Spherically symmetric spacetimes

2.1 Spherically symmetric spacetimes

The study of the global properties of solutions to the spherically symmetric Einstein–Vlasov system was initiated by Rein and Rendall in 1990. They chose to work in coordinates where the metric takes the form

2 2μ(t,r) 2 2λ(t,r) 2 2 2 2 2 ds = − e dt + e dr + r (dθ + sin θ dϕ ),

where $t ∈ ℝ$ , $r ≥ 0$ , $θ ∈ [0,π]$ , $ϕ ∈ [0, 2π]$ . These are called Schwarzschild coordinates. Asymptotic flatness is expressed by the boundary conditions

lim λ(t,r) = lim μ (t,r) = 0, ∀t ≥ 0. r→ ∞ r→∞

A regular centre is also required and is guaranteed by the boundary condition

λ (t,0) = 0.

With

x = (r sin φcos θ,rsinφ sinθ,r cosφ )

as spatial coordinate and

j j j λ x-⋅ p-x- v = p + (e − 1) r r

as momentum coordinates, the Einstein–Vlasov system reads

μ− λ v ( x ⋅ v μ−λ ∘ -------) x ∂tf + e ∘-------2-⋅ ∇xf − λt-----+ e μr 1 + |v|2 --⋅ ∇vf = 0, (31 ) 1 + |v| r r e− 2λ(2rλr − 1 ) + 1 = 8πr2 ρ, (32 ) − 2λ 2 e (2rμr + 1 ) − 1 = 8πr p. (33 )

The matter quantities are defined by

∫ ∘ -------- ρ(t,x ) = 1 + |v|2f(t,x,v)dv, (34 ) ∫ℝ3 ( x-⋅ v)2 ----dv---- p(t,x ) = 3 r f(t,x,v)∘1 -+--|v|2. (35 ) ℝ

Let us point out that this system is not the full Einstein–Vlasov system. The remaining field equations, however, can be derived from these equations. See [88

] for more details. Let the square of the angular momentum be denoted by $L$ , i.e.

L := |x |2|v|2 − (x ⋅ v)2.

A consequence of spherical symmetry is that angular momentum is conserved along the characteristics of Equation (31 ). Introducing the variable

w = x ⋅-v, r

the Vlasov equation for $f = f (t,r,w, L)$ becomes

where

∘ -------------- E = E (r,w, L) = 1 + w2 + L∕r2.

The matter terms take the form

∫ ∞ ∫ ∞ ρ(t,r) = π- Ef (t,r,w, L) dw dL, (37 ) r2 −∞ 0 π ∫ ∞ ∫ ∞ w2 p(t,r) = -2 --f (t,r,w, L )dw dL. (38 ) r −∞ 0 E

Let us write down a couple of known facts about the system (32

, 33

, 36

, 37

, 38

). A solution to the Vlasov equation can be written as

where $R$ and $W$ are solutions to the characteristic system

dR- = e(μ−λ)(s,R)----W------, (40 ) ds E (R,W, L ) dW L ----= − λt(s,R )W − e(μ−λ)(s,R )μr (s,R )E (R,W, L ) + e(μ− λ)(s,R)--3-----------, (41 ) ds R E (R, W, L )

such that the trajectory $(R (s,t,r,w,L )$ , $W (s,t,r,w, L ),L )$ goes through the point $(r,w,L )$ when $s = t$ . This representation shows that $f$ is non-negative for all $t ≥ 0$ and that $f ≤ ∥f0∥∞$ . There are two known conservation laws for the Einstein–Vlasov system: conservation of the number of particles,

∫ ∞ (∫ ∞ ∫ ∞ ) 4π2 eλ(t,r) f (t,r,w, L) dL dw dr, 0 −∞ 0

and conservation of the ADM mass

Let us now review the global results concerning the Cauchy problem that have been proved for the spherically symmetric Einstein–Vlasov system. As initial data we take a spherically symmetric, non-negative, and continuously differentiable function $f0$ with compact support that satisfies

This condition guarantees that no trapped surfaces are present initially. In [88

] it is shown that for such an initial datum there exists a unique, continuously differentiable solution $f$ with $f (0) = f0$ on some right maximal interval $[0,T )$ . If the solution blows up in finite time, i.e. if $T < ∞$ , then $ρ(t)$ becomes unbounded as $t → T$ . Moreover, a continuation criterion is shown that says that a local solution can be extended to a global one provided the $v$ -support of $f$ can be bounded on $[0,T )$ . (In [88

] they chose to work in the momentum variable $v$ rather than $w, L$ .) This is analogous to the situation for the Vlasov–Maxwell system where the function $Q (t)$ was introduced for the $v$ -support. A control of the $v$ -support immediately implies that $ρ$ and $p$ are bounded in view of Equations (34

, 35

). In the Vlasov–Maxwell case the field equations have a regularizing effect in the sense that derivatives can be expressed through spatial integrals, and it follows [46 ] that the derivatives of $f$ also can be bounded if the $v$ -support is bounded. For the Einstein–Vlasov system such a regularization is less clear, since e.g. $μr$ depends on $ρ$ in a pointwise manner. However, certain combinations of second and first order derivatives of the metric components can be expressed in terms of matter components only, without derivatives (a consequence of the geodesic deviation equation). This fact turns out to be sufficient for obtaining bounds also on the derivatives of $f$ (see [88

] for details). By considering initial data sufficiently close to zero, Rein and Rendall show that the $v$ -support is bounded on $[0,T )$ , and the continuation criterion then implies that $T = ∞$ . It should be stressed that even for small data no global existence result like this one is known for any other phenomenological matter model coupled to Einstein’s equations. The resulting spacetime in [88

] is geodesically complete, and the components of the energy momentum tensor as well as the metric quantities decay with certain algebraic rates in $t$ . The mathematical method used by Rein and Rendall is inspired by the analogous small data result for the Vlasov–Poisson equation by Bardos and Degond [12 ]. This should not be too surprising since for small data the gravitational fields are expected to be small and a Newtonian spacetime should be a fair approximation. In this context we point out that in [89

] it is proved that the Vlasov–Poisson system is indeed the nonrelativistic limit of the spherically symmetric Einstein–Vlasov system, i.e. the limit when the speed of light $c → ∞$ . (In [96 ] this issue is studied in the asymptotically flat case without symmetry assumptions.) Finally, we mention that there is an analogous small data result using a maximal time coordinate [101

] instead of a Schwarzschild time coordinate. In these coordinates trapped surfaces are allowed in contrast to the Schwarzschild coordinates.

The case with general data is more subtle. Rendall has shown [95 ] that there exist data leading to singular spacetimes as a consequence of Penrose’s singularity theorem. This raises the question of what we mean by global existence for such data. The Schwarzschild time coordinate is expected to avoid the singularity, and by global existence we mean that solutions remain smooth as Schwarzschild time tends to infinity. Even though spacetime might be only partially covered in Schwarzschild coordinates, a global existence theorem for general data would nevertheless be very important since it is likely that it would constitute a central step for proving weak cosmic censorship. Indeed, if this coordinate system can be shown to cover the domain of outer communications and if null infinity could be shown to be complete, then weak cosmic censorship would follow. A partial investigation for general data in Schwarzschild coordinates was done in [92 ] and in maximal-isotropic coordinates in [101 ]. In Schwarzschild coordinates it is shown that if singularities form in finite time the first one must be at the centre. More precisely, if $f (t,r,w, L) = 0$ when $r > ε$ for some $ε > 0$ , and for all $t$ , $w$ , and $L$ , then the solution remains smooth for all time. This rules out singularities of the shell-crossing type, which can be an annoying problem for other matter models (e.g. dust). The main observation in [92 ] is a cancellation property in the term

λtw + eμ−λμrE

in the characteristic equation (41 ). In [101 ] a similar result is shown, but here also an assumption that one of the metric functions is bounded at the centre is assumed. However, with this assumption the result follows in a more direct way and the analysis of the Vlasov equation is not necessary, which indicates that such a result might be true more generally. Recently, Dafermos and Rendall [33 ] have shown a similar result for the Einstein–Vlasov system in double-null coordinates. The main motivation for studying the system in these coordinates has its origin from the method of proof of the cosmic censorship conjecture for the Einstein–scalar field system by Christodoulou [31 ]. An essential part of his method is based on the understanding of the formation of trapped surfaces [28 ]. The presence of trapped surfaces (for the relevant initial data) is then crucial in proving the completeness of future null infinity in [31 ]. In [32 ] the relation between trapped surfaces and the completeness of null infinity was strengthened; a single trapped surface or marginally trapped surface in the maximal development implies completeness of null infinty. The theorem holds true for any spherically symmetric matter spacetime if the matter model is such that “first” singularities necessarily emanate from the center (where the notion of “first” is tied to the casual structure). The results in [92 ] and in [101 ] are not sufficient for concluding that the hypothesis of the matter needed in the theorem in [32 ] is satisfied, since they concern a portion of the maximal development covered by particular coordinates. Therefore, Dafermos and Rendall [33 ] choose double-null coordinates which cover the maximal development, and they show that the mentioned hypothesis is satisfied for Vlasov matter.

In [93 ] a numerical study was undertaken of the Einstein–Vlasov system in Schwarzschild coordinates. A numerical scheme originally used for the Vlasov–Poisson system was modified to the spherically symmetric Einstein–Vlasov system. It has been shown by Rein and Rodewis [94 ] that the numerical scheme has the desirable convergence properties. (In the Vlasov–Poisson case convergence was proved in [106 ]; see also [40 ]). The numerical experiments support the conjecture that solutions are singularity-free. This can be seen as evidence that weak cosmic censorship holds for collisionless matter. It may even hold in a stronger sense than in the case of a massless scalar field (see [29 , 31 ]). There may be no naked singularities formed for any regular initial data rather than just for generic data. This speculation is based on the fact that the naked singularities that occur in scalar field collapse appear to be associated with the existence of type II critical collapse, while Vlasov matter is of type I. This is indeed the primary goal of their numerical investigation: to analyze critical collapse and decide whether Vlasov matter is type I or type II.

These different types of matter are defined as follows. Given small initial data no black holes are expected to form and matter will disperse (which has been proved for a scalar field [27 ] and for Vlasov matter [88 ]). For large data, black holes will form and consequently there is a transition regime separating dispersion of matter and formation of black holes. If we introduce a parameter $A$ on the initial data such that for small $A$ dispersion occurs and for large $A$ a black hole is formed, we get a critical value $Ac$ separating these regions. If we take $A > Ac$ and denote by $mB (A )$ the mass of the black hole, then if $m (A ) → 0 B$ as $A → A c$ we have type II matter, whereas for type I matter this limit is positive and there is a mass gap. For more information on critical collapse we refer to the review paper by Gundlach [50 ].

For Vlasov matter there is an independent numerical simulation by Olabarrieta and Choptuik [75 ] (using a maximal time coordinate) and their conclusion agrees with the one in [93 ]. Critical collapse is related to self similar solutions; Martín-García and Gundlach [68 ] have presented a construction of such solutions for the massless Einstein–Vlasov system by using a method based partially on numerics. Since such solutions often are related to naked singularities, it is important to note that their result is for the massless case (in which case there is no known analogous result to the small data theorem in [88 ]) and their initial data are not in the class that we have described above.

We end this section with a discussion of the spherically symmetric Einstein–Vlasov–Maxwell system, i.e. the case considered above with charged particles. Whereas the constraint equations in the uncharged case do not involve any problems to solve once the distribution function is given (and satisfies Equation (43 )), the charged case is more challenging. In [73 ] it is shown that solutions to the constraint equations do exist for the Einstein–Vlasov–Maxwell system. In [72 ] local existence is then shown together with a continuation criterion, and in [71 ] the regularity theorem in [92 ] is generalized to this case.