2. The Reduced Action.

The dynamics of a generic system containing matter fields interacting with gravity is encoded in the Einstein-Hilbert action

 

S = S_g + S_m + S_B

$$\frac{1}{16 \pi G_N} \int _V d^4x \sqrt{g} {\cal R} ^{(4)} + \int... ...g}\, {\cal L} _m + \frac{1}{8 \pi G_N} \int _B d^3x \sqrt{h}\, {\cal K} \quad .$$ = (3)

The notation is as follows: G_N stands for Newton's constant, g is the determinant of the four dimensional spacetime metric, ${\cal R}^{(4)}$ is the corresponding Ricci scalar, ${\cal L}_{m}$ is the matter field Lagrangian density, ${\cal K}$ is the extrinsic curvature of the three dimensional boundary, (B), of the four-dimensional region V of integration, and h is the determinant of the three-dimensional metric on the boundary. The presence of the surface term allows the field equations to be obtained from a variational principle in such a way that only the metric on the boundary is held fixed.

The actual physical system under consideration consists of two static, spherically symmetric, spacetimes ${\cal M}_1$ and ${\cal M}_2$ glued together along a time-like manifold $\Sigma$ which represents the world history of the shell.

Our choice for ${\cal L}_{m}$ is

$${\cal L}_{m} = \Lambda _{in}$$ (4)

in ${\cal M}_1$,

$${\cal L}_{m} = \Lambda _{out}$$ (5)

in ${\cal M}_2$, and

$${\cal L}_{m} = \delta ( \Sigma ) {\cal L} _{sh}$$ (6)

along the shell, where

$${\cal L} _{sh} = - \rho \quad .$$ (7)

As noted in Section I, $\Lambda _{in}$ and $\Lambda _{out}$ are the two cosmological constants, and $\rho$ is the shell energy density. Furthermore, one could easily extend the content of the action integral by adding the electromagnetic term $- ( 16 \pi ) ^{-1} F _{\mu \nu} F ^{\mu \nu}$ to ${\cal L}_{m}$. This addition would lead to the Reissner-Nordström-de Sitter solution in ${\cal M}_1$ and ${\cal M}_2$. However, we shall not consider this case here, as its discussion would only obscure the simplicity of the approach that we wish to illustrate.

Solving Einstein's equations in ${\cal M}_1$, we obtain the Schwarzschild-de Sitter solution

$$ds _{1} ^{2} = - A _{in} d T ^{2} + \frac{1}{A _{in}} d r ^{2} + r ^{2} d \Omega ^{2} \quad ,$$ (8)

where

$$A _{in} = 1 - \frac{2 M _{in}}{r} + \frac{\Lambda _{in}}{3} r ^{2}$$ (9)

and T, r, $\vartheta$, $\varphi$ are the usual Schwarzschild coordinates. Similarly, in ${\cal M}_2$

$$ds _{2} ^{2} = - A _{out} d t ^{2} + \frac{1}{A _{out}} d r ^{2} + r ^{2} d \Omega ^{2} \quad ,$$ (10)

where

$$A _{out} = 1 - \frac{2 M _{out}}{r} + \frac{\Lambda _{out}}{3} r ^{2} \quad .$$ (11)

The evolution of the shell, against this fixed background, is described by the radius $R(\tau )$ and by the lapse $N(\tau)$ which characterize the shell intrinsic geometry according to equation (2). This dynamics can be obtained by reducing the action (3) to a functional depending only on $N(\tau)$ and $R(\tau )$. Our procedure of reduction follows closely the discussion of ref.[7] (see also ref.[6]); one element of novelty consists in the inclusion of the lapse function $N(\tau)$, which is essential to derive the Hamiltonian constraint that one expects in a reparametrization invariant theory.

To begin with, let us specify the volume V of integration. As shown in fig.1, the volume is bounded by two space-like surfaces T = T _i and T = T _f, and by two time-like surfaces: r = R ₁ and $r = R ( \tau ) > R _{1}$ in ${\mathcal{M}} _{1}$, and by t = t _i, t = t _f, $r = R (\tau)$ and $r = R _{2} > R (\tau)$ in ${\cal M}_2$. T₁, t₁, T₂, t₂, R₁, R₂ are constants. Furthermore, we assume, for the moment, that V lies in a region of spacetime where the Killing vectors of the metric (8-10), $\partial _T$ and $\partial _t$ respectively, are both time-like. Later on, we shall give a supplementary rule for the case in which the above condition is not satisfied.

  

Figure: Graphical representation of the integration volume with two dimensions suppressed: T_i (T_f) and t_i (t_f) label the initial (final) surfaces which are the space-like boundaries of the integration volume. R₁ and R₂ represent the time-like boundaries, and $R(\tau )$ parametrizes the shell radius which separates the interior domain from the exterior one.

Evaluating the induced metric on both side of the shell, and comparing with eq.(2), we find the implicit dependence of the Schwarzschild time variables on the parameter $\tau$,

$$\frac{dt}{d\tau} = \dot{t} = \frac{\beta _{out}}{A _{out}} \quad ,$$ (12)

where

$$\beta _{out} = \sqrt{N^2 A _{out} + \dot{R}^2}$$ (13)

and

$$\frac{dT}{d\tau} = \dot{T} = \frac{\beta _{in}}{A _{in}} \quad ,$$ (14)

where

$$\beta _{in} = \sqrt{N^2 A _{in} + \dot{R}^2} \quad .$$ (15)

Note that the $\beta$-functions defined above contain a sign ambiguity[7], which is resolved by requiring that $\beta$ be positive (negative) if the outer normal to $\Sigma$ points toward increasing (decreasing) values of r.

Our immediate objective, for the remainder of this section, is to obtain the reduced form of the action integral, and we shall do so by discussing each individual term separately. The explicit form of $S^{\Sigma}_m$, the contribution of the shell to S_m, is given by

$$S_m^{\Sigma} = 4 \pi \int _{\tau _i} ^{\tau ^f} \rho \left( R \left( \tau \right) \right) R(\tau) ^2 d \tau \quad$$ (16)

supplemented by an equation of state relating the energy density $\rho$ to the tangential pressure p. Such a relation is usually obtained from the conservation equation for the shell stress-energy tensor,

$$\dot{\rho} = - 2 ( \rho - p ) \frac{\dot{R}}{R} \quad ,$$ (17)

and we shall discuss some simple special cases of this equation later on in the text.

Next, in order to calculate $S^{\Sigma}_g$, the contribution of the shell to S_g in the action integral, we first introduce Gaussian normal coordinates near the shell

$$ds^2 = g _{\tau \tau } \left( \tau, \eta \right) d \tau ^2 + d \eta ^2 + r \left( \tau, \eta \right) ^2 d \Omega ^2 \quad .$$ (18)

As usual, the triplet ($\tau$, $\vartheta$, $\varphi$) specifies a point on the shell, while $\eta$ represents the geodesic distance off the shell. One has $\eta > 0$ in ${\cal M}_2$, and $\eta < 0$ in ${\cal M}_1$. Furthermore,

$$g _{\tau \tau } \left( \tau, 0 \right) - N(\tau) ^2$$ = (19)

$$r \left( \tau, 0 \right) R( \tau ) \quad .$$ = (20)

One can now start the evaluation of $S^{\Sigma}_g$:

$$S^{\Sigma}_g = \frac{1}{16 \pi G_N} \int _{- \epsilon} ^{+ \... ...\int d \vartheta d \varphi \sqrt{g}\, {\cal R} ^{(4)} \quad .$$ (21)

Here $\epsilon$ is an arbitrary small positive number, which at the end of the calculation is set to zero. The Ricci scalar ${\cal R}^{(4)}$ can be expressed as

$${\cal R} ^{(4)} = {\cal R} ^{(3)} - \left( K _{ij} K ^{ij} + K^2 \right) - 2 \frac{\partial K}{\partial \eta} \quad$$ (22)

where K_ij stands for the extrinsic curvature of the hyper-surface of constant $\eta$, and ${\cal R}^{(3)}$ is the Ricci scalar constructed from the three-dimensional metric g _ij on the hyper-surface of constant $\eta$. The relation between K _ij and the normal derivative of the three-metric is simply

$$K _{ij} = \frac{1}{2} \frac{\partial g _{ij}}{\partial \eta} \quad .$$ (23)

Evaluating these quantities, one eventually obtains

$$K _{\vartheta \vartheta} = \frac{K _{\varphi \varphi}}{\sin ^2 \vartheta} = \frac{R \beta _{in}}{N} \quad \mathrm{for} \quad \eta = - \epsilon$$ (24)

$$K _{\vartheta \vartheta} = \frac{K _{\varphi \varphi}}{\sin ^2 \vartheta} = \frac{R \beta _{out}}{N} \quad \mathrm{for} \quad \eta = + \epsilon$$ (25)

and

$$K _{\tau \tau} = - 2 \frac{\ddot{R}N}{\beta _{in}} - \frac{N^2}{\beta _{in}} \frac{d A _{in}}{d R} + 2 \frac{\dot{R}\dot{N}}{\beta _{in}} \quad \mathrm{for} \quad \eta = - \epsilon$$ (26)

$$K _{\tau \tau} = - 2 \frac{\ddot{R}N}{\beta _{out}} - \frac{N^2}{\beta _{out}} \frac{d A _{out}}{d R} + 2 \frac{\dot{R}\dot{N}}{\beta _{out}} \quad \mathrm{for} \quad \eta = + \epsilon .$$ (27)

All of the above leads to the following expression of $S^{\Sigma}_g$

$$S^{\Sigma}_g = \frac{1}{2 G_N} \int _{\tau _i} ^{\tau _f} d ... ...ot{R} \dot{N}}{N} \right) \right \rceil ^{in} _{out} \quad .$$ (28)

In eq.(28) the notation $[ \dots \rceil ^{in} _{out}$ means, as usual, $[ \dots ] \rceil _{in} - [ \dots ] \rceil _{out}$.

The contributions of ${\cal M}_1$ and ${\cal M}_2$ to S_g, which we denote by S¹_g and S²_g respectively, are easily evaluated since

$${\cal R} ^{(4)} _{in/out} = 4 \Lambda _{in/out} \quad .$$ (29)

This reduces the form of $S^{1\,(2)}_g$ to a volume integral of the same type as $S^{1\,(2)}_m$.

Finally, the surface term in the action integral, which eliminates the second derivative term $\ddot{R}$ present in eq.(28), can be written as

$$S_B = \frac{1}{8 \pi G} \int d^3x \sqrt{h} \nabla _{\mu} n ^{\mu} \quad ,$$ (30)

where $n ^{\mu}$ is the unit normal to the boundary three-surface, and h is the determinant of the metric of the three-surface. Again, following the same steps as in ref.[7], one obtains

$$S^{\Sigma}_B - \frac{1}{2 G_N} \int _{\tau _i} ^{\tau _f}d\tau\, {d\over d\tau... ... ^{-1} \left( \frac{\dot{R}}{\beta} \right) \right \rceil ^{in} _{out} \right\}$$ =

$$- \frac{1}{2 G_N} \left[ R^2 \dot{R} \tanh ^{-1} \left( \frac{\dot{R}}{\beta} \right) \right. +$$ =

$$\qquad \qquad \qquad \left . + \frac{R^2 }{ A} \left( \frac{A \dd... ...- \frac{ A \dot{R} \dot{N}} {N\beta} \right) \right \rceil ^{in} _{out} \quad .$$ (31)

The end result of the above calculations is the following expression of the action integral

$$S = \int ^{\tau _f} _{\tau _i} L d \tau + \quad\mathrm{boundary terms} \quad ,$$ (32)

where

$$L = \left[ - \frac{R \dot{R}}{G_N} \tanh ^{-1} \left( \frac... ...R}{G_N} \left( \beta _{in} - \beta _{out} - \kappa R N \right)$$ (33)

and $\kappa \equiv 4 \pi \rho G_N$. The additional ``boundary terms'' in eq.(32) collectively refer to contributions which are simply proportional to $\int ^{t_2} _{t_1} dt$ or $\int ^{T_2} _{T_1} dt$. Their explicit form is irrelevant for our purposes. Indeed, since the intrinsic dynamics of the shell depends only on the interior and exterior geometry, both of which are fixed, and not on the spacetime volume V chosen, we define the shell effective action by subtracting the boundary terms in eq.(32) [6]. Therefore, the promised form of the reduced, or effective, action is given by

$$S _{eff} = \int ^{\tau _f} _{\tau _i} L d \tau \quad .$$ (34)

As a consistency check, in the next section we shall verify that varying this reduced action yields the Israel equation of motion (1) for the shell.