Arnowitt-Deser-Misner formalism

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Author: Dr. Stanley Deser, Lauritsen Laboratory , California Institute of Technology, US and The Martin A. Fisher School of Physics, Brandeis University, US.

ADM Formalism

S. Deser

Physics Department, Brandeis University, Waltham, MA and California Institute of Technology, Pasadena CA, USA

This approach to General Relativity, and more generally to gauge theories, was developed to emphasize its field-theoretic, rather than geometric, content. In particular, the dynamics of the gravitational field, viewed as a Hamiltonian system, was displayed in a way that could lead to quantization. A contemporary review of the dozen papers in the ADM series, is still the best source [1]. Here, we merely excerpt some of the basic concepts underlying the program. A separate entry describes ADM energy, which has become the standard measure for asymptotically flat, and was later extended to asymptotically (Anti)de Sitter spacetimes [2].

Here, a precise determination of the independent dynamical modes of the gravitational field is arrived at when the theory has been cast into canonical form and consequently involves the minimal number of variables specifying the state of the system. At this level, one will have all the relevant information about the field's behavior in familiar form. The canonical formalism, involving only the minimal set of variables (which will turn out to be four), is also essential to the quantization program, since it yields directly simple Poisson bracket (P.B.) relations among these conjugate, unconstrained, variables. Two essential aspects of canonical form are: (1) that the field equations are of first order in the time derivatives; and (2) that time has been singled out so that the theory has been recast into 3+1 dimensional form. These two features are characteristic of Hamilton (or P.B.) equations of motion, in contrast to the Lagrange equations. The first requirement may be achieved in general relativity, since its Lagrangian may be written in a form linear in the time derivatives ('Palatini' form). The type of variable fulfilling the second requirement will be seen also to possess a natural geometrical interpretation.

The use of the Palatini Lagrangian and of 3+1 dimensional notation does not, of course, impair the general covariance of the theory under arbitrary coordinate transformations. In possessing this covariance, general relativity is precisely analogous to the parameterized form of mechanics in which the Hamiltonian and the time are introduced as a conjugate pair of variables of a new degree of freedom. When in parameterized form, a theory is invariant under an arbitrary re-parameterization, just as general relativity is invariant under an arbitrary change of coordinates. The action of general relativity will thus be seen to be in "already parameterized" form. The well-known relations between the usual canonical form and the parameter description will thus provide the key for deriving the desired canonical form for the gravitational field. We will therefore begin with a brief review of parameterized particle mechanics. Then, the Lagrangian of general relativity will be cast into Palatini and 3+1 dimensional form, and the geometrical significance of the variables will be discussed. We will see then that relativity has a form identical to parameterized mechanics. To complete the analysis, we obtain the canonical variables and their relations as well as the P.B. equations of motion. Once canonical form is reached, the physical interpretation of quantities involved follows directly. Thus, the canonical variables themselves represent the independent excitations of the field (and hence provide the basis for defining gravitational radiation in a coordinate-independent way). Further, the numerical value of the Hamiltonian for a particular state of the system provides the primary definition of total energy. It is also possible to set up the analysis of gravitational radiation in a fashion closely analogous to electrodynamics by introducing a suitable definition of the wave zone. In this region, gravitational waves propagate as free radiation, independent of the strong field interior sources. The waves obey ordinary (flat-space) wave equations and consequently satisfy superposition. The Poynting vector may also be defined invariantly in the wave zone. In contrast, the Newtonian-like parts of the metric cannot be determined within the wave zone; they depend strongly on the interior non-linearities.

We begin with a brief analysis of the relevant properties of the parameter formalism For simplicity, we first deal with a system of a finite number $M$ of degrees of freedom. Its action may be written as

(1)

$I = \int^{t{_1}}_{t{_2}} dt L = \int^{t{_1}}_{t{_2}} dt \left( \sum^M_{i=1} p_i\dot q_i - H(p,q) \right)$

where $\dot q \equiv dq/dt$ and the Lagrangian has been expressed in a form linear in the time derivatives. (This will be referred to as the first-order form since independent variation of $p_i$ and $q_i$ gives rise to the first-order equations of motion.) The maximal information obtainable from the action arises when not only $p_i$ and $q_i$ are varied independently, but $t$ is also varied and endpoint variations are allowed. Postulating that the total $\delta I$ is a function only of the endpoints $[\delta I = G(t_1) - G(t_2)]$ leads to: (1) the usual Hamilton equations of motion for $p_i$ and $q_i$ ; (2) conservation of energy $(d H/dt = 0)$ ; and (3) the generating function

(2)

$G(t) = \sum_i p_i \delta q_i - H \delta t$

Here $\delta q_i = \delta_0 q_i + \dot q_i \delta t$ where $\delta_0 q_i$ denotes the independent ("intrinsic") variation of $q_i$ . The generating function can easily be seen to be the conventional generator of canonical transformations. Thus $G_q = \sum_i p_i\, \delta q_i$ generates changes $q_i \rightarrow q_i + \delta q_i ,\ p_i \rightarrow p_i$ while $G_t = - H \delta t$ generates the translation in time. That is, for $G_q$ one has $[q_j , G_q ] = \sum_i [q_j,p_i]\delta q_i =\delta q_j$ , where [ $A,B$ ] means the Poisson bracket (P.B.), and for $G_t$ one has $[q_i , G_t] = - [q_i,H]\delta t = -\dot q_i \delta t$ by the P.B. form of the equations of motion. The above elementary discussion may be inverted to show that, for the action of (1), if every variable occurring in $H$ is also found in the $p\dot q$ term, then the theory is in canonical form and $p_i$ and $q_i$ obey the conventional P.B. relations. The motion of the system (1) is described in terms of one independent variable $t$ (the "coordinate"). The action may be cast, as is well known, into parameterized form, in which the time is regarded as a function $q_{M+1}$ of an arbitrary parameter $\tau$ :

$I = \int^{\tau_1}_{\tau_2} d\tau \, L_\tau \equiv \int^{\tau_1}_{\tau_2} d\tau \left[ \sum^{M+1}_{i=1} p_i q^\prime_i \right] \; .$

Here, $q^\prime \equiv dq/d\tau$ , and the constraint equation $p_{M+1} + H(p, q) =0$ holds. One may equally well replace this constraint by an additional term in the action:

(3)

$I = \int^{\tau_1}_{\tau_2} d\tau \left[ \sum^{M+1}_{i=1} p_i q^\prime_i - NR \right]$

where $N(\tau )$ is a Lagrange multiplier. Its variation yields the constraint equation $R(p_{M+1}, p, q) = 0$ , which may be any equation with the solution (occurring as a simple root) $p_{M+1} = -H$ . The theory as cast into form (3) is now generally covariant with respect to arbitrary coordinate transformations $\bar \tau = \bar \tau (\tau )$ , bearing in mind that $N$ transforms as $dq/d\tau$ . The price of achieving this general covariance has been not only the introduction of the $(M + 1)$ st degree of freedom, but, more important, the loss of canonical form, due to the appearance of the Lagrange multiplier $N$ in the "Hamiltonian", $H^\prime \equiv NR$ . ( $N$ occurs in $H^\prime$ but not in $\sum^{M+1}_{i=1} p_iq^\prime_i$ .) A further striking feature which is due to the general covariance of this formulation is that the "Hamiltonian" $H^\prime$ vanishes by virtue of the constraint equation. This is not surprising, since the motion of any particular variable $F(p, q)$ with respect to $\tau$ is arbitrary, i.e., $F^\prime$ may be given any value by suitable recalibration $\tau \rightarrow \bar\tau$ . As we shall see, the Lagrangian of general relativity may be written in precisely the form of (3). We will, therefore, be faced with the problem of reducing an action of the type (3) to canonical form (1). The general procedure consists essentially in reversing the steps that led to (3). If one simply inserts the solution, $p_{M+1} = -H$ , of the constraint equation into (3), one obtains

(4)

$I = \int d\tau \left[ \sum^M_{i=1} p_i q^\prime_i - H(p,q) q^\prime_{M+1} \right] .$

All reference to the arbitrary parameter $\tau$ disappears when $I$ is rewritten as

(5)

$I = \int dq_{M+1} \left[ \sum^M_{i=1} p_i (dq_i/dq_{M+1})-H \right]$

which is identical to (1) with the notational change $q_{M+1} \rightarrow t$ . Equation (5) exhibits the role of the variable $q_{M+1}$ as an "intrinsic coordinate". By this is meant the following. The equation of motion for $q_{M+1}$ is $q^\prime_{M+1} = N(\partial R/\partial q_{M+1})$ from (3). Also, none of the dynamical equations determine $N$ as a function of $\tau$ . Thus $N$ and hence $q_{M+1}$ , are left arbitrary by the dynamics (though, of course, a choice of $q_{M+1}$ as a function of $\tau$ fixes $N$ ). One is therefore free to choose $q_{M+1} (\tau )$ to be any desired function and use this function as the new independent variable (parameter): $q_i = q_i( q_{M+1})$ , $p_i = p_i( q_{M+1})$ , $i = I \ldots M$ . The action of (5), and hence the relations between $q_i$ , $p_i$ , and $q_{M+1}$ are now independent of $\tau$ . They are manifestly invariant under the general "coordinate transformation" $\bar\tau = \bar\tau (\tau )$ (for the simple reason that $\tau$ itself no longer appears). The choice of $q_{M+1}$ as the independent variable thus yields a manifestly $\tau$ -invariant formulation and gives an "intrinsic" specification of the dynamics. This is in contrast to the original one in which the trajectories of $q_1 \ldots q_{M+1}$ are given in terms of some arbitrary variable $\tau$ (which is extraneous to the system).

In practice, we shall arrive at the intrinsic form (5) from (4) in an alternate way. Since the relation between $q_{M+1}$ and $\tau$ is undetermined, we are free to specify it explicitly, i.e., impose a "coordinate condition". If, in particular, this relation is chosen to be $q_{M+1}= \tau$ (a condition which also determines $N$ ), the action (4) then reduces (5) with the notational change $q_{M+1} \rightarrow \tau$ ; the non-vanishing Hamiltonian only arises as a result of this process.

This simple analysis has shown that the way to reduce a parameterized action to canonical form is to insert the solution of the constraint equations and to impose coordinate conditions. Further, the imposition of coordinate conditions is equivalent to the introduction of intrinsic coordinates.

In field theory it will prove more informative to carry out this analysis in the generator. We exhibit here the procedure in the particle case: The generator associated with the action of (3) is

(6)

$G = \sum^{M+1}_{i=1} p_i \delta q_i - NR \delta\tau$

Upon inserting constraints, the generator reduces to

(7)

$G = \sum^M_{i=1} p_i \delta q_i - H \delta q_{M+1} .$

Imposing the coordinate condition $q_{M+1} = t$ then yields (2). From this form, one can immediately recognize the $M$ pairs of canonical variables and the non-vanishing Hamiltonian of the theory.

One can, of course, perform the above analysis for a parameterized field theory as well. Here the coordinates appear as four new field variables $q^{M+\mu} = x^\mu (\tau^\alpha )$ , and there are four extra momenta $p_{M+\mu}(\tau^\alpha )$ conjugate to them. Four constraint equations are required to relate these momenta to the Hamiltonian density and the field momentum density, and correspondingly, there are four Lagrange multipliers $N_\mu (\tau ^\alpha)$ for a field.

The usual action integral for general relativity

(8)

$I = \int d^4x {\mathcal L} = \int d^4x\, \sqrt{-g} R$

yields the Einstein field equations when one considers variations in the metric (e.g., $g_{\mu\nu}$ or the density $\tilde g^{\mu\nu} = \sqrt{-g} g^{\mu\nu}$ ). These Lagrange equations of motion are then second-order differential equations. It is our aim to obtain a canonical form for these equations, that is, to put them in the form $\dot q = \partial H/\partial p,\; \dot p = -\partial H/\partial q$ . As a preliminary step, we will restate the Lagrangian so that the equations of motion have two of the properties of canonical equations: they are first-order equations; and they are solved explicitly for the time derivatives. The second property will be obtained by a 3 + 1 dimensional breakup of the original four-dimensional quantities, as will be discussed below. The first property is insured by using a Lagrangian linear in first derivatives. In relativity, this consists in regarding the Christoffel symbols $\Gamma_{\mu\ \nu}^{\ \alpha}$ as independent quantities in the variational principle. Thus, one may rewrite (8) as

(9)

$I = \int d^4 x \tilde{\rm g}^{\mu\nu} R_{\mu\nu} (\Gamma )$

where

(10)

$R_{\mu\nu} (\Gamma ) \equiv \Gamma_{\mu\ \nu,\alpha}^{\ \alpha} - \Gamma_{\mu\ \alpha, \nu}^{\ \alpha} + \Gamma_{\mu\ \nu}^{\ \alpha} \Gamma_{\alpha\ \beta}^{\ \beta} - \Gamma_{\mu \ \beta}^{\ \alpha} \Gamma_{\nu\ \alpha}^{\ \beta}.$

Note that these covariant components $R_{\mu\nu}$ of the Ricci tensor do not involve the metric but only the affinity $\Gamma_{\mu\ \nu}^{\ \alpha}$ . Thus, by varying $\tilde g^{\mu\nu}$ , one obtains directly the Einstein field equations

(11)

$R_{\mu\nu} = 0 .$

These equations no longer express the full content of the theory, since the relation between the now independent quantities $\Gamma_{\mu\ \nu}^{\ \alpha}$ and $g_{\mu\nu}$ is still required. This is obtained as a field equation by varying $\Gamma_{\mu\ \nu}^{\ \alpha}$ . [We use units such that 16 $\pi\gamma c^{-4} = 1 = c$ , where $\gamma$ is the Newtonian gravitational constant; electric charge is in rationalized units. Latin indices run from 1 to 3, Greek from 0 to 3, and $x^0 = t$ . Derivatives are denoted by a comma or the symbol $\partial_\mu$ .]

The three-dimensional quantities appropriate for the Einstein field are (as will be discussed in detail later)

(12)

$\begin{array}{lcl} &&g_{ij} \equiv {}^4g_{ij} , \quad\quad N \equiv (-{}^4g^{00})^{-1/2} , \quad\quad N_i \equiv {}^4g_{0i}\\ &&\pi^{ij} \equiv \sqrt{-{}^4g} ({}^4\Gamma_{p\ q}^{\ 0}-g_{pq} {}^4\Gamma_{r \ s}^{\ 0} g^{rs}) g^{ip}g^{jq} . \end{array}$

Here and subsequently we mark every four-dimensional quantity with the prefix $^4$ , so that all unmarked quantities are understood as three-dimensional. In particular, $g^{ij}$ in (12) is the reciprocal matrix to $g_{ij}$ . The full metric $^4g_{\mu\nu}$ and $^4g^{\mu\nu}$ may, with (12), be written

(13)

${}^4g_{00} = - (N^2 - N_iN^i)$

where $N^i = g^{ij}N_j$ , and

(14)

$\begin{array}{lcl} &&{}^4g^{0i} = N^i/N^2, \quad\quad {}^4g^{00} = -1/N^2,\\ &&{}^4g^{ij} = g^{ij} - (N^iN^j/N^2). \end{array}$

One further useful relation is

(15)

$\sqrt{-^4g} = N\sqrt{g} .$

In terms of the basic quantities of (12), the Lagrangian of general relativity becomes

(16)

$\begin{array}{lcl} \ {\mathcal L} = \sqrt{-{}^4g} \ {}^4R &=&-g_{ij} \partial_t\pi^{ij}-NR^0 - N_iR^i\\ && - 2 (\pi^{ij} N_j - \textstyle{\frac{1}{2}} \pi N^i + N^{|i}\sqrt{g} )_{,i} \end{array}$

where

(17)

$\begin{array}{lcl} R^0 &\equiv& - \sqrt{g} [{}^3R + g^{-1} (\textstyle{\frac{1}{2}}\pi^2 - \pi^{ij}\pi_{ij} )]\\ R^i &\equiv& - 2 \pi^{ij}\,_{|j} . \end{array}$

The quantity $^3R$ is the curvature scalar formed from the spatial metric $g_{ij}, \; _|$ indicates the covariant derivative using this metric, and spatial indices are raised and lowered using $g^{ij}$ and $g_{ij}$ . (Similarly, $\pi \equiv \pi^i\,_i$ .) As in the electromagnetic example, we have allowed second-order space derivatives to appear by eliminating such quantities as $\Gamma_i\,^k\,_j$ in terms of $g_{ij,k}$ .

One may verify directly that the first-order Lagrangian (16) correctly gives rise to the Einstein equations. One obtains

(18)

$\partial_t g_{ij} = 2Ng^{-1/2}(\pi_{ij} - \textstyle{\frac{1}{2}} g_{ij}\pi ) + N_{i|j} + N_{j|i}$

(19)

$\begin{array}{lcl} \partial_t \pi^{ij}& = & - N\sqrt{g} (^3\!R_{ij} - \textstyle{\frac{1}{2}} g^{ij}\, {}^3\!R) + \textstyle{\frac{1}{2}} Ng^{-1/2} g^{ij} (\pi^{mn}\pi_{mn} -\frac{1}{2}\pi^2 )\\ && -2Ng^{-1/2} (\pi^{im}\pi_m^{\ \ j} -\textstyle{\frac{1}{2}}\pi\pi^{ij}) + \sqrt{g} (N^{|ij} - g^{ij} N^{|m}_{\ \ \ |m}) \\ && + (\pi^{ij} N^m)_{|m} - N^i_{\ \ |m}\pi^{mj} - N^j_{\ \ |m}\pi^{mi} \end{array}$

(20)

$R^\mu(g_{ij},\pi^{ij}) = 0 .$

Equation (18), which results from varying $\pi^{ij}$ , would be viewed as the defining equation for $\pi^{ij}$ in a second-order formalism. Variation of $N$ and $N_i$ yields equations (20), which are the ${}^4G^0_{\ \mu}\equiv {}^4R^0_{\ \mu}-\frac{1}{2}\delta^0_{\ \mu}\ {}^4R=0$ equations, while equations (19) are linear combinations of these equations and the remaining six Einstein equations $(^4G_{ij}=0)$ .

Before proceeding with the reduction to canonical form, it is enlightening to examine, from a geometrical point of view, our specific choices (12) of three-dimensional variables. Geometrically, their form is governed by the requirement that the basic variables be three-covariant under all coordinate transformations which leave the $t$ =const surfaces unchanged. Any quantities which have this property can be defined entirely within the surface (this is clearly appropriate for the 3+1 dimensional breakup). One fundamental four-dimensional object which is clearly also three-dimensional is a curve $x^\mu(\lambda)$ which lies entirely within the 3-surface, i.e., $x^0 (\lambda)$ = const. The vector $v^\mu\equiv dx^\mu/d\lambda$ tangent to this curve is therefore also three-dimensional. The restriction that the curve lie in the surface $t$ =const is then $v^0 = 0$ , and conversely any vector $V^\mu$ , with $V^0 = 0$ is tangent to some curve in the surface. Three such independent vectors are $V^\mu_{(i)} = \delta^\mu_i$ . Given any covariant tensor $A_{\mu\ldots\nu}$ , its projection onto the surface is then $V^\mu_{(i)}\ldots V^\nu_{(j)}A_{\mu\ldots\nu} = A_{i\ldots j}$ . Thus, the covariant spatial components of any four-tensor form a three-tensor which depends only on the surface (in contrast to the contravariant spatial components which are scalar products with gradients rather than tangents, and hence depend also on the choice of spatial coordinates in the immediate neighborhood of the surface). This accounts for the choice of $g_{ij}$ , rather than $^4\!g^{ij}$ . In contrast, $N$ and $N_i$ do not have the desired invariance and, in fact, by choosing coordinates such that the $x^i$ = const lines are normal to the surface, one obtains $N_i = 0$ . (If $x^0$ is arranged to measure proper time along these lines, one has also $N = 1$ .)

The quantity which plays the role of a momentum is more difficult to define within the surface, since it refers to motion in time leading out of the original $t =$ const surface. Such a quantity is, however, provided by the second fundamental form $K_{ij}$ , which gives the radii of curvature of the $t =$ const surface as measured in the surrounding four-space. These "extrinsic curvatures" describe how the normals to the surface converge or diverge, and hence determine the geometry of a parallel surface at an infinitesimally later time. Since $K_{ij}$ describes a geometrical property of the $t =$ const surface, as imbedded in four-space, it again does not depend on the choice of coordinates away from the surface. This may also be seen from a standard definition, $K_{ij} = - n_{(i;j)}$ , which expresses $K_{ij}$ as the covariant spatial part of the tensor $n_{(\mu;\nu )}$ (the four-dimensional covariant derivative of the unit normal, $n_\mu= - N\delta^0_\mu$ to the surface). For convenience in ultimately reaching canonical form, we have chosen, instead of $K_{ij}$ , the closely related variable $\pi^{ij} = - \sqrt{g} (K^{ij} - g^{ij} K)$ . Thus, the geometrical analysis defines $g_{ij}$ and $\pi^{ij}$ as suitable quantities, unaffected by the choice of coordinates later in time, while $N$ and $N_i$ describe how the coordinate system will be continued off the $t =$ const surface.

Returning to the field equations (18-20), we may now analyze them from the point of view of the initial value problem. If one specifies $g_{ij}, \; \pi^{ij}$ and $N,\; N_i$ initially, it is clear that the equations uniquely determine $g_{ij}$ and $\pi^{ij}$ at a later time, while $N$ and $N_i$ remain undetermined then. Since the latter merely express the continuation of the coordinates, the intrinsic (coordinate-independent) geometry of space-time is determined uniquely by an initial choice of $g_{ij}$ and $\pi^{ij}$ . This choice is restricted, however, by the four constraint equations (20) which relate these twelve variables at the initial time. Subject to these compatibility conditions, then, the ( $g_{ij}, \; \pi^{ij}$ ) constitute a complete set of Cauchy data for the theory.

The maintenance in time of the constraints is guaranteed by the Bianchi identities $(^4\!G_{\mu \ , \nu}^{\ \nu}\equiv 0)$ . Hence

(21)

${}^4\!G_{\mu \ ,0}^{\ 0} = - {}^4\!G_{\mu \ ;i}^{\ i} - {}^4\!G_\mu^{\ \nu}\ {}^4\!\Gamma_{\nu\ 0}^{\ 0} + {}^4\!G_\nu^{\ 0} \ {}^4\!\Gamma_{\mu \ 0}^{\ \nu} .$

Thus, by virtue of the dynamic equations $^4\!G_{ij}=0$ (and consequently of their spatial derivatives) at $t = 0$ , the constraints $^4\!G_\mu^{\ 0}=0$ hold at all times if they hold initially.

While the twelve variables ( $g_{ij}, \; \pi^{ij}$ ) constitute a complete set of Cauchy data, they do not provide a minimal set (which the canonical formalism will eventually give to be two pairs, corresponding to two degrees of freedom). We may now count the number of minimal variables. Of the twelve $g_{ij}, \; \pi^{ij}$ , we may eliminate four by using the constraint equations (20). There will correspondingly be four "Bianchi" identities among the twelve equations of motion (18) and (19). As we have seen, $N$ and $N_i$ determine the continuation of the coordinate system without affecting the intrinsic geometry ( i.e., the physics of the field). For every choice of $N$ and $N_i$ as functions of the remaining eight Cauchy data (which represents a choice of coordinate frame), there will result four equations stating that the time derivatives of four of the remaining eight ( $g, \; \pi$ ) variables vanish. [More precisely, a choice of coordinate frame is made by specifying four functions $q^\mu$ of ( $g_{ij}, \; \pi^{ij}$ ) as the coordinates $x^\mu$ . The equations for $\partial_tq^\mu\; (=\delta^\mu_0 )$ then determine $N$ and $N_i$ .] Thus, after these coordinate conditions are imposed, we are left with four dynamic equations of the form $\partial_t u_a = f_a(u) \; (a = 1, 2, 3, 4)$ . These equations govern the motion of a system of two degrees of freedom. This is to be expected, since the linearized gravitational field is a massless spin two field, and the self-interaction of the full theory should not alter such kinematical features as the number of degrees of freedom.

To summarize,the Einstein Lagrangian of (16) becomes, up to total divergences:

(22)

${\mathcal L} = \pi^{ij} \partial_t g_{ij} - NR^0 - N_iR^i ,\quad\quad R^\mu= R^\mu(g_{ij}, \pi^{ij}) .$

Equation (22) is thus precisely in the form of a parameterized theory's Lagrangian as in (3). This form just expresses the invariance of the theory with respect to transformations of the four coordinates $x^\mu$ and hence the $x^\mu$ are parameters in exactly the same sense that $\tau$ was in the particle case. That the $N$ and $N_i$ are truly Lagrange multipliers follows from the fact that they do not appear in the $pq^\prime$ (i.e., $\pi^{ij} \partial_tg_{ij}$ ) part of ${\mathcal L}$ . Their variation yields the four constraint equations $R^\mu=0$ . The "Hamiltonian" ${\mathcal H}^\prime \equiv NR^0 + N_iR^i$ vanishes due to the constraints. The true non-vanishing Hamiltonian of the theory will arise only after the constraint variables have been eliminated and coordinate conditions chosen. The analysis leading to the canonical form is carried out in the next section.

We are now in a position to cast the general theory into canonical form. The geometrical considerations were useful in obtaining the Lagrangian in the form (22), which we recognized as the Lagrangian of a parameterized field theory corresponding to (3).

The reduction of (22) to the canonical form analogous to (1) requires an identification of the four extra momenta to be eliminated by the constraint equations (20). To this end we consider the generator arising from (22):

(23)

$G = \int d^3x [\pi^{ij} \delta g_{ij} + T_{\ \mu}^{0 \ \prime} \delta x^\mu] .$

The $T_{\ \mu}^{0 \ \prime}\delta x^\mu$ term comes from the independent coordinate variations. However, $T_{\ \mu}^{0 \ \prime}$ vanishes as a consequence of the constraint equations. For example, $T_{\ 0}^{0 \ \prime} = - NR^0 -N_iR^i = 0$ . When the constraints are inserted, $G$ reduces to

(24)

$G = \int d^3x \ \pi^{ij} \delta g_{ij}$

[corresponding to (7)] where four of the twelve ( $g_{ij}, \; \pi^{ij}$ ) are understood to have been expressed in terms of the rest by solving $R^\mu= 0$ for them. This elimination exhausts the content of the constraint equations. Finally, as in the particle case, coordinate conditions (now four in number) must be chosen and this information inserted into (24), leaving one now with only four dynamical variables ( $\pi^A, \; \phi_A$ ). If, in fact, the generator at this stage has the form

(25)

$G = \int d^3x \left[\sum^2_{A=1} \pi^A \delta\phi_A + {\mathcal I}^0_{\ \mu}(\pi^A, \phi_A ) \delta x^\mu\right]$

then the theory is clearly in canonical form with $\pi^A$ and $\phi_A$ as canonical variables and $\int {\mathcal I}^0_{\ \mu} \delta x^\mu$ the generator of translations $\delta x^\mu$ . In (25), ${\mathcal I}^0_{\ \mu}[\pi^A, \phi_A]$ has arisen in the elimination of the extra momenta $p_{M+\mu}$ , by solving the constraint equations, and the $x^\mu$ , now represent the four variables chosen as coordinates $q_{M+\mu}$ .

In order to achieve the form (25) for relativity, it is useful to be guided by the linearized theory. Here one must treat the constraint equations to second order since our general formalism shows that the Hamiltonian arises from them. Through quadratic terms, equations (20) may be written in the form

(26)

$\begin{array}{rcl} g_{ij,ij} -g_{ii,jj}& = &{\mathcal P}_2\,\!^0[g_{ij},\pi^{ij}]\\ -2\pi^{ij}\,\!_{,j}& = & {\mathcal P}_2\,\!^i[g_{ij},\pi^{ij}] \end{array}$

where ${\mathcal P}_2^{\ 0}$ and ${\mathcal P}_2^{\ i}$ are purely quadratic functions of $g_{ij}$ and $\pi^{ij}$ . These equations determine one component of $g_{ij}$ and three components of $\pi^{ij}$ in terms of the rest. The content of equations (26) can be seen more easily if one makes the following linear orthogonal decomposition on $g_{ij}$ and $\pi^{ij}$ . For any symmetric array $f_{ij} = f_{ji}$ one has

(27)

$f_{ij} = f_{ij}^{\ \ TT} + f_{ij}^{\ \ T} + (f_{i,j} + f_{j,i})$

where each of the quantities on the right-hand side can be expressed uniquely as a linear functional of $f_{ij}$ . The quantities $f_{ij}^{\ \ TT}$ are the two transverse traceless components of $f_{ij}(f_{ij}^{\ \ TT}{}_{,j} \equiv 0, \; f_{ij}^{\ \ TT} \equiv 0)$ . The trace of the transverse part of $f_{ij}$ , i.e., $f^T$ , uniquely defines $f_{ij}^{\ \ T}$ according to

(28)

$f_{ij}^{\ \ T} \equiv \textstyle{\frac{1}{2}} [\delta_{ij} f^T -(1/\nabla^2)f^T_{\ \ ,ij}]$

(and clearly, $f_{ij}^{\ \ T}{}_{,j} = 0, \; f_{ii}^{\ \ T} = f^T$ . The operator $1/\nabla^2$ is the inverse of the flat space Laplacian, with appropriate boundary conditions. The longitudinal parts of $f_{ij}$ reside in the remaining part, $f_{i,j} + f_{j,i}$ . Decomposing $f_i$ into its transverse and longitudinal (curl-less) parts, one has $f_i = f_i\,\!^T + \textstyle{\frac{1}{2}} f^L\,\!_{,i} \; (f_i\,\!^T\,\!_{,i} \equiv 0)$ . The remainder then becomes $f_i\,\!^T\,\!_{,j} + f_j\,\!^T\,\!_{,i} + f^L\,\!_{,ij}$ . One may express $f_i, \; f^T$ and $f_{ij}\,\!^{TT}$ in terms of $f_{ij}$ by

(29)

$\begin{array}{lcl} f_i &=& (1/\nabla^2) [f_{ij,j} - \textstyle{\frac{1}{2}}\, (1/\nabla^2)f_{kj,kji}]\\ f^T &=& f_{ii} - (1/\nabla^2)f_{ij,ij}\\ f_{ij}\,\!^{TT} &=& f_{ij} - f_{ij}\,\!^T [f_{mn}] - \{ f_{i,j}[f_{mn}] + f_{j,i}[f_{mn}] \} \end{array}$

The foregoing orthogonal decomposition for a symmetric tensor is just the extension of the usual decomposition of a vector into longitudinal and transverse parts employed in electromagnetic theory.

Returning to (26), one has

(30)

$\begin{array}{rcl} - \nabla^2 g^T &=& {\mathcal P}_2\,\!^0 \\ - 2\nabla^2 (\pi^{iT} + \pi^L\,\!_{,i})&=& {\mathcal P}_2\,\!^i . \end{array}$

To first order, one sees that $g^T$ and $\pi^i$ vanish. These structures begin, therefore, at second order where $g^T = - (1/\nabla^2){\mathcal L}_{1{\rm in}}$ and $-2 (\pi^{iT} + \pi^L\,\!_{,i}) = (1/\nabla^2){\mathcal I}^{0i}_{1{\rm in}}$ . Here ${\mathcal H}_{1{\rm in}}$ , and ${\mathcal I}^{0i}_{1{\rm in}}$ are obtained from ${\mathcal P}^0_2$ and ${\mathcal P}^i_2$ by setting $g^T$ and $\pi^i$ equal to zero there. They are just the linearized theory's Hamiltonian and field momentum densities. The analysis has thus shown that the constraint equations can be solved for $g^T$ and $\pi^i$ (in terms of the remaining variables) as the four extra momenta. To see explicitly that ${\mathcal H}_{1{\rm in}}$ , and ${\mathcal I}^{0i}_{1{\rm in}}$ generate the appropriate time and space translations, one must return to the generator. Inserting the orthogonal decomposition (27) for both $g_{ij}$ and $\pi^{ij}$ into (24), one obtains

(31)

$G = \int d^3x [\pi^{ijTT} \delta g_{ij}\,\!^{TT} + \pi^{ijT} \delta g_{ij}\,\!^T + 2 (\pi^i\,\!_{,j} +\pi^j\,\!_{,i})\delta g_{i,j}] .$

The cross-terms in (31) have vanished due to the orthogonality of the decomposition (e.g., $\int d^3x\pi^{ijTT}\delta g_{i,j} = -\int d^3x \pi^{ijTT}\,\!_{,j} \delta g_i = 0$ ). We have also used here the fact that taking the variation of a quantity does not alter its transverse or longitudinal character in such a linear breakup and that the derivatives commute with the variation. Equation (31) may be brought to the desired form by further integration by parts and addition of a total variation:

(32)

$G = \int d^3x \{ \pi^{ijTT} \delta g_i\,\!^{TT} - (-\nabla^2 g^T) \delta [-(1/2\nabla^2)\pi^T] + \; [-2\nabla^2 (\pi^{iT} + \pi^L\,\!_{,i})]\} \delta g_i$

Equation (32) is now in the form of (7). The final step in reduction to canonical form is to impose coordinate conditions. The structure of (32) suggests that one choose as coordinate conditions

(33)

$\begin{array}{lcl} t&=& - (1/2\nabla^2)\pi^T\\ x^i &=& g_i \; . \end{array}$

Alternately, these coordinate conditions can be written in more conventional form by eliminating $\pi^T$ and $g_i$ via (29):

(34)

$\begin{array}{rcl} \pi^{ii}\,\!_{,jj} - \pi^{ij}\,\!_{,ij} &=& 0 \\ g_{ij,j} &=& 0 \; . \end{array}$

One can see that these coordinate conditions are acceptable by looking at those of the field equations that involve $\partial_tg_i$ and $\partial_t\pi^T$ . The linear part of equation (18) gives, as the equations for the longitudinal part of $g_{ij}$ ,

(35)

$\partial_t (g_{i,j} + g_{j,i}) = N_{i,j} + N_{j,i}\; .$

The Lagrange multipliers $N_i \equiv g_{0i}$ are functions determined only when coordinate conditions are imposed and must vanish at infinity where space is flat. Inserting (33) into (35) gives, consistent with the boundary conditions, $N_i = 0$ everywhere. Similarly, from (19), one has

(36)

$\partial_t [ - (1/2\nabla^2 ) \pi^T ] = N \; .$

Condition (33) implies $N\equiv (-g_{00})^{-1/2}=1$ , again consistent with the required asymptotic limit.

Alternately, one can see that equations (34) are physically appropriate coordinate conditions by a direct comparison with the known results of linearized theory. Thus, as mentioned above, ${\mathcal H}_{\rm 1in} = - \nabla^2g^T$ and ${\mathcal I}^{0i}_{\rm 1in} = - 2\nabla^2 (\pi^{iT} + \pi^L\,\!_{,i})$ are the linearized theory's Hamiltonian and momentum densities and so their coefficients in the generator (32) must be $\delta t$ and $\delta x^i$ respectively, in order that the form (25), be reproduced.

Since the generator is now

(37)

$G = \int d^3x [\pi^{ijTT} \,\delta g_{ij}\,\!^{TT} - {\mathcal H}_{\rm 1in} (\pi^{ijTT}, g_{ij}\,\!^{TT})\delta t + {\mathcal I}^{0i}_{\rm 1in} (\pi^{ijTT}, g_{ij}\,\!^{TT}) \delta x^i]$

the linearized theory has been put into canonical form, with $g_{ij}\,\!^{TT}$ and $\pi^{ijTT}$ as the two canonically conjugate pairs of variables.

We will now see the usefulness of the linearized theory in suggesting the choice of canonical variables for the full theory. Since the identification is made from the bilinear part of the Lagrangian $\pi^{ij} \partial_t g_{ij}$ , which is the same as for the linearized theory, the greater complexity of the full theory, i.e., its self-interaction, is to be found only in the non-linearity of the constraint equations. Even in the constraint equations, the linearized theory will guide us in choosing $g^T$ and $\pi^i$ as the four extra momenta to be solved for.

The full theory can now easily be put into canonical form. The generator of (32) is, of course, also correct for the full theory since it comes from the bilinear part of the Lagrangian. The constraint equations (20) now read (in a coordinate system to be specified shortly)

(38)

$\begin{array}{rcl} -\nabla^2 g^T &=& {\mathcal P}^0 [g_{ij}\,\!^{TT}, \pi^{ijTT}; \; g^T, \pi^i ; \; g_i , \pi^T ]\\ -2\nabla^2 (\pi^{iT} + \pi^L \,\!_{,i})& = &{\mathcal P}^i [g_{ij}\,\!^{TT}, \pi^{ijTT}; \; g^T, \pi^i ; \; g_i , \pi^T ] \end{array}$

where ${\mathcal P}^\mu$ are non-linear functions of $g_{ij}$ and $\pi^{ij}$ . One can again solve these (coupled) equations (at least by a perturbation-iteration expansion) for $g^T$ and $\pi^i$ . Thus, one can again choose $-\nabla^2g^T$ and $-2\nabla^2 (\pi^{iT} + \pi^L\,\!_{,i})$ as the four extra momenta to be eliminated. We denote the solutions of equations

(39)

$\begin{array}{rcl} -\nabla^2 g^T &=& -{\mathcal I}^0\,\!_0 [g_{ij}\,\!^{TT}, \pi^{ijTT}, \; g_i , \pi^T ]\\ -2\nabla^2 (\pi^{iT} + \pi^L \,\!_{,i}) &=& -{\mathcal I}^0\,\!_i [g_{ij}\,\!^{TT}, \pi^{ijTT}, \; g_i , \pi^T ] \; . \end{array}$

These equations are the counterpart of $p_{M+1} = -H$ in the particle case.

As we have seen, the four constraint equations are maintained in time as a consequence of the other field equations. Hence, after inserting equations (39) into (18, 19), one finds that four of these twelve (those for $\partial_t g^T$ and $\partial_t\pi^i$ ) are "Bianchi" identities, leaving eight independent equations in the twelve variables $g_{ij}\,\!^{TT}, \pi^{ijTT}, \; g_i , \pi^T$ and $N$ , and $N_i$ . These equations are linear in the time derivatives of the first eight variables.

We now impose the coordinate conditions (33, 34) which determine $\pi^T$ and $g_i$ . The $\partial_t g_i$ and $\partial_t\pi^T$ equations become determining equations for $N$ and $N_i$ [the full theory's analog of (35) and (36)]. $N$ and $N_i$ are no longer 1 and 0 respectively, but now become specific functionals of $g_{ij}\,\!^{TT}$ and $\pi^{ijTT}$ , which could (in principle) be calculated explicitly. In the last four equations, then, $N$ and $N_i$ may, in principle, be eliminated, leaving a system of four equations involving only $g_{ij}\,\!^{TT}$ and $\pi^{ijTT}$ , and linear in their time derivatives. We will now see that this reduced system is in Hamiltonian form.

The generator (32) reduces to canonical form [with coordinate, conditions (33, 34) imposed and constraints (39) inserted]:

(40)

$G = \int d^3x \, [ \pi^{ijTT} \; \delta g_{ij}\,\!^{TT} + {\mathcal I}^0\,\!_0 \delta t + {\mathcal I}^0\,\!_i \delta x^i ]$

while the Lagrangian now becomes

(41)

${\mathcal L} = \pi^{ijTT} \; \partial_t g_{ij}\,\!^{TT} + {\mathcal I}^0\,\!_0 \; .$

It can be shown that the solutions ${\mathcal I}^0\,\!_\mu$ of the constraint equations do not depend explicitly on the coordinates $x^\mu$ of (33, 34) (see III). This is not unexpected, since the variables $g_{ij}$ and $\pi^{ij}$ appearing on the right-hand side of (38) do not depend explicitly on the coordinates in this frame. (Thus, only $g_{i,j} = x^i\,\!_{,j} = \delta^i\,\!_j$ and $\pi^T = -2\nabla^2 t = 0$ appear in $g_{ij}$ and $\pi^{ij}$ .)

With the generator now in canonical form, we can immediately write down the fundamental equal time P.B. relations for $g_{ij}\,\!^{TT}$ and $\pi^{ijTT}$ . These are

(42)

$\begin{array}{rcl} \ [g_{ij}\,\!^{TT} ({\mathbf x}), \pi^{mnTT}({\mathbf x}^\prime )]&=& \delta^{mn}\,\!_{ij} ({\mathbf x} -{\mathbf x}^\prime )\\ \ [ g_{ij}\,\!^{TT} ({\mathbf x}), g_{mn}\,\!^{TT}({\mathbf x}^\prime )] \; = 0 &=& [\pi^{ijTT} ({\mathbf x}), \pi^{mnTT} ({\mathbf x}^\prime )]\; . \end{array}$

The $\delta^{mn}\,\!_{ij} ({\mathbf x})$ in (42) is a conventional Dirac $\delta$ -function modified in such a way that the transverse-traceless nature of the variables on the left-hand side is not violated. Note that the definition of this modified $\delta$ -function does not depend on the metric; it is symmetric, transverse, and traceless on each pair of indices:

(43)

$\begin{array}{lcl} &&\delta^{mn}\,\!_{ij} = \delta^{nm}\,\!_{ij} = \delta^{mn}\,\!_{ji} = \delta^{ij}\,\!_{mn}\\ &&\quad\quad\delta^{mm}\,\!_{ij} = 0 = \delta^{mn}\,\!_{ii}\\ &&\quad\quad\quad \delta^{mn}\,\!_{ij,j} = 0 \; . \end{array}$

From the form of equations (40, 41), one also has the P.B. equations of motion:

(44)

$\begin{array}{lcl} \partial_t g_{ij}\,\!^{TT} &=& [g_{ij}\,\!^{TT}, H] =\delta H/\delta\pi^{ijTT}\\ \partial_t \pi^{ijTT} &=& [\pi^{ijTT}, H] = -\delta H/\delta g_{ij}\,\!^{TT} \end{array}$

where $H \equiv - \int d^3x {\mathcal I}^0\,\!_0$ is the Hamiltonian. The last equalities in equations (44) follow from equations (42). That (42) and (44) are consistent with the Lagrangian equations obtained by varying (41) is now immediate. Corresponding to the time translation equations (44), one also has for spatial displacements

(45)

$\begin{array}{lcl} \partial_k g_{ij}\,\!^{TT} = [P_k , g_{ij}\,\!^{TT}] = -\delta P_k/ \delta\pi^{ijTT} \\ \partial_k \pi^{ijTT} = [P_k , \pi^{ijTT}] = \delta P_k/\delta g_{ij}\,\!^{TT} \end{array}$

where $P_k \equiv + \int {\mathcal I}^0\,\!_k d^3x$ is the total momentum operator. With the canonical momentum $P_k\,\!^c \equiv \int d^3x {\mathcal I}^0\,\!_k\,\!^c \equiv -\int d^3x \pi^{mnTT} g_{mn}\,\!^{TT}\,\!_{,k}$ , equations (45) are obvious.

This completes the (compressed) analysis of General Relativity a la ADM. Subsequent work has applied it to a manifold set of problems, ranging from post-Newtonian motion, gravitational radiation and its detection, to cosmology and supergravity.

This article summarizes original collaborative research with R. Arnowitt and C.W. Misner. Its preparation was supported by NSF grant PHY 07-57190 and DOE-FG-02-92ER40701.

REFERENCES

1. R. Arnowitt, S. Deser, and C.W. Misner, The Dynamics of General Relativity in Gravitation: An introduction to current research, L. Witten, ed (Wiley NY 1962) ; reprinted as gr-gc/0465109.

2. L.F. Abbott and S. Deser, Stability of Energy with a Cosmological Constant. Nucl Phys., B195, 76, (1982).

Invited by:

Dr. Riccardo Guida, Institut de Physique Théorique, CEA, IPhT; CNRS, Gif-sur-Yvette, France

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Categories: Physics | Space-Time and Gravitation