
\documentstyle[czech,a4,12pt]{article}


\title{Diskrétní lineariární aproximace rovinného volnì visícího lana}



\date{30.5.2001}

%\baselineskip=18pt
%\textheight=23cm
%\textwidth=15cm

%\baselinestretch{1.3}

%\newcommand{\xc}{\mathord{\mathop{x}\limits_{(c)}}}
%\newcommand{\hT}{\mathord{\mathop{h}\limits_{T}}}




\begin{document}

\maketitle


\section{Zadání}

Naleznìte základní vlastní frekvenci dvou linearizovaných systémù tvoøených $N$ stejnými
hmotnými tyèkami resp. hmotnými body a nehmotnými závìsy jež pro
velká $N$ aproximují v rovinì se pohybující za jeden konec
povìšené homogenní lano.

\section{Poznámky}

Oznaèíme-li $\psi_i$ úhel, který svírá $i$-tá tyèka se svislicí, je
poloha jejího konce dána vztahem
$$\vec {x_i} =a\sum_{k=1}^i[\sin(\psi_i),\cos(\psi_i)].$$
Sestavte lagrangián a linearizací jej uveïte do tvaru
$$L =  A_{ij} ~\dot{\psi_i}~\dot{\psi_j} ~-~  B_{ij}~ {\psi_i}~{\psi_j}.$$
Protože pohybové rovnice mají tvar
$$  A_{ij} ~\ddot{\psi_j}~ +~  B_{ij} ~{\psi_j} = 0,$$
mùžeme z vlastních hodnot matice ${\bf B}^{-1}{\bf A}$ urèit
(pøevrácenou hodnotu kvadrátu) vlastní frekvence. Vlastní vektor
pak urèuje tvar stojaté vlny.

Pøi sestavování lagrangiánu mùžete využít vztah
$T = m/2 [\vec u . \vec v + 1/3 (\vec u - \vec v )^2]$
urèující kinetickou energii homogenní tyèky hmotnosti $m$ jejíž konce se pohybují rychlostmi
$\vec u$ a $\vec v$.




\end{document}



We shall start from the geometrodynamical approach to a spherically
symmetric geometry as described in \cite{KK}. Up to very few exceptions (noted in
the following) we adopt the same notation and natural units
$G=c=1$.

In geometrodynamics, a spherically symmetric spacetime is
generated by evolving a spherically symmetric three-dimensional
Riemannian space $\Sigma$ with the line element described by two
functions, $\Lambda (r)$ and $R(r)$, of a radial variable $r$:
\begin{equation}\label{1}
d \sigma ^2 = \Lambda ^2 (r) dr^2 + R^2(r) d \Omega ^2.
\end{equation}
Here $d\Omega ^2 = d \theta ^2 + \sin ^2 \theta d \varphi ^2$,
$R(r)$ is the curvature radius of the $2$-sphere $r=$ constant; under
transformations of $r$, $R(r)$ behaves as a scalar, $\Lambda (r)$
as a scalar density. The range of the radial variable $r$ depends
on the problem we wish to discuss. If null dust falls into a
primordial Schwarzschild black hole (which thus existed before an
``irradiation'' by dust), variable $r\in (-\infty, +\infty)$
covers both (``right'' and ``left'') infinities, whereas if an
infalling ``ball'' of null dust creates a Schwarzschild black
hole, the point $r = 0$ is the center of the ball, and $r \in <0,
+\infty)$. As discussed in the Introduction, our main interest will
be in treating the slab of null dust falling in or out with
Minkowski space inside. Then again $r \in <0, +\infty).$
Clearly, appropriate boundary conditions must be formulated and
boundary terms analyzed depending on the physical situation
considered.

If a spherically symmetric spacetime is foliated by spherically
symmetric leaves $\Sigma$, labelled by a time parameter $t\in
I\!R$, the metric functions $\Lambda$ and $R$ depend also on $t$.
The leaves are related by the lapse function $N(r,t)$, and by the
only non-vanishing component of the shift vector $N^r (r,t)$.
Modulo boundary terms, the \textit{vacuum} dynamics of the
spherically symmetric gravitational field follows from the
canonical form of the ADM action, integrated over $\theta$ and
$\varphi$. It reads (cf. \cite{KK}, Eqs. (31)-(33)):
\begin{equation}\label{2}
S^G = \int dt \int dr (P_\Lambda \dot{\Lambda} + P_R \dot{R}
-NH^G - N^r H_r^G),
\end{equation}
where the gravitational super-Hamiltonian
\begin{equation}\label{3}
H^G = R^{-1}P_RP_\Lambda +\frac{1}{2} R^{-2} \Lambda P^2_{\Lambda}
+ \Lambda ^{-1} RR'' - \Lambda ^{-2}RR' \Lambda ' + \frac{1}{2}\Lambda ^{-1} R'^{2}
- \frac{1}{2} \Lambda,
\end{equation}
and supermomentum
\begin{equation}\label{4}
H^G_r = P_RR' - \Lambda P'_\Lambda,
\end{equation}
the dots and primes denoting the derivatives with respect to
variables $t$ and $r$. The momenta, $P_R$ and $P_\Lambda$,
obtained by differentiating the ADM spacetime action with respect
to the velocities $\dot{\Lambda}$ and $\dot{R}$, read
\begin{equation}\label{5}
P_\Lambda = -N^{-1} R (\dot{R} - R' N^r),
\end{equation}
\begin{equation}\label{6}
P_R = - N^{-1} \left[\Lambda(\dot{R} - R'
N^r)+R(\dot{\Lambda}-(\Lambda N^r)') \right].
\end{equation}
Inversely,
\begin{equation}\label{7}
\dot{\Lambda} = -NR^{-2}(RP_{R} - \Lambda P_{\Lambda}) + (\Lambda
N^r)',
\end{equation}
\begin{equation}\label{8}
\dot{R} = -NR^{-1} P_{\Lambda} + R'N^r.
\end{equation}

A general spacetime action for null dust has the form \cite{BiKu}
\begin{equation}\label{9}
S^{ND} = \int dt \int d^3x L^{ND},
\end{equation}
where the Lagrangian is given by
\begin{equation}\label{10}
L^{ND} = -\frac{1}{8\pi} \sqrt{|g|} g^{\alpha\beta}\rho k_\alpha
k_\beta,
\end{equation}
with
\begin{equation}\label{11}
k_\alpha = W_iZ^i_{,\alpha},
\end{equation}
$i=1,2,3$ is a summation index.
We inserted the factor $(4\pi)^{-1}$ into (10) as compared with
the general action introduced in \cite{BiKu}, so as to obtain
simplest expressions after the integration over $\theta$ and
$\varphi$ is made. Radiation density $\rho$ (denoted
by $M$ in \cite{BiKu}) is, in any case, not defined uniquely unless
a preferred unique normalization of the null vector field is
chosen (see \cite{BiKu}, for details). Equations of motion of null
dust follow from varying the action (9) - (11) with respect to
scalars $\rho$, $Z^i$, and $W_i$. As compared with a general case,
the Vaidya spherically symmetric metric is much simpler. The
incoming null vector field $k_\alpha$ is hypersurface orthogonal,
and we thus put
\begin{equation}\label{12}
k_\alpha = - \Phi_{,\alpha},
\end{equation}
where scalar function $\Phi = \Phi(r,t)$ is the only remaining
non-trivial variable $Z^i$. Putting the corresponding parameter
$W=-1$, we have chosen an affine parametrization along the rays to
which $k_\alpha$ is tangent \cite{BiKu}. Assuming the radiation
density to depend on $r$ and $t$ only, $\rho = \rho (r,t)$, we can
integrate over $\theta$ and $\varphi$ in the action (9), (10),
(12), arriving at the reduced action of the form
\begin{equation}\label{13}
S^{ND} = \int dt \int dr L^{ND},
\end{equation}
where
\begin{eqnarray}\label{14}
L^{ND} = -\frac{1}{2}\Lambda R^2 \rho \left[
N\Lambda^{-2}(\Phi')^2 - N^{-1} (\dot{\Phi}^2 - 2N^r \dot{\Phi}\Phi' + (N^r\Phi ')^2) \right
].
\end{eqnarray}
Define the momentum of the null dust by $\Pi = \delta L^{ND}/\delta
\dot{\Phi}$ so that
\begin{equation}\label{15}
\Pi = N^{-1} \Lambda R^2 \rho (\dot{\Phi} - N^r \Phi'),
\end{equation}
inversely,
\begin{equation}\label{16}
\dot{\Phi} = (\Lambda R^2 \rho)^{-1} N\Pi + N^r \Phi '.
\end{equation}
The action (13) - (14) is easily converted to the canonical
form:
\begin{equation}\label{17}
S^{ND} = \int dt \int dr (\Pi \dot{\Phi} - NH^{ND} - N^r
H_r^{ND}),
\end{equation}
where the null dust super-Hamiltonian
\begin{equation}\label{18}
H^{ND} = (2\Lambda)^{-1} \left[ \Pi ^2 (\rho R^2)^{-1} + \rho R^2
\Phi^{'2} \right],
\end{equation}
and the supermomentum
\begin{equation}\label{19}
H^{ND}_r = \Pi \Phi '.
\end{equation}
Notice that the null dust action is very similar to the action of
a massless scalar field $\widetilde{\Phi}$ with a conjugate momentum
$\widetilde{\Pi}$ (\cite{KK}, Eqs. (197) and (198)):
\begin{equation}\label{20}
H^{SF} = (2 \Lambda)^{-1} \left[ \widetilde{\Pi}^2 R^{-2} + R^2
\widetilde{\Phi}^{'2}\right],
\end{equation}
\begin{equation}\label{21}
H_r^{SF} = \widetilde{\Pi}\widetilde{\Phi}'.
\end{equation}
In the case of null dust, however, the super-Hamiltonian can be
simplified by excluding the scalar function $\rho$ which plays
a role of a Lagrange multiplier in (17). From the condition
\begin{equation}\label{22}
\frac{\delta S^{ND}}{\delta \rho} = 0
\end{equation}
we find
\begin{equation}\label{23}
\rho^2 = \Pi^2 (R^2 \Phi')^{-2},
\end{equation}
so that
\begin{equation}\label{24}
\rho = R^{-2}\left|\frac{\Pi}{\Phi'}\right|,
\end{equation}
because we require $\rho$ to be positive. Substituting for $\rho$
back to the super-Hamiltonian (18), we arrive at a simple
expression for the null dust super-Hamiltonian
\begin{equation}\label{25}
H^{ND} = \Lambda ^{-1} |\Pi \Phi '|.
\end{equation}
Such a result was to be expected since even in a general case of a
null dust the super-Hamiltonian can be expressed in terms of
supermomenta as follows \cite{BiKu}:
\begin{equation}\label{26}
H^{ND} = (g^{ab} H^{ND}_a H^{ND}_b)^\frac{1}{2}.
\end{equation}
Regarding (1), (19) and (25), we verify that the general
relation (26) is indeed satisfied.

The total super-Hamiltonian and supermomentum are then (Eqs.
(3), (4), (24), (25))
\begin{eqnarray}\label{27}
H = H^G + H^{ND} = R^{-1} P_R P_\Lambda + \frac{1}{2}
R^{-2}\Lambda P^2 _\Lambda \nonumber \\
+ \Lambda ^{-1} RR'' - \Lambda ^{-2} RR'
\Lambda ' + \frac{1}{2} \Lambda ^{-1} R'^2 - \frac{1}{2} \Lambda +
\Lambda^{-1}|\Pi \Phi '|.
\end{eqnarray}
\begin{equation}\label{28}
H_r = H_r^G + H_r ^{ND} = P_R R' - \Lambda P' _\Lambda + \Pi \Phi
'.
\end{equation}

\section{Geometrodynamics of Vaidya spacetimes}

The simplest form of the Vaidya metric describing incoming null
dust is obtained by using the incoming null coordinate $V$,
together with the Schwarzscild curvature coordinate $R$ (see Eq.
(xx):
\begin{equation}\label{29}
ds^2 = -\left[ 1-\frac{2M(V)}{R} \right] dV^2 + 2dV dR + R^2d\Omega
^2
\end{equation}
Here $M(V)$- the ``advanced mass'', measured at ${\cal J}^-$ may be an
arbitrary, non-decreasing function of the ``advanced-time''
coordinate $V$.

Let us foliate a Vaidya spacetime by spherically symmetric
spacelike leaves $\Sigma$, labelling the leaves by a time parameter
$t\in I\!R$, and using a radial label $r$ on the leaves.
We can substitute the foliation
\begin{equation}\label{30}
V = V(t,r),\,\,\,\,\,\,\,\,  R = R(t,r)
\end{equation}
into the Vaidya element (x), and using the abbreviation
\begin{equation}\label{31}
F = (1 - 2M/R),
\end{equation}
we find
\begin{eqnarray}\label{32}
ds^2 = - (F \dot{V}^2 - 2 \dot{V}\dot{R}) dt^2 \nonumber \\
+ (-F \dot{V}V' +
\dot{V}R' + V'\dot{R}) 2dt dr \nonumber \\
+ (-FV'^2 + 2V'R') dr^2 + R^2 d\Omega ^2,
\end{eqnarray}
where the dots and commas denote $\partial/\partial t$ and $\partial/\partial
r$. Comparing (32) with the general ADM form of the spherically
symmetric line element,
\begin{equation}\label{33}
ds^2 = - \left[ N^2 - \Lambda ^2 (N^r)^2 \right] dt^2 + 2 \Lambda
^2 N^r dt dr + \Lambda ^2 dr^2 + R^2 d\Omega ^2 ,
\end{equation}
we can express function $\Lambda$, and the lapse and shift
functions in the form:
\begin{equation}\label{34}
\Lambda ^2 = - F V'^2 + 2V'R',
\end{equation}
\begin{equation}\label{35}
N = \frac{\dot{V} R' - V' \dot{R}}{(-FV'^2 + 2V'R')^{\frac{1}{2}}}
= \frac{\dot{V}R' - V'\dot{R}}{\Lambda},
\end{equation}
\begin{equation}\label{36}
N^r = \frac{-F\dot{V}V' + \dot{V}R' + V'\dot{R}}{-FV'^2 + 2V'R'} =
\frac{-F\dot{V}V' + \dot{V}R' + V'\dot{r}}{\Lambda ^2}.
\end{equation}

Returning now back to the expression (5) for the momentum
$P_\Lambda$, we may substitute for $\Lambda, N$, and $N^r$ to
obtain
\begin{equation}\label{37}
P_\Lambda = -R(-R' + FV') (-FV'^2 + 2V'R')^{-\frac{1}{2}}.
\end{equation}
>>From the last equation we can express $V'$ regarding (34) in the
form
\begin{equation}\label{38}
-V' = R^{-1} F^{-1} \Lambda
P_\Lambda - F^{-1}R'.
\end{equation}
Returning back to Eq. (34) we can calculate function $F$, Eq.
(31), in terms of the canonical data:
\begin{equation}\label{39}
F = \left( \frac{R'}{\Lambda} \right) ^2 - \left(
\frac{P_\Lambda}{R}\right)^2.
\end{equation}
Substituting for $F$ into Eq.(38), we obtain $V'$ entirely in
terms of the canonical data. Finally, combining Eqs. (31) and
(39) we can express the mass $M$ as a function of the canonical
data,
\begin{equation}\label{40}
M = \frac{1}{2}R^{-1}P^2_\Lambda - \frac{1}{2}\Lambda ^{-2}RR'^2 +
\frac{1}{2}R.
\end{equation}
Notice that the equations (39) and (40) are of exactly the
same form as in the vacuum Schwarzschild geometry. They can also be
interpreted in a similar way. Equation (40) enables one to learn
the mass function $M[V(r,t)]$ from the canonical data on any piece
of a spacelike hypersurface $\Sigma$. Equations (38), (39)
give the difference of the advanced times $V(r_1)$ and $V(r_2)$
between any two points of $\Sigma$.

In vacuum spherically symmetric spacetimes one can find a
canonical transformation to a new set of variables in terms of
which the constraints remarkably simplify \cite{KK}. The
transformation replaces the canonical variables $(\Lambda, P_\Lambda , R, P_R)$
by new canonical variables $(M, P_M, R, {\cal P}_R)$
where $M$ is given by Eq. (40),
\begin{equation}\label{41}
P_M = R^{-1} F^{-1} \Lambda P_\Lambda,
\end{equation}
\begin{equation}\label{42}
R = R,
\end{equation}
\begin{eqnarray}\label{43}
{\cal P}_R = P_R - \frac{1}{2}R^{-1} \Lambda P_\Lambda -
\frac{1}{2}R^{-1}F^{-1}\Lambda P_\Lambda \nonumber\\
-R^{-1}\Lambda ^{-2}F^{-1}\left[ (\Lambda P_\Lambda)' (RR')-(\Lambda
P_\Lambda)(RR')' \right],
\end{eqnarray}
where $F$ is given in terms of ``old'' canonical variables by
(39). This transformation was shown \cite{KK} to be canonical
without employing constraints or dynamical equations. We can make
easily sure that it remains a canonical transformation also on the
extended phase space which includes the null dust canonical
variables $\Phi,\Pi$.

The only point we have to check is that the boundary terms vanish
as in the case of primordial vacuum Schwarzschild black holes
analyzed in \cite{KK}. We have thus to pay attention to the
behaviour of the canonical variables $\Lambda,R,P_\Lambda,P_R$ at
the origin and at spatial infinity in the case of a slab of
radiation collapsing into Minkowski space, and at both the right
and left infinities if a slab collapses into a primordial
Schwarzschild black hole. Since slabs of radiation do not extend
to spatial infinities, there our fall-off conditions are identical
to those considered in \cite{KK}. Therefore, we assume the canonical
variables at infinities to behave as follows:
\begin{eqnarray}\label{44}
&\!&\Lambda (t,r) = 1 + M_{\pm} (t)|r|^{-1}+O (|r|^{-(1+\epsilon)}),\nonumber\\
&\!&\ R(t,r) = |r| + O (|r|^{-\epsilon}), \nonumber \\
&\!&\ P_\Lambda (t,r)= O (|r|^{-\epsilon}), \nonumber \\
&\!&\ P_R(t,r) = O (|r|^{-(1+\epsilon)}),
\end{eqnarray}
where $O(r^{-i})$ means that this term falls off as $r^{-i}$,
whereas its j-th spatial derivatives as $r^{-(i+j)}$;
the absolute values appear since in the case of primordial black
holes $r\in I\!R$. The mass term can depend on $t$, $M_+$
(resp. $M_-$) is the Schwarzschild mass at the right (resp. left)
infinity. The Schwarzschild time $T$ along a slice $\Sigma$ is
assumed to behave as
\begin{equation}\label{45}
T(r) = T_\infty (t)+ O(r^{-1}),
\end{equation}
whereas the Lagrange multipliers in the action, the lapse and
shift, satisfy
\begin{equation}\label{46}
N = N_\pm (t) + O (|r|^{-\epsilon}), \;\;\;\;\;\; N^r =
O(|r|^{-\epsilon}).
\end{equation}
Since advanced time $V$ is given in terms of canonical coordinates
by Eqs. (38),(39), the fall-off conditions (44) imply
\begin{equation}\label{47}
V = |r| + 2M_\infty \ln \left( \frac{r}{2M_\infty} \right) + T_\infty (t) +
O(r^{-\epsilon}),
\end{equation}
so that
\begin{equation}\label{48}
V' = 1 + 2M_\infty \frac{1}{|r|}+ O(|r|^{-1-\epsilon}).
\end{equation}
In case of the radiation falling into flat space only right
infinity occurs but in this case we need the boundary conditions
at the origin. At the origin of Minkowski space the boundary
conditions are the same as in \cite{HaKi}. The 4-metric is flat,
the 3-metric must be smooth. The hypersurface $\Sigma$ must meet
the origin parallelly to $T =$ constant hypersurfaces to avoid
conical singularities. Therefore, we require at $r\rightarrow 0$
\begin{equation}\label{49}
T' (r) = 0, \;\;\;\;\;\; R = r + O (r^2),
\end{equation}
\begin{equation}\label{50}
V = T + R = T_0 + r, \;\;\;\;\; V' = 1.
\end{equation}
The boundary terms, given by the expression (95) in \cite{KK},
\begin{equation}\label{51}
\mathcal{B} = \frac{1}{2} R \delta R \ln \left| \frac{RR' + \Lambda P_\Lambda}{RR' + \Lambda
P_\Lambda}\right|,
\end{equation}
evaluated at the boundaries, can also be written in terms of the
advanced time variable $V$ and function $F$ as
\begin{equation}\label{52}
\mathcal{B} = \frac{1}{2} R\delta R \ln \left|\frac{V'FR - 2 RR'}{V' FR}\right|.
\end{equation}
Regarding the boundary conditions (44) - (50), which imply
$\delta R \sim O (|r|^{-\epsilon})$ at infinity(ies) and  $\delta R \sim O(r
^2)$ at the origin, we easily see that the boundary terms vanish in
all the situations we are dealing with.


The constraints (27) and (28) acquire now much simpler form:
\begin{equation}\label{53}
  \Lambda H = - F^{-1}M'R' - FP_M \mathcal {P}_R + |\Pi\Phi'|,
\end{equation}
\begin{equation}\label{54}
H_r = \mathcal {P}_RR' + P_MM' + \Pi\Phi',
\end{equation}
where $F$ is again given by (31), but $M(r)$ and $R(r)$ are now
understood as new canonical variables. (Notice that in (53) we
scaled $H$ by $\Lambda$.) From Eq. (38) we find the advanced
time coordinate in terms of new canonical variables also in a
simplified form:
\begin{equation}\label{55}
-V' = P_M - F^{-1}R'.
\end{equation}

As we shall see below, we may assume $\Pi\Phi '>0$, and thus
delete the absolute value in (53). Subtracting then (53) from
(54), we obtain
\begin{equation}\label{56}
H_r - \Lambda H = (\mathcal{P}_R + F^{-1}M')(R' + F P_M).
\end{equation}
Since
\begin{equation}\label{57}
P_M = -(\frac{M_{,T}}{M_{,R}}F^{-1})T',
\end{equation}
where $T$ is the Schwazschild time (cf. Eq. (xx)), the
second factor in (56) is generally non-zero, $R' + FP_M \neq 0$.
Hence, the constraint equation $H_r - \Lambda H = 0$
implies
\begin{equation}\label{58}
\mathcal{P}_R + F^{-1}M' = 0.
\end{equation}
Substituting from here back into the momentum constraint $H_r =
0$, we get
\begin{equation}\label{59}
-M'V' + \Pi\Phi' = 0,
\end{equation}
where $V'$ is determined by (55).

The action can thus be written as
\begin{eqnarray}\label{81}
&\!&\ S = \int dt \int _0^\infty dr \left[  P_M \dot{M} + \mathcal{P}_R \dot{R} +
\Pi\dot{\Phi} -N^M (-M' V' + \Pi \Phi ') - N^R ({\mathcal P}_R + F^{-1} M')  \right]\nonumber\\
&\!&\ - \int dt M_\infty \dot{T}_\infty,
\end{eqnarray}
where $V'$ and $F$ are determined in terms of canonical variables
by Eqs. (55) and (31). In (60) we added the well-known
Regge-Teitelboim boundary term into the action, which acquires the
form $M_\infty \dot{T}_\infty$ after the parametrization at
infinity, as it is analyzed in detail in \cite{KK}.
%\subsection{...}
(There would be another term of this type, corresponding to the left infinity,
if null dust is collapsing into a primordial Schwarzschild black
hole.)


The action (60) can still be simplified by introducing the
advanced time variable as a canonical coordinate. We shall achieve
this in two steps. First introduce new canonical momenta conjugate
to $M$ and $R$ by
\begin{equation}\label{82}
\Pi _M = -V' = P_M - F^{-1}R',
\end{equation}
\begin{equation}\label{83}
\Pi _R = {\cal P}_R + F^{-1} M'.
\end{equation}
Substituting into the Liouville form in (60), we find
\begin{eqnarray}\label{84}
\int ^{\infty}_{0} dr \left[ P_M \delta M + {\cal P}_R \delta R + \Pi \delta
\Phi \right] = \int ^{\infty}_{0} dr \left[ \Pi _M \delta M + \Pi _R
\delta R +  \Pi \delta \Phi \right] \nonumber \\
+ \int ^{\infty}_{0} dr \left[ F^{-1} (R' \delta M - M' \delta R)
\right].
\end{eqnarray}
One can check by a short calculation that
\begin{eqnarray}\label{85}
F^{-1} (R'\delta M - M'\delta R) = (\frac{1}{2} R \delta R \ln F)'
- \delta (\frac{1}{2} R'R \ln F),
\end{eqnarray}
so that transformation (61),(62) is indeed canonical,
since with our boundary conditions the terms in (64) at the
boundaries drop out.

Action (60) then takes the form
\begin{eqnarray}\label{86}
S = \int dt \int ^\infty _0 \left[ \Pi _M \dot{M} + \Pi _R \dot{R}
+ \Pi \dot{\Phi} \right] \nonumber\\
- N^M (\Pi _M M' + \Pi \Phi ') - N^R \Pi _R - \int dt M_\infty
\dot{T}_\infty.
\end{eqnarray}

Noticing that $\Pi _M = -V'$ (Eq. 61)), we shall now
introduce $V$ as a new canonical variable by putting
\begin{equation}\label{87}
V = - \int ^r_0 d\widetilde{r}  \Pi _M + T_0,
\end{equation}
and prove that its conjugate momentum is
\begin{equation}\label{88}
\Pi _V = - M'.
\end{equation}
To show this, let us first notice that from the last relation
follows
\begin{equation}\label{89}
M(r) = -\int ^r_0 d\widetilde{r} \Pi _V (\widetilde{r}),
\end{equation}
so that
\begin{equation}\label{90}
\delta M(r) = -\int ^r_0 d\widetilde{r} \delta \Pi _V (\widetilde{r}).
\end{equation}
Next, we use the ``per partes'' trick (analogous to Eq. (135)
in\cite{KK})
\begin{equation}\label{91}
\int ^\infty _0 dr \left[ \left(\int ^r_\infty d \widetilde{r} V'\right) \times
\int ^r_0 d\widetilde{r} \delta \Pi _V(\widetilde{r})
\right] ' = 0,
\end{equation}
which implies
\begin{equation}\label{92}
\int ^\infty _0 d r \{ \left[ V'(r) - (V(\infty))' \right] \int
^r_0 d\widetilde{r} \delta \Pi _V (\widetilde{r}) \}
= - \int ^\infty _0 dr \left[ V (r) - V(\infty) \right] \delta \Pi
_V (r).
\end{equation}
Taking into account the boundary conditions (47) and
(48), we have
\begin{equation}\label{93}
V'(\infty) = 1, \;\;\;\;    V(r) \sim  r + 2M_\infty \ln
(r/2M_\infty)+ T_\infty.
\end{equation}
Hence, using (69), (70) and (72), the Liouville
form associated with the action (65) can be expressed as
follows:
\begin{eqnarray}\label{94}
&\!&\int ^\infty _0 dr \left[ \Pi _M \delta M + \Pi _R \delta R + \Pi
\delta\Phi \right] - M_\infty \delta T_\infty = \nonumber \\
&=& \int ^\infty _0 dr \left[ -V \delta \Pi _V + \Pi _R \delta R +
\Pi\delta \Phi \right] \nonumber \\
&-& \int _0^ \infty dr \delta M (r) - V (\infty) \delta M_\infty -
M_\infty
\delta T_\infty = \nonumber \\
&\!&\int^\infty_0 dr \left[ \Pi_V \delta V + \Pi _R \delta R +
\Pi\delta\Phi\right]  \\
&-& \int _0   ^\infty dr \left[ (\Pi_V V) + \delta M(r) - r\delta
M_\infty - 2 M_\infty \ln (r/2M_\infty) \delta M_\infty -
T_\infty\delta M_\infty - M_\infty \delta T_\infty
\right]. \nonumber
\end{eqnarray}
%\begin{eqnarray}\label{94}
%\int ^\infty _0 dr \left[ \Pi _M \delta M + \Pi _R \delta R + \Pi
%\delta\Phi \right] - M_\infty \delta T_\infty = \nonumber \\
%= \int ^\infty _0 dr \left[ -V \delta \Pi _V + \Pi _R \delta R +
%\Pi\delta \Phi \right] \nonumber \\
%- \int _0^ \infty dr \delta M (r) - V (\infty) \delta M_\infty - M_\infty
%\delta T_\infty = \nonumber \\
%\int^\infty_0 dr \left[ \Pi_V \delta V + \Pi _R \delta R +
%\Pi\delta\Phi\right] \nonumber \\
%- \int _0   ^\infty dr \left[ (\Pi_V V) + \delta M(r) - r\delta
%M_\infty- 2 M_\infty \ln (r/2M_\infty) \delta M_\infty -
%T_\infty\delta M_\infty - M_\infty \delta T_\infty \right].
%\end{eqnarray}
Since the last five terms can be written as
\begin{eqnarray}\label{95}
\delta \{ - \int ^\infty_0 dr \left[ \Pi_V V + M(r) \right] -
rM_\infty- M^2_\infty (\ln (r/2M_\infty)
 + \frac{1}{2}) - M_\infty
T_\infty \},
\end{eqnarray}
they can be omitted from the action. Hence, the transformation (66),
(67) is indeed a canonical transformation, and the advanced time
coordinate is a canonical variable.


%\subsection
N\v{E}CO O SKAL\'ARN\'{I}M CHARAKTERU V?\\
In this way we thus arrive at the action
\begin{eqnarray}\label{96}
S = \int dt \int ^\infty_0 dr \left[ \Pi _V \dot{V} + \Pi _R
\dot{R} + \Pi \dot{\Phi} - N^V (V' + \Pi\Phi '/\Pi _V) - N^R \Pi
_R \right].
\end{eqnarray}
The pair of functions $N^V$ and $N^R$ form a vector w.r. to transformation of
coordinates of  background. The function $N^V H_V + N^R H_R$ generates
evolution in the direction described by the vector $(N^V, N^R)$.
The constraints acquired now much simpler form than they had with
the super-Hamiltonian and supermomentum (27) and (28).
However, it is only the second constraint, $\Pi _R = 0$, which is
of the desired form of the vanishing canonical momentum conjugate
to the embedding variable $R$. To achieve the same simplification
with the embedding variable $V$ in the whole phase space does not
appear to be feasible.

%\subsection
PETR \\
THEREFORE, IT IS ... AT THIS POINT THAT WE CONFINE TO THE
CONSTRAINT SURFACE AND EXPRESS THE ACTION IN TERMS OF THE MATTER
VARIABLES.

We introduce the scalar matter variable $\zeta\in I\!R$,
closely related to the spacetime scalar $\Phi$, and the
radiation-field density $\mu(\zeta) \in I\!R ^+$ in the matter
variable space $Z$. The variable $\zeta$ is the label which
designates the individual infinitesimal slabs (shells) of the
radiation of which the entire (thick) slab is formed. At any given
slice $\Sigma$, we can put
\begin{equation}\label{97}
\Phi (r) = \zeta (V(r)).
\end{equation}
The infinitesimal slab $(\zeta,\zeta + d\zeta)$ in the material
space is associated with the advanced time by function $v(\zeta)$,
the inverse of which is $\zeta(v)$. With $\zeta$ fixed the
relation (76) gives a particular $r$ at which the given slab
``crosses'' the slice $\Sigma$. Since $v(\zeta)$ and $\zeta(v)$
are inverse, $\zeta(v(x)) = x$ for all $x$; hence a change of
function $v(\zeta) \rightarrow v(\zeta ) + dv (\zeta)$ is
accompanied by the change $\zeta (v) \rightarrow \zeta (v) + d \zeta
(v)$, where
\begin{equation}\label{98}
d\zeta (v(x)) = -\frac{1}{v_\zeta (v(x))} dv (x),
\end{equation}
with $v_\zeta = dv/d\zeta = (d\zeta /dv)^{-1}$. A total change of
the variable $\Phi$ in (76) is then given due to the change of the
material function $v(\zeta)$ (resp. $\zeta(v))$ plus the change of
the embedding spacetime variable $V(r)$. As a result, we obtain
\begin{equation}\label{99}
d\Phi (r) = - \frac{1}{v_\zeta(\zeta(V(r))} dv (\zeta(V(r)) +
\frac{1}{v_\zeta(\zeta(V(r))}dV(r).
\end{equation}
Since $v(\zeta) = V(r)$ (cf.(76)), we also have $v_\zeta d\zeta = V'(r)
dr$ so that
\begin{equation}\label{100}
dr = (v_\zeta/V') d\zeta.
\end{equation}

The second material variable, the density $\mu(\zeta)$, is defined
by the relations
\begin{equation}\label{101}
M(V) = \int_{-\infty} ^V d \widetilde{V}\widetilde{\mu}
(\zeta (\widetilde{V})) = \int _Z d\zeta v_\zeta
\widetilde{\mu}(\zeta) = \int _Z \mu (\zeta) d\zeta.
\end{equation}
Now we turn to the constraint surface at which
\begin{equation}\label{102}
V' + \Pi \Phi '/\Pi _V = 0.
\end{equation}
Since by (67) and (80) we have
\begin{equation}\label{103}
\Pi _V - M'(V(r)) =  M_{,V} V'(r) = (\mu/v_\zeta)V',
\end{equation}
whereas from (76) we have $\Phi ' = (1/v_\zeta)V'$, we find
\begin{equation}\label{104}
\Pi = \mu V'
\end{equation}

With both constraints  being satisfied the non-vanishing terms in
the Liouville form in the action (75) turn out be
\begin{eqnarray}\label{105}
&\!&\int ^\infty _0 dr \left[ \Pi _V \delta V + \Pi \delta \Phi
\right] \nonumber \\
&=& \int _Z d \zeta (v_\zeta /V' ) \{ - (\mu/v_\zeta)V'
\delta V \nonumber \\
&+&  \mu V' \left[ - (1/v_\zeta) \delta v (\zeta) +
(1/v_\zeta)\delta V \right] \} \nonumber \\
&=& \int _Z d\zeta \mu (\zeta) \delta v (\zeta),
\end{eqnarray}
where we have used Eqs. (78) - (83).

Thus, finally we converted the complete action along the
constraint surface to the simple form
\begin{equation}\label{106}
S = \int dt \int _Z d\zeta \mu (\zeta) \dot{v} (\zeta)
\end{equation}
in which only the material variables enter. All embedding
variables dropped out.





%References:

\begin{thebibliography}{99}
\bibitem{KK} K. V. Kucha\v{r}, Phys. Rev. \textbf{D50}, 3961 (1994)
\bibitem{BiKu}J. Bi\v{c}\'ak and K. V. Kucha\v{r}, Phys. Rev.
\textbf{D56}, 4878 (1997)
\bibitem{HaKi}P. H\'aj\'{\i}\v{c}ek and C. Kiefer, Nucl. Phys. \textbf{B}
... (2001)

\end{thebibliography}




%\begin{eqnarray}\label{94}
%&\!&\int ^\infty _0 dr \left[ \Pi _M \delta M + \Pi _R \delta R + \Pi
%\delta\Phi \right] - M_\infty \delta T_\infty = \nonumber \\
%&=& \int ^\infty _0 dr \left[ -V \delta \Pi _V + \Pi _R \delta R +
%\Pi\delta \Phi \right] \nonumber \\
%&-& \int _0^ \infty dr \delta M (r) - V (\infty) \delta M_\infty -
%M_\infty
%\delta T_\infty = \nonumber \\
%&\!&\int^\infty_0 dr \left[ \Pi_V \delta V + \Pi _R \delta R +
%\Pi\delta\Phi\right] \nonumber \\
%&-& \int _0   ^\infty dr \left[ (\Pi_V V) + \delta M(r) - r\delta
%M_\infty- 2 M_\infty \ln (r/2M_\infty) \delta M_\infty -
%T_\infty\delta M_\infty - M_\infty \delta T_\infty \right].
%\end{eqnarray}





\end{document}


\bibitem{}

\bibitem{}
\bibitem{b10} Sasakawa T. and Hor ‡ek J. 1982 J.Phys. B{\bf 15} L169-174



Because the super-Hamilton $H(r)$ generates the time evolution, it
is instructive to calculate its Poisson brackets with canonical
variables. After some calculations we find
\begin{equation}\label{60}
\{M(r), H(r')\} = \left[ -\Lambda^{-3}R' H_r + \Lambda ^{-3}
\pi\phi' (R' + FP_M) \right] \delta (r,r')
\end{equation}
Again, in the vacuum case $\pi=0$, and the supermomentum
constraint $H_r=0$ guarantees that $M(r)$ is a constant of motion.
Under the presence of infalling dust the mass $M$ is changing in
accordance with (3.27).  We also easily find
\begin{equation}\label{61}
\{M(r), H_r(r')\} = M'(r) \delta(r,r'),
\end{equation}
as expected for a spatial scalar. Similarly, we can calculate the
Poisson brackets of the super-Hamiltonian and supermomentum with
the null dust momentum $\pi$, to obtain
\begin{equation}\label{62}
\{\pi(r), H(r')\} = -\frac{1}{\Lambda(r')}\pi(r')\frac{\partial}{\partial
r'}\delta(r',r),
\end{equation}
\begin{equation}\label{63}
\{\pi(r), H_r(r')\} = -\pi(r')\frac{\partial}{\partial
r'}\delta(r',r).
\end{equation}
Since $\pi (r)$, being a momentum conjugate to a scalar $\phi$, is
a scalar density, the derivatives of $\delta$-function appear on
the r.h.s. of (3.29) and (3.30). It is easy to express also the
Poisson brackets
\begin{equation}\label{64}
\{\phi,H\} = \Lambda^{-1}\phi'\delta(r,r'),
\end{equation}
and
\begin{equation}\label{65}
\{\phi,H_r\} = \phi'\delta (r,r').
\end{equation}
Using these results for the Poisson brackets with the
super-Hamiltonian and supermomentum, we can find the change of the
canonical data when the hypersurface is displaced by the proper
time $N(r)$ in the normal direction and shifted by the tangential
vector $N^r(r)$.
For example,
\begin{equation}\label{66}
\dot{\phi} = \{\phi, \int dr (NH + N^r H_r)\} = (N\Lambda^{-1} +
N^r) \Phi'.
\end{equation}
Let us chose $t=T, r = R$, where $T, R$ are the Schwarzschild
coordinates, and put $\Phi = v(T,R)$ in accordance with (1.22) and
(3.24). Substituting for $N, \Lambda$ and $N^r$ from the Vaidya
metric in the Schwarzschild coordinates (1.1) and (1.2). We see
that Eq. (3.33) reads
\begin{equation}\label{67}
\frac{\dot{\Phi}}{\Phi'} = \frac{\dot{v}}{v'} =
\frac{\dot{M}}{M'},
\end{equation}
where dots and primes are derivatives with respect to the
Schwarzschild time $T$ and curvature radius $R$. Taking the
derivative of (3.34) with respect to $R$, and using Eq. (3.18) for
$v'$ with $P_M = 0, R' = 1$ (which is a consequence of the
Schwarzschild slicing), we recover the only non-trivial
combination of the field equations (1.5) (which follows from
(1.3)).

Alternatively, expressing Eq. (3.33) in with $t = T_E, r = R$, and
comparing with  the Vaidya metric in Eddington coordinates (1.26),
in which
\begin{eqnarray}\label{68}
N = (1 + \frac{2M}{R})^{-\frac{1}{2}}, \;\;\;\; \Lambda = (1 +
\frac{2M}{R})^\frac{1}{2}\nonumber \\
N^r = \frac{2M}{R} (1 + \frac{2M}{R})^{-1},
\end{eqnarray}
we find (3.33) to imply
\begin{equation}\label{69}
\dot{v} = v'
\end{equation}
or
\begin{equation}\label{70}
\dot{M} = M'
\end{equation}
where now dots and primes denote derivatives with respect to
Eddington coordinates. Indeed, the condition (3.37) is the only
implication of the field equations which has to be satisfied (cf.
Eq. (1.27), where $M$ is an arbitrary function of $T_E + R.)$
Analogously we may discuss the other Hamilton equations....
Analogously, using Eqs. (3.27) and (3.28), we find
\begin{equation}\label{71}
\dot{M} = - N\Lambda ^{-3}R' H_r + N\Lambda^{-3} \pi\phi '(R' +
FP_M)+ N^rM'(r).
\end{equation}
Assuming $H_r = 0$, and taking into account (3.26) and (3.33) we
just regain the result (3.34). Therefore, we again find the
evolution of the mass in accordance with Eq. (1.5) if we put $t=T,
r=R$, and in accoradance with (1.27) for $t=T_E$ and $r=R$.


In the same way, we may calculate the other Poisson brackets,
however, no other results than those discussed above arise.
Finally, following \ref{K} consider the canonical transformation
$(\tau _- \infty, M, P_M)\rightarrow (\widetilde{T}_S,
P_{\widetilde{T}})$ given by
\begin{equation}\label{72}
\widetilde{T}(r) = \tau _\infty - \int d\overline{r}P_M
(\overline{r}),
\end{equation}
\begin{equation}\label{73}
P_{\widetilde{T}}(r) = -M'(r).
\end{equation}
One can prove by the same procedure as used in \ref{K} that
(3.39), (3.40) is a canonical transformation, so that $(R, P_R, \widetilde{T}, \phi,
\pi)$form a canonical chart on the extend phase space.

In terms of these new canonical variables the super-Hamiltonian
and supermomentum (3.16), (3.17)) turn into
\begin{equation}\label{74}
\Lambda H = F^{-1} P_{\widetilde{T}}R' + F \widetilde{T}' P_R +
\pi\phi',
\end{equation}
\begin{equation}\label{75}
H_r  + P_R R' +\widetilde{T}' P_{\widetilde{T}} + \pi\phi ',
\end{equation}
where
\begin{equation}\label{76}
F(r) = 1 + \frac{2}{R(r)}\int d\overline{r}
P_{\widetilde{T}}(\overline{r}).
\end{equation}
The corresponding constraints $\Lambda H = 0, H_r = 0$ can easily
be solved for $P_{\widetilde{T}}$ and $P_R$ either directly or by
using the preceding results. Comparing (3.40) and (3.21) we
obtain\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; $P_R - F^{-1}P_{\widetilde{T}} = 0,$ \\
or
\begin{equation}\label{77}
FP_R = P_{\widetilde{T}}
\end{equation}
Substituting back into the scaled super-Hamiltonian (3.42) we
convert it, as expected for null dust (cf. Eq.(2.26), into the
supermomentum (3.42). As before, if we choose $v = \phi$ (cf. Eqs.
(1.22), (3.24)) so that Eqs. (3.26) and (3.40) imply
\begin{equation}\label{78}
P_{\widetilde{T}} = -\pi.
\end{equation}
Finally, using then directly the momentum constraint (3.42) we can
express the momentum $P_R$ in the form
\begin{equation}\label{79}
P_R = (R')^{-1} \pi (\widetilde{T}' - \phi ').
\end{equation}
Therefore, in contrast to the case of massless scalar field
considered in \ref{Ro}, with the null dust we are able to solve
the constraints with respect to $P_{\widetilde{T}}$ and $P_R$ in
terms of the remaining canonical variables. It is easy to see that
if Eqs. (3.45) and (3.46) are satisfied, both constraints and
relations (3.18) and (3.44) are satisfied as well. Imposing the
coordinate and slicing conditions $r = R, T' = 0$ (which are
equivalent to the BCMN gauge conditions - cf. ()) we obtain
\begin{eqnarray}
%\centreline
P_R(R) = - \pi (R) \phi ' (R) \nonumber
\end{eqnarray}
in exactly the same form as for massless scalar field. The
Hamiltonian for the reduced theory is simply
\begin{equation}\label{80}
H_{red} \left[ \phi , \pi \right] = -\int dr P_T = \int dr \pi =
\int dr M'
\end{equation}
\textit{Note}\\
Perhaps $T$ is in fact spacetime scalar - in contrast to the
scalar field case.

The reason appears to be the following: we saw that for the null
dust $\Lambda H(r) = H_r$. Therefore,\\
$\{ T(r), \Lambda H (\overline{r}) \} = \{ T(r), H_r (\overline{r}) \}
= \{ T(r), P_R(\overline{r})R' (\overline{r}) + T' (\overline{r})P_T (\overline{r})
+ \pi (\overline{r})\phi ' (\overline{r}) \} = T'(\overline{r}) \delta  (r,
\overline{r}),$\\
if I am not mistaken.

In the scalar field case\\
$\Lambda H = F^{-1}P_T R' + FP_RT' + \frac{1}{2} (R^{-2} \pi ^2 + R^2 \phi
'^2)$\\
but $F(r) = 1 + \frac{2}{R(r)}\int d\overline{r}
P_T(\overline{r})$\\
and this $F$ with $P_T$ causes troubles when calculating $\{ T(r), \Lambda H (\overline{r})\} = ... \Theta(r -
\overline{r})$\\












































































Fig. 1                                       Table 1






\epolynomials $P_

