All Questions
Tagged with adm-formalism differential-geometry
18
questions
1
vote
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
0
votes
0
answers
97
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Extrinsic curvature, Gauss equation and Christoffel symbol contribution
This question is in the context of geometry of hypersurfaces and ADM formalism. In a $4$-dimensional manifold, we define a $3$-hypersurface with space-like tangent basis $e_a$, $a=1,2,3$, and a normal ...
- 175
1
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0
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44
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Scalar curvature in ADM Formalism (coordinate to coordinate-free transition)
I am attempting to express the scalar curvature in a coordinate-independent manner. Following the works of Bojowald, Thiemann, we have:
$$ {}^{(4)}R= {}^{(3)}R+K_{a b}K^{a b}- (K_a^a)^2 - 2\nabla_a v^...
- 403
0
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1
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201
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What diffeomorphism does the Hamiltonian constraint generate?
Consider the Hamiltonian constraint $\mathcal H(x)$ in the ADM formalism. What diffeomorphism does this generate?
- 742
2
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0
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51
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Why no cosmological constant in momentum constraint?
In the ADM formalism of general relativity, one decomposes the Einstein equations in (3+1) dimensions. More explicitely, if the Einstein equations are given by
$$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\...
- 854
2
votes
1
answer
141
views
Static Spacetime = no cosmological constant?
I stumbled over a strange result, which cannot be true: In the (3+1)-formulation of general relativity, one considers a metric of the type
$$g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=(-\alpha^{2}+\...
- 854
1
vote
0
answers
115
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Different definitions of lapse functions?
I stumbled over two different definitions of the lapse function in the ADM formalism and wanted to convince myself that they are the same by using a simple example. However, I get different results, ...
- 854
2
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0
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81
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How to prove ADM mass is independent of the foliation considered?
ADM mass for an asymptotically-flat
spacetime can be defined as (Poisson, Eric. A relativist's toolkit. p.147):
$$M=-\frac{1}{8 \pi} \lim _{S_t \rightarrow \infty} \oint_{S_t}\left(k-k_0\right) \sqrt{\...
- 95
0
votes
1
answer
391
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Extrinsic Curvature expression (ADM Formalism)
I'm reading The ADM Formalism chapter of Baez's book Gauge Fields, Knots and Gravity and on page 429 we have the expression
$$ K_{ij}=\frac{1}{2}N^{-1}(\dot{q}_{ij}- {}^3\nabla_i N_j - {}^3\nabla_j ...
- 403
1
vote
1
answer
72
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Basis vector as "Array" choice in tensor calculus / GR, and 3+1 decomposition
In differential geometry and general relativity, once we have chosen a basis on our spacetime, say $ \{t,r,\theta, \phi \} $, we can represent every tensor as an "array" of numbers, so a &...
- 163
4
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0
answers
165
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Why do the definitions of ADM-energy, -linear momentum and -mass make sense?
In asymptotically flat spacetimes, the ADM-energy, linear momentum and mass are defined as
$$E:= \frac{1}{16\pi}\lim_{r\to\infty} \int_{S^2_r}\sum_{i,j}\partial_ig_{ij}-\partial_jg_{ii}\frac{x^j}{r}\...
- 113
2
votes
0
answers
145
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Frozen Formalism Problem
Before stating my question, let me say what I do understand:
In the ADM formalism, the Hamiltonian density of the gravitational field can be written as,
$$\mathcal{H} = h n + H_a N^a$$
where n is the ...
- 696
5
votes
0
answers
204
views
Maxwell's Equations In Curved Spacetime Derived Via ADM Formalism VS Differential Forms Discrepancy
I' m trying to understand how the electromagnetic potential for diagonal Bianchi IX models when an electromagnetic field is aligned with one of the three axis is computed. The metric in question is
\...
- 93
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