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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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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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 $\...
Bowen Zhao's user avatar
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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 ...
hyriusen's user avatar
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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^{\...
polology's user avatar
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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 ...
LolloBoldo's user avatar
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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 ...
Fred's user avatar
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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^...
Powder's user avatar
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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_{...
dennis's user avatar
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BRST structure functions in gravity?

In the classical Hamiltonian BRST formalism, there arise structure functions $\Omega^{\beta_1...\beta_n}_{\alpha_1...\alpha_{n+1}}$ ($n\geq0$) --- see https://inspirehep.net/literature/221897 for ...
dennis's user avatar
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ADM decomposition incompatible with black hole?

In establishing ADM or 3+1 decomposition, one starts with choosing a foliation $\Sigma_t$ where t is a scalar function and $\Sigma_t$ is demanded to be a spacelike slice, i.e. with time-like normal ...
Bowen Zhao's user avatar
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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?
dennis's user avatar
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Composition of diffeomorphisms in the ADM formalism?

In the ADM formalism there are 4 constraints: $C_\mu(x)$, which are known as the Hamiltonian and spatial diffeomorphism constraints. In the quantum theory, $C_\mu(x)$ are the generators of ...
dennis's user avatar
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Are the Hamiltonian and spatial diffeomorphism constraints satisfied at all times for GR?

The ADM formulation of GR allows the Einstein equations to be recast as an initial value problem. According to Sec 3.3 of these notes the outline of the procedure is as follows: Pick a 3-metric $h_{...
dennis's user avatar
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357 views

How to derive gravitational path integral from the Hamiltonian operator formalism?

How does one derive the gravitational path integral $\int [dg]\exp(iS_{\text{EH}}/\hbar)$ from the Hamiltonian operator formalism? The connection between the Hamiltonian operator formalism and the ...
dennis's user avatar
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Conjugate variables in gravity?

We know that in the traditional quantum mechanics the conjugate variables are position and momentum, but what is known about the elusive quantum gravity? It came to my mind that if there is something ...
Hulkster's user avatar
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Is Loop quantum gravity an unadulterated quantisation of general relativity, or does it have additional assumptions?

I was reading this Phys.SE answer written by user346. At the end of point 3, they say they've only made a change of canonical variables from the ADM formalism to get the Ashtekar formalism. Then point ...
Ryder Rude's user avatar
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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}+\...
B.Hueber's user avatar
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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}+\...
B.Hueber's user avatar
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Showing the ADM-Mass of Schwarzschild-Spacetime

Can someone show that the ADM-Mass of Schwarzschild is identical with the Mass-Parameter?
Mac Menders's user avatar
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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 ...
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1 answer
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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 ...
Joeseph123's user avatar
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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 ...
B.Hueber's user avatar
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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 ...
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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, ...
B.Hueber's user avatar
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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 $\...
nodumbquestions's user avatar
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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{\...
P11P's user avatar
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3 votes
1 answer
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How compute the mass of AdS-Schwarzschild by ADM mass formula?

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 : $$ ds^2 =-f dt^2 +\frac{dr^2}{f} +r^2 d\sigma^2_{d-1} $$ ...
Mojtaba's user avatar
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3 votes
1 answer
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About characteristics of smearing function

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 ...
Chunhui's user avatar
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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 ...
Chunhui's user avatar
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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 ...
Powder's user avatar
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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 ...
QuantumEyedea's user avatar
2 votes
1 answer
148 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 ...
TomS's user avatar
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1 answer
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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 &...
Johnny's user avatar
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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 ...
Chandrahas's user avatar
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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 ...
Tom's user avatar
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3 votes
1 answer
654 views

Do the Misner-Sharp mass and ADM mass asymptote to one another for static spacetimes

I know for asymptotically flat spacetimes, one can define the Arnowitt-Deser-Misner (ADM) mass of the spacetime $$M_{ADM} = \frac{1}{16\pi} \lim_{r\to\infty} \int d^2\sigma r^a \gamma^{cd}\left( \...
physics_researcher's user avatar
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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 (...
aceituna's user avatar
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1 answer
493 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 ...
Tom's user avatar
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1 answer
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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,...
Tom's user avatar
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4 votes
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150 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}\...
aceituna's user avatar
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2 votes
2 answers
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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 ...
2 votes
0 answers
141 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 ...
Nathanael Noir's user avatar
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3 answers
156 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\...
Thorstein's user avatar
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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 ...
Terry Walton's user avatar
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0 answers
194 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 \...
Topology21's user avatar
3 votes
2 answers
417 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 $\Sigma$. ...
James J's user avatar
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2 votes
2 answers
811 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}...
James J's user avatar
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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 ...
Carlo Palazzi's user avatar
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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 ...
Bilbo's user avatar
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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 ...
KatherinD's user avatar