Questions tagged [adm-formalism]
This tag contains questions relating to ADM formalism i.e., Arnowitt-Deser-Misner formalism which is a Hamiltonian formulation of General Relativity that plays an important role in canonical quantum gravity and numerical relativity.
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ADM Formalism for the Effective String Theory
We will consider the following effective action of string theory in leading order of $\alpha'$:
$$S=\frac{1}{2\kappa^2_0}\int d^{D}X\sqrt{-G}e^{-2\Phi}\left[R-2\Lambda-\frac{1}{12}H_{\mu\nu\lambda}H^{\...
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Extrinsic Curvature in a conformally-flat spacetime that is also asymptotically-flat spacetime
I would appreciate if someone can confirm or correct my understanding of extrinsic-curvature (as in the ADM 3+1 decomposition of spacetime) when dealing with a conformally-flat spacetime.
(I updated ...
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Do the Komar/ADM mass equations also hold in 2+1D?
All definitions I have come across for the ADM mass require asymptotic flatness, which always is defined for 4 dimensional spacetimes. I was wondering if these formulae in 3+1D hold in 2+1D aswell?
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What is the determinant of the Wheeler-DeWitt metric tensor constructed from spatial metrics in ADM formalism?
The Hamiltonian constraint of General relativity has the following form
\begin{align}
\frac{(2\kappa)}{\sqrt{h}}\left(h_{ac} h_{bd} - \frac{1}{D-1} h_{ab} h_{cd} \right)p^{ab} p^{cd} - \frac{\sqrt{h}}{...
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Solving the Hamiltonian constraint equation in General Relativity
The constraint equation of general relativity reads as follows
\begin{align}
\frac{(2\kappa)}{\sqrt{h}}\left(h_{ac} h_{bd} - \frac{1}{D-1} h_{ab} h_{cd} \right)p^{ab} p^{cd} - \frac{\sqrt{h}}{(2\...
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ADM mass calculation for the BTZ black hole
Considering a non-rotating and non-charged 2+1 dimensional black hole, known as the BTZ black hole which obtained by adding a negative cosmological constant $\Lambda=-\frac{1}{l^2},l\ne0$ to the ...
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Partial integration of the Gibbons-Hawking-York boundary term
In https://arxiv.org/abs/1402.6334 on page 16 in their Eq. (5.21), they go from the equation
$$S_G+S_\chi\approx\int d^2x\sqrt{-g}XR+\int dt\sqrt{-\gamma}XK$$
to the equation
$$=\int d^2x \sqrt{-g}\...
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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[\...
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Is the mass of curved space, additional mass?
According to Einstein, mass, say in the form of matter, curves space. It is the curvature of space that gives rise to gravity. Now I have heard there is an energy associated with the curvature of ...
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Relation between the Wheeler–DeWitt equation and string theory
Can we derive the Wheeler–DeWitt equations from string theory? Since they are both quantum gravity theory.
A simple way seems to be the following logic:
The Wheeler–DeWitt equation is the canonically ...
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Spinfoams/LQG Inequivalence
I know up to some degree how spinfoam models and LQG work, but there are some details that i still miss since i have still a naif knowledge.
In the literature it as often said that an open problem is ...
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Observor in 3+1 decomposition
I'm a bit puzzled about the concepty of observer in 3+1 or ADM decomposition in GR.
The decomposition is typically described as starting with a scalar field $t$, whose spacelike level surface $\...
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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 ...
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Lapse and shift choice meaning in asymptotic Minkowski space in ADM formalism
In A Relativist's toolkit by Poisson the defining of the ADM mass starts by introducing an asymptotic Lorenztian frame and in the asymptotic portion of the hypersurface the flow vector becomes:
$t^{\...
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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 ...
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