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Tagged with adm-formalism constrained-dynamics
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Solving the Hamiltonian constraint equation in General Relativity
The constraint equation of general relativity reads as follows
\begin{align}
\frac{(2\kappa)}{\sqrt{h}}\left(h_{ac} h_{bd} - \frac{1}{D-1} h_{ab} h_{cd} \right)p^{ab} p^{cd} - \frac{\sqrt{h}}{(2\...
- 682
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Lagrange Multipliers in ADM formalism
I'm following this lecture notes (https://javierrubioblog.com/wp-content/uploads/2017/08/adm1.pdf) on ADM formalism. After getting action as (see Eq.(5.41))
$$
S=\int\mathrm{d}^4xN\sqrt{\gamma}\left[\...
- 139
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What diffeomorphism does the Hamiltonian constraint generate?
Consider the Hamiltonian constraint $\mathcal H(x)$ in the ADM formalism. What diffeomorphism does this generate?
- 742
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Composition of diffeomorphisms in the ADM formalism?
In the ADM formalism there are 4 constraints: $C_\mu(x)$, which are known as the Hamiltonian and spatial diffeomorphism constraints. In the quantum theory, $C_\mu(x)$ are the generators of ...
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Are the Hamiltonian and spatial diffeomorphism constraints satisfied at all times for GR?
The ADM formulation of GR allows the Einstein equations to be recast as an initial value problem. According to Sec 3.3 of these notes the outline of the procedure is as follows:
Pick a 3-metric $h_{...
- 742
2
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Topology change and Canonical Formulation?
In the ADM formulation of general relativity it is assumed that the spacetime topology is $\Bbb{R}\times \Sigma$. Suppose I wanted to consider spacetimes that undergo topology change with foliation ...
- 749
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How to derive the Hamiltonian of general relativity (ADM formalism without surface terms)?
Given that $$ds^{2} =−N^{2}dt^{2} +h_{ij}(dx^{i} +N^{i}dt)(dx^{j}+N^{j}dt)$$
$$S=\int dt d^{3} x\sqrt{h} N(^{3}R+K_{ij}+K^{ij}-K^{2})$$
where $^{3}R$ is the Ricci scalar of $hij$, $h$ the ...
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