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Tagged with constrained-dynamicsgeneral-relativity
38 questions
0votes
1answer
95views
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\...
- 814
1vote
1answer
238views
The problem of time in classical general relativity
It is well known that the Hamiltonian of General Relativity is a linear combination of constraints. This poses a challenge in quantum gravity. If a state $\psi$ solves the constraints ($\hat C_\alpha \...
- 153
17votes
1answer
445views
What is the full algebra of BRST-invariant observables for general relativity?
The Hamiltonian formulation of general relativity - either in the ADM formalism or in Ashtekar variables - is straightforwardly a gauge theory. While the BRST formalism has primarily been developed to ...
- 131k
5votes
2answers
320views
Dirac procedure for Wheeler De Witt equation
After computing the Hamiltonian constraint and the momentum constraint in general relativity the Hamiltonian constraint is turned into an operator equation and solved in a manner similar to a ...
- 1,552
0votes
1answer
245views
What diffeomorphism does the Hamiltonian constraint generate?
Consider the Hamiltonian constraint $\mathcal H(x)$ in the ADM formalism. What diffeomorphism does this generate?
- 790
0votes
1answer
121views
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_{...
- 790
0votes
0answers
148views
Variational formulation for the Kerr solution
The critical points of the Einstein-Hilbert action, if one allows axial symmetry, are Kerr-type solutions of the Einstein field equations, in which a parameter $\alpha$ interpreted as the angular ...
- 41
2votes
1answer
79views
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 ...
- 919
0votes
1answer
563views
Hamiltonian constraint
I am having some difficulties regarding this wikipedia article. I can't understand why the action (for the harmonic oscilator) is written as it is. That is, why is the Hamiltonian followed by a $\...
- 998
3votes
1answer
221views
Constructing a field theory action for the point particle in curved space
The point particle action in the Hamiltonian formalism is $$ S = \int d\tau \Big( -p_\mu \dot{x}^\mu - \frac{e}{2}(g^{\mu\nu} p_\mu p_\nu - m^2) \Big) \ ,\tag{1} $$ where I explicitly displayed the ...
- 367
2votes
2answers
314views
If quantum gravity is a TQFT, why isn't the Wheeler-De Witt equation satisfied automatically?
It is often said that QG is a topological QFT: given a bordism between $D$-manifolds $\Sigma_1$ and $\Sigma_2$, QG assigns a unitary between the Hilbert spaces associated with $\Sigma_1$ and $\Sigma_2$...
- 1,515
1vote
1answer
162views
How to derive infinitesimal gauge transformations from constraints?
I am reading some papers about quantizing the gravitational fields, for example, here, here, and here. Since the classical actions for gravitational fields are singular, they contain some constraints. ...
- 105
2votes
1answer
188views
Triangulation of the Hamiltonian constraint in Loop quantum gravity
Im trying to obtain regularized (and triangulated) version of Hamiltonian constraint in the LQG. However, one step remains unclear to me. I am starting with the Euclidean Hamiltonian: $H_E=\frac{2}{\...
4votes
1answer
520views
Hamilton-Jacobi-Einstein equation
I have been looking at the Hamiltonian formalism of GR for some time and recently stumbled across the Hamilton-Jacobi-Einstein equation: $$\frac{1}{\sqrt{g}} (\frac{1}{2}g_{pq}g_{rs} - g_{pr}g_{qs}) \...
- 469
0votes
1answer
668views
Why Hamiltonian of gravity is zero?
In paper Topological Gravity as the Early Phase of Our Universe there's statement: Hamiltonian of gravity would vanish by time reparameterization invariance. How to derive such result?
- 5,757
