=& -(r-b) \sin^2 \theta\\ {\Gamma^\varphi}_{\varphi\theta} &=& \tan \theta \end{eqnarray}$$

$$\begin{eqnarray} {\Gamma^r}_{rr} &=& \frac{b'(r) r - b(r)}{2r(r - b(r))}\\ {\Gamma^\theta}_{r\theta} &=& {\Gamma^\varphi}_{r\varphi} = r^{-1} \end{eqnarray}$$

4.1.2. Riemann tensor

$${R^t}_{rtr} =$$

4.1.3. Ricci tensor

4.1.3. Ricci scalar

$$R = -2(1 - \frac{b(r)}{r}) \left[ \Phi''(r) + (\Phi'(r))^2- \frac{b'(r)}{r(r-b(r))} - \frac{ b'(r) r + 3b(r) - 4r}{2r(r - b(r)} \Phi'(r) \right]$$

4.2. In proper time coordinates

4.2.1. Christoffel symbols

4.2.2. Riemann tensor

4.2.3. Ricci tensor

4.2.3. Ricci scalar

5. Symmetries

The Morris-Thorne wormhole, being a static and spherically symmetric solution, has 3 Killing vectors. In both coordinates, it is the Killing vectors $\partial_t$, $\partial_\theta$ and $\partial_\varphi$.

6. Stress-energy tensor

7. Curves

In Schwarzschild coordinates

The geodesic equation

In proper time coordinates

The geodesic equation

8. Equations

In Schwarzschild coordinates

The wave equation

In proper time coordinates

The wave equation

9. Causal structure

The Morris-Thorne wormhole is globally hyperbolic and has no singularities.

10. Asymptotic structure

11. Energy conditions

12. Limits and related spacetimes

In the case $r(l) = \sqrt{l^2 - a^2}$ for some $a > 0$, the Morris-Thorne wormhole reduces to the Ellis-Bronnikov drainhole.

13. Misc.

Bibliography