Yesterday I came across a fairly recent paper of Galloway, Schleich, and Witt on the non-existence of trapped surfaces and geons in (2+1) dimensional gravity. The non-existence of trapped surfaces in (2+1) GR was first established by Ida in a 2000 paper. In this post I will give a proof of this fact, and use this as an excuse to present the "double null foliation" formulation for general relativity that was used by, among others, Christodoulou's recent work on dynamical formation of black holes, and Klainerman and Nicolo's re-presentation of the stability of Minkowski space. By working in (2+1) instead of the customary (3+1) situation, the notations simplify and it is easier to see how the double null formalism proceeds.
Double null foliation and adapted null frame
Let $(M^{(1+n)}, g)$ be an $(n+1)$ dimensional Lorentzian manifold. A double null foliation of (a subset of) $M$ consists of a pair of functions $u,\underline{u}$ that
- solve the eikonal equation $g(du,du) = g(d\underline{u},d\underline{u}) = 0$ and
- intersect transversely.
We will denote by $S_{u,\underline{u}}$ the (doubly-parametrised) intersections of the level sets of $u$ and $\underline{u}$.
Define $L,\underline{L}$ to be the null geodesic fields associated to $u,\underline{u}$, we compute their inner product to be \begin{equation} \Omega^{-2} := - g_{ij}L^i \underline{L}^j \end{equation} and we define the normalised null vectors to be $\ell:= \Omega L$ and $\underline{\ell} := \Omega\underline{L}$. Note that \[ g_{ij}\ell^i \underline{\ell}^j = -1 \qquad \ell(\underline{u}) = \underline{\ell}(u) = -\Omega^{-1} \] and that for an arbitrary vector field $latex X$, we have \[ g(X,\ell) = \Omega g(X,L) = \Omega X(u) \qquad g(X,\underline{\ell}) = \Omega g(X,\underline{L}) = \Omega X(\underline{u})\] which implies by a direct computation which I omit here that
By an adapted null frame we will refer to the set of $(n+1)$ (locally defined) vector fields $\{\ell,\underline{\ell}, e_1, \ldots, e_{n-1}\}$ such that $e_i(u) = e_i (\underline{u}) = 0$ and $g(e_i,e_j) = \delta{ij}$. The $e_i$ are "tangential vector fields". The double null formalism is basically the method of moving frames using this particular choice of null frame. For Lorentzian manifolds this leads to some simplifications, as we will demonstrate below in the case of the (2+1) dimensional manifold. (For the (3+1) case with vanishing Ricci curvature, a complete description of the geometry can be found in Klainerman and Nicolo, The evolution problem in general relativity.)
(As a side remark, in the geometric analysis of hyperbolic PDEs, another useful foliation is that single-null set-up based on a single solution $u$ to the eikonal equation and a time-like foliation $t$. This construction is used in, e.g. the original proof of stability of Minkowski space by Christodoulou and Klainerman, the recent work by Christodoulou on the formation of shocks in relativistic fluids, and by Klainerman and Rodnianski on low-regularity local well-posedness of Einstein's equations.)
Null connection coefficients and null structure equations
Hereon we specialise to the (2+1) dimensional case. Here, the level sets $S_{u,\underline{u}}$ are one dimensional, and we will assume that they are compact (and hence topologically circles). By a choice of orientation there is then a unique vector field, which we will write $\theta$, that is unit length and is tangent to the level sets. The null frame will now be denoted by $\{\ell,\underline\ell,\theta\}$. For convenience we will denote \begin{equation} \omega := \ell(\log\Omega) \qquad \underline{\omega} := \underline{\ell}(\log\Omega) \qquad \vartheta := \theta(\log\Omega) \end{equation} as we shall see these capture half of the connection coefficients. In the moving frame method, we need to know the Ricci rotation coefficients of the frame. We can summarise it in the following, where the quantities $\zeta,\chi,\underline\chi$ are considered to be defined by the third and fourth equations, and their compatibility with the other expressions results from Proposition 2 above. \begin{equation} \begin{aligned} \nabla_{\ell}\ell &= \omega \ell \newline \nabla_{\underline\ell}\underline\ell &= \underline\omega \underline\ell \newline \nabla_{\theta}\ell &= - \zeta\ell + \chi\theta \newline \nabla_{\theta}\underline\ell &= \zeta\ell + \underline\chi\theta \newline \nabla_{\ell}\theta &= (\vartheta - \zeta)\ell \newline \nabla_{\underline\ell}\theta &= (\vartheta + \zeta)\underline\ell \newline \nabla_{\ell}\underline\ell &= - \omega\underline\ell + (\vartheta - \zeta)\theta \newline \nabla_{\underline\ell}\ell &= - \underline\omega\ell + (\vartheta + \zeta)\theta \newline \nabla_{\theta}\theta &= \underline\chi\ell + \chi\underline\ell \end{aligned} \end{equation}
(Remark: in (3+1) or higher dimensions, things that multiply $\theta$ should be replaced by tensorial quantities. More precisely, we see that $\vartheta$ and $\zeta$ should be one-forms, and $\chi, \underline\chi$) should be replaced by symmetric two tensors.)
The quantities $\chi,\underline\chi$ carry special meaning: they are the null expansion factors associated to $\ell,\underline\ell$ respectively. Their physical interpretation is this: look at a point $p\in S_{u,\underline{u}}$ and a small neighborhood around it on the level surface. "Flow" the neighborhood with either $\ell$ or $\underline\ell$. $\chi$ measures the rate of change of the infinitesimal "volume" of $S_{u,\underline{u}}$ near $p$. So if $\chi$ is positive, the local volume is expanding, and the rays corresponding to $\ell$ are diverging. And if $\chi$ is negative, the local volume is shrinking, and the rays corresponding to $\ell$ are converging.
We say that a particular circle $S_{u,\underline{u}}$ is "trapped" if both $\chi,\underline\chi$ are negative along the entire circle; we say it is marginally trapped if one is negative and the other 0. A trapped surface is the hallmark of a black hole.
Next we need to derive the null structure equations. To do so we use the definition for the Riemann curvature tensor \begin{equation} R(X,Y)Z = \nabla_{[X,Y]}Z - [\nabla_X,\nabla_Y]Z \end{equation} to develop propagation equations for the connection coefficients in terms of curvature quantities. In higher dimensions we next decompose the Riemann curvature into the Ricci and Weyl curvatures: the Ricci curvature we "prescribe" using Einstein's equation, while the Weyl curvature themselves solve propagation equations derived using the Bianchi identities. But in (2+1), Weyl vanishes identically. So we are left with the following equations \begin{equation} \begin{aligned} \mathrm{Ric}(\ell,\ell) &= - \chi^2 - \ell(\chi) + \omega\chi \newline \mathrm{Ric}(\underline\ell,\underline\ell) &= - \underline{\chi}^2 - \underline\ell(\underline\chi) + \underline\omega\underline\chi \newline \mathrm{Ric}(\ell,\theta) &= -2\omega(\vartheta + \zeta) - 2 \zeta\chi - \ell(\vartheta + \zeta) \newline \mathrm{Ric}(\underline\ell,\theta) &= -2\underline\omega(\vartheta - \zeta) + 2\zeta\underline\chi - \underline\ell(\vartheta - \zeta) \newline \mathrm{Ric}(\ell,\underline\ell) + \frac12 \mathrm{Ric}(\theta,\theta) &= -2 \omega\underline\omega -2\zeta^2 + (\vartheta-\zeta)(\vartheta+\zeta) - \ell(\underline\omega) - \underline\ell(\omega) \newline \mathrm{Ric}(\theta,\theta) &= \begin{cases} -2\theta(\vartheta + \zeta) - 2(\vartheta + \zeta)^2 + 2\chi\underline\chi +2 \underline\ell(\chi) + 2\underline\omega\chi\newline -2\theta(\vartheta - \zeta) - 2(\vartheta - \zeta)^2 + 2\chi\underline\chi + 2\ell(\underline\chi) + 2 \omega\underline\chi\end{cases} \end{aligned} \end{equation}
Observing that $\ell(\underline\omega) - \underline\ell(\omega) = -2\zeta\vartheta$, the second to last equation can be used as null propagation equation for both $\omega$ and $\underline\omega$. The two expressions on the RHS of the last equation are in fact equal.
The first two of the null structure equations above are called Raychaudhuri's identities. Morally speaking, they show that gravity is "attractive" (under the null energy condition).
Energy condition
We shall assume that the dominant energy condition is satisfied on our manifold. This means that we require the Einstein tensor $G = \mathrm{Ric} - \frac12 R g$ to evaluate to a non-negative value when given as input two future directed causal vectors. In particular, $G(\ell,\underline\ell) \geq 0$. A direct computation shows that $G(\ell,\underline\ell) = \frac12\mathrm{Ric}(\theta,\theta)$.
Non-existence of black holes in (2+1) dimensions
In contrast to the (3+1) dimensional case, in (2+1) dimensions it is in fact not possible to have a nondegenerate outermost trapped surface. That is the content of Ida's theorem. We can, however, approach it from a more dynamical point of view using the null structure equations above. From now on let us assume that we are working on some foliation such that $\underline\chi$ is always strictly negative.
Now, while the values of $\chi,\underline{\chi}$ depend on the precise foliation chosen, the signs of those two quantities are only dependent on the circle $S_{u,\underline{u}}$ itself. This follows from the last of the equations defining the null coefficients. (Redefining the double null foliation so that $S_{u,\underline{u}}$ remains fixed for one particular parameter value pair, we see that necessarily $\ell$ and $\underline\ell$ only changes by positive scaling.)
An immediate corollary is that black holes cannot exist.
Another way to read the result is that trapped regions cannot form dynamically.
Starting with $S$, we choose two families of null vectors $\ell,\underline\ell$ such that $\chi = 0$ and $\underline\chi < 0$ and $g(\ell,\underline\ell) = -1$. By rescaling $\ell \mapsto e^\varphi\ell$ and $\underline\ell\mapsto e^{-\varphi}\underline\ell$ we can force the corresponding $\zeta$ to be constant along $S$. Now, develop $\ell$ and $\underline\ell$ geodesically to define the surfaces $\{u = 0\}$ and $\{\underline{u} = 0\}$. Along $\{u = 0\}$, choose level surfaces for $\underline{u}$ by transporting $\ell(\underline{u}) = -1$. From these level surfaces we can develop our double null foliation.
The key is that for this foliation, along $S$ we have \begin{equation} -\ell(\chi)|_S = \mathrm{Ric}(\ell,\ell) \geq 0 \end{equation} and \begin{equation} \underline\ell(\chi)|_S = \mathrm{Ric}(\theta,\theta) + 2\zeta^2 \geq 0 \end{equation} where we used that $\Omega = 1$ along $S$ so that $\vartheta = 0$ there, and that by construction $\theta\zeta = 0$. Hence if $\mathrm{Ric}(\ell,\ell) + \mathrm{Ric}(\theta,\theta) > 0$, we can find $D\subset \mathbb{R}_+\times\mathbb{R}_+\subset \mathbb{R}^2$ such that $\bar{D} \ni (0,0)$ and $\chi|_{S_{u,\underline{u}}} < 0$ for every $(u,-\underline{u})\in D$.
