The Einstein-Vlasov system in spherical symmetry II: spherical perturbations of static solutions
We reduce the equations governing the spherically symmetric perturbations of static spherically symmetric solutions of the Einstein-Vlasov system (with either massive or massless particles) to a single stratified wave equation , with containing second derivatives in radius, and integrals over energy and angular momentum. We identify an inner product with respect to which is symmetric, and use the Ritz method to approximate the lowest eigenvalues of numerically. For two representative background solutions with massless particles we find a single unstable mode with a growth rate consistent with the universal one found by Akbarian and Choptuik in nonlinear numerical time evolutions.
- I Introduction
- II Background equations
- III Spherical perturbation equations
- IV Perturbation spectrum
- V Conclusions
i.1 The Einstein-Vlasov system
The Vlasov-Einstein system describes an ensemble of particles of identical rest mass, each of which follows a geodesic. The particles interact with each other only through the spacetime curvature generated by their collective stress-energy tensor, whereas particle collisions are neglected.
For massive particles, this is a good physical model of a stellar cluster. For either massive or massless particles, the Einstein-Vlasov system also serves as a well-behaved toy model of matter in general relativity. In particular, spherically symmetric solutions of the Einstein-Vlasov system with small data are known to exist globally in time for massive ReinRendall1992 () and massless particles Dafermos2006 (). Self-similar spherically symmetric solutions with massless particles have been analyzed in JMMGundlach2002 () and RendallVelazquez2011 (). The existence of spherically symmetric static solutions with massive particles was proved in ReinRendall2000 (), and there is numerical evidence that at least some are stable within spherical symmetry AndreassonRein2006 (). Spherically symmetric static solutions with massless particles were analysed and constructed numerically in paperI (), and we investigate their linear perturbations here. See also AndreassonLRR2011 () for a review and additional references.
i.2 Motivation for this paper
This is the second paper in a series motivated by Akbarian and Choptuik’s AkbarianChoptuik () (from now on, AC) recent study of numerical time evolutions of the massless Einstein-Vlasov system in spherical symmetry. AC found two apparently contradictory results:
I) Taking several 1-parameter families of generic smooth initial data and fine-tuning the parameter to the threshold of black hole formation, AC found what is known as type-I critical collapse: in the fine-tuning limit the time evolution goes through an intermediate static solution. The lifetime of this static solution increases with fine-tuning to the collapse threshold as
where is the parameter of the family, its value at the black-hole threshold, and is the proper time at the centre, in units of the total mass of the critical solution. (1) implies the existence of a single unstable mode growing as . AC found that was approximately universal (independent of the family), with value , and that the metric of the intermediate static solution was also approximately universal (up to an overall length and mass scale). In particular, its compactness was in the range and its central redshift was in the range .
II) Conversely, constructing static solutions by ansatz, AC found that these covered much larger ranges of and , but that each one was at the threshold of collapse. That is, adding a small generic perturbation to the static initial data and evolving in time with their nonlinear code, they found that for one sign of the perturbation the perturbed static solution collapsed while for the other sign it dispersed. They found that was in the narrow range , compatible with the value above.
Result II suggests that in spherical symmetry with massless particles, the black hole-threshold coincides with the space of static solutions. If so, then each static solution would have precisely one unstable mode (with its sign deciding between collapse and dispersion), with all other modes either zero modes (moving to a neighbouring static solution) or purely oscillating.
One aspect of Result I, namely that the spacetime of the critical solution is universal, would imply that this universal solution has one unstable mode (as before), but that all its other modes (including those tangential to the black hole threshold) are decaying ones, so that the attracting manifold of the critical solution is precisely the black hole threshold. Indeed, this is the familiar picture of type-I critical collapse in other matter-Einstein systems. However, this is in apparent contradiction to Result II.
In the first paper in this series paperI () (from now Paper I), we used a symmetry of the massless spherically symmetric Einstein-Vlasov system to reduce its number of independent variables from four to three. We then numerically constructed static solutions with compactness in the range . Based on this, we conjectured that the apparent contradiction above is resolved by the critical solution seen in fine-tuning generic initial data being universal only to leading order, and that this leading order is selected by the way in which it is approached during the evolution of generic smooth initial data.
To make further progress, it seems essential to analyse the spectrum of linear perturbations directly. This is the programme of the current paper. In contrast to the static solutions investigated in Paper I, their perturbations do not simplify significantly for , and hence all of our analysis, except for the numerical examples in Sec. IV.5, will be for .
i.3 Plan of the paper
In order to make the presentation self-contained and to establish notation, we review some material from Paper I in Sec. II. We begin in Sec. II.1 by presenting the equations of the time-dependent spherically symmetric Einstein-Vlasov system. We do this in a form in which the massless particle limit is regular and leads to a reduction of the number of independent variables. We discuss static solutions in Sec. II.2, and the massless limit, for both the time-dependent and static case, in Sec. II.3.
In Sec. III we then derive the spherical perturbation equations. In Sec. III.1 we perturb the Vlasov and Einstein equations about a static background, splitting the perturbation of the Vlasov distribution function into parts and that are even and odd, respectively under reversing time. In Sec. III.2 we change independent variables from momentum to energy, as we did for the background solutions. We quickly dispense with static perturbations in Sec. III.3, and in Sec. III.4 we reduce the perturbed Vlasov and Einstein equations to a single integral-differential equation . In Sec. III.5 we dispense with the relatively trivial perturbations on regions of phase space where the background solution is vacuum. We state the massless limit in Sec. III.6.
In Sec. IV we attempt to find the spectrum of eigenvalues. In Sec. IV.1 we identify a positive definite inner product with respect to which is symmetric. In Sec. IV.2 we rewrite the Hamiltonian as where is bounded, giving us at least a lower bound on . We then switch to an approximation method, the Ritz method, which we review in Sec. IV.3. In Sec. IV.4 we specify some properties of the function space in which to look for perturbation modes, that is eigenfunctions of . In Sec. IV.5 we pick two specific background solutions that we obtained numerically in Paper I and use the Ritz method numerically. We find values of in agreement with AC.
Sec. V contains a summary and outlook. Throughout the paper, defines , and we use units such that .
Ii Background equations
ii.1 Field equations in spherical symmetry
We consider the Einstein-Vlasov system in spherical symmetry, with particles of mass . We write the metric as
To fix the remaining gauge freedom we set . The Einstein equations give the following equations for the first derivatives of the metric coefficients:
where , and are the radial pressure, energy density and radial momentum density, measured by observers at constant . The fourth Einstein equation, involving the tangential pressure , is a combination of derivatives of these three, and is redundant modulo stress-energy conservation.
The Vlasov density describing collisionless matter in general relativity is defined on the mass shell of the cotangent bundle of spacetime, or in coordinates. We define the square of particle angular momentum
In spherical symmetry this is conserved, leaving to . In order to obtain a reduction of the Einstein-Vlasov system in the limit , we replace by the new independent variable
The Vlasov equation for is
where we have defined the shorthand
The non-vanishing components of the stress-energy tensor are
where we have introduced the integral operator (acting to the right)
ii.2 Static solutions
In the static metric
with Killing vector , the particle energy
is conserved along particle trajectories. In order to obtain a reduction of the static Einstein-Vlasov system in the limit , we replace by
Hence we have
where we have defined
The general static solution of the Vlasov equation, for simplicity with a single potential well, is then given by
(If forms more than one potential well, could be different in each of them Schaeffer1999 ().)
The static Einstein equations are
where we have defined the integral operator (acting to the right)
and the shorthand
The second equality in (24) shows that is the radial particle velocity, expressed in units of the speed of light and measured by static observers. (Note that by definition .) The fact that (24) can be rewritten as
shows that is an effective potential for the radial motion of the particles with given constant “energy” and angular momentum squared . In particular determines the radial turning points for all particles with a given and .
We define as the value of at its one local maximum in . We define as the other value of for which takes the same value. We also define as the value of at its one local minimum in . Intuitively, is the “lip” of the effective potential for particles of a given , and its “bottom”. We define as the upper boundary in of the support of and as the lower boundary. We define and as the values where or .
For massless particles, particles with all values of move in the same potential , and all other quantities we have just defined do also not depend on . See Fig. 1 for an illustration.
Any particles with would be unbound, and if such particles were present in a static solution with the ansatz (20) they would be present everywhere and hence the total mass could not be finite. We must therefore have for . must either have compact support in , or fall off as sufficiently rapidly for to be finite. Both assumptions are assumed implicitly in the definition (23) of when we formally extend the upper integration limits in and to .
By contrast, real values of require , and when changing from to , the condition must be imposed explicitly as a limit to the integration range in (23). Hence needs to be defined for all (it can be zero in part of that range), but not all of that range contributes to the integral , and hence to the stress-energy tensor, at every value of .
ii.3 Reduction of the field equations for
Because is conserved, the field equations already do not contain . Furthermore, and appear in the coefficients of the field equations only in the combination . In the limit , these coefficients become independent of . As first used in JMMGundlach2002 (), the integrated Vlasov density
in place of . We have reduced the Einstein-Vlasov system to a system of integral-differential equations in the independent variables only. A solution of the reduced system represents a class of solutions , all with the same spacetime but different matter distributions.
in place of and
in place of . Each solution of the reduced system represents a class of solutions . We now define and as the limits of support of .
In Paper I, we conjectured that the space of spherically symmetric static solutions with massless particles is a space of single functions of one variable, subject to certain positivity and integrability conditions. This single function can be taken to be any one of , or .
As a postscript to Paper I, we note here that a mathematically similar result holds for Vlasov-Poisson system for a subclass of static solutions, namely the “isotropic” ones, where the Vlasov function is assumed to be a function of energy only. With the isotropic ansatz the Vlasov density uniquely defines the Newtonian gravitational potential and vice versa DeJonghe (). Physically, however, the two results are different, as the relativistic result with massless particles holds for all solutions, depending on both energy and angular momentum.
Iii Spherical perturbation equations
iii.1 Perturbation ansatz
In the study of the linear perturbations of a static solution we return to the general case . We make the perturbation ansatz
where is a static solution of the Einstein-Vlasov system, and where is given by (17). We define to be even in and to be odd. In particular, as must be at least once continuously differentiable () to obey the Vlasov equation in the strong sense, its odd-in- part must vanish at , or
We now expand to linear order in the perturbations. The linearised Vlasov equation splits into the pair
where we have defined the differential operator
Note, from (8), that the time-dependent Vlasov equation in the fixed static spacetime is . By construction, obeys , and hence , as we have already used to construct static solutions. The even-odd split of the perturbed Vlasov equation into the pair (35,36) reflects the fact that reversing both and together must leave the field equations invariant.
The perturbed Einstein equations are
We have used the background Einstein equations to eliminate all integrals of in favour of and . The integrability condition for between (39) and (40) is identically obyed modulo the perturbed Vlasov equation (35,36), and hence the perturbed Einstein equation (40) is redundant [just as (5) is in the nonlinear equations].
iii.2 Change of variable from to
To simplify the perturbation equations, we change independent variables from to , with again given by (17). From the resulting transformation of partial derivatives, we need only the following identities:
where we have defined the shorthand
with given by (26). is the coordinate speed corresponding to the physical speed .
on its own appears in the field equations only acting on the metric and its perturbations, in which case the term denoted by in (42) is irrelevant. On the matter perturbations, acts only in the combination . From now on is understood to mean .
With (43) the perturbed Vlasov equation becomes
Equivalently, the stress-energy perturbations are given by
Finally, the boundary condition on becomes
iii.3 Static perturbations
Static perturbations can be obtained more directly by linear perturbation of the nonlinear static equations. The infinitesimal change , , to a neighbouring static solution gives
iii.4 Reduction to a single stratified wave equation
We can uniquely split any time-dependent perturbation that admits a Fourier transform with respect to into a static part (with frequency ) and a genuinely non-static part (with frequencies ). For genuinely non-static perturbations we can uniquely invert by dividing by in the Fourier domain.
where we have defined the integral-differential operators (acting to the right)
Note that , , do not commute with each other, while commutes with all of them because their coefficients are independent of .
In this compact notation it is easy to see that by row operations we can reduce the system (56) to the upper diagonal form
Hence we have reduced the time-dependent problem to the single integral-differential equation
The coefficients , , and commute with each other but not with the operators or . and do commute with each other because of the boundary condition (53).
It is convenient to split the time evolution operator into a kinematic and a gravitational part as
Hence we can write (63) as
This form stresses that (69) is a stratified wave equation, in the sense that for fixed and we have a wave equation in , with coefficients also depending on and , while different values of and are also coupled through the double integral over and , which represents the gravitational interactions between particles of different momenta at the same spacetime point. Intuitively, the characteristic speeds of (69) are because any matter perturbation is propagated simply at the velocity of its constituent particles.
We note in passing that in our notation the static perturbations are solutions of
Given a solution of the master equation (63), in any genuinely nonstatic perturbation can be reconstructed from as
iii.5 Preliminary classification of non-static perturbations
The right-hand side of (69) is generated by perturbations at all values of and at at given spacetime point , but because of the overall factor it only acts on perturbations at those values of and where , that is, where background matter is present.
This suggests a preliminary classification of all perturbations into
stellar modes, with support on the region in space that lies inside the potential well and where also has support;
bottom modes, with support inside the potential well and for values of below the support of ;
middle modes, with support inside the potential well and for values of above the support of but below the top of the potential well;
outer modes, with support outside the potential well and for values of below the top of the potential well;
top modes, with values of above the top of the potential well.
This classification of modes is illustrated in Fig. 1 for the massless case, where all values of experience the same effective potential . Obviously, the middle modes do not exist if the potential well is filled to the top (), and the bottom modes do not exist if it is filled to the bottom ().
Particles contributing to the bottom, middle, top and outer modes move freely in the effective potential set by the static background solution, without experiencing a gravitational back-reaction. We may call them the trivial modes. Mathematically, the trivial modes obey (69) in a region of space where the right-hand side vanishes. In that region they can be solved in closed form. To do this, we define the “tortoise radius”
This gives us
and hence (69) becomes
for , subject to the Dirichlet boundary conditions
where are the left and right turning points given by . Hence we can write down the general local solution of (69) with in d’Alembert form as
subject to the boundary conditions (75).
With for the trivial modes, (71) becomes , and hence
We note also that must be non-negative for physical trivial modes, as we cannot subtract particles from a vacuum region.
The trivial modes (except for the outer modes) induce stellar modes through gravitational interactions, or mathematically through the right-hand side of (69). This means that for a complete solution of the problem, we need to solve for stellar modes including an arbitrary driving term generated by the other modes, that is
where describes the stellar modes and is the sum of the top, middle and bottom modes. (The outer modes do not couple to the stellar modes at all.)
iii.6 Massless case
In the massless case, we define the integrated matter perturbations in the obvious way:
The reduced perturbation equations with are obtained from the ones in the general case by replacing and with and , and with and , and with . The coefficients , , and all become independent of . In particular, all particles now move in the same effective potential .
Iv Perturbation spectrum
iv.1 Inner product
Starting from (III.4), we rewrite the gravitational part of the Hamiltonian more explicitly as
where we have defined the shorthands
We have defined for later use: note that . For clarity we have not written out function arguments, but recall that acting on anything gives a function of and only, that , , , and hence that and .
We now construct an inner product with respect to which is a symmetric operator. Consider first solutions , with support only where , that is the modes we have characterised as trivial. Then only is non-vanishing, and the perturbed Vlasov equation reduces to the free wave equation (74). The obvious inner product is therefore the well-known one for the free wave equation, that is
where is an arbitray weight. Expressing this in terms of , we have
If we interchange the integration over with the double integration over and , we obtain
which is explicitly symmetric in and .
Consider now solutions , with support only where , that is stellar modes. Assuming further that wherever (as will be the case for our examples), we must then make the choice
(up to a positive constant factor), that is
where for the last term we have used integration by parts in .
We now define on all of phase space by (91) on the support of , and an arbitrary positive , for example , everywhere else.
iv.2 Quadratic form of the Hamiltonian
Consider the operator
It is easy to see that is a projection operator, in the sense that
We also have
and so is symmetric,
As a projection operator, can only have eigenvalues and . The corresponding eigenspaces and are orthogonal, as
for any two vectors , .
Hence every normalisable vector can be uniquely split into two vectors, one from each eigenspace, that is
with . We write this statement as
From (95) we see that
while all other matrix elements of and all of are non-trivial.
Using integration by parts in in (93) and the boundary conditions
and so is antisymmetric,
Similarly, if we define
Hence we have
We can also write as
where we have defined the shorthand coefficient
Recall that we have assumed that , and hence . Evidently , and from (21) we see that , so we have . From
we then obtain a (negative) lower bound on the spectrum of , namely . This is not sharp as .
Using also that and that commutes with multiplication by any function of , we have