Smooth metrics can hide thin shells

This note is motivated by the dynamical generalization of a flat “defect wormhole” metric that appears in Klinkhamer (2023a); *Klinkhamer:2023sau; *Klinkhamer:2023avf; *Klinkhamer:2023nok, which has attracted some attention lately Wang (2023); Ahmed (2023). The construction of the static metric is rather straightforward. Consider the line element for flat spacetime, written in spherical coordinates (I employ the $(-,+,+,+)$ signature throughout):

$$ds^{2}=-dt^{2}+dr^{2}+r^{2}d^{2}\Omega,$$

(1)

where $d^{2}\Omega:=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ is the standard line element on the unit 2-sphere. Now perform a coordinate transformation defined by the relation:

$$r=\sqrt{\lambda^{2}+\rho^{2}},$$

(2)

where $\lambda$ is assumed to be a positive real parameter and $\rho\in\{-\infty,\infty\}$. One obtains the line element Klinkhamer (2023a, b):

$$ds^{2}=-dt^{2}+\frac{\rho^{2}}{\lambda^{2}+\rho^{2}}d\rho^{2}+(\lambda^{2}+\rho^{2})d^{2}\Omega,$$

(3)

which describes a spacetime with topology $R\times R\times S^{2}$ and two asymptotically flat regions, one in the limit $\rho\rightarrow\infty$, and another in the limit $\rho\rightarrow-\infty$. A direct evaluation for generic $\rho$ yields a vanishing Riemann tensor, so that one might naively interpret Eq. (3) as describing a flat wormhole with a throat at $\rho=0$.

Upon adding a time dependence $\lambda\rightarrow\lambda(t)$ in Eq. (2), one obtains the dynamical generalization of Eq. (3):

	$\displaystyle ds^{2}=$	$\displaystyle-\left[1-\frac{[\lambda(t)\lambda^{\prime}(t)]^{2}}{\lambda^{2}+\rho^{2}}\right]dt^{2}+\frac{2\lambda(t)\lambda^{\prime}(t)}{\lambda(t)^{2}+\rho^{2}}dtd\rho$		(4)
		$\displaystyle+\frac{\rho^{2}}{\lambda(t)^{2}+\rho^{2}}d\rho^{2}+\left[\lambda(t)^{2}+\rho^{2}\right]d^{2}\Omega.$

Since the Riemann tensor vanishes for generic $\rho$, one might naively imagine that the geometry described by Eq. (4) is everywhere flat, but if that were the case, the function $\lambda(t)$ (the areal radius of the throat $\rho=0$) would be unconstrained by the Einstein equation. Such a feature would be disastrous for general relativity, rendering the initial value problem ill-posed, in conflict with accepted theorems, for instance Thm. 10.2.2 of Wald (1984), which establishes that given smooth initial data, there exists a unique spacetime that is a solution to the Einstein equation and is continuously dependent on the initial data.

So what gives? It turns out that the smoothness of the metric tensor components is deceptive—a closer inspection of Eq. (3) at $\rho=0$ reveals hints of the deception. In particular, $r(\rho)$ in Eq. (2) is not bijective in a neighborhood of $\rho=0$ containing both positive and negative values of $\rho$, and the $d\rho^{2}$ term in Eq. (3) vanishes at $\rho=0$, rendering the metric degenerate there, as pointed out in Klinkhamer (2023a); *Klinkhamer:2023sau; *Klinkhamer:2023avf (though degenerate metrics can in some cases indicate coordinate singularities, at $\theta=0$ for instance). A more suggestive hint comes from the observation Mukohyama that the expansion scalar for a null radial geodesic congruence changes sign as the congruence passes through the throat $\rho=0$; the same property is used to show that spherically symmetric wormholes violate energy conditions (see Morris and Thorne (1988) and Sec. 13.4.2 of Visser (1995)). Indeed, the mystery unravels when explicitly computing limits of the extrinsic curvature tensor $K_{\mu\nu}:=\tfrac{1}{2}\pounds_{n}(g_{\mu\nu}-n_{\mu}n_{\nu})$ for constant $\rho$ surfaces (which are timelike), where $\pounds_{n}$ is the Lie derivative with respect to $n^{\mu}$, the unit normal vector to a surface of constant $\rho$:

	$\displaystyle\lim_{\rho_{+}\rightarrow 0}K_{\mu\nu}$	$\displaystyle=\text{diag}(0,0,\lambda,\lambda\sin^{2}\theta)$		(5)
	$\displaystyle\lim_{\rho_{-}\rightarrow 0}K_{\mu\nu}$	$\displaystyle=\text{diag}(0,0,-\lambda,-\lambda\sin^{2}\theta).$

$\rho_{+}\rightarrow 0$ indicates the limit for $\rho>0$, and $\rho_{-}\rightarrow 0$ indicates the limit for $\rho<0$. The extrinsic curvature tensor for the family of constant $\rho$ surfaces is discontinuous in $\rho$ at $\rho=0$, signaling the presence of a distributional contribution to the curvature (see Israel (1966) and Sec. 3.7 of Poisson (2004)). The Einstein tensor acquires a nonvanishing term proportional to $\delta(\rho)$, and it follows from the Einstein equation that the line element (3) is sourced by a (necessarily exotic) thin shell at $\rho=0$.

One can draw two conclusions from this result. The first is that the line elements (3) and (4) describe thin shell wormholes Visser (1989, 1995), which have been studied extensively in classical general relativity (see Alcubierre (2017) and references therein) and as toy models in quantum gravity Redmount and Suen (1993); *Redmount:1993ue. The second is the eponymous conclusion that metric tensors with smooth components can hide thin shell sources—one should check for the presence of thin shells at hypersurfaces on which a metric becomes degenerate.

Acknowledgements — I thank S. Mukohyama, R. A. Matzner, and F. R. Klinkhamer for clarifying remarks and feedback.