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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### 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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### 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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### 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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### 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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### 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 \...
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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 $\Sigma$. ...
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### 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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### 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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### 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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### 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 ...
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### 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 ...
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 ...
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 $A_{\mu}^{i}$ ...