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Black hole interacting with matter as a simple dynamical system

Recently, a variational principle has been derived from Einstein-Hilbert and a matter Lagrangian for the spherically symmetric system of a dust shell and a black hole. The so-called physical region of the phase space, which contains all physically meaningful states of the system defined by the variational principle, is specified; it has a complicated boundary. The principle is then transformed to new variables that remove some problems of the original formalism: the whole phase space is covered (in particular, the variables are regular at all horizons), the constraint has a polynomial form, and the constraint equation is uniquely solvable for two of the three conserved momenta. The solutions for the momenta are written down explicitly. The symmetry group of the system is studied. The equations of motion are derived from the transformed principle and are shown to be equivalent to the previous ones. Some lower-dimensional systems are constructed by exclusion of cyclic variables, and some of their properties are found.


I. INTRODUCTION
The spherically symmetric gravitating shell has been used as a simplified quantum model at many occasions. One can mention the quantum effects of domain walls in the early Universe [1], quantum aspects of gravitational collapse [2], quantum theory of black holes [3] or Hawking evaporation of black holes [4].
The equations of motion for the shells has been derived by Dautcourt [5] and transformed to a geometric (i.e. "gauge invariant") form by Israel [6]. However, for a quantum theory, one needs a variational principle rather than dynamical equations. Such variational principle has as yet been just guessed from the dynamical equations, Refs. [7] and [8], or from some intermediate variational principle, Refs. [4] and [9]. Quite a number of different Hamiltonians have resulted and the corresponding quantum theories have not been unitarily equivalent. There are two sources of the ambiguity.
The first source of ambiguity is the invariance of the general relativity with respect to any coordinate transformations on one hand, and the property of Hamiltonians to generate dynamics with respect to a particular time coordinate on the other. For example, the choice of the proper time along the shell as such a coordinate leads to the coshp Hamiltonian, Refs. [7] and [10], the Schwarzschild time coordinate inside the shell to the square-root Hamiltonian, Ref. [2], and the Schwarzschild time coordinate outside the shell to a merely implicitly determined Hamiltonian of Refs. [4] and [9]. An attempt to work out a gauge invariant theory for at least some class of the guessed Hamiltonians is Ref. [8]; based on a technical assumption that the super-Hamiltonian is quadratic in the momenta, it gives a unique action principle.
The second source of ambiguity is the fact that equations of motion do not determine the corresponding variational principle uniquely in general. This ambiguity can be removed by a direct derivation of the variational principle for the shell from the Einstein-Hilbert and shell matter Lagrangian. Three such derivations exist: Refs. [11], [12] and [13]; they all lead to the same variational principle for the same system. In Refs. [12] and [13], the reduction of the second order formalism to the spherically symmetric case is done first, followed by a reduction to dynamical variables of the shell alone and by the transformation to the second order formalism. In Ref. [11], a first and second order formalism for a general shell and gravitational field is derived (no symmetry); this second order formalism is reduced by the spherical symmetry and to shell variables in Ref. [14].
We are then left with the following dilemma. The super-Hamiltonian of Ref. [8] is simple and amenable to the existing gauge invariant quantization methods (like e.g., Refs. [17] and [18]). The super-Hamiltonian of Refs. [12], [14] and [13] follows from the Einstein-Hilbert and shell-matter Lagrangian, but it is extremely complicated: it contains nested square roots and hyperbolic functions, and for a good measure, it is formulated in coordinates that diverge on horizons. It would be difficult to quantize.
A natural question then arise: what is the relation between the guessed and simple variational principle of Ref. [8] and the derived but complicated one of Refs. [12], [14] and [13]? If the simple principle were equivalent to the complicated one, we could use the simple principle and forget the complicated one. However, the answer turns out to be rather surprising (see Ref. [15]): the two descriptions are only locally, but not globally equivalent. The local equivalence explains, why the equations of motion are the same. For the study of the problem, a geometric approach of Ref. [16] has been applied. All gauge invariant properties of a constrained system are encoded in two objects: the constraint manifold Γ and the presymplectic form Ω on Γ. The equivalence of two such systems, (Γ 1 , Ω 1 ) and (Γ 2 , Ω 2 ) is then a well-defined mathematical concept independent of a choice of extended phase space, a choice of constraint functions or a choice of gauge.
In this situation, it seems natural to look for a transformation of the variational principle of Refs. [12], [14] and [13] to a better set of coordinates, and this will be the main topic of the present paper. The method will again be based on the gauge invariant description (Γ, Ω) of reparametrization invariant systems. We shall perform the transformation in three steps; to motivate the steps, we take into account the symmetry of the system, the structure of the so-called Cartan form Θ, which is defined by dΘ := Ω, and the topology of the space (Γ, Ω). The topological problem involved here is the existence of transversal surfaces in Γ; such surfaces are nowhere tangential to the dynamical trajectories. Functions whose levels are transversal are not only helpful, if one looks for nice coordinates, because they simplify Θ. Quite generally, they have to do with the existence of Hamiltonian and with the possibility to give the quantum dynamics the form of Schrödinger equation (cf. Ref. [16]). We shall, therefore, also look for transversal surfaces in a more systematic way. Finally, we shall find that the structure of the phase space of the spherically symmetric shell is not simple. Only a proper subset, which we call physical region, contains physical states of the shell. Its nature can be roughly specified as follows. All points inside it correspond to the system consisting of a shell of positive mass and radius, interacting with a black hole. The boundary of the region is complicated; it contains shells with zero radius, shells with zero rest mass (which cannot, however, be regarded as shells made of light-like matter), as well as self-gravitating isolated shells with positive radius and mass-these are the points at the boundary that are physically meaningful. The points outside the region that describe shells with positive radius and mass correspond to systems consisting of the shell and a negative mass source. A conclusion seems to be that the system will be difficult to quantize even if the new variables has made it algebraically simple. One problem will be to keep the spectra of observables contained within the physical region. Another problem will be to construct a unitary dynamics, because the classical dynamics breaks down at the boundary of the physical region.
The plan of the paper is as follows. In Sec. II, we collect the relevant results of the previous papers, mainly Refs. [14] and [13]. The variational principle that we describe allows also for an internal degree of freedom of the shell; it depends, therefore, of two additional variables in comparison with Refs. [12] and [14]: the proper time along the shell trajectory and the rest mass of the shell. We determine the constraint manifold Γ and the Cartan form Θ that follow from the variational principle. We study the physical region.
In Sec. III, we introduce three new functions on the phase space that are regular at the horizons and that will replace the momentum P of the shell and the two Schwarzschild times T ± that diverge there. We describe their physical and geometrical meaning, using also some results of previous papers. In Sec. IV, we specify a new set of coordinates on Γ, and show that the transformation from the old to the new coordinates is differentiable and invertible. We also find the ranges of these coordinates in the physical region. Sec. V collects and extends the results about symmetries of the system that were obtained in Ref. [15] and expresses the symmetry transformations in the new coordinates; the form of the transformation simplifies. The new times are cyclic coordinates as the old have been. In Sec. VI, we transform the Cartan form into new coordinates. The square roots and hyperbolic functions disappear, and an equivalent form becomes regular at the horizons so that it can be smoothly matched accross them. However, the Cartan form is still complicated and this motivates further transformations. The final result thereof can be described as follows. The extended phase space is (R 8 ,Ω). The natural coordinates of R 8 form a Darboux chart for Ω. The equation of the constraint surface Γ in R 8 is polynomial in all momenta and the presymplectic form dΘ is the pull-back ofΩ to Γ.
In Sec. VII, we study the polynomial constraint. We show that it defines two disjoint submanifolds in R 8 . One, denoted by Γ 6 , is six-dimensional and lies well outside the physical region. The other, denoted by Γ 7 , is seven-dimensional and intersects the physical region. In the intersection of Γ 7 with the physical region, we find two different foliations by transversal surfaces. They are not globally transversal, however, because none of them is intersected by all dynamical trajectories. The problem is that each dynamical trajectory starts or finishes at the (R = 0)-subset of the boundary of the physical region. The two different Hamiltonians coresponding to the two foliation are written down explicitly; they contain second and third roots.
In Sec. VIII, we repeat the study for some important lower-dimensional cases; they can be obtained by exclusion of cyclic coordinates from the original system. In particular, a (dynamical) black hole and the shell without its internal degree of freedom, the shell in the field of a fixed black hole external field, or the isolated self-gravitating shell (with the flat spacetime inside) are considered.

II. DESCRIPTION OF THE SYSTEM
In this section, we shall collect some results of Refs. [8], [14] and [13] so that the paper becomes relatively self-consistent. We shall also describe and study the physical region.
A spherically symmetric thin-shell spacetime solution of Einstein equations can be constructed as follows. Consider two Schwarzschild spacetimes M 1 and M 2 with Schwarzschild masses E 1 and E 2 . Let Σ 1 be a timelike hypersurface in M 1 and Σ 2 be one in M 2 . Let Σ 1 divide M 1 into two subspacetimes, M 1+ and M 1− , and similarly Σ 2 divide M 2 into M 2+ and M 2− . As everything is spherically symmetric, all spacetimes are effectively twodimensional; we chose fixed time and space orientation in the two-dimensional Schwarzschild spacetimes so that future and past as well as right and left are unambiguous; let then M 2+ and M 1+ be right with respect to M 2− and M 1− . Let Σ 1 and Σ 2 be isometric; then the spacetime M 1− can be pasted together with the spacetime M 2+ along the boundaries Σ 1 and Σ 2 . The result is a shell spacetime M s . Given a shell spacetime, we shall leave out the indices 1 and 2, having right (left) energy E + (E − ), shell trajectory Σ, and the right (left) subspacetime M + (M − ). Thus, M + = M 2 ∩ M s and M − = M 1 ∩ M s . In Refs. [14], [15] and [8], the Schwarzschild masses E ǫ , ǫ = ±1 were assumed to be positive.
Each subspacetime M ǫ has the metric where and it is split up by its horizons into four quadrants: we denote by Q I that which is adjacent to the right infinity, Q II to the left infinity, Q III to the future singularity and Q IV to the past singularity. Suitable notation (introduced in Ref. [14]) enables us to write all formulas in a form valid in any quadrant: we define four sign functions a ǫ and b ǫ distinguishing the quandrants of M ǫ : a ǫ := sgnF ǫ , and b ǫ equals to +1 (−1) in the past (future) of the event horizon in the spacetime (M ǫ , g ǫ ). We also include the dust internal degree of freedom, M, and the proper time T along the shell trajectory using the results of Ref. [13]. Then the total action for the dust and gravity can be rewritten in the form where ν is a Lagrange multiplier, is the super-Hamiltonian, sh a x := (e x + ae −x )/2 for any a = ±1 and x ∈ (−∞, ∞). The momentum P conjugate to the radial coordinate R can be defined as follows (for more detail, see Ref. [14]). Let (n ǫ , m ǫ ) be the orthonormal dyad at a shell point in the subspacetime (M ǫ , g ǫ ), the vector n ǫ being tangential to the shell and future oriented, m ǫ being right oriented; we call it shell dyad. Let further the so-called Schwarzschild dyad (n Sǫ , m Sǫ ) be defined at each point of the shell in the spacetime (M ǫ , g ǫ ) by the requirement that n Sǫ be timelike future oriented, m Sǫ be right oriented and that one of the vectors be tangential to the (R = const)-curves. Clearly, in each quadrant, we have a different formula for the components of the Schwarzschild dyad; with respect to the Schwarzschild coordinates, Then, P ǫ /R is the hyperbolic angle between the Schwarzschild and shell dyads: The extended phase space of the system is eight-dimensional, split up into 16 disjoint sectors, each sector being a pair of quadrants, one chosen from the left and one from the right subspacetimes. Each of these sectors can be covered with the coordinates P , R, E + , T + , E − , T − , M and T. The constraint surface that we call Γ is defined by the constraint equation C = 0 in the extended phase space. It does not intersects all 16 sectors, but it is split up into at most (if E − > M) non-empty intersections of the sectors with it. 1 These can be covered by the system of seven coordinates P , R, E + , T + , E − , T − and T. The following coordinates are assumed to have non-trivial ranges: The pull back of the Liouville form of the action (2) to the constraint surface Γ in these coordinates is simply where M is now a function of the seven coordinates defined by The form (5) is called Cartan form of the system; it contains all information about the dynamics and about the Poisson brackets (see, e.g. Refs. [19] or [15]). The present paper starts from these formulas and quantities. The first new observation that we make is, that all these formulae remain valid also for the values E + = 0 or E − = 0, if the notation and conventions are adapted a little. These cases are interesting because they describe an isolated self-gravitating shell (with the flat spacetime inside). Indeed, the metric of the flat spacetime is covered by formula (1). However, the flat spacetime does not contain any horizon and is not split into quadrants. Still, a piece of the flat spacetime can be used in four distinct ways in the contruction of the shell spacetime: either it lies to the left (E − = 0) or to the right (E + = 0) of the shell, and, in each case, the R coordinate can either increase to the right or to the left (R is always spacelike in the flat spacetime). We can, therefore, formally define the E-spacetime for E > 0 as before to be the Kruskal spacetime of Schwarzschild mass E with four quadrants, and for E = 0 as consisting of two topologically separated quadrants Q I and Q II , each isometric to the flat spacetime, but with oposite time and space orientations: in Q I , the radial ("Schwarzschild") coordinate R increases to the right and the "Schwarzschild" time T to the future; in Q II , R increases to the left and T to the past. It is amusing to observe that the validity of the formulae (2) and (3) can then be extended to vanishing E's, if we just use the values a ǫ = 1 everywhere, b ǫ = +1 in Q I and b ǫ = −1 in Q II . This works because the components of the Schwarzschild dyad that are crucial for the derivation of Eq. (3) in Ref. [14] retain their form in the quadrants Q I and Q II of the E-spacetime for E = 0, and there are no quadrants Q III and Q IV . Similarly, the sign relation (14) of Ref. [8], which is basic for the form of the dynamical equations, remains valid with our orientation and quadrant conventions. Observe that the presence of two topologically separated subspacetimes is not observable and does not imply any physical assertion; similarly, the orientation of coordinates is a mere coordinate convention without any physical meaning.
A very important but rather embarrasing observation is now that a formally impeccable shell spacetime can be pasted together from Schwarzschild spacetimes of any mass, even a negative one. The metric is still given by Eq. (1) and the spacetime has a similar global structure as the flat one: there are no horizons and there are two possible orientations of how it can be built in. Again, all equations remain valid, if we accept the same sign, quadrant and orientation conventions as for the flat spacetime and define the (two-quadrant) E-spacetime for E < 0 analogously to what we have done for E = 0. The corresponding shell dynamics is then again given by the action (2).
Let us quickly discuss the dynamical equations following from the action (2) under these conventions. There are three conservation laws, resulting from varying the action with respect to the cyclic coordinates T + , T − and T. The variations with respect of the conserved momenta and P with the subsequent simplification by the constraint (6) yield: The equation determiningṖ can be obtained either by varying the action with respect to R, or by differentiating the constraint equation C = 0 with respect to t followed by substitution for all other t-derivatives from Eqs. (7)- (10). Let us rewrite the constraint equation in the form Eqs. (7)-(11) represent a complete system of dynamical equations for the shell. We derive some important consequences from the dynamical equations. Eqs.
Theorem 1 Let E ǫ be non-positive and not larger than E −ǫ for some value of ǫ. Then the timelike (connected) center (R = 0) curve lies in the subspacetime M ǫ on the ǫ-side of the shell.
To prove the theorem, we observe that E ǫ ≤ E −ǫ implies Thus, the left-hand side of Eq. (17) can only be positive, if τ ǫ = −ǫ.
The assumption E ǫ ≤ 0 of the theorem implies that M ǫ is the two-quadrant E-spacetime with τ ǫ > 0 in the quadrant Q I and τ ǫ < 0 in the quadrant Q II ; this follows from the definition of τ ǫ and from the positivity of F ǫ (in fact, for E ǫ ≤ 0, we have F ǫ ≥ 1). Hence, to the right (M + ) of the shell, there is a part of Q II with the center to the right, and to the left of the shell (M − ), there must be Q I with the center to the left, Q. E. D. An interpretation of the theorem is that the shell spacetime containing one or two negative-mass subspacetimes is unphysical: there must be at least one negative mass source somewhere outside the shell. The theorem also says that a shell spacetime constructed from one flat and one positive-mass subspacetimes must have a regular center inside the flat subspacetime.
To summarize: The action (2) generates a regular dynamics in the enlarged phase space defined just by For E + > 0 and E − > 0, it describes a shell interacting with a black hole of mass E − (or E + , depending on from where we observe the shell). For E + > 0 and E − = 0 (or E + = 0 and E − > 0), it describes a self-gravitating shell with flat spacetime inside. If any of E + and E − are negative, it describes the shell interacting with a negative mass source. Only the points with are physically sensible; the subset specified by the inequalities (18) and (19) is the physical region of the phase space. There seems to be nothing about the variational principle (2) that would help us to enforce the validity of the Eqs. (19) in any natural, automatic way.

III. RADIAL AND KRUSKAL MOMENTA
From the definition of the functions T ± and P as given in the previous section, it follows that they are singular at the horizons: T ǫ at R = 2E ǫ and P at R = 2E − as well as at R = 2E + . In this section, we introduce three functions that are well-defined on the whole constraint surface to replace T ± and P . Another replacement of T ± and P with similar properties has been tried in Ref. [14]. There, the functions were constructed from the Kruskal coordinates; however, the Kruskal coordinates make no sense for flat spacetime and so the important case E − = 0 could not be incorporated. In this subsection, we remove this problem.
Let us define the new functions q, [T 1 ] andT 1 by and the auxiliary functions u ǫ and v ǫ by  10) and (20), we find that hence, q is a kind of radial momentum, and it is regular everywhere along dynamical trajectories. The meaning of the functionsT 1 and [T 1 ] can be inferred from that of T ǫ 1 , which are defined by Eqs. (21) and (22) lead then to for each ǫ = ±1. Let us limit ourselves to the physical region and distinguish two cases. E ǫ = 0. Then, T ǫ 1 = T ǫ in both quadrants of the E-spacetime, so b ǫ T ǫ 1 is the time coordinate of the inertial system of the Minkowski spacetime M ǫ in which the center of mass of the shell is in rest. E ǫ > 0. Then, it is possible to introduce Kruskal coordinates U ǫ and V ǫ in the Kruskal spacetime M ǫ by and by the requirement that both functions U ǫ and V ǫ increse to the future, where κ is the well-known monotonous function defined on the interval (−1, ∞) by its inverse, At each point of the shell trajectory U ǫ = U ǫ (t) and V ǫ = V ǫ (t), the so-called Kruskal momentum, P ǫ K , was defined in Ref. [14] by In Ref. [15], four further functions u ǫ and v ǫ were defined by where κ ǫ := κ(−U ǫ V ǫ ).
The following relation between the pairs u ǫ , v ǫ and R, E ǫ was shown to hold in Ref. [15] (Eqs. (32) and (33)): A comparison with Eqs. (13) and (14) shows that our functions u ǫ and v ǫ defined by Eqs. (23) coincide with the functions u ǫ and v ǫ of Ref. [15], if the dynamical equations are satisfied. Eq. (26) then shows that u ǫ and v ǫ vanish at horizons and that their logarithms diverge there. It then also follows for the functions T ǫ 1 along dynamical trajectories that We can see that the functions T ǫ 1 are directly determined by the geometry of the dynamical trajectory in the spacetime M ǫ and are regular everywhere along the trajectory.

IV. A REGULAR COORDINATE SYSTEM
In this section, we describe a coordinate system that is regular at the horizons, and at the same time remains meaningful for the special values E ǫ = 0 of Schwarzschild masses.
As such coordinates on Γ, we suggest the seven functions q, R, M, T, [E],T 1 , and [T 1 ]. Let us show that the Jacobian of the transformation from these coordinates to P , R, E + , T + , E − , T − and T is non-zero. To calculate the determinant, we observe that the four columns R, T + , T − and T of the Jacobian For the derivatives involved here, Eqs. (6) and (20) yield immediately: Substituting this in Eq. (28) and using again Eq. (6), we arrive at the simple result which holds in each sector of Γ. It follows that the transformation is regular within each sector and can be inverted. In fact, it is not difficult to find the functions defining the inverse transformation. First, we show thatĒ in each sector. To this aim, we substitute for q and M from Eqs. (6) and (20) into Eq. (29) and use the identities In this way, we obtain Now, the equality follows immediately from Eq. (6). The next non-trivial part of the inverse transformation is the relation The substitution for q and M gives where η = ±1. After some rearrangement and applying Eqs. (30), we have Then, Eq. (6) leads immediately to and this implies Eq. (31). The rest of the inverse transformation is easy: the functions R and T are the same in both sets of variables, and hence, The two inequalities E + ≥ 0 and E − ≥ 0 are equivalent to [E] 2 ≤ 4Ē 2 . Eq. (29) implies that this inequality is, in turn, equivalent to From Eq. (34), it follows that Eq. (35) shows that the inequality (37) is not weaker than (34). Hence, all information is contained in the following inequality:

V. SYMMETRY OF THE SYSTEM
In this section, we collect the results of Ref. [15] on the symmetry of the shell action and describe the action of the symmetry transformations on the new variables.
In Ref. [15], we have found that there is a continuous symmetry group generated by an arbitrary function of E + and E − . A simple generalization of the argument given in Ref. [15] implies that the following finite transformation of the variables E ǫ , u ǫ , v ǫ andP ǫ , which were used there, is a symmetry of the system: the other variables being invariant, where Λ(E + , E − ) is an arbitrary smooth function of two variables (the factor 1/4 is introduced for convenience). In addition to this continuous infinitely dimensional group, there were two reflections, which we denote here by σ 1 and σ 2 . σ 1 is a time reflection defined by and σ 2 is a left-right reflection defined by It is clear that M is invariant with respect of the whole group; hence, q is invariant with respect to the Λ-transformation for any Λ(E + , E − ), it changes sign by the time reflection σ 1 and is invariant with respect to σ 2 . The function R was written in terms of u ǫ and v ǫ in Ref. [15] in the form Realising that 4u Using this formula, one finds easily that The time reflection σ 1 changes the signs of both times: but the left-right reflection σ 2 acts as follows: From the physical meaning of M and T, it follows that they are invariant with respect to the continuous transformation for any Λ(E + , E − ) and of the left-right reflection σ 2 , whereas the time reflection σ 1 must give However, as T is a new cyclic coordinate, the new system has larger continuous symmetry: we can extend Λ to an arbitrary function of the three variables E + , E − , and M, and define the action on T as follows: Let us postpone the proof that this extended Λ-transformation is a symmetry to Sec. VI. It generates three independent constant shifts of the cyclic variablesT 1 , [T 1 ] and T that are different in different shell spacetimes. We can see from these results that our new variables transform particularly simply. This was, in fact, the idea that helped to find the variables in the first place.

VI. TRANSFORMATIONS OF THE CARTAN FORM
Let us transform the Cartan form Θ in each sector of Γ to the variables q, R, M, T, [E], T 1 and [T 1 ] and check that it can then be extended smoothly accross the boundaries of the sectors.
If we substitute for P , T + , T − and M, we obtain The logarithms can be rearanged as follows: The total differential of a function (that diverges badly at the horizons) can be left out. We obtain where we must substitute for E + , E − ,Ē, u ǫ and v ǫ from Eqs. (32), (29) and (23). This form of Θ has been obtained in Ref. [15] (without the matter term); there, however, u ǫ and v ǫ were regarded as independent variables. Observe that Eqs. (23), (29) and (32) imply so that u ǫ and v ǫ are not independent functions for E ǫ = 0. Now, we can show that the last term in Θ is regular: we just rewrite it using Eq. (42): Then, applying the well-known identity [AB] =Ā[B] +B[A] that holds for jumps and averages of any two functions A and B, and after using Eq. (29), we arrive at This, together with Eq. (41) gives Θ as a function of new variables. It is a complicated one, so we have to transform the coordinates further in order to get a simple expression.
The following question will give some direction to this search for simplicity: is the variable [T 1 ] suitable for the role of time? More concretely, the time levels in the constraint surface should be transversal (this is shown in Ref. [10]). A surface is transversal, if dynamical trajectories intersect it transversally and only once. We can see if the surface [T 1 ] = const has this property as follows. The direction of motion on the constraint surface coincides with the direction of degeneration of the presymplectic form dΘ; hence, the pull-back dΘ f of dΘ to any transversal surface f = const must be non-degenerate. If the surface is 2ndimensional, then the 2n-form dΘ f ∧ . . . ∧ dΘ f must be everywhere non-zero. In our case, n = 3. Let us calculate Eqs. (41) and (44) imply that where A i , i = 1, . . . , 6 are some functions of the variables R, q, [E], and M. It is clear, therefore, that and the points where the six-form vanishes coincide with those where A 1 vanishes. A 1 can be calculated from Eq. (44) with the result Thus, A 1 vanishes for Do these points lie in the physical region that is given by inequality (38)? Let us substitute the value of [E] 2 into Eq. (38) and rewrite the result in the form This inequality implies inequality (35); hence, all zero points of A 1 lie in the physical region. We conclude that [T 1 ] = const is no transversal surface. Let us try to shift [T 1 ] by a function dependent on the variables R, q, [E] and M: Our motivation for choosing such a form is that T 2 is then transformed by the Λ-symmetry (39) in the same way as [T 1 ] is, and that it is also a cyclic coordinate. Substitution for [T 1 ] into dΘ changes the coefficient A 1 at dR ∧ dq in dΘ T 2 by Comparing A 1 , Eq. (45), with ∂Ē/∂R calculated from Eq. (29), we observe that which is always positive; thus, one possible shift is If we perform this transformation in Θ, we find out easily that another shift, miraculously cancels some pesky cross terms in dΘ. Indeed, shifting the 'times' in the Cartan form, we have: Using Eqs. (41) and (44), we obtain The last three terms do not contain other variables than q, R and M and so the terms with We have not found any simple physical or geometrical interpretation of the new times T 2 and T 3 ; observe that the shifts (46) and (47) "mix the sides". The fact that the last three terms in Θ contain only three variables means that they can be simplified to just one term. For example, This suggests the second step of our transformation: We arrive so at the final shape of the Cartan form whereĒ is given by Eq. (29) in which q and R are expressed in terms of p and x; such expressions can be obtained by solving Eqs. (48) and (49) for q and R. Consider Eq. (48). If we keep M fixed, p is an increasing function of q in the whole interval (−∞, ∞), for Hence, the function maps the range (−∞, ∞) of q onto the range (−∞, ∞) of p in a bijective way, and it possesses a unique inverse. One can explicitly write down this inverse using second and third roots; this will makeĒ a complicated function of p, x, M and [E] (we shall write down this function later, cf. Eq. (63)). Eq. (50) implies that and Eq. (29) gives Thus, the surface T 3 = const is not transversal at [E] = 0, but T 2 = const and T = const are transversal everywhere. The Cartan form (50) is not yet very simple, becauseĒ is a complicated function of p, x, [E] and M. What we can still do is to extend the phase space to an eight-dimensional manifold with the cordinatesĒ, [T 2 ], [E],T 2 , M, T, p and x, and to express the constraint between these variables as a polynomial. Indeed, applying Eq. (49), we can rewrite Eq. (29) as follows squaring Eq. (48) and substituting the above expression for q 2 into the result, we obtain This is a constraint that is polynomial in the momenta p,Ē, [E] and M. The inequalities defining the ranges of the variablesĒ, [T 2 ], [E],T 2 , M, T, p and x in the physical region are also simplified: Let us, finally, find the transformation of the variables p, x, M, T,Ē, T 2 , [E] and T 3 by the symmetry group. The results are as follows. We have for the Λ-transformation, for the time reflection σ 1 , and for the left-right reflection σ 2 , Thus, the simple form of the symmetry transformations is preserved by the shifts (46) and (47) as well as by transformations (48) (51) is clearly invariant of the group. Thus, the action of the symmetry group is well-defined in the whole phase space and it preserves the physical region.

VII. THE CONSTRAINT SURFACE
In this section, we are going to study the properties of the constraint (51). We are interested in the topology of the constraint surface; we shall investigate the salient question of whether or not any spurious solutions have been unintentionally included during the process of transforming the constraint to the polynomial form; and we shall look for solutions of the constraint (51) with respect to some momenta.
Our new variational principle reads where The constraint surface is trivial in the direction of the three times T 2 , T 3 and T, so we can limit ourselves to its projection to the five-dimensional space spanned byĒ, [E], M, p and x. Let us observe that C 1 depends onĒ, [E] and x through a function that we shall call B, and that the gradient of B is non-zero. It is, therefore, advantageous, to write C 1 as a composed function, Let us first look for the singular points of the constraint surface (that is, where the gradient of C 1 vanishes). We have As all three derivatives must vanish at a singular point, all such points are determined by the equations All points satisfying Eqs. (59) are solutions of Eq. (51); however, the physical ranges (52) of the variablesĒ, [E] and x do not allow B to be negative. We conclude that the constraint surface is regular (smooth) in the physical region (52). The topology of the constraint surface can be found in the shortest way, if we consider C 1 as a function C 1 (B) of B keeping all other variables constant. Studying Eqs. (54) and (58) we can easily see that C 1 (B) decreases in the interval (−∞, −M 2 /2) from ∞ to 9p 2 , it increases in the interval (−M 2 /2), M 2 /2), from 9p 2 to 9p 2 + 2M 6 and it decreases again in the interval (M 2 /2, ∞) from 9p 2 +2M 6 to −∞. It follows that there is only one solution for p = 0 (because C 1 (B) = 9p 2 for B = M 2 ). As C 1 (M 2 ) = 9p 2 > 0, the solution must satisfy the inequality B > M 2 . This solution depends smoothly on the values of all other variables. For p = 0, there are, however, two solutions: one with B = M 2 , which is a continuous extension of the previous p = 0 case, and a new one with B = −M 2 /2, which appears when the local minimum at −M 2 /2 touches the B-axis. This shows that the constraint surface consists of two components: one is a seven-dimensional smooth manifold, let us denote it by Γ 7 , that lies in the region B ≥ M 2 , and one is a six-dimensional smooth manifold, let us denote it by Γ 6 , that is defined by Eqs. (59).
Next, let us study the solvability of the constraint equation (51) with respect to the conserved momenta. The unique solution of the constraint (51) with respect to B at Γ 7 is given by where f (p, M) := 1 2 3 9p 2 + M 6 + 3p 9p 2 + 2M 6 + 1 The equation (51) has, in fact, three independent solutions, but only one of them is real. For p = 0, B reaches its minimum M 2 on Γ 7 ; thus, at Γ 7 , Given B, one can solve uniquely forĒ and there is also a unique solution for [E] 2 . The solution forĒ isĒ Similarly, we can find the solution of the constraint (51) with respect to M. Consider C 1 as a function and then there is only one non-negative solution. A reasonable solution is, therefore, unique, it depends continuously of the other variables and it lies at Γ 7 . We can write it in the form This solution can, of course, be also expressed by means of square and third roots, but only if one employs complex numbers. We can see that squaring of M introduced non-physical solutions with negative M; we have to choose the positive branch. Observe that all points at the surface Γ 7 must satisfy the inequalities (62) and (64). The equations of motion that follow from the action (53) can be written as followṡ It is easy to derive the radial equation from them. On Γ 6 , we have only a static solutions: x =Ṫ =ṗ =Ṫ 2 =Ṫ 3 = 0.
Using the constraint (51) to exclude p, we obtain the radial equation on Γ 7 : Let us compare it with Eq. (13). Eqs. (65) and (49) imply that whereas Eqs. (55), (49) and (14) give Hence, the radial equations (71) and (13) are equivalent, and the dynamics of the new action on Γ 7 coincides with that of the old action. Γ 6 consists of the unintentionally added unphysical solutions, at least on the phase space defined by Eqs. (18), but nothing is added in the physical region. Finally, we shall study the question of the monotonicity of the time functions T 2 , T 3 and T along the dynamical trajectories at Γ 7 . Eqs. (57) and (58) show that the right-hand sides of Eqs. (66) and (68) cannot vanish, and we have for ν 1 > 0: Hence, T 2 and T are good times and, as we have seen above, the constraint can be solved uniquelly for the corresponding momenta,Ē and M. As for T 3 , Eq. (67) implies thatṪ 3 > 0 for [E] > 0,Ṫ 3 = 0 for [E] = 0 andṪ 3 < 0 for [E] < 0. The solvability with respect to [E] is also only partial: there is a unique positive and a unique negative solution. Thus T 3 is a good time only in some special cases, as the next section will show.

VIII. SOME INTERESTING SPECIAL CASES
In this section, we shall remove some degrees of freedom and describe the resulting simpler models in terms of the new variables.

A. Dust degrees of freedom removed
Here, we remove the variables M and T and return to the system considered in Ref. [14]. It has two degrees of freedom: the black hole mass E − and the position x of the shell. To this aim, we first choose a particular value of M and demote it so to a mere parameter. In this way, a submanifold Γ 7M of Γ 7 emerges. Second, we take a quotient of Γ 7M by the T-curves (see Ref. [15] for more detail of this exclusion of a cyclic variable). The system that results has the action where C 1 is given by Eq. (54) as before, only M is a parameter now. The equations of motion comprise Eq. (65) withoutṀ = 0, as well as Eqs. (66), (67), (69) and (70).
The properties of this system are analogous to that of the original one. In particular, the constraint is regular everywhere on Γ 7M because the derivative (56) and (58) cannot both simultaneously vanish, T 2 is a good time variable and the constraint equation is uniquely solvable forĒ.

B. Black hole degree of freedom removed
Here, we demote the variable E − to a parameter and remove the corresponding cyclic variable T 2 /2 − T 3 ; the remaining two degrees of freedom are the internal energy M and the position x of the shell. The system obtained in this way will describe the dynamics of the dust shell in the field of a fixed black hole if E − > 0 or in the Minkowski space if E − = 0. One returns so to the system considered in Ref. [13]; the procedure, in different coordinates, has been performed in Ref. [15].
To begin with, we transform variables as follows: Then, we choose a particular value for E − ; this defines the black hole mass, and formally, a six-dimensional submanifold Γ 7E of Γ 7 . Finally, we take the quotient Γ 7E /T − of Γ 7E by the T − -curves. The result is the action where C 2 = 9p 2 − (B 2 − M 2 )(2B 2 + M 2 ), and The variable T is a good time and C 2 = 0 is of course solvable for M exactly as in the general case. The variable T + 2 is, for large values of the parameter E − , not a good time, because it is a combination T 2 /2 + T 3 of T 2 that increases along all dynamical trajectories and T 3 that increases along those with E + > E − and decreases along those with E + < E − . One can show thatṪ + 2 is positive for E + − E − > 0 and negative for E + − E − < −M and that there is no surface for E − > M that would be transversal to both the dynamical trajectories and the orbits of the continuous subgroup. 3 The derivative of C 2 with respect to E + changes sign somewhere if E − is sufficiently large and so there are more that one, or no solution of the constraint with respect to E + .
There is one prominent exception: suppose that E − = 0. Then, E + − E − is positive if E + is. Hence, in the physical region (where E + > 0), T + 2 is a good time and the constraint has a unique solution for E + there, namely where f is defined by Eq. (61).
One can also remove all four variables T, M, E − and T − 2 so that only one degree of freedom, the position x of the shell remains, but it is not difficult to work out the properties of the system using the results obtained in this section, and we left it as an easy exercise to the reader.
The special case E − = 0 is also interesting, because it is a part of the boundary of the physical region given by inequalities (52) on Γ 7 . The variational principle (53) does not break down at the boundary, that is, it defines a regular dynamics there. From this, it must clearly follow that the action defines a regular dynamics even at those points of Γ 7 that do not satisfy the conditions (52). We have studied the meaning of this dynamics and of these points in Sec. II.

IX. ACKNOWLEDGMENTS
Important discussions with Karel V. Kuchař are acknowledged. This work was supported in part by the Swiss Nationalfonds, and by the Tomalla Foundation, Zurich.