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A curved temporal chart gives a local reduced Hamiltonian

Statement

Let $m \geq 0$ and $c > 0$ . Assume a temporal product chart with lapse-shift data $(N, β, h)$ as in Temporal chart with lapse, shift and positive slice metric, all four coordinates measured in metres. Write $g_{a b} = g (\partial_{a}, \partial_{b})$ for $a, b \in {0, 1, 2, 3}$ and let $g^{a b}$ denote the entries of the inverse matrix; then $p^{♯} : = \sum_{a, b = 0}^{3} g^{a b} p_{b} \partial_{a}$ is the unique vector satisfying $g (p^{♯}, w) = p (w)$ for every tangent vector $w$ . Put $T = {t \in R : c t \in I}$ . In every reduced expression below, $N$ , $β^{i}$ , $h_{i j}$ and $h^{i j}$ mean their pullbacks to $T \times Ω$ : $N (c t, q)$ , $β^{i} (c t, q)$ , $h_{i j} (c t, q)$ and $h^{i j} (c t, q)$ , respectively. Thus $κ$ , $p_{0}$ and $H$ are functions of $(t, q, p)$ on the massive reduced domain, or on its nonzero-spatial-momentum part when $m = 0$ ; the spacetime metric and shell covector are evaluated at $(c t, q)$ . On the future shell $C_{m} = \frac{1}{2} (g^{a b} p_{a} p_{b} + m^{2} c^{2}) = 0$ , define

$κ = \sqrt{m^{2} c^{2} + h^{i j} p_{i} p_{j}}, p_{0} = β^{i} p_{i} - N κ, H (t, q, p) = - c p_{0} = c (N κ - β^{i} p_{i}) .$

For $m = 0$ , require $p_{i} \neq 0$ . Let $J = [t_{1}, t_{2}]$ satisfy $t_{1} < t_{2}$ and $[c t_{1}, c t_{2}] \subset I$ , where $I$ is the first-coordinate interval of the temporal product chart; require the path $t \mapsto (c t, q (t))$ to stay in the chart. Take $q^{i}, p_{i}$ smooth on a neighborhood of $J$ and vary them by smooth one-parameter families that keep the paths in the chart, fix $q$ at both endpoints and vary $p$ freely; in the massless case the paths and sufficiently small variations stay in $p_{i} \neq 0$ . Then the gauge $x^{0} = c t$ reduces the first-order action to $\int_{t_{1}}^{t_{2}} (p_{i} {\dot{q}}^{i} - H) d t$ , and its variational equations on the interior of $J$ are exactly ${\dot{q}}^{i} = H_{p_{i}}$ , ${\dot{p}}_{i} = - H_{q^{i}}$ . The construction is local to this chart and intervals transverse to its time slices; no global curved-space gauge or quotient is asserted.

Facts & Assumptions

Given: A smooth metric in the specified lapse-shift form, $N > 0$ , positive definite $h$ , $m \geq 0$ , $c > 0$ , and the future shell.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Assume countable choice ACω, meaning every sequence of nonempty sets has a choice function; no full Axiom of Choice is assumed.

[F1]

Temporal chart with lapse, shift and positive slice metric: $g = - N^{2} (d x^{0})^{2} + h_{i j} (d x^{i} + β^{i} d x^{0}) (d x^{j} + β^{j} d x^{0})$ , with $x^{0}$ temporal and $h > 0$ .

[F2]

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model supplies the fixed smooth time-oriented Lorentzian four-manifold and metric with signature $(-, +, +, +)$ .

[F3]

First-order shell action yields massive and null geodesics: the first-order equations follow by independent variation.

[F4]

Canonical Poisson brackets and local Hamiltonian flow: in canonical coordinates, variational equations are Hamilton's equations with the stated sign convention.

[F5]

A regular level set is an embedded submanifold: a regular shell level is embedded.

[F6]

Pairing of an observer with future causal vectors: for a future unit observer $n$ and nonzero future-causal $p^{♯}$ , $p (n) = g (p^{♯}, n) < 0$ .

Proof

1.1givenA1F1F2F6algebra

Use the supplied smooth tangent/cotangent structures under [A1] and the Lorentzian temporal-chart data [F1,F2]. Expand the metric matrix and invert its block form. Direct multiplication yields $g^{00} = - N^{- 2}$ , $g^{0 i} = β^{i} / N^{2}$ , $g^{i j} = h^{i j} - β^{i} β^{j} / N^{2}$ . Therefore $g^{a b} p_{a} p_{b} = - \frac{(p_{0} - β^{i} p_{i})^{2}}{N^{2}} + h^{i j} p_{i} p_{j} .$ The shell equation gives $p_{0} - β^{i} p_{i} = \pm N κ$ . The future normal $n = N^{- 1} (\partial_{0} - β^{i} \partial_{i})$ has $p (n) = (p_{0} - β^{i} p_{i}) / N < 0$ by [F6] for a nonzero future-causal shell covector, so the future branch is the negative root $p_{0} = β^{i} p_{i} - N κ$ . Positive $h$ implies κ>0 except the excluded massless zero momentum.

2.1givenF1F5step 1.1

The fibre derivative of $C_{m}$ with respect to $p_{0}$ is $g^{0 b} p_{b} = κ / N > 0$ on this branch. Thus the root is smooth and the shell is locally a graph over $(t, q^{i}, p_{i})$ ; [F5] also gives the embedded regular-level structure.

2.2givenF3step 1.1

On $x^{0} = c t$ , the canonical term is $p_{0} c + p_{i} {\dot{q}}^{i} = p_{i} {\dot{q}}^{i} - H$ . The constraint term vanishes on the solved shell, so the reduced action has the claimed integrand.

3.1givenF4step 2.1step 2.2∎

Applying the fixed-endpoint first-variation equations to the reduced canonical action gives ${\dot{q}}^{i} = \partial H / \partial p_{i}$ , ${\dot{p}}_{i} = - \partial H / \partial q^{i}$ ; by [F4] these are Hamilton's equations for this time-dependent chart Hamiltonian. Smooth root and $N > 0$ ensure the gauge parameter $t = x^{0} / c$ is valid on the stated interval; nothing here makes it global outside the chart.

Depends on

Dependency tree · 2 levels

Sources

Eric Gourgoulhon, 3+1 Formalism and Bases of Numerical Relativity, complete lecture notes (arXiv:gr-qc/0703035)