Nariai solution

*When a is small, ƒ(r) has two zeros at positive values of r, which are the location of the black hole and cosmological horizon respectively. As the parameter a increases, keeping the cosmological constant fixed, the two positive zeros come closer. At some value of a, they collide.

Approaching this value of a, the black hole and cosmological horizons are at nearly the same value of r. But the distance between them doesn't go to zero, because ƒ(r) is very small between the two zeros, and the square root of its reciprocal integrates to a finite value. If the two zeros of ƒ are at R + ε and R − ε taking the small ε limit while rescaling r to remove the ε dependence gives the Nariai solution.*

If I plot f(r) I can find a solution for f(r)=0 in which both horizons collide.


There are three distinct but related solutions of Einstein field equations with positive cosmological constant:

  1. Nariai solution (N);

  2. extreme Schwarzschild–de Sitter (SdS) solution;

  3. near-extreme SdS black hole.

(1) could be seen as an asymptotic geometry of the (2) or as an approximation of the metric between two horizons of (3).

Nariai solution is the direct product, $\mathrm{SdS}_2 \times S_2$, of 2D de Sitter spacetime and 2D sphere, the radius of $S_2$ “fiber” is the same at each point of $\text{dS}_2$. This is a homogeneous spacetime, all its points are equivalent, there is no horizon collision here. Inertial observer placed at any point of spacetime would notice two component cosmological horizon, equal distance away from her. Both components of horizon are the same and there is $Z_2$ symmetry that exchanges them.

To discuss the other solutions let us first write the SdS solution: $$ ds^{2}=-f(r)\,dt^{2}+{dr^{2} \over f(r)}+r^{2}(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}), $$ where $$ f(r)= 1 - \frac{2 M }{r} - \frac{\Lambda}{3}r^2. $$ If the mass parameter $M$ is small enough ($3 M< \Lambda^{-\frac 12}$), the function $f(r)$ has 2 zeroes (at $r_1$ and $r_2$) the smaller radius corresponding to black hole event horizon and the larger is the cosmological horizon. (Note, that the black hole in this universe introduces a dedicated reference frame so the cosmological horizon of this reference frame is singled out as being a Killing horizon).

Near-extreme SdS black hole (this is described in the Wikipedia page on SdS). If the zeroes of $f(r)$ are close to each other ($3 M = \Lambda ^{-\frac12} - \epsilon$) the function $f(r)$ remains small on the interval $(r_1,r_2)$ which means that the metric component $g_{rr}$ is large and the metric distance between the two horizons: $$ d = \int ds = \int_{r_1}^{r_2} \sqrt{g_{rr}} dr = \int_{r_1}^{r_2} f(r)^{-\frac12} dr $$ approaches finite value as the $r_2 - r_1$ shrinks. So, while in the near extreme case $r$ varies very little between horizons there is a lot of space between them, and again no horizon collision! Approximately the patch between horizons then could be described as a patch of Nariai solution. This could be seen by noticing that the $(t,r)$ part of the metric after rescaling of variables becomes the $\text{dS}_2$ metric up to $O(|r_2-r_1|)$, and the $r^2 d \Omega^2$ is almost constant there. Away from horizons the global structure of SdS is of course quite different from N.

Extreme SdS solution is obtained when the $M$ parameter is chosen so that function $f(r)$ has a double zero at some $r=\rho$ (this happens when $3 M = \Lambda ^{-\frac12}$). This is not a black hole at all! Instead, the $g_{tt}$ is positive while $g_{rr}$ is negative everywhere except $r=\rho$, so $r$ is a time coordinate while $t$ is spatial (similar to the Schwarzschild black hole interior only now it is for all values of $r$). The proper time for stationary observer (coordinates $t$, $\theta$, $\phi$ are fixed) from $r=\rho-\epsilon$ to $r=\rho+\epsilon$ is infinite. Nariai spacetime asymptotics is recovered near $r=\rho$, with $t$ as spatial coordinate of $\text{dS}_2$ with “flat” slicing and $r$ becoming “time” (it is better to also introduce a new time variable $T$ with $dT=\sqrt{-g_{rr}}\, dr$). So properly one has to consider two regions: the Kantowski–Sachs cosmological spacetime that starts with black hole-like singularity and asymptotically approaches Nariai solution in infinite time (this is $r<\rho$ patch), and region interpolating between Nariai and $\text{dS}_4$ (this is $r>\rho$ patch). Horizons here are cosmological and dynamical but in the asymptotic Nariai region they again have the same properties as Nariai horizons.


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