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# 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 scalar

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

## 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 $\partial_\varphi$.

## 9. Causal structure

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

## 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.