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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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 ...
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51 views

ADM formalism, closed timelike curves and chronological protection conjecture

Is it correct that based on the premise of foliation by Cauchy surfaces the ADM fomalism restricts the set of solutions for general relativity to causal manifolds and therefore excludes closed ...
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1answer
38 views

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 &...
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30 views

Normal vector and time-function in the initial value formulation of General relativity

I am trying to follow the initial-value and Hamiltonian formalism sections in Wald but I am confused about the definitions used to define hyper-surfaces. I am hoping that the conventions and variables ...
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35 views

Particle velocity measured by eulerian observer

I am trying to understand equation 7.18 in the book "Relativistic Hydrodynamics" by Luciano Rezzolla and Olindo Zanotti. Here is the setup. Let $n^\mu=\frac{1}{\alpha}(1,-\beta^i)$ be the ...
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39 views

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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1answer
95 views

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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1answer
33 views

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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29 views

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}\...
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110 views

Hamiltonian formalism of General Relativity Textbook

I've been reading Wald's book on General Relativity and in appendix $E_{2}$ it discusses the Hamiltonian formalism of General Relativity.I would like to understand it more, can you recommend me a ...
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80 views

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 ...
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3answers
113 views

Setting $N=1$ and $N^a=0$ in the Einstein-Hilbert action

In the ADM formalism of general relativity, one obtains a $3+1$ split of spacetime by setting $$\mathrm d s^2=(-N^2+N_a N^a) \,\mathrm d t^2 + 2N_a\,\mathrm d t\,\mathrm d x^a + q_{ab} \,\mathrm d x^a\...
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54 views

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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125 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 \...
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2answers
210 views

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 $\Sigma$. ...
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2answers
369 views

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}...
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115 views

Transformation of ADM parameters under diffeomorphisms

I am trying to prove the invariance of the ADM formalism under (infinitesimal) diffeomorphisms. I have checked Wald and other textbooks on the subject but have been unable to find expressions for how ...
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86 views

Hamilton equations of motion for matter fields coupled to general relativity in ADM formalism

Do you know what are the Hamiltonian formalism analogs of the Klein-Gordon equation and/or the Maxwell equations in general relativity? Showing how these equations of motion for matter in the ...
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126 views

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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109 views

Canonical transformation of spacetime or under field theory formalism

Is there a standard way to define Canonical Transformations in the case of field theory or spacetime? I saw that under ADM formalism it is possible to define a generating function: $$ G(t)= \sum_i p_i ...
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1answer
1k views

ADM mass of a black hole and mass of the associated matter

The ADM formalism gives a definition for the energy (Hamiltonian) of a static, asymptotically flat spacetime. This energy can be equated to the mass of the matter (for example, a black hole) which ...
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1answer
582 views

Hamiltonian formulation of general relativity in ADM formalism

I'm working on ADM formalism, and this is part of my Hamiltonian $$H_{2}=2 \int d^3 x \quad p^{a b}D_{a}N_{b} = 2 \int d^3 x \quad p^{ab}h_{bc}(\partial_a N^{c}+\Gamma^{c}_{ab}N^{d}).$$ I need to ...
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3answers
893 views

A couple of questions on the ADM formalism in general relativity

I've been reading up on the ADM formalism in general relativity and have been stuck on a couple of concepts. The first is to do with the foliation of spacetime into space-like hypersurfaces. I ...
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436 views

Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one "end"?

The question is: What is the mathematical and/or physical basis for saying that a (static) spacetime manifold with more than one asymptotically flat region at infinity ("end") has a distinct ...
2
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1answer
630 views

How does a spatial covariant derivative act on tensors that are not purely spatial?

I have a possibly dumb question on ADM formalism. Starting with a metric in ADM form \begin{equation} ds^2 = -N^2dt^2 + q_{ij}(dx^i + N^idt)(dx^j + N^jdt) \end{equation} where $i,j$ only run over the ...
5
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1answer
149 views

Derivation of metric of spacetime with a point source in 2+1 dimension using ADM formalism

In "Quantum Gravity in 2+1 dimension" by S Carlip, Sec 3.1 (where the metric of a spacetime with a point source is derived, using the ADM formalism), equation 3.8 states that (this is the ...
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1answer
2k views

ADM Hamiltonian formalism and Quantum gravity

is there a Hamiltonian reformultion of gravity ?=? if so if we use the usual Quantization scheme we can not we quantizy the gravity ?? in terms of a Gauge Theory with the potential $ A_{\mu}^{i} $ ...
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428 views

Calculation of the non-Gaussity parameter for primordial cosmological perturbations by the ADM Formalism

Maldacena has used the ADM Formalism in one of his papers (http://arxiv.org/abs/astro-ph/0210603) in computing the the three point correlation function (i.e the non-Gaussianity) parameter for ...