Wormholes
Wormholes are a theoretical construct that allow the connection of otherwise distant points of a spacetime. While probably not physical (or at least measurable), they still remain quite an interesting feature of general relativity and related theories, having implications reaching to most mathematical and physical areas of the theory.
1. Definition
There's no good definition of wormholes that cover all the salient cases well, either being too broad or too narrow, but here's a few definitions and heuristics for them.
The most basic definition of a wormhole is a spacetime such that some spacelike region has a handle. To simplify things, let's consider a globally hyperbolic spacetime. If for some $(n-1)$ manifold $\Sigma$, the spacetime has the topology
$$\mathcal M = (\Sigma \# T^{n-1}) \times \mathbb R$$with $T^{n-1}$ a torus of the same dimension as the spacelike hypersurface. The torus is generally considered to be the throat of the wormhole itself. This definition has two main flaws :
- It only works for intra-universe wormholes (wormholes connecting a spacetime to itself), and leaves out inter-universe wormholes (connecting two different spacetimes), which are an important class of them.
- It does not take into account geometry at all. For instance the toroidal universe $T^2 \times \mathbb R$ can be expressed as $(S^2 \#T^2) \times \mathbb R$, but it does not have important features of a wormhole such as a throat.
An alternative definition for spherically symmetric spacetimes is to consider the generic metric
$$ds^2 = -f(t,l) dt^2 + g(t,l) dl^2 + r^2(t,l) d\Omega^2$$Then a spherically symmetric spacetime is considered a wormhole if the radius function $r$ possesses at least one regular minimum, corresponding to the throat of the wormhole, and we also ask that $\lim_{l\to \pm \infty} r = \infty$.
This definition has the opposite issues : it only applies to inter-universe wormholes, the two universes are of trivial topology, only one wormhole is possible, only spherical symmetry is allowed and it is entirely dependant on the coordinate chart.
If we start with a (not necessarily connected) spacetime foliated in spacelike hypersurfaces, of the form $\mathcal M = \Sigma \times \mathbb R$ or $\Sigma \times S$ for some constant topology $\Sigma$, a simple way to do this is by surgery : take two connected finite open sets (with the same topology) $U_1$ and $U_2$, such that $U_1 \cap U_2 = \varnothing$ and then remove them from the spacelike hypersurface, $\Sigma \setminus (U_1 \cup U_2)$. This leaves a manifold with two boundaries, $\partial U_1$ and $\partial U_2$. If we identify those boundaries, the resulting spacelike hypersurface has a wormhole near the boundaries.
This definition isn't the best because it only deals with the topology, and even then it becomes quickly pathological : we can excise a ball from Minkowski space and glue it back on. But it's an easy way to generate a lot of varying wormhole topologies with some care taken.
An important local property of wormholes is how they affect geodesic congruences, focusing them upon entry before defocusing them. This is a property that is true of wormholes of topologies as different as the Morris-Thorne wormhole, a space with a handle or the bag-of-gold spacetime.
2. The Morris-Thorne wormhole
The Morris-Thorne wormhole is one of the simplest example, simply a spherically symmetric connection between two copies of $\mathbb{R}^{n-1}$.
$$ds^2 = -f(t,l) dt^2 + dl^2 + r^2(t,l) d\Omega^2$$As can be seen from the coordinates, if we allow them to run their full course (we have $t$ and $l$ ranging over $\mathbb{R}$), it has the topology $\mathbb{R}^2 \times S^2$, and in general $\mathbb{R}^2 \times S^{n - 2}$ in $n$ dimensions. Those are called the proper radial length coordinates, as the coordinate $l$ (assuming some foliation along $t$) is indeed just the proper radial distance.
The metric components have to fulfill the following conditions :
- The metric has no horizons
- It is asymptotically flat
For now, let's focus on the static wormhole, which is the most commonly studied one.
$$ds^2 = -f(l) dt^2 + dl^2 + r^2(l) d\Omega^2$$A few more important coordinates we can use for this are the Schwarzschild coordinates :
$$ds^2 = -e^{2\phi_\pm (r)} dt^2 + \frac{dr^2}{1 - b_\pm(r) / r} + r^2 d\Omega^2$$So called because they have roughly the same form as the Schwarzschild metric. Unlike the proper radial length coordinates, these require two coordinate patches, one for each side of the wormhole (as the metric becomes otherwise degenerate for $r = 0$).
The non-zero components of the Christoffel symbols are
\begin{eqnarray} {\Gamma^t}_{tl} &=& {\Gamma^t}_{lt} = \frac{f'}{2f}\\ {\Gamma^l}_{tt} &=& \frac{f'}{2} \\ {\Gamma^l}_{\theta\theta} &=& -rr'\\ {\Gamma^l}_{\varphi\varphi} &=& -rr' \sin^2(\theta)\\ {\Gamma^\theta}_{\theta l} &=& {\Gamma^\theta}_{l\theta} = \frac{r'}{r}\\ {\Gamma^\theta}_{\varphi\varphi} &=& -\sin(\theta) \cos(\theta)\\ {\Gamma^\varphi}_{\varphi l} &=& {\Gamma^\varphi}_{l\varphi} = \frac{r'}{r}\\ {\Gamma^\varphi}_{\varphi\theta} &=& {\Gamma^\varphi}_{\theta\varphi} = \cot(\theta) \end{eqnarray}This leads to the following non-zero Riemann tensor components :
\begin{eqnarray} x \end{eqnarray}Matter distribution
No matter the shape of the wormhole, it will violate the null energy condition.
no go theorem
3. The Ellis-Bronnikov drainhole
One of the simplest example of the Morris-Thorne wormhole is the Ellis-Bronnikov drainhole. While not quite entirely what the original paper on the topic was discussing, this has come to describe a Morris-Thorne wormhole where $f = 1$ and $r^2 = l^2 + a^2$, $a$ some parameter corresponding to the radius of the wormhole throat.
$$ds^2 = - dt^2 + dl^2 + (l^2 + a^2) d\Omega^2$$This simplifies the analysis considerably :
4. The Morris-de Sitter wormhole
It can be interesting to consider the case of a Morris-Thorne-type wormhole with a cosmological constant, as it is one of the simplest model of an expanding wormhole.
In a local patch of Schwarzschild-type coordinates, we're interested in a typical Morris-Thorne wormhole :
$$ds 2 = -e {2\Phi(r)} dt^2 + \frac{dr^2}{1-b(r)/r} + r^2 d\Omega^2$$with the usual orthonormal basis :
\begin{eqnarray} e_t &=& e^{-\Phi}\\ e_r&=& \sqrt{1-b/r}\\ e_\theta &=& r^{-1}\\ e_\phi &=& (r \sin \theta)^{-1} \end{eqnarray}What we want is for the solution to obey the Einstein field equations with a cosmological constant, that is,
\begin{eqnarray} \rho(r) &=& \frac{b'}{r^2} - \Lambda\\ \tau(r) &=& \frac{b}{r^3} - 2 (1 - \frac{b}{r} \frac{\Phi'}{r} - \Lambda)\\ p(r) &=& (1 - \frac{b}{r}) \left[ \Phi'' + (\Phi')^2 - \frac{b'r-b}{2r^2 (1 - b/r)} \Phi' - \frac{b'r - b}{2r^3(1-b/r)} + \frac{\Phi'}{r} \right] + \Lambda \end{eqnarray}5. Cut and paste wormholes
The simplest method to obtain intra-universe wormhole (at least as far as calculations are involved, the mathematical structures aren't quite as forgiving) is the cut and paste wormhole : remove two diffeomorphic regions $S_1, S_2$ from the spacetime, and then identify their boundaries $\partial S_1, \partial S_2$. This require a fair bit of mathematical legwork as it requires both a proper definition of the procedure and proving quite a lot of GR theorems which may not work properly here as most of GR has a lot of assumptions on the smoothness of the metric. Most of this can be found in this article here, so as to separate a bit the fairly long build up of this formalism. We can then jump straight to the matter of building an actual cut and paste wormhole.
Spherical wormholes in Minkowski space
The simplest case of a cut and paste wormhole is to simply pick Minkowski space with two spheres removed and identified. That is, take Minkowski space $M = \mathbb{R}^{1,n-1}$, with two balls throughout time $\mathcal{S}_1$, $\mathcal{S}_2$ of topology $S^{n-2} \times \mathbb{R}$. We have two embeddings for the surfaces generated by those balls
$$f_i(t, \varphi^a) = (t, r_i(t) x^a(\varphi^a) + q^a(t) )$$with $r(t)$ the radius at the time $t$, $q^a(t)$ the position of the center of the sphere, and $x^a(\phi^a)$ the usual mapping of hyperspherical coordinates.
\begin{eqnarray} x^1 &=& \cos(\varphi^1) \end{eqnarray}To simplify computations, we'll only consider the $2$ and $3$ dimensional case here :
\begin{eqnarray} x &=& \cos(\varphi)\\ y &=& \sin(\varphi) \end{eqnarray} \begin{eqnarray} x &=& \sin(\theta) \cos(\varphi)\\ y &=& \sin(\theta) \sin(\varphi)\\ z &=& \cos(\theta) \end{eqnarray}We can define a level-set function simply using the (signed) distance from the surface
$$\phi^\pm(t, \vec{x}) = \pm \left[ |\vec{x} - \vec{q}(t)|^2 - r^2(t) \right]$$A normal vector to this surface is
$$n^\mu = (- \left[ (x_a + q_a)\dot{q}^a + \dot{r} \right], (x_a - q_a))$$The induced metric on this surface is, for the spacelike components, simply the sphere metric
$$\bar{g}_{ab} = \delta_{ab} r^2 \prod_{i=1}^{a-1} \sin^2 \varphi^i$$The timelike component is
$$\bar{g}_{tt} = -1 + \| x^a \dot{r} + \dot{q}^a \|$$We can interpret the second term as the velocity of a point on the mouth at the point $x$. For the hypersurface to be timelike, we'll then require that $\| x^a \dot{r} + \dot{q}^a \| < 1$, in other word the wormhole mouth must always move slower than the speed of light.
The cross terms are
$$\bar{g}_{ta} = r \dot{r} x^c_{,a} x_c + r x^c_{,a} \dot{q}_c$$There are many possible gluing functions, although the requirement of isometry between the two induced metrics remove some of those. To keep things tractable, we'll only consider a few specific cases later on.
An important matter for solving the geodesic equation is to find out how vectors are mapped from one mouth to the other.
Static wormholes
The simplest case is to simply consider two wormholes at rest with respect to each other, not rotating or expanding. This is just
$$f_i(t, \varphi^a) = (t, r_i x^a(\varphi^a) + q^a )$$with the induced metric $\bar{g}_{tt} = -1$, $\bar{g}_{ta} = 0$, and the spatial part simply the metric of a sphere. The isometry condition then implies :
\begin{eqnarray} -1 &=& -\dot{b}^2\nonumber\\ \delta_{ab} r^2 \prod_{i=1}^{a-1} \sin^2(\varphi^i) &=& \delta_{ab} r^2 \prod_{i=1}^{a-1} \sin^2(\xi(\varphi^i))\sum_{b}^{} \xi^a_{,b}(\varphi^i) \xi^a_{,b}(\varphi^i) \end{eqnarray}$b$ has a trivial solution, and $\xi$ is just the set of isometries on $S^n$, with solutions
\begin{eqnarray} b(t) &=& \pm t + t_0\nonumber\\ \xi(\varphi) &\in& \mathrm{O}(n-1)\nonumber \end{eqnarray}The simplest case is $b(t) = t$ and $\xi(\varphi) = P$, the parity transformation ($\theta \to \theta + \pi$ in the $2+1$ dimensional case).
We're gonna need some formula of the distance between the two mouthes.
The twin paradox case
In this setting, one of the wormhole mouth moves with respect to the other, in some configuration. The simplest one is simply to have one mouth moving in one direction at constant speed. While not the best scenario (it means that it is infinitely far in the past and will collide with the other mouth in the future), it is quite simple and is still a functioning example of a time machine. The two embeddings are :
\begin{eqnarray} f_1(t, \varphi^a) &=& (t, r x^a(\varphi^a)) f_2(t, \varphi^a) &=& (t, r x^a(\varphi^a) + x^a_0 - v^a t) \end{eqnarray}For simplicity, all movement will be done with respect to $x$. The two mouthes are originally separated by $x_0$ and are approaching each other at speed $v_x$ (v_x > 0$).
6. Smooth interuniverse wormholes
Making a smooth version of an interuniverse wormhole is somewhat challenging due to the lack of any obvious symmetry one could use for it, but it still remains somewhat possible.
Wormhole in a spherical universe
The simplest example that can be furnished of a smooth interuniverse wormhole is simply a wormhole in an otherwise spherical universe, as the sphere is the identity element of the connected sum.
$$M = S^n \# T^n = T^n$$So that we can simply use a spacetime with the topology of a torus, for instance with coordinates $(\theta, \varphi)$ in $2+1$ dimensions. The simplest such example would be the Clifford torus
$$ds^2 = -dt^2 + d\theta^2 + d\varphi^2$$Not the best example, as there is no real presence of a mouth (the radius has no minimum). To remedy this, we simply change the scale. If we assume that the spacetime is rotationally invariant under $\varphi$, and $\theta$ describes the "radial distance" from the wormhole (let's say $(0,\pi)$ corresponds to the inside of the mouth while $(\pi, 2\pi)$ is the rest of the universe), we simply want that distance to diminish inside the wormhole. Keeping things otherwise symmetrical, this is
$$ds^2 = -dt^2 + f(\theta) d\theta^2 + d\varphi^2$$The conditions on $p$ being that it is roughly $1$ at $3\pi/2$ and has a minimum at $\pi/2$. An example of such a function is
$$f(\theta) = 1 - \frac{\alpha}{2}(1 + \sin(\theta))$$which is $1$ at $\theta = 3\pi/2$ and $1 - \alpha$ at $\theta = \pi/2$. For $\alpha \in (0,1)$, this is a proper
Asymptotically Minkowski spacetime
In $2+1$ dimensions, an important observation is that the connected sum of the torus and the plane is simply a punctured torus, $T \setminus \{ p \}$. Hence a wormhole of the appropriate topology can be concocted from there.
The simplest metric one can put on the punctured torus is simply the flat metric, although this will obviously not correspond to what we'd like ideally, which is an asymptotically Minkowski spacetime.
Misner investigated the topic of wormholes, related to Wheeler's geometrodynamics. To do this with simple enough coordinates, he considered the bispherical coordinate system, which has the benefit of spanning two spheres rather simply. As a reminder, bispherical coordinates in Euclidian space are $(\sigma, \tau, \phi)$
\begin{eqnarray} x &=& a \frac{\sin(\sigma)}{\cosh(\tau) - \cos(\sigma)} \cos(\phi)\\ y &=& a \frac{\sin(\sigma)}{\cosh(\tau) - \cos(\sigma)} \sin(\phi)\\ z &=& a \frac{\sinh(\tau)}{\cosh(\tau)- \cos(\sigma)} \end{eqnarray} \begin{eqnarray} \sigma &=& \arccos(\frac{R^2 - a^2}{Q})\\ \tau &=& \operatorname{arcsinh} (\frac{2az}{Q})\\ \phi &=& \arctan(\frac{y}{x}) \end{eqnarray}The Aichelburg-Schein timehole
It is difficult to make a realistic model for intra-universe wormholes, simply due to the fact that we do not really have a realistic model for a two-body problem in general relativity (at least in $3+1$ dimensions). Two free spherical bodies cannot be held in static equilibrium (they'd simply attract each other), and they cannot simply orbit each other in a simple manner. Hence any such model will have to move away from such notions in some way.
The first Aichelburg-Schein timehole model takes inspirations from the Israel-Khan system of two singularities (point particles), held in equilibrium by a strut between them. The starting point is to consider the generic form of axisymmetric spacetimes, the Weyl metric
$$ds^2 = -e^{2\lambda(r,z)} dT^2 + e^{2\left[ \nu(r,z) - \lambda(r,z) \right]} (dr^2 + dz^2) + r^2 e^{-2\lambda(r,z)} d\varphi^2$$If we consider the vacuum solution, the Einstein field equations reduce to
\begin{eqnarray} \nabla^2 \lambda = \partial_r^2 \lambda + \partial_z^2 \lambda + r^{-1} \partial_r \lambda &=& 0\\ \partial_r \nu &=& r \left[ (\partial_r \lambda)^2 - (\partial_z \lambda)^2 \right]\\ \partial_z \nu &=& 2r\partial_r \lambda \partial_z \lambda \end{eqnarray}Hence given any solution to $\lambda$ we get a solution to $\nu$.
There are many possible vacuum solutions of the Weyl metric, but the one that will interest us here is the Schwarzschild metric in Weyl coordinates. Due to $\lambda$ obeying the Laplace equation, it is common to compare Weyl solutions to Newtonian potential solutions. In the case of the Schwarzschild metric, $\lambda$ has the form of a rod of mass $m$ and length $2m$ :
$$\lambda = \frac{1}{2} \ln(\frac{R_1 + R_2 - 2m}{R_1 + R_2 + 2m})$$with
\begin{eqnarray} R_1 &=& (r^2 + (z-m)^2)^\frac{1}{2}\\ R_2 &=& (r^2 + (z+m)^2)^\frac{1}{2} \end{eqnarray}which gives
$$\nu = \frac{1}{2} \ln(\frac{(R_1 + R_2)^2 - 4m^2}{4R_1R_2})$$While the Newtonian parallel is common, there are very little links between the actual source of a Weyl metric and the Newtonian source of $\lambda$,
In general there is no correspondence between the geometry of the source for a Weyl metric, and the geometry of the Newtonian source from which it is generated.
which is useful to keep in mind.
It can then be checked that the coordinate transform
\begin{eqnarray} \rho &=& \frac{1}{2} (R_1 + R_2 + 2m)\\ \cos(\theta) &=& \frac{1}{2m} (R_2 - R_1) \end{eqnarray}leads to the canonical (exterior) Schwarzschild metric
$$ds^2 = - (1 - \frac{2m}{\rho}) dt^2 + (1 - \frac{2m}{\rho})^{-1} d\rho^2 + \rho^2 (d\theta^2 + \sin^2 \theta d\varphi^2)$$Last updated : 2019-05-21 10:51:15