ADM formulation and geometrodynamics

   

This is the third post of a series of four posts concerned with the canonical quantization of general relativity.

 

ADM formulation and geometrodynamics

Spacetime is naturally assumed to be globally hyperbolic which means that it is diffeomorphic to the direct product $\mathbb{R}\times \Sigma$ where $\Sigma$ is a three-dimensional smooth manifold.

We consider then a foliation of the spacetime manifold given by the spatial Cauchy  hypersurfaces $\Sigma_t$ of constant time $t$.  Let $n^{\mu}$ be the unit normal vector field to the hypersurfaces  $\Sigma_t$ which is given explicitly by
\begin{eqnarray}
n^{\mu}=-N\frac{\partial t}{\partial x_{\mu}}~,~n_{\mu}n^{\mu}=-1.
\end{eqnarray}
The normalization $N$ is the lapse function which measures the rate of change of the proper time with respect to the coordinate time $t$ as one moves normally to the hypersurfaces $\Sigma_t$. It  is given explicitly by
\begin{eqnarray}
N=-g_{\mu\nu}t^{\mu}n^{\nu}.\label{lapse}
\end{eqnarray}
The time flow in this foliated spacetime will be given by a vector field $t^{\mu}$ which satisfies $t^{\mu}\nabla_{\mu}t=1$, i.e.
 \begin{eqnarray}
t^{\mu}=\frac{\partial x^{\mu}}{\partial t}.
\end{eqnarray}
We decompose $t^{\mu}$ into its normal and tangential parts with respect to the hypersurface  $\Sigma_t$ as
 \begin{eqnarray}
t^{\mu}=Nn^{\mu}+N^{\mu}~,~N^{\mu}=N^ie_i^{\mu}.
\end{eqnarray}
The $e_i^{\mu}$ are  tangent vectors to the hypersurface  $\Sigma_t$ given by
\begin{eqnarray}
e_i^{\mu}=\frac{\partial x^{\mu}}{\partial y^i}.
\end{eqnarray}
The $y^i$ are coordinates on the hypersurface $\Sigma_t$, i.e. the coordinates $x^{\mu}$ are split as $x^{\mu}\longrightarrow y^{\mu}=(t,y^i)$, and $N^i$ is the so-called shift vector which measures the shift of the local spatial coordinate system as one moves normally to the hypersurfaces $\Sigma_t$. It is  given by
\begin{eqnarray}
N^{\mu}=h^{\mu}~_{\nu}t^{\nu}.\label{shift}
\end{eqnarray}
The three-dimensional metric $h_{ij}$ is the induced metric on the  hypersurface $\Sigma_t$ given explicitly by
\begin{eqnarray}
h_{ij}=g_{\mu\nu}e_i^{\mu}e_j^{\nu}=h_{\mu\nu}e_i^{\mu}e_j^{\nu}~,~h_{\mu\nu}=g_{\mu\nu}+n_{\nu}n_{\nu}.
\end{eqnarray}
We define the inverse metric $h^{ij}$ in the usual way, viz $h_{ij}h^{jk}=\delta_i^k$.  More precisely, we compute (using also $h^{\mu\nu}=g^{\mu\nu}-n^{\mu\nu}$) the result $h_{\mu\nu}h^{\nu\alpha}=\delta_{\mu}^{\alpha}+n_{\mu}n^{\alpha}$.

We compute immediately that
\begin{eqnarray}
dx^{\mu}&=&\frac{\partial x^{\mu}}{\partial t} dt+\frac{\partial x^{\mu}}{\partial y^i}dy^i\nonumber\\
&=&t^{\mu}dt+e_i^{\mu}dy^i\nonumber\\
&=&(N dt) n^{\mu}+(dy^i+N^i dt) e_i^{\mu}.
\end{eqnarray}
Also
\begin{eqnarray}
ds^2&=&g_{\mu\nu}dx^{\mu}dx^{\nu}\nonumber\\
&=&g_{\mu\nu}\bigg[N^2dt^2n^{\mu}n^{\nu}+(dy^i+N^idt)(dy^j+N^jdt)e_i^{\mu}e_j^{\nu}\bigg]\nonumber\\
&=&-N^2dt^2+h_{ij}(dy^i+N^idt)(dy^j+N^jdt).\label{4dmetric1}
\end{eqnarray}
The ADM metric is then given explicitly by (with $N_i=h_{ij}N^j$)
\begin{eqnarray}
g_{\mu\nu}=\left( \begin{array}{cc}
-N^2+N^iN_i &  N_j \\
N_i & h_{ij}
\end{array} \right).
\end{eqnarray}
The inverse ADM metric $g^{\mu\nu}$ is then given by
\begin{eqnarray}
g^{\mu\nu}=\left( \begin{array}{cc}
-\frac{1}{N^2} &  \frac{1}{N^2}N^j \\
  \frac{1}{N^2}N^i & h^{ij}-\frac{1}{N^2}N^iN^j
\end{array} \right).
\end{eqnarray}
We conclude that all information about the original four-dimensional metric $g_{\mu\nu}$ is contained in the lapse function $N$, the shift vector $N^i$ and the three-dimensional metric $h_{ij}$. The lapse and the shift $N$ and $N^i$ are not dynamical variables but only Lagrange multipliers yielding under their respective variation the so-called Hamiltonian and diffeomorphism constraints which satisfy a Dirac algebra of first class constraints. A particular choice of $N$ and $N^i$, i.e. a particular choice of  foliation plays the role of a gauge fixing condition (called the time gauge) for the diffeomorphism group. In other words, invariance under general coordinate transformations which form the group of diffeomorphism  is not lost but only fixed and in fact the diffeomorphism invariance of the theory is still precisely encoded in the Dirac algebra of the first class constraints of the theory.

Next we would like to rewrite the Hilbert-Einstein Lagrangian density  in terms of the three-dimensional quantities $N$,  $N^i$ and $h_{ij}$ and then compute the Hamiltonian density.

First we compute
\begin{eqnarray}
\sqrt{-g}d^4x=N\sqrt{h}d^4y.
\end{eqnarray}
A central object in the discussion of how the hypersurfaces $\Sigma_t$ are embedded in the four-dimensional spacetime manifold ${\cal M}$ is the extrinsic curvature $K_{\mu\nu}$. This is given essentially by $1)$ comparing the normal vector $n_{\mu}$ at a point $p$ and the parallel transport of the normal vector $n_{\mu}$ at a nearby point $q$ along a geodesic connecting $q$ to $p$  on the hypersurface $\Sigma_t$ and then $2)$ projecting the result onto the hypersurface $\Sigma_t$. The first part is clearly given by the covariant derivative whereas the projection is done through the three-dimensional metric tensor. Hence the extrinsic curvature must be defined by
\begin{eqnarray}
K_{\mu\nu}&=&-h_{\mu}^{\alpha}h_{\nu}^{\beta}\nabla_{\alpha}n_{\beta}\nonumber\\
&=&-h_{\mu}^{\alpha}\nabla_{\alpha}n_{\nu}.
\end{eqnarray}
In the second line of the above equation we have used $n^{\beta}\nabla_{\alpha}n_{\beta}=0$ and $\nabla_{\alpha}g_{\mu\nu}=0$. We can check that $K_{\mu\nu}$ is symmetric and tangent, viz
\begin{eqnarray}
K_{\mu\nu}=K_{\nu\mu}~,~h^{\alpha}_{\mu}K_{\alpha\nu}=K_{\mu\nu}.\label{admex}
\end{eqnarray}
 The next goal is to compute in terms of the three-dimensional quantities the scalar curvature $R$.  We start from (where $G$ is the Einstein tensor $G_{\mu\nu}=R_{\mu\nu}-Rg_{\mu\nu}/2$)
\begin{eqnarray}
R&=&-Rg_{\mu\nu}n^{\mu}n^{\nu}\nonumber\\
&=&-2(R_{\mu\nu}-G_{\mu\nu})n^{\mu}n^{\nu}\nonumber\\
&=&-2R_{\mu\nu}n^{\mu}n^{\nu}+R_{\mu\nu\alpha\beta}h^{\mu\alpha}h^{\nu\beta}.
\end{eqnarray}
We compute
\begin{eqnarray}
R_{\mu\nu\alpha\beta}h^{\mu\alpha}h^{\nu\beta}&=&h_{\beta\rho}R_{\mu\nu\alpha}~^{\rho}h^{\mu\alpha}h^{\nu\beta}\nonumber\\
&=&g^{\beta\eta}g^{\kappa\sigma}\big(h_{\kappa}^{\mu}h_{\eta}^{\nu}h_{\sigma}^{\alpha}R_{\mu\nu\alpha}~^{\rho}h_{\rho}^{\theta}\big)h_{\theta\beta}\nonumber\\
&=&g^{\beta\eta}g^{\kappa\sigma}\big(^{(3)}R_{\kappa\eta\sigma}~^{\theta}+K_{\kappa\sigma}K_{\eta}^{\theta}-K_{\eta\sigma}K_{\kappa}^{\theta}\big)h_{\theta\beta}\nonumber\\
&=&g^{\kappa\sigma}\big(^{(3)}R_{\kappa\eta\sigma}~^{\theta}+K_{\kappa\sigma}K_{\eta}^{\theta}-K_{\eta\sigma}K_{\kappa}^{\theta}\big)h_{\theta}^{\eta}\nonumber\\
&=&^{(3)}R+K^2-K_{\mu\nu}K^{\mu\nu}.
\end{eqnarray}
In the third line we have used the first Gauss–Codacci relation. Next we compute
\begin{eqnarray}
R_{\mu\nu}n^{\mu }n^{\nu}
&=&\nabla_{\mu}(Kn^{\mu}+n^{\nu}\nabla_{\nu}n^{\mu})-K_{\mu\nu}K^{\mu\nu}+K^2.
\end{eqnarray}
The first term is a total divergence and hence it can be neglected. We get then the so-called ADM Lagrangian density
\begin{eqnarray}
{\cal L}_{\rm ADM}&=&\sqrt{-g}R\nonumber\\
&=&\sqrt{h} N(^{(3)}R-K^2+K_{\mu\nu}K^{\mu\nu}).\label{ADM}
\end{eqnarray}
In the above equation $K^2=(h_{\mu\nu}K^{\mu\nu})^2$. The extrinsic curvature $K_{\mu\nu}$ is the covariant analogue of the time derivative of the metric. Indeed, by using the concept of the Lie derivative we can show after some more steps that  (where $D_{\mu}$ is the three-dimensional covariant derivative)
\begin{eqnarray}
K_{\mu\nu}
&=&-\frac{1}{2N}\big(\dot{h}_{\mu\nu}-D_{\mu}N_{\nu}-D_{\nu}N_{\mu}\big).
\end{eqnarray}
It is straightforward now to compute the conjugate momentum  $\Pi^{\mu\nu}$ corresponding to the metric $h_{\mu\nu}$. We find
\begin{eqnarray}
\Pi_{\mu\nu}&=&\frac{{\cal L}_{\rm ADM}}{\partial \dot{h}_{\mu\nu}}\nonumber\\
&=&-\sqrt{h}(K_{\mu\nu}-Kh_{\mu\nu}).
\end{eqnarray}
From this identity we can show that $h_{\mu\nu}\Pi^{\mu\nu}=2\sqrt{h}h_{\mu\nu}K^{\mu\nu}$.

The ADM Hamiltonian density is then given by
\begin{eqnarray}
{\cal H}_{\rm ADM}&=&\dot{h}^{\mu\nu}\Pi_{\mu\nu}-{\cal L}_{\rm ADM}\nonumber\\
&=&-\sqrt{h}N^{(3)}R+2D_{\mu}N_{\nu}.\Pi^{\mu\nu}+\frac{N}{\sqrt{h}}(\Pi_{\mu\nu}\Pi^{\mu\nu}-\frac{1}{2}\Pi^2)\nonumber\\
&=&\sqrt{h}N\bigg[-^{(3)}R+\frac{\Pi_{\mu\nu}\Pi^{\mu\nu}}{h}-\frac{\Pi^2}{2h}\bigg]+\sqrt{h}N_{\nu}\bigg[-2D_{\mu}\bigg(\frac{\Pi^{\mu\nu}}{\sqrt{h}}\bigg)\bigg]\nonumber\\
&=&\sqrt{h}NH_0+\sqrt{h}N_{i}H^{i}.
\end{eqnarray}
In the last two lines we have dropped a total divergence since it only leads to a  boundary term which is assumed to be negligible for large spatial surfaces encompassing spacetime. Indeed, the corresponding ADM Lagrangian density takes  the form
\begin{eqnarray}
{\cal L}_{\rm ADM}
&=&\dot{h}_{\mu\nu}\Pi^{\mu\nu}-\sqrt{h}NH_0-\sqrt{h}N_{i}H^{i}.
\end{eqnarray}
The Hamiltonian is then obtained by integrating the Hamiltonian density over the hypersurface $\Sigma_t$. We get
  
\begin{eqnarray}
H_{\rm ADM}&=&\int d^3y{\cal H}_{\rm ADM}\nonumber\\
&=& \int d^3y \sqrt{h}NH_0+\int d^3y \sqrt{h}N_{i}H^{i}\nonumber\\
&=&H(N)+D(\vec{N}).
\end{eqnarray}
Finally by varying the Lagrangian density with respect to the lapse function $N$ and the shift vector $N^{\mu}$ we obtain the Hamiltonian and diffeomorphsim first class constraints given respectively by
\begin{eqnarray}
H_0\equiv -^{(3)}R+\frac{\Pi_{\mu\nu}\Pi^{\mu\nu}}{h}-\frac{\Pi^2}{2h}=0.
\end{eqnarray}
And
\begin{eqnarray}
H_i\equiv -2D_{\mu}\bigg(\frac{\Pi^{\mu\nu}}{\sqrt{h}}\bigg)=0.
\end{eqnarray}
This vanishing should be properly understood not as identically vanishing but as weakly vanishing in the sense of Dirac, i. e. it vansihes only on physical states not any state. We have then the constraints

\begin{eqnarray}
H(N)\simeq 0.
\end{eqnarray}
And
\begin{eqnarray}
D(\vec{N})\simeq 0.
\end{eqnarray}
The Hamiltonian constraint  $H(N)\simeq 0$ constrains the Hamiltonian (and it generates the time flow of the theory which connects different hypersurfaces $\Sigma_t$) whereas the diffeomorphism constraint $D(\vec{N})\simeq 0$ constrains the momentum of the theory (and generates diffeomorphism transformations on the hypersurfaces $\Sigma_t$ themselves). These constraints are first class which means that they do close under the Poisson brackets, i.e. their Dirac algebra is given by \cite{Thiemann:2006cf}
\begin{eqnarray}
&&\{D(\vec{N}),D(\vec{N}^{\prime})\}=8\pi GD({\cal L}_{\vec{N}}\vec{N}^{\prime})\nonumber\\
&&\{D(\vec{N}),H({N}^{\prime})\}=8\pi G H({\cal L}_{\vec{N}}{N}^{\prime})\nonumber\\
&&\{H({N}),H({N}^{\prime})\}=8\pi G D(q^{-1}(NdN^{\prime}-N^{\prime}dN)).
\end{eqnarray}
In the above equation  ${\cal L}$ is the Lie derivative and $q$ is the pullback metric, i.e. $q\equiv h$. This algebra is universal in the sense that it encodes in a precise sense the diffeomorphism invariance of the theory (despite the explicit choice of the foliation, i.e. the explicit choice of the lapse function $N$ and the shift vector $N^i$ which should only be viewed as a gauge fixing choice).

In fact any theory characterized by invariance under general coordinate transformations will contain Hamiltonian and diffeomorphism constraints satisfying precisley the above Dirac algebra (this always comes about from the arbitrary nature of the foliation and the associated arbirary choice of the lapse function $N$ and shift vector $N^i$  which necessarily appear as Lagrange multipliers in the action with singular Legender transformation).

We remark that the Hamiltonian vansihes also weakly which is also another universal property of theories with diffeomorphism invariance which is the fact that there is no Hamiltonian in the dynamics of these theories (since there is no time really!!)  but only Hamiltonian constraint.

In summay, the phase space of the Hamiltonian formulation of general relativity consists  therefore of all pairs $(h_{ij},\Pi^{kl})$ where the extrinsic curvature $K^{kl}$ stands in place of the momentum $\Pi^{kl}$ through the relation $\Pi_{\mu\nu}=-\sqrt{h}(K_{\mu\nu}-Kh_{\mu\nu})$. The fundamental Poisson bracket is given by
\begin{eqnarray}
\{h_{ij}(t,\vec{y}_1),\Pi^{kl}(t,\vec{y}_2)\}= \delta^3(\vec{y}_1-{y}_2)\delta_{i}^k\delta_j^l.
\end{eqnarray}
The starting point of the canonical quantization program of geometrodynamics is then the commutation relations
\begin{eqnarray}
[\hat{h}_{ij}(t,\vec{y}_1),\hat{\Pi}^{kl}(t,\vec{y}_2)]= i\hbar \delta^3(\vec{y}_1-\delta{y}_2)\delta_{i}^k\delta_j^l.
\end{eqnarray}
The operators  $\hat{h}_{ij}(t,\vec{y}_1)$ and $\hat{\Pi}^{kl}(t,\vec{y}_2)$ are defined on physical states $\Psi(h_{ij})$ by
\begin{eqnarray}
&&\hat{h}_{ij}\Psi(h_{ij})={h}_{ij}\Psi(h_{ij})\nonumber\\
&&\hat{\Pi}^{kl}\Psi(h_{ij})=-i\hbar\frac{\delta}{\delta h_{kl}}\Psi(h_{ij}).
\end{eqnarray}
The physical states $\Psi(h_{ij})$ are thoses states in the Hilbert space which are annihilated by the Hamiltonian and diffeomorphism constraints which are also implemented as  operators, namely

\begin{eqnarray}
\hat{H}(N)\Psi(h_{ij})= 0.
\end{eqnarray}
And
\begin{eqnarray}
\hat{D}(\vec{N})\Psi(h_{ij})= 0.
\end{eqnarray}