8. The FGG-method and the Hamiltonian constraint

In this appendix we wish to discuss the relationship between our variational procedure and the FGG-approach. In this connection, it seems worth emphasizing that our equations of motion were obtained by extremizing the action S_eff with respect to arbitrary variations of the functions $N(\tau)$ and $R(\tau )$, vanishing at the end points, while the temporal boundary was held fixed. This variational procedure is consistent with our minisuperspace approach to the intrinsic dynamics of the shell, according to which there should be no reference to the internal, or external time coordinate, or to the particular volume of integration chosen, since the latter is just a convenient device to arrive at the equations of motion for the shell. This represents a departure from the FGG-method[7], which is based on a different variational procedure. In that approach, the evolution of the shell is parametrized by the external time, so that it seems natural to consider variations in which the initial and final external time t_i and t_f are held fixed. This, however, means that $\tau _f$ is not fixed, but varies according to

$$\delta \tau _f = \frac{ \int ^{\tau _f} _{\tau _i} \left\... ...ot{R} \frac{\partial F}{\partial \dot{R}}\right) _f } \quad .$$ (100)

This equation is obtained by taking the variation of eq.(12)

$$t_f - t_i = \int ^{\tau _f} _{\tau _i} F \left( R, \dot{R}, N\right) d \tau \quad ,$$ (101)

where

$$F \equiv \frac{\beta _{out}}{A _{out}} = \frac{\sqrt{N^2 A _{out} + \dot{R} ^2}}{A _{out}} \quad .$$ (102)

Since now $\tau _f$ is not fixed, the variation of S _eff yelds additional contributions

 

$$\delta S \int ^{\tau _f} _{\tau _i} \left\{ \left[ \frac{\partial L}{\part... ...elta R(\tau) + \frac{\partial L}{\partial N} \delta N (\tau ) \right\} d \tau +$$ =

$$\qquad \qquad \qquad \qquad + L \left( \tau _f \right) \delta \tau _f + \left( \frac{\partial L}{\partial \dot{R}} \right) _f \delta R_f \quad ,$$ (103)

where

$$\delta R_f = - \dot{R} \left( \tau _f \right) \delta \tau _f \quad .$$ (104)

Inserting eqs.(100) and (104) into eq.(103), we obtain

$$\delta S \int ^{\tau _f} _{\tau _i} \left\{ \left[ \frac{\partial L}{\part... ...( \frac{\partial f}{\partial \dot{R}} \right) \right] \delta R(\tau) \right . +$$ =

$$\qquad \qquad \qquad + \left . \left( \frac{\partial L}{\partial ... ... \frac{\partial F}{\partial N} \right) \delta N (\tau ) \right\} d \tau \quad ,$$ (105)

where

$$\xi _f = \left( F - \dot{R} \frac{\partial F}{\partial \dot{R}} \right) _f$$ (106)

and $H_f \equiv H (\tau _f)$. Demanding that $\delta S$ vanishes, results now in the following equations of motion

$$\frac{\partial L}{\partial N} - \frac{H_f}{\xi _f} \frac{\partial F}{\partial N} = 0$$ (107)

$$\frac{\partial L}{\partial R} - \frac{d}{d \tau} \left( \... ...\partial F}{\partial \dot{R}} \right) \right] = 0 \quad .$$ (108)

Now, Eq.(107) does not include an acceleration term; again, it represents a constraint which can be rewritten as

$$\frac{H}{N} + \frac{H_f}{\xi _f} \frac{\partial F}{\partial N} = 0 \quad .$$ (109)

Eq.(108), on the other hand, gives

$$\frac{d}{d\tau} \left( \frac{H}{N} \right) - \frac{H_f}{\x... ...\partial F}{\partial \dot{R}} \right) \right] = 0 \quad ,$$ (110)

so that the solution of both equations (109-110) is H=0. In fact, setting H=0 in eq.(109) we find H_f = 0, which, once inserted in eq.(110), implies that the Hamiltonian constraint H=0 is preserved in time, which is our result.