Uniqueness and Nonuniqueness in the Einstein Constraints

The conformal thin-sandwich (CTS) equations are a set of four of the Einstein equations, which generalize the Laplace-Poisson equation of Newton’s theory. We examine numerically solutions of the CTS equations describing perturbed Minkowski space, and find only one solution. However, we find two distinct solutions, one even containing a black hole, when the lapse is determined by a fifth elliptic equation through specification of the mean curvature. While the relationship of the two systems and their solutions is a fundamental property of general relativity, this fairly simple example of an elliptic system with nonunique solutions is also of broader interest.

Figure 1

Solutions of the standard (four) conformal thin-sandwich equations vs the conformal amplitude. As the critical amplitude ${\tilde{\mathcal{A}}}_{c,4}\approx 2.01$ is approached, the conformal factor grows to infinity.

Figure 2

Solutions of the extended (five) conformal thin-sandwich equations vs conformal amplitude. The conformal factor remains finite as the critical amplitude ${\tilde{\mathcal{A}}}_{c,5}\approx 0.304$ is approached, as confirmed by the inset [$\delta \tilde{\mathcal{A}}={\tilde{\mathcal{A}}}_{c,5}-\tilde{\mathcal{A}}$, $\delta \psi ={\psi }_{\mathrm{crit}}-\max (\psi )$].

Figure 3

Maximum of the conformal factor, minimum of the physical lapse $N={\psi }^6\tilde{N}$, and maximal magnitude of the shift vs conformal (unphysical) amplitude. Solid/dashed curves represent the lower/upper branch. The insets show enlargements around the critical point; circles/diamonds denote individual solutions along the lower/upper branch.

Figure 4

Energy $E$ vs unphysical amplitude $\tilde{\mathcal{A}}$ and physical amplitude ${\mathcal{A}}_{\max }$. The left inset shows an enlargement around the critical point. The right inset plots the data of Fig. 3 vs ${\mathcal{A}}_{\max }$.