All Questions
Tagged with adm-formalism metric-tensor
7
questions
1
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108
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What is the determinant of the Wheeler-DeWitt metric tensor constructed from spatial metrics in ADM formalism?
The Hamiltonian constraint of General relativity has the following form
\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}}{...
- 682
1
vote
1
answer
151
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Star Radius in the Oppenheimer-Snyder metric using ADM formalism
I'm working with gravitational collapse models, in particular with the Oppenheimer-Snyder model.
Short list of the assumptions for those unfamiliar with the model, you have a spherical symmetric ...
- 1,611
5
votes
1
answer
141
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Canonical Commutation relations in gravity
The canonical commutation relations in gravity are sometimes written
$$
[\gamma_{ij}(x),\pi^{kl}(y)]=\frac{i\hbar}{2}(\delta_i^k\delta_j^l+\delta_i^l\delta_j^k)\delta^3(x-y),\tag{0}
$$
where $\gamma_{...
- 742
1
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How to compute the scalar $^{(4)}R_{\mu\nu} \; ^{(4)}R^{\mu\nu}$ in the ADM formalism in General Relativity?
In the Arnowitt-Deser-Misner (ADM) formalism in General Relativity, the line element takes the form
$$
ds^2 = - N^2 dt^2 + \gamma_{ij} ( N^i dt + dx^i) (N^j dt + dx^j) \ ,
$$
where $\gamma^{ij}$ is ...
- 3,719
3
votes
2
answers
453
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ADM formulation of GR derivative on the 3-metric
In the ADM formalism where the projector is given by ${P^\mu}_\alpha={\delta^\mu}_\alpha+n^\mu{n}_\alpha$ and $n^\alpha$ is a future pointing normal vector to the constant time hypersurface $\Sigma$. ...
- 311
2
votes
2
answers
857
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Determinant of ADM metric
I am studying inflation and for the calculation of the bispectrum we are using the ADM formalism where the metric is the following form:
$$g_{\mu\nu}=\begin{bmatrix}-N^2+N^iN_i&N_i\\N_i&h_{ij}...
- 311
1
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0
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How to derive the Hamiltonian of general relativity (ADM formalism without surface terms)?
Given that $$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 ...
- 27