Static Spherically Symmetric Einstein-Vlasov Bifurcations of the Schwarzschild Spacetime

Abstract

We construct a one-parameter family of static and spherically symmetric solutions to the Einstein-Vlasov system bifurcating from the Schwarzschild spacetime. The constructed solutions have the property that the spatial support of the matter is a finite, spherically symmetric shell located away from the black hole. Our proof is mostly based on the analysis of the set of trapped timelike geodesics and of the effective potential energy for static spacetimes close to Schwarzschild. This provides an alternative approach to the construction of static solutions to the Einstein-Vlasov system in a neighbourhood of black hole vacuum spacetimes made in Rein (in Mathematical proceedings of the cambridge philosophical society. Cambridge University Press, Cambridge, vol 115, pp 559–570, 1994).

A Study of the Geodesic Motion in the Exterior of Scwharzschild Spacetime

We present a detailed study of the geodesic motion in the exterior region of a fixed Schwarzschild spacetime. We will classify the geodesics based on the study of the effective energy potential. Such a classification is of course classical and we refer to [8, Chapter 3] or [23, Chapter 33] for more details. In this section, we prove Proposition 1.

• First, note that $E^{Sch}_\ell$ is a cubic function in $\displaystyle \frac{1}{r}$. Its derivative is given by

  $$\begin{aligned} \forall r> 2M\, , \quad {E^{Sch}_\ell }'(r) = \frac{2}{r^4}\left( Mr^2 -\ell r + 3M\ell \right) . \end{aligned}$$

  Three cases are possible :

• 1. $\displaystyle {E^{Sch}_\ell }'$ has two distinct roots $\displaystyle r_\mathrm{max}^\mathrm{Sch}<r_\mathrm{min}^\mathrm{Sch}$, where $r_\mathrm{max}^\mathrm{Sch}$ and $r_\mathrm{min}^\mathrm{Sch}$ correspond respectively to the maximiser and the minimiser of $\displaystyle {E^{Sch}_\ell }$. They are given by

  $$\begin{aligned}&r_\mathrm{max}^\mathrm{Sch}(\ell ) = \frac{\ell }{2M}\left( 1 - \sqrt{1-\frac{12M^2}{\ell }}\right) , \\&r_\mathrm{min}^\mathrm{Sch}(\ell ) = \frac{\ell }{2M}\left( 1 + \sqrt{1-\frac{12M^2}{\ell }}\right) . \end{aligned}$$

  The extremums of ${E^{Sch}_\ell }$ are given by

  $$\begin{aligned}&E^\mathrm{min}(\ell ) = E^{Sch}_\ell (r_\mathrm{min}^\mathrm{Sch}(\ell )) = \frac{8}{9}+\frac{\ell -12M^2}{9Mr_\mathrm{min}^\mathrm{Sch}(\ell )}, \end{aligned}$$

  (A.1)
  $$\begin{aligned}&E^\mathrm{max}(\ell ) = E^{Sch}_\ell (r_\mathrm{max}^\mathrm{Sch}(\ell )) = \frac{8}{9}+\frac{\ell -12M^2}{9Mr_\mathrm{max}^\mathrm{Sch}(\ell )}. \end{aligned}$$

  (A.2)

  In fact, we compute

  $$\begin{aligned} 1 - \frac{2M}{r_\mathrm{max}^\mathrm{Sch}(\ell )} = 1 - \frac{4M^2}{\ell \left( 1 - \sqrt{1 - \frac{12M^2}{\ell }} \right) } = \frac{2}{3} - \frac{1}{3}\sqrt{1 - \frac{12M^2}{\ell }}, \end{aligned}$$

  and

  $$\begin{aligned} 1 + \frac{\ell }{r_\mathrm{max}^\mathrm{Sch}(\ell )^2} = 1 + \frac{\ell }{36M^2}\left( 1 + \sqrt{1 - \frac{12M^2}{\ell }}\right) ^2. \end{aligned}$$

  Therefore,

  $$\begin{aligned} E_\ell ^\mathrm{Sch}(r_\mathrm{max}^\mathrm{Sch}(\ell ))&= \left( 1 - \frac{2M}{r_\mathrm{max}^\mathrm{Sch}(\ell )}\right) \left( 1 + \frac{\ell }{r_\mathrm{max}^\mathrm{Sch}(\ell )^2} \right) \\&= \left( \frac{2}{3} + \frac{\ell }{18M^2} + \frac{\ell }{18M^2}\sqrt{1 - \frac{12M^2}{\ell }}\right) \left( \frac{2}{3} - \frac{1}{3}\sqrt{1 - \frac{12M^2}{\ell }}\right) \\&=\frac{8}{9}+\frac{\ell -12M^2}{9Mr_\mathrm{max}^\mathrm{Sch}(\ell )}, \end{aligned}$$

  where the last expression is obtained by straightforward computations. We do the same thing for $E^\mathrm{max}(\ell )$. This case occurs if and only if $\displaystyle \ell >12M^2$.

• 2. $\displaystyle {E^{Sch}_\ell }'$ has one double root at $r_c = 6M$. The extremum of $E^{Sch}_\ell$ is given by

  $$\begin{aligned} E_\ell ^c = \frac{8}{9}. \end{aligned}$$

  This case occur if and only if $\ell = 12M^2$.

• 3. $\displaystyle {E^{Sch}_\ell }'$ has no real roots. Then, $\displaystyle {E^{Sch}_\ell }$ is monotonically increasing from 0 to 1. This case occurs if and only if $\ell < 12M^2$.

We refer to Fig. 1 in the introduction for the shape of the potential energy in the three cases.

• Note that by the mass shell condition (2.27), we have

  $$\begin{aligned} E^2 \ge E_\ell ^\mathrm{Sch}(r) \end{aligned}$$

  for any timelike geodesic moving in the exterior region. In particular,

  $$\begin{aligned} E^2 \ge E^\mathrm{min}(\ell ) \ge \frac{8}{9} \end{aligned}$$

  along any geodesic with $\ell >12M^2$. Therefore, we obtain a lower bound on E:

  $$\begin{aligned} E \ge \sqrt{\frac{8}{9}} \end{aligned}$$

  for geodesics with $\ell >12M^2$.

• Now, we claim that the trajectory is a circle of radius $r_0^\mathrm{Sch} > 2M$ if and only if

  $$\begin{aligned} E^{Sch}_\ell (r_0^\mathrm{Sch}) = E^2 \quad \text {and}\quad {E^{Sch}_\ell }'(r_0^\mathrm{Sch}) = 0. \end{aligned}$$

  (A.3)

  Indeed, if the motion is circular of radius $r_0^\mathrm{Sch}$, then $\displaystyle \forall \tau \in {\mathbb {R}}\,:\, r(\tau ) = r_0^\mathrm{Sch}$. Thus,

  $$\begin{aligned} \forall \tau \in {\mathbb {R}} \,,\, w(\tau ) := \dot{r}(\tau ) = 0 \quad \text {and}\quad \dot{w}(\tau ) = 0. \end{aligned}$$

  Besides, it is a solution to the system (2.35)–(2.36). Therefore

  $$\begin{aligned} \forall \tau \in {\mathbb {R}} \,,\, {E^{Sch}_\ell }'(r_0^\mathrm{Sch}) = {E^{Sch}_\ell }'(r(\tau )) = 0. \end{aligned}$$

  By (2.27), we have

  $$\begin{aligned} \forall \tau \in {\mathbb {R}} \,,\, w(\tau )^2 + {E^{Sch}_\ell }(r(\tau )) = E^2. \end{aligned}$$

  In particular,

  $$\begin{aligned} {E^{Sch}_\ell }(r_0^\mathrm{Sch}) = E^2. \end{aligned}$$

  Now, let us suppose that there exists $r_0^\mathrm{Sch}>2M$ such that

  $$\begin{aligned} E^{Sch}_\ell (r_0^\mathrm{Sch}) = E^2 \quad \text {and}\quad {E^{Sch}_\ell }'(r_0^\mathrm{Sch}) = 0. \end{aligned}$$

  By the above assumption, we have $w = 0$ and ${E^{Sch}_\ell }'(r_0^\mathrm{Sch}) = 0$ so that the point $(r_0^\mathrm{Sch}, 0)$ is a stationary point $\forall \ell \ge 12 M^2$. Furthermore,

• 1. $r_0^\mathrm{Sch}= 6$, if $\ell = 12M^2$,

• 2. $r_0^\mathrm{Sch}\in \left\{ r_\mathrm{min}^\mathrm{Sch}(\ell ), r_\mathrm{max}^\mathrm{Sch}(\ell )\right\}$, if $\ell > 12M^2$.

The circular orbits are thus characterised by (A.3).

• We consider now the case $\ell >12M^2$ and the equation

  $$\begin{aligned} E^2 = E^{Sch}_\ell (r). \end{aligned}$$

  (A.4)

  Let $\displaystyle (\ell , E)\in \Big ]12M^2, \infty \Big [\times \Big ]\frac{8}{9}, \infty \Big [$. Four cases may occur:

• 1. If $\displaystyle E^2 = E_\ell ^\mathrm{min}$ or $\displaystyle E^2 = E_\ell ^\mathrm{max}$, then $r_\mathrm{min}^\mathrm{Sch}$ or $r_\mathrm{max}^\mathrm{Sch}$ satisfy (A.3), so that they are double roots. Besides, we note that $\ell = \ell _{ub}$ where

  $$\begin{aligned} \ell _{ub}(E) := \frac{12M^2}{1-4\alpha -8\alpha ^2-8\alpha \sqrt{\alpha ^2+\alpha }} \quad \quad \alpha := \frac{9}{8}E^2 - 1 \end{aligned}$$

  (A.5)

  satisfies $E^2 = E_\ell ^\mathrm{min}$. In fact, we solve the equation below for $\ell$

  $$\begin{aligned} \frac{8}{9} + \frac{\ell - 12M^2}{9\frac{\ell }{2}\left( 1 + \sqrt{1-\frac{12M^2}{\ell }}\right) } = E^2. \end{aligned}$$

  We make the following change of variables

  $$\begin{aligned} \alpha := \frac{9}{8}E^2 - 1\quad \text {and}\quad X := \sqrt{\frac{\ell - 12M^2}{\ell }}. \end{aligned}$$

  The equation becomes

  $$\begin{aligned} \frac{X^2}{4\left( 1 + X \right) }= \alpha , \end{aligned}$$

  which is easily solvable for X. We can then obtain $\ell _{ub}(E)$. Similarly, we obtain $\ell _{lb}$ defined by

  $$\begin{aligned} \ell _{lb}(E) := \frac{12M^2}{1-4\alpha -8\alpha ^2+8\alpha \sqrt{\alpha ^2+\alpha }}, \end{aligned}$$

  (A.6)

  which solves the equation $E^2 = E_\ell ^\mathrm{max}$.

• 2. If $\displaystyle E^2 > E_\ell ^\mathrm{max}$, then two cases occur

  • (a) $E^2 < 1$. The equation (A.4) has one simple root $r_2^\mathrm{Sch}(E, \ell )> r_\mathrm{min}^\mathrm{Sch}(\ell )$

  • (b) $E^2 \ge 1$. The equation (A.4) has no positive roots. The trajectories in this case are similar to the trajectories in case 3 (where $\ell <12M^2$).

  • 3. $\displaystyle E^2 \in \Big ]E_\ell ^\mathrm{min}, E_\ell ^\mathrm{max}\Big [$. Again, two cases occur

    • (a) $E^2 < 1$. Then, the equation (A.4) admits three simple positive roots $r_i^\mathrm{Sch}(E, \ell )$

      $$\begin{aligned} r_0^\mathrm{Sch}(E, \ell )< r_\mathrm{max}^\mathrm{Sch}(\ell )< r_1^\mathrm{Sch}(E, \ell )< r_\mathrm{min}^\mathrm{Sch}(\ell ) < r_2^\mathrm{Sch}(E, \ell ). \end{aligned}$$

      (A.7)

    • (b) $E^2 \ge 1$. The equation (A.4) admits two simple positive roots $r_i^\mathrm{Sch}(E, \ell )$

      $$\begin{aligned} r_0^\mathrm{Sch}(E, \ell )< r_\mathrm{max}^\mathrm{Sch}(\ell )< r_1^\mathrm{Sch}(E, \ell ) < r_\mathrm{min}^\mathrm{Sch}(\ell ) . \end{aligned}$$

      (A.8)

• We consider now the case $\ell \le 12M^2$. Three cases may occur:

• 1. If $\displaystyle E^2 = \frac{8}{9}$ and $\ell = 12M^2$, then the equation (A.4) has one triple positive root $\displaystyle r_c = 6M$.

• 2. If $0<E^2<1$, then the equation (A.4) has one simple positive root $r_1^\mathrm{Sch}(E, \ell )$.

• 3. If $E^2 \ge 1$, then the equation (A.4) no positive roots.

• Based on the above cases, we define the following parameters sets

  $$\begin{aligned} {\mathcal {A}}_{circ}&:= \left\{ \left( \sqrt{\frac{8}{9}}, 12M^2\right) \right\} \bigcup \left\{ (E, \ell )\in \Big ]\sqrt{\frac{8}{9}}, \infty \Big [\times \Big ]12M^2, \infty \Big [\,:\, E^2 < 1,\quad \ell = \ell _{ub}(E) \right\} \nonumber \\&\quad \bigcup \left\{ (E, \ell )\in \Big ]\sqrt{\frac{8}{9}}, \infty \Big [\times \Big ]12M^2, \infty \Big [\,:\, \ell = \ell _{lb}(E) \right\} , \end{aligned}$$

  (A.9)
  $$\begin{aligned} {\mathcal {A}}_\mathrm{bound}&:= \left\{ (E, \ell )\in \Big ]\sqrt{\frac{8}{9}}, \infty \Big [\times \Big ]12M^2, \infty \Big [\,:\, E^2<1 ,\quad \ell _{lb}(E)<\ell < \ell _{ub}(E) \right\} , \end{aligned}$$

  (A.10)
  $$\begin{aligned}&{\mathcal {A}}_{unbound} := \left\{ (E, \ell )\in \Big ]\sqrt{\frac{8}{9}}, \infty \Big [\times \Big ]12M^2, \infty \Big [\,:\, E^2 \ge 1 ,\quad \ell > \ell _{lb}(E) \right\} , \end{aligned}$$

  (A.11)
  $$\begin{aligned} {\mathcal {A}}_{abs}&:= \left\{ (E, \ell )\in \Big ]\sqrt{\frac{8}{9}}, \infty \Big [\times \Big ]12M^2, \infty \Big [\,:\, E^2< 1, \quad \ell < \ell _{lb}(E) \right\} \nonumber \\&\quad \bigcup \left\{ (E, \ell )\in \Big ]0, 1\Big [\backslash \sqrt{\frac{8}{9}}\times \left[ 0, 12M^2 \right] \right\} \end{aligned}$$

  (A.12)

• Now, we determine the nature of orbits (circular, bounded, unbounded, “absorbed by the black hole”) in terms of the parameters $(E, \ell )$ as well as the initial position and velocity. Let $\ell \in [0, \infty [$, $\displaystyle {{\tilde{r}}}\in ]2M, \infty [$ and $\displaystyle {{\tilde{w}}}\in {\mathbb {R}}$. We compute

  $$\begin{aligned} E = \sqrt{E^{Sch}_\ell ({{\tilde{r}}}) + {{\tilde{w}}}^2}. \end{aligned}$$

• 1. If $(E, \ell )\in {\mathcal {A}}_\mathrm{bound}$, then there exists $r_i^\mathrm{Sch} := r_i^\mathrm{Sch}(E, \ell ), i\in \left\{ 0, 1, 2\right\}$ solutions of (A.4) and satisfying (A.7). Now recall that

  $$\begin{aligned} E^{Sch}_\ell (r) \le E^2, \quad \forall r>2M. \end{aligned}$$

  (A.13)

  This implies that ${{\tilde{r}}}$ must lie in the region $\displaystyle \Big ]2M, r_0^\mathrm{Sch}\Big ]\cup [r_1^\mathrm{Sch}, r_2^\mathrm{Sch}]$. Two cases are possible

• either the geodesic starts at some point in $]2M, r_0^\mathrm{Sch}]$ and reaches the horizon $r= 2M$ in a finite proper time,

• or the geodesic is trapped between $r_1^\mathrm{Sch}$ and $r_2^\mathrm{Sch}$.

• 2. If $(E, \ell )\in {\mathcal {A}}_{unbound}$, then there exists $r_i^\mathrm{Sch} := r_i^\mathrm{Sch}(E, \ell ), i\in \left\{ 0, 1, \right\}$ solutions of (A.4) and satisfying (A.8). By (A.13), ${{\tilde{r}}}$ must lie in the region $]2M, r_0^\mathrm{Sch}]\cup [r_1^\mathrm{Sch}, \infty [$. Therefore, two cases are possible

• either the geodesic starts at some point in $]2M, r_0^\mathrm{Sch}]$ and reaches the horizon $r= 2M$ in a finite proper time,

• or the geodesic stars at some point in $[r_1^\mathrm{Sch}, \infty [$, hits the potential barrier at $r_1^\mathrm{Sch}$ and goes to back to infinity (case of negative initial radial velocity); or the geodesic stars at some point in $[r_1^\mathrm{Sch}, \infty [$ and goes to infinity (case of positive initial radial velocity).

• 3. If $(E, \ell )\in {\mathcal {A}}_{abs}$, then the equation (A.4) has at most one positive root $r_0^\mathrm{Sch}$. Two cases are possible:

• The geodesic starts at some point in $]2M, r_0^\mathrm{Sch}]$ and reaches the horizon in a finite proper time.

• The geodesic starts at some point in $]2M, \infty [$ negative initial radial velocity and reaches the horizon in a finite time.

• 4. If $(E, \ell )\in {\mathcal {A}}_{circ}$, then the equation (A.4) admits one triple root if $\displaystyle E = \sqrt{\frac{8}{9}}$ or a double root. In this case, the geodesic is a circle if and only if it starts at the point $r_c\in \left\{ 6M, r_\mathrm{min}^\mathrm{Sch}(\ell ), r_\mathrm{max}^\mathrm{Sch}(\ell )\right\} .$