Global structure of the critical solution

3.3 Global structure of the critical solution

In adapted coordinates, the metric of the critical spacetime is of the form $e−2τ$ times a regular metric. From this general form alone, one can conclude that $τ = ∞$ is a curvature singularity, where Riemann and Ricci invariants blow up like $4τ e$ (unless the spacetime is flat), and which is at finite proper time from regular points in its past. The Weyl tensor with index position $Cabcd$ is conformally invariant, so that components with this index position remain finite as $τ → ∞$ . This type of singularity is called “conformally compactifiable” [198 ] or “isotropic” [93 ]. For a classification of all possible global structures of spherically symmetric self-similar spacetimes see [107 ].

The global structure of the scalar field critical solution was determined accurately in [157 ] by assuming analyticity at the centre of spherical symmetry and at the past light cone of the singularity (the self-similarity horizon, or SSH). The critical solution is then analytic up to the future lightcone of the singularity (the Cauchy horizon, or CH). Global adapted coordinates x and $τ$ can be chosen so that the regular centre r = 0, the SSH and the CH are all lines of constant x, and surfaces of constant $τ$ are never tangent to x lines. (A global $τ$ is no longer a global time coordinate.) This is illustrated in Figure 4 .

Approaching the CH, the scalar field oscillates an infinite number of times but with the amplitude of the oscillations decaying to zero. The scalar field in regular adapted coordinates $(x, τ)$ is of the form

where $Freg(τ )$ , $Fsing(τ)$ and $H (τ)$ are periodic with period $Δ$ , and the SSH is at x = 0. These functions have been computed numerically to high accuracy, together with the constants $K$ and $𝜖$ . The scalar field itself is smooth with respect to $τ$ , and as $𝜖 > 0$ , it is continuous but not differentiable with respect to x on the CH itself. The same is true for the metric and the curvature. Surprisingly, the ratio m/r of the Hawking mass over the area radius on the CH is of order 10^–6 but not zero (the value is known to eight significant figures).

As the CH itself is regular with smooth null data except for the singular point at its base, it is not intuitively clear why the continuation is not unique. A partial explanation is given in [157 ], where all DSS continuations are considered. Within a DSS ansatz, the solution just to the future of the CH has the same form as Equation (31 ). $Freg(τ )$ is the same on both sides, but $Fsing(τ)$ can be chosen freely on the future side of the CH. Within the restriction to DSS this function can be taken to parameterise the information that comes out of the naked singularity.

There is precisely one choice of $Fsing(τ)$ on the future side that gives a regular centre to the future of the CH, with the exception of the naked singularity itself, which is then a point. This continuation was calculated numerically, and is almost but not quite Minkowski in the sense that m/r remains small everywhere to the future of the SSH.

All other DSS continuations have a naked, timelike central curvature singularity with negative mass. More exotic continuations including further CHs would be allowed kinematically [42 ] but are not achieved dynamically if we assume that the continuation is DSS. The spacetime diagram of the generic DSS continued solution is given in Figure 3 .

Figure 3: The spacetime diagram of all generic DSS continuations of the scalar field critical solution, from [157 ]. The naked singularity is timelike, central, strong, and has negative mass. There is also a unique continuation where the singularity is replaced by a regular centre except at the spacetime point at the base of the CH, which is still a strong curvature singularity. No other spacetime diagram is possible if the continuation is DSS. The lines with arrows are lines of constant adapted coordinate x, with the arrow indicating the direction of $∂∕∂ τ$ towards larger curvature.