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Spacetime gluing

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An important process in the creation of spacetimes is the so-called gluing (aka cutting and pasting or adjunction), which consists of identifying regions of one or more spacetimes together to obtain a new spacetime. This can be used to construct a variety of topologically complex spacetimes using fairly simple metrics, including wormholes. Despite being a useful tool, there does not seem to be much in the ways of general resources on the topic, so this article will focus on how to construct such spacetimes rigorously, what their properties are, how to perform analysis on them, etc.

The name can actually refer to two different processes, which we will both see : identifying the boundaries of manifolds with boundaries, and identifying open sets of a manifold. The first kind is more commonly used and has the most literature on the topic, which is why we will mostly focus on it.

1. Cutting and pasting manifolds by identifying boundaries

The most common type of manifold gluing is done with manifolds with boundaries. The basic idea behind cutting and gluing using this method is to remove open sets, define some diffeomorphism $h$ between the boundaries left behind (called the gluing function), and then taking a quotient of the resulting space using the equivalence relation defined by $h$. Most of the informations from this section come from C.T.C. Wall's Differential topology[1].

We'll be interested in a rather general case where we can either glue a manifold to itself or different manifolds together, but for now we'll consider the case of two manifolds, which is slightly simpler and also the most common one used. So for now, we'll consider two manifolds of the same dimension $M_1$ and $M_2$.

The cutting part is fairly simple : consider open sets $S_i \subset M_i$. We'll cut $S_i$ from $M_i$ by simply considering the resulting manifold with boundaries $M_i \setminus S_i$, with the new boundary $\partial S_i$. The gluing will then occur along those boundaries. But first, we need to show some properties of boundaries in manifolds. Consider some smooth hypersurface $\mathcal{S}$ of a manifold, with some smooth embedding $i : \mathcal{S} \to M$. First, we need to define the notion of the normal bundle. As we're only really concerned with fairly simple manifolds, we can just use the definition for Riemannian manifolds. For a manifold with a Riemannian metric $g$, :

In the case of a hypersurface, this leads us to a vector bundle of dimension $1$, as any vector of $T\mathcal{S}$ cannot be normal to itself (since the metric is Riemannian), and we therefore have an $(n-1)$-dimensional basis for every $T_p\mathcal{S} \subset T_pM$.

This leads us to define the notion of a tubular neighbourhood :

To prove the required theorem for tubular neighbourhood, we'll require some intermediate theorems. First, some useful lemmas on metric spaces (since the class of manifolds we're interested in always admits a Riemannian metric, it is indeed always a metric space).

If we assume further that $\mathcal{S}$ is orientable, we can also use this stronger theorem

This allows us the notion of a collared neighbourhood.

While this is quite a general theorem, in our case we'll stick to the case of cutting orientable pieces from the manifold, in which case the process is fairly simple : remove the piece, the collared neighbourhood corresponds to the tubular neighbourhood, appropriately rescaled.

Since manifolds are $T_5$ spaces, we can guarantee that there exists tubular neighbourhoods of two disjoint subsets which are also disjoint, hence so are their collared neighbourhoods. To show this, simply consider the two boundaries as separated sets $\partial S_1$ and $\partial S_2$. By the $T_5$ property of our manifold, there are two disjoint open sets $U_1$ and $U_2$ associated to each boundary. Consider the manifolds that they both constitute and in each of those, construct a collared neighbourhood. We therefore have two disjoint collared neighbourhoods.

Now that we have those elements, we can define the gluing of those two boundaries in the following way :

Quotient spaces and topology

We're going to need a bit of a refresher on quotient spaces here. Given a topological space $X$, consider an equivalence relation $\sim$ on it, ie

\begin{equation} \sim = \{ \langle x, y \rangle | x, y \in X \} \end{equation}

such that we have the equivalence axioms, ie it is reflexive, symmetric and transitive. We define the quotient of $X$ by $\sim$ as $X^* = X / \sim$ via a surjection $\pi$

\begin{eqnarray} \pi : X &\to& X^*\\ x &\mapsto& \pi(x) = [x] \end{eqnarray}

where $[x]$ is the equivalence class of $x$ by $\sim$. In other words, $[x] = \{ y \in X | y \sim x \}$. The quotient topology on $X^*$ is the topology for which $\pi$ is continuous, ie, $U \subset X^*$ is open if $\pi^{-1}(U)$ is open.

Given all the conditions that we defined earlier, it can then be shown that the gluing of a manifold with boundaries can be given a manifold structure. First, let's consider the points outside of the boundary, ie $\text{Int}(M)$. As the equivalence relation here is simply equality, we simply have $\pi(p) = \{ p \}$. We can use the projection function on singletons $p_1(\{ p \}) = p$ here to define an atlas. For every $p \in \text{Int}(M)$, take every coordinate patch $U_{p, \alpha}$ such that $p \in U_{p, \alpha}$ with a map

\begin{equation} \varphi_{p, \alpha} : U_{p, \alpha} \to \mathbb{R}^n \end{equation}

Those maps either already exist or are restrictions of the atlas to the interior of $M$ and $\mathbb{R}_{\geq 0} \times \mathbb{R}^{n-1}$. We can easily adapt those charts to $\bar{M}$ via $\bar{U}_{p, \alpha} = \pi(U_{p, \alpha})$

\begin{eqnarray} \bar{\varphi}_{p, \alpha} : \bar{U}_{p, \alpha} &\to& \mathbb{R}^n\\ q &\mapsto& \bar{\varphi}_{p, \alpha}(q) = \varphi_{p, \alpha}(p_1(q)) \end{eqnarray}

The charts and their transitions stem naturally from that definition. We can then define $\bar{U}_{p, \alpha}$ to define the topology of $\bar{M}$ outside of the junction. Here we need to define some additional charts and their transitions.

First, consider the manifold structure of the boundary itself. $\partial S_i$ has an $(n-1)$-dimensional atlas of the form $\{ U_{i,\alpha}, \phi_{i,\alpha} \}$. The two collared neighbourhoods also define two functions $c_i : \partial S_i \times [0,1)$. We can first separate our collared neighbourhood according to the boundary atlas,

\begin{eqnarray} C_{i,\alpha} = \{ p \in M | \exists q \in U_{i,\alpha}, \exists r \in [0,1), c_i(q, r) = p \} \end{eqnarray}

As we have a homeomorphism $h$ mapping the two boundaries, every open set $U_{1,\alpha}$ has an equivalent $U_{2, \alpha} = h(U_{1, \alpha})$. From this, we define the neighbourhood

\begin{eqnarray} C_{\alpha} = C_{1, \alpha} \cup C_{2, \alpha} \end{eqnarray}

We can then use the collared neighbourhood to define the map

\begin{equation} \begin{array}{r@{}l} c : \partial S \times D^1 &{}\to M\\ (p, r) &{}\mapsto c(p,r) = \begin{cases} \pi(c_1(p,r)) & r \geq 0 \\ \pi(c_2(h(p),r)) & r \leq 0 \end{cases} \end{array} \end{equation}

Combined with our atlas of the boundary, we get the atlas $(C_\alpha, c_\alpha)$, with

\begin{eqnarray} c_\alpha : C_\alpha &\to& \mathbb{R}^n\\ p &\mapsto& c_\alpha(p) = (\phi_{\alpha}(p_1(c^{-1}(p))), p_2(c^{-1}(p))) \end{eqnarray} Show uniqueness of diff structure

A property we will need later on is that the cut and paste procedure is commutative with a cut of a sphere, at least as far as the topology goes. In other words,

\begin{equation} \text{SK}((M \setminus S^n), \mathcal{S}, h) = \text{SK}(M, \mathcal{S}, h) \setminus S^n \end{equation}

First we have that for any connected manifold, any cut $M \setminus S^n$

The connected sum

A very common example of manifold cutting and pasting is the connected sum, which is to take two disconnected manifolds, removing a ball from each component before identifying them. Such a procedure is usually denoted by

\begin{equation} M = M_1 \# M_2 \end{equation}

In our formalism, this is the cut and paste procedure of $M = M_1 \sqcup M_2$, cutting out a sphere in each connected component before gluing their boundaries. The most common definition of the connected sum is, considering two manifolds $M_1$ and $M_2$, with two embeddings

\begin{equation} \iota_i : D^n \to M_i \end{equation}

such that $\iota_1$ preserves the orientation and $\iota_2$ reverses it. The connected sum is then performed by considering

\begin{equation} (M_1 \setminus \iota_1(0)) \sqcup (M_2 \setminus \iota_2(0)) \end{equation}

with the identification of $\iota_1(tu)$ and $\iota_2((1 - t)u)$ for every unit vector of $D^n$ and $t \in (0,1)$.

We can show this definition to be equivalent to the gluing along boundaries by considering a deformation retraction on our disks onto their boundaries, by simply considering the map

\begin{eqnarray} r : D^n \times [0, 1] &\to& S^n\\ (r, \phi^i, x) &\mapsto& ((1 - x) r + x, \phi^i) \end{eqnarray}

Another procedure we will commonly need for spacetimes is cutting two balls but this time from the same connected component of a manifold. In which case we have the following theorem :

This leads to the useful fact that gluing two spheres of $\mathbb{R}^n$ give us a manifold homeomorphic to the punctured torus $T^n \setminus \{ p \}$.

2. Gluing by identifying open sets

This type of gluing is basically just a redefinition of a manifold using two other manifolds in the standard way, by combining their coordinate charts into a single atlas. While much more simple in that aspect, it is on the other hand riskier to use as it may quickly give rise to pathological manifolds, such as non-Hausdorff manifolds.

3. Cutting and pasting fiber bundles and their sections

As we are most interested in the metric tensor for spacetimes, it will be useful to see what happens to sections of vector bundles in general after gluing. To study this, first let's consider two broad categories of maps : the maps to and from our manifold.

First is the category of maps $\varphi : F \to M$, from some manifold $F$ to our spacetime manifold $M$. We define the glued map resulting from it, $\bar{\varphi} : F \to \bar{M}$, by $\bar{\varphi} = \pi \circ \varphi$. $\bar{\varphi}$ is continuous if, for $U \subset \bar{M}$ an open set, $\varphi^{-1} \circ \pi^{-1} (U)$ is also open.

Take again as before the glued manifold $M$ constructed from gluing two boundaries $\partial S_1$, $\partial S_2$, $M = \operatorname{SK}(M_0, \partial S_1, \partial S_2, h)$.

By the definition of a glued map, this is only possible if, for all $p, q \in M$ such that $\pi(p) = \pi(q)$, we have $\phi(p) =\phi(q)$.

This glued tangent bundle is the tangent bundle of the glued manifold. To show this, let's consider the definition of the tangent bundle, ie, for $U \subset M$, there is a local vector trivialization

\begin{eqnarray} \psi_U : \pi^{-1}(U) &\to& \mathbb{R}^n \times \mathbb{R}^n\\ (p, v) &\mapsto& \end{eqnarray} [prove that a glued vector field made of two existing vector fields is continuous if h maps the boundaries together]

4. Smoothing out discontinuities

Before we study the case of spacetimes with $C^0$ metrics, it is useful to consider the case where the metric given is simply smooth. This can be done in all cases by considering a smoothing procedure over the junction patch.

5. Distributions on manifolds

As we're cutting and pasting together manifolds with a metric associated to them, it is not guaranteed that the resulting spacetime will have a metric of any great regularity. The only condition that we'll ask is that the metric obtained be $C^0$. Due to this, a lot of theorems go out of the window, as most of general relativity is proven with at least a $C^2$ metric in mind, meaning that a lot has to be proven from the ground up again.

First, we need to provide an appropriate definition for the derivatives of a $C^0$ metric. As usual for this sort of things, we'll use weak derivatives.

5.1 Tensor distributions

Tensor distributions are defined much in the same way as scalar ones. First, we need to define the concept of test fields.

Tensor distributions are then defined as one would expect : a $\mathscr{T}^s_r$ tensor distribution is a linear functional on the space of $T^r_s$ tensor densities.

\begin{array}{r@{}l} T : \mathscr T^r_s(M) &{}\to \mathbb{R}\\ \mathfrak{t} &{}\mapsto T(\mathfrak{t}) \end{array}

and similarly, the space of $T^s_r$ tensors can be embedded in $\mathscr{T}^r_s$ by the map

\begin{equation} T[\mathfrak{t}] = \int {T^{abc...}}_{\alpha\beta\gamma...} {\mathfrak{t}^{\alpha\beta\gamma...}}_{abc...} \end{equation}

As can be guessed from the appearance of measures, we'll stick to the case of orientable manifolds so as to not get in the murky waters of integrating on non-orientable manifolds, although overall, as long as our region of interest is itself orientable (in this case, the collared neighbourhood around the junction), things should be about the same in the non-orientable case.

The Geroch-Traschen class of metrics is then comprised of locally bounded $C^0$ metrics which admit an inverse that is also locally bounded and with a first weak derivative that is locally square integrable. In this case, the Christoffel symbols

\begin{equation} {\Gamma^a}_{bc} = \frac 12 g^{ad} (g_{bd,c} + g_{cd, b} - g_{bc,d}) \end{equation}

is the product of a $C^0$ function and a locally square-integrable distribution, hence a square-integrable distribution itself, and the Riemann tensor

\begin{equation} {R^\rho}_{\mu\nu\sigma} = 2{\Gamma^\rho}_{\mu[\sigma,\nu]} + 2{\Gamma^\rho}_{\lambda[\nu}{\Gamma^\lambda}_{\sigma]\mu} \end{equation}

is the sum of a distribution and a locally integrable function which can then be embedded as a distribution.

5.2 Thin-shell formalism

Given all this, we are now prepared to actually perform the junction of two spacetime boundaries. In our case, let's assume that, at our joined collar neighbourhood $c = c_1 \cup c_2$ (we'll rename the two regions $c_+$ and $c_-$ for reasons that will become clear later on), we have a metric on the original spacetime such that

$$g(\partial c_+) = g_(\partial c_-)$$

So as to ensure continuity.

6. The geodesic equation

It can be tricky to analyze properly distributional spacetimes, as almost every theorem regarding analysis on spacetimes assume at least a $C^{1-}$ (locally Lipschitz continuous derivative) metric, while our metric is simply $C^0$. Proving such things as the existence and uniqueness of geodesics around the points of discontinuity of the connection then requires some care.

6.1 Filippov differential inclusion

Ordinary differential equation theory is ill-equipped to deal with the geodesic equation in $C^0$ spacetimes, even the Carathéodory theorem, as what we are dealing with here is a system of differential equations of the form

$$\dot{y}(\lambda) = F(y(\lambda))$$

with $y = (x^\mu, u^\mu)$ and $F = (u, -\Gamma^\mu_{\alpha \beta}(x) u^\alpha u^\beta )$. The system is therefore discontinuous in $F$ with respect to $u$. The proper setting to discuss such equation is Filippov's differential inclusion theory, which deals with set-valued functions, such that our differential equation becomes

$$\dot{y}(\lambda) \in F(y(\lambda))$$

In the case of a continuous function, set-valued function versions of a function $f$ simply become $f_{\text{set}}(x) = \{ f(x) \}$, the singleton of the value at that point, as we'll see later on, in which case things become essentially identical to ordinary differential equation theory. In the case of a discontinuous function though, we'll deal with a range of values corresponding roughly to the values of the function near that point. The set is referred to as its essential convex hull :

$$\mathcal{F}\left[F\right](y) = \bigcap_{\delta > 0} \bigcap_{\mu(S) = 0} \overline{\operatorname{co}}(F(B(y, \delta) \setminus S))$$

$B$ is a ball centered around the point $y$ with radius $\delta$, from which we remove $S$, a subset of measure $0$, and we then take the convex hull of our function on that set, and then take the intersection for every value of $\delta$ and $S$. We're essentially taking the smallest interval of values around our point.

$\mathcal{F}$ obeys a few interesting properties which help with its analysis :

1. if $f,g$ are locally bounded functions, then $$\mathcal{F}\left[ f + g \right] \subset \mathcal{F}\left[ f \right] + \mathcal{F}\left[ g \right] $$
2. if $f$ is continuous, then $$\mathcal{F}\left[ f \right] = \{ f(x) \}$$
3. if $g : \mathbb{R}^m \to \mathbb{R}^{p \times n}$ is a continuous function, and $f : \mathbb{R}^m \to \mathbb{R}^n$ locally bounded, then $$\mathcal{F}\left[ g f \right](x) = g(x) \mathcal{F}\left[ f \right](x) $$

The most important function we'll have to use for this purpose is the Heaviside theta function $\theta$, for which we have, at the discontinuity,

\begin{equation} \mathcal{F}(\theta)(0) = \left[ 0, 1 \right] \end{equation}

and the same will be true for every composite function $\theta(f(x))$ at the zeros of $f$.

6.2 The geodesic inclusion

With the tools previously developped, we now have that the differential inclusion to solve is

\begin{equation} \dot{y}(\lambda) \in F(y(\lambda)) \end{equation}

or, decomposed into $y = (x^\mu, u^\mu)$,

\begin{eqnarray} \dot{x}^\mu &\in& \left\{ u^\mu \right\}\\ \dot{u}^\mu &\in& \mathcal{F}\left[\mathcal{-\Gamma^\mu_{\alpha \beta}(x) u^\alpha u^\beta}\right] \end{eqnarray}

While proving the existence of a solution is important, uniqueness would also help. Fortunately, we are considering here some fairly specific $C^0$ metrics. From what we have seen, the Christoffel symbols decompose as

$$\Gamma^\mu_{\alpha \beta}(x) = \xi^+({\Gamma^\sigma}_{\mu\nu})^+ + \xi^-({\Gamma^\sigma}_{\mu\nu})^-$$

As the connection is $C^0$ on both sides, every term here is locally bounded, so that we may simply write, at the junction

$$\mathcal{F}\left[ \Gamma^\mu_{\alpha \beta}\right] (c) = \left[ 0, 1 \right] {\Gamma^\sigma}_{\mu\nu}(c)$$

and the connection being continuous outside of the junction, we simply have $\left\{ {\Gamma^\sigma}_{\mu\nu}(x) \right\}$ there.

6.3. Geodesics around the junction

Locally, we can always use the geodesic equation as if it were in the original spacetime around a small enough section of the geodesic, except at the point of the junction itself. Crossing the junction will therefore imply a geodesic acting normally in $c_+$, converging to the junction $\mathcal{S}$ where it will have a particular position and momentum $(x, p)$, equivalent on the other side to another position $(x', p')$ in $c_-$, before continuing on its way. To do this, we need to consider the coordinate system around the junction and perform two coordinate transforms, from $c_+$ to the junction, and from the junction to $c_-$.

Let's consider a curve $\gamma$ lying entirely in $c_+$, converging to $\mathcal{S}$, ie,

\begin{equation} \lim_{\lambda \to \lambda_c} \gamma(\lambda) \in \mathcal{S} \end{equation}

While we can define a point $p$ of the boundary (as we started with a manifold with boundaries after all), the tangent will not work out here as the curve is not defined beyond $\lambda_c$, and all we have is the left derivative

\begin{equation} \lim_{\lambda \to \lambda_c} \gamma'(\lambda) \end{equation}

Let's consider a coordinate system $\{ x_+^\mu \}$ on $c_+$ and $\{ y^\mu \}$ at the junction, and denote $x^\mu(\lambda)$ the coordinates of $\gamma$ in $c_+$ and $y^\mu$ the coordinates in the junction. As the transition from one coordinate system to another is a diffeomorphism, we have that

\begin{equation} J^\nu_\mu (\lim_{\lambda \to \lambda_c} \dot{x}^\mu(\lambda)) = \lim_{\lambda \to \lambda_c} J^\nu_\mu(\dot{x}^\mu(\lambda)) = y^\nu(\lambda_c) \end{equation}

Prove that if $\gamma'(\lambda) \to u$ at the boundary of $g^+$, then $\gamma'(\lambda) = J u$ at the junction

7. Spacetime gluing and causality

One useful thing to consider is the case of the gluing of spacetimes is the case where the initial spacetime $M$ is globally hyperbolic, so that $M \cong \mathbb{R} \times \Sigma$, with the possibility to foliate it by some time function $t : M \to \mathbb{R}$ such that $t^{-1}(T) = \Sigma_t \cong \Sigma$, from which we cut out two timelike tubes $S = \mathbb{R} \times \sigma$. In other words, for every Cauchy hypersurface $\Sigma_t$, we have that $\Sigma_t \cap S = \sigma_t \cong \sigma$.

This process can be used for instance for the production of thin-shell wormholes. The question then becomes, under what circumstances is the new spacetime also globally hyperbolic, or more generally of any causal class?

Our gluing function becomes much simpler here. We have here

\begin{eqnarray} h : \mathbb{R} \times \partial \sigma &\to& \mathbb{R} \times \partial \sigma\\ (t, x) &\to& (\alpha(t, x), \beta(t,x)) \end{eqnarray}

$\alpha$ here corresponds to the identification of points of equal time, while $\beta$ is the matching of the spatial boundary.

One of the most important condition we have is time-orientability. Our manifold is time-orientable if there exists a nowhere-vanishing continuous timelike vector field $X$, and more generally if for every curve, parallel transport here is that, if for any timelike vector field $X$, we have $\langle dh, X \rangle < 0$, then the resulting manifold is not time-orientable. To show this, simply consider any curve $\gamma$ lying entirely in $M \setminus \{ S_1, S_2 \}$ except for its endpoints connecting two identified points $p_i \in \partial S_i$. As $M$ is time-orientable, any timelike vector $V$ propagated along $\gamma$ will remain of the same time-orientation. Once the manifold is oriented though, our curve forms a loop and therefore the manifold will fail to be time-orientable if the mapping of $V$ via $h$ reverses its time orientation. In other words, it will fail to be time-orientable if, for our timelike vector field $X$ and curve $\gamma$,

\begin{eqnarray} g(h_* X(p_1), X(p_2)) < 0 \end{eqnarray}

In other words,

If we restrict our attention to time-orientable manifolds, another problem becomes : under what circumstances can the resulting manifold be foliated by spacelike hypersurfaces? The simplest case is obviously to consider

\begin{eqnarray} h(t, x) = (\alpha(x), \beta(t,x)) \end{eqnarray}

Bibliography

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Last updated : 2023-04-14 15:42:10