4. Effective Dynamics

As discussed in the Introduction, shell-dynamics covers a wide range of physical situations, from gravitational collapse into black holes, to wormholes and inflationary bubbles. Several analytical and graphical methods have been proposed to deal with the effective dynamics of a shell[13]. From our present perspective, the essential idea and technical steps can be summarized thus: let us rewrite the Hamiltonian constraint, taking explicitly into account the sign multiplicity of the $\beta$ functions,

$$R \left( \sigma _{in} \beta _{in} - \sigma _{out} \beta _{out} \right) = \kappa R^2 \quad .$$ (44)

Squaring this expression, we obtain

$$\sigma _{in} = {\rm Sgn} \left[ \kappa \left( A _{in} - A _{out} + \kappa ^2 R^2 \right) \right]$$ (45)

$$\sigma _{out} = {\rm Sgn} \left[ \kappa \left( A _{in} - A _{out} - \kappa ^2 R^2 \right) \right] \quad .$$ (46)

These relations determine the signs of the $\beta$ functions along the shell trajectory.

Squaring twice the constraint (44), one arrives at a simple expression for the equation of motion of the shell

$$\dot{R} ^2 + V(R) = 0 \quad ,$$ (47)

where

 

$$- \frac{ \left( A _{in} - A _{out} - \kappa ^2 R^2 \right) ^2 } { 4 \kappa ^2 R^2 } + A _{out}$$ V(R) =

$$- \frac{ \left( A _{in} - A _{out} + \kappa ^2 R^2 \right) ^2 } { 4 \kappa ^2 R^2 } + A _{in}$$ =

$$- \frac{ \left[ \left( A _{in} + A _{out} - \kappa ^2 R^2 \right) ^2 - 4 A _{in} A _{out} \right] } { 4 \kappa ^2 R^2 } \quad .$$ = (48)

Remarkably, the radial evolution of the shell is governed by an equation which is equivalent to the classical energy equation of a unit mass particle moving in a potential V(R) with zero energy: classically allowed paths require $\dot{R} ^2 > 0$, i.e. V(R) < 0. Turning points correspond to $\dot{R} = 0$, i.e., V(R) = 0. When V(R) > 0 one has $\dot{R} ^2 < 0$, and one speaks of a forbidden (or Euclidean) trajectory. Thus, using eq.(47), one can analyze the motion of the shell by specifying, case by case, the interior and exterior geometry (A_in, A_out), as well as the shell matter content ($\rho (R)$) [13]. The analysis is facilitated by plotting, on an energy diagram, the potential curve V(R), together with the horizon curves (A=0) and the curves corresponding to the vanishing of the $\beta$ functions. Presently, we simply make some general observations following from the analysis of eqs.(45-48):

1.

turning points (i.e. zeros of the potential) exist only if A_in > 0 and A_out > 0;

2.

if A>0, $\beta$ can change sign only along a forbidden path; if A<0, $\beta$ can change sign along a classical path.

Furthermore, in the course of the above analysis, one must keep in mind that the equations of motion were obtained by considering trajectories which lie in regions where both Killing vectors $\partial _T$ and $\partial _t$ are time-like (i.e., A _in > 0 and A _out > 0). When the above conditions are not satisfied, our choice of integration volume is no longer appropriate, since the time variable (t or T) becomes a space-like coordinate, and does not have a regular behavior at the horizons (i.e., when A=0). With hindsight, however, since our effective action correctly leads to Israel's equation of motion H=0, and makes no reference to the integration volume, we assume that the expression for S _eff is valid for every shell trajectory when supplemented by a working prescription suggested in Ref.[7]: when one of the factor A becomes negative, i.e. when $\dot{R}/\beta > 1$, the replacement $\tanh ^{-1} ( \dot{R}/\beta ) \rightarrow \tanh ^{-1} ( \beta/\dot{R})$ should be understood in eq.(33).

The dynamics of the shell can also be analyzed directly in terms of the Hamiltonian (37)

$$H = - \frac{R}{G_N} \left( \beta _{in} - \beta _{out} - \kappa N R \right) \quad .$$ (49)

However, the usefulness of the above expression is limited by the fact that it is given in implicit form, i.e., it contains the velocity $\dot{R}$ which, in turn, is a function of the phase space variables (R,P_R) obtained by inverting eq.(36)

$$P_R = - \frac{R}{G_N} \left[ \tanh ^{-1} \left( \frac{\dot{R}}{\beta} \right) \right \rceil ^{in} _{out} \quad .$$ (50)

Therefore, our immediate task is to recast the Hamiltonian in a form which is more amenable to physical applications, an example of which will be discussed at the end of this section.

>From equation (50), it follows

$$\frac{\partial P_R}{\partial \dot{R}} = - \frac{R}{G_N} \le... ...) = \frac{\kappa N R^2}{G_N \beta _{in} \beta _{out}} \quad ,$$ (51)

where the Hamiltonian constraint has been used. If $\beta$ changes sign, then P_R is not a monotonic function of $\dot{R}$, and its inverse is defined only in those intervals in which eq.(51) is either positive or negative. As shown before, $\beta$ can change sign along a classical path when A<0 (in this case $\tanh ^{-1} (\beta / \dot{R})$ vanishes), or on a forbidden path, when A>0. In a forbidden region, $\dot{R}$ is imaginary, in which case we define the Euclidean momentum by analytic continuation of eq.(50)

$$P_R^E = - \frac{R}{G_N} \left[ \tan ^{-1} \left( \frac{\dot{R} _E}{\beta _E} \right) \right \rceil ^{in} _{out} \quad ,$$ (52)

with

$$\frac{\pi}{2} < \arctan \left( \frac{\dot{R}_E}{\beta _E} \right) < \frac{\pi}{2} \quad ,$$ (53)

and $\beta _E \equiv \sqrt{N^2 A - \dot{R} ^2 _E}$. Evidently, the above definition is not unique. For instance, the choice of the interval $[-\pi , 0]$ (as in ref.[7]), leads to a non-vanishing P_E at turning points when the $\beta$'s have opposite sign there. On the other hand, our choice (53) has the disadvantage that P_E is not continuous when $\beta$ changes sign, i.e. $\tanh ^{-1} (\beta / \dot{R}) \rightarrow \pi/2$ for $\beta _E \rightarrow 0^+$, whereas $\tanh ^{-1} (\beta / \dot{R}) \rightarrow - \pi/2$ for $\beta _E \rightarrow 0^-$. In terms of spacetime diagrams, the vanishing of $\beta _E$ corresponds to an Euclidean trajectory which jumps from a region of the Penrose diagram where the normal to the shell points, say, towards increasing values of r, to a region where the normal points toward decreasing values of r. Therefore, we will limit our considerations only to those systems for which the $\beta$ functions have a definite sign. Presumably, some singular cases such as the quantum mechanical nucleation of wormholes, which involve Euclidean trajectories along which $\beta$ changes sign, may be dealt with in terms of pseudo-manifolds [7], or degenerate vierbeins [14].

Coming back to eq.(50), we can rewrite it as

$$\cosh \left( \frac{G_N P_R}{R} \right) = \frac{\beta _{in} \beta _{out} - \dot{R} ^2}{N^2 \sqrt{A _{in} A _{out}}} \quad .$$ (54)

Then, combining this equation with the implicit form of the Hamiltonian (49), one can eliminate the $\beta$'s and $\dot{R}$, obtaining finally the promised form of the Hamiltonian

$$H = \kappa N R^2 - \frac{NR}{G_N} \left[ A _{in} + A _{ou... ...\left( \frac{G_N P}{R} \right) \right] ^{\frac{1}{2}} \quad .$$ (55)

This expression actually corresponds to the case A_in > 0, A_out > 0. One can proceed in a similar fashion in the other cases. A more complete account of the results is given in Appendix B. Here, however, just to give a sense of the applicability of this Hamiltonian formulation, we mention a particular form of the Hamiltonian (55) which has already appeared in the literature [4]. It corresponds to a shell of dust (p=0, i.e., $\rho \sim R^{-2}$) separating a Minkowski (interior) spacetime ( A _in = 1, $\sigma _{in} = +1$), from a Schwarzschild (exterior) one ( A _out = 1 - 2 G_N M_out/R, $\sigma _{out} = +1$). In this case, the matching equation becomes (in the gauge N=1)

$$\frac{R}{G_N} \left( \beta _{in} - \beta _{out} \right) = m \quad ,$$ (56)

where m is a constant representing the rest mass of the shell. In the light of our present formulation, one recognizes the l.h.s. of eq.(56) as the Hamiltonian of the shell,

$$H = \frac{R}{G_N} \left[ 2 - \frac{2 G_N M_{out}}{R} - 2 \... ...eft( \frac{G_N P_R}{R} \right) \right] ^{\frac{1}{2}} \quad .$$ (57)

This expression coincides, up to a constant term and a sign, with the special form that our eq.(55) takes under the above hypotheses ( A _in = 1, A _out = 1 - 2 G_N M_out/R, $\sigma _{in} = \sigma _{out} = +1$, $\rho \sim R^{-2}$, N=1), and leads to the same dynamics. In Ref.([5]), however, a rather different, and equally arbitrary construction is performed. In this new interpretation of the model, (see also ref.[9,11]), after squaring eq.(56), one identifies M_out, instead of m, as the numerical value of the shell proper Hamiltonian which now reads

$$H = m \cosh \left( \frac{P_R}{m} \right) - \frac{G_N M^2_{out}}{2 R} \quad .$$ (58)

Note the inequivalence of the two effective Hamiltonians (57) and (58) which underscores the arbitrariness of the above constructions in the absence of a coherent and unified approach to shell dynamics.