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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Spinfoams and LQG

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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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 $\... 0 votes 0 answers 66 views 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 ... • 133 2 votes 0 answers 48 views 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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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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How to extract mass from the tt component of the metric

Given an asymptotically flat metric, what is the general recipe in order to obtain the mass from the $tt$ component? In the case of the Schwarzschild metric it is obvious but is there a recipe that ...
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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}+\... • 834 2 votes 0 answers 194 views Showing the ADM-Mass of Schwarzschild-Spacetime Can someone show that the ADM-Mass of Schwarzschild is identical with the Mass-Parameter? • 149 1 vote 0 answers 55 views Reference for Classical probability theory implemented on General Relativity I am looking for a reference for something like : a probability density put on the ADM phase space, thereby making the metric probabilistic in the classical probability sense. But not exactly this. I ... 2 votes 1 answer 63 views 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 2 votes 0 answers 154 views Gauge freedom in (3+1) general relativity In the ADM formalism of general relativity, in which one rewrites Einstein's field equations in terms of evolution and constraint eqautions of the 3-metric and the extrinsic curvature, after fixing ... • 834 2 votes 0 answers 57 views ADM equations with cosmological constant Does anyone know a reference in which the ADM equations for general relativity (the equations for the spatial metric and conjugate momentum or extrinsic curvature using the lapse and shift in the 3+1 ... 1 vote 0 answers 99 views 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, ... • 834 2 votes 1 answer 80 views Is the Hamiltonian a pure boundary term in linearised gravity? It's well-known that in general relativity, the Hamiltonian consists purely of a boundary term: the so-called ADM Hamiltonian. This is because the bulk term is an integral of the constraint operator \... • 1,395 2 votes 0 answers 71 views 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{\...
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I want to compute the mass of AdS schwarzschild by ADM mass formula but I could not find where I am wrong. AdS schwarzschild line element is : $$ds^2 =-f dt^2 +\frac{dr^2}{f} +r^2 d\sigma^2_{d-1}$$ ...
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I met the word smearing function (or test function) when I was learning ADM formalism in GR books. What makes me scratch my head is the reason of introducing such a smearing function when we calculate ...
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Functional derivative acts on covariant derivative

I'm confusing about how functional derivatives act on a covariant derivative. I'm doing such a calculation: In ADM formalism, let $h_{ij}(x)$ be the spatial metric while $\pi^{ij}(x)$ is its momentum ...
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Relation between the canonical symplectic form on phase space and the Hamiltonian in GR

I am working on the Hamiltonian formulation of the Einstein equations of motion in General Relativity, where the aim is to find the Hamiltonian generating the dynamics from the Einstein equations (...
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Jacobi Matrix between Cartesian and Schwarzschild coordinates

Let $\mathcal P$ be a photon at position $\vec x =(x,y,z)$ with 3-velocity $\vec v=(v_x,v_y,v_z)$, where both are given in local Cartesian coordinates. I want to follow the photons geodesic by ...
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Direction of normal vector on spacelike foliation

Let $(\Sigma_t)_{t\in\mathbf{R}}$ be a spacelike hypersurface foliation of our spacetime $(\mathcal{M},g_{\mu\nu})$. Let $n^\mu$ be the future-directed unit normal to the hypersurface $\Sigma_t$. Thus,...
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Dynamic equation for the geometrodynamic field momentum in ADM formalism

I'm currently working through MTW's Gravitation and I've reached Chapter 21. I've spent quite a bit of time trying to reproduce Equation 21.115 and have managed to produce 5 out of the six terms in ...
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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 \...
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 $\Sigma$. ...