general relativity - Lorentz invariance of the 3 + 1 decomposition of spacetime

Why is allowed decompose the spacetime metric into a spatial part + temporal part like this for example

$$ds^2 ~=~ (-N^2 + N_aN^a)dt^2 + 2N_adtdx^a + q_{ab}dx^adx^b$$

($N$ is called lapse, $N_a$ is the shift vector and $q_{ab}$ is the spatial part of the metric.)

in order to arrive at a Hamiltonian formulation of GR? How is a breaking of Lorentz invariance avoided by doing this ?

Sorry it this is a dumb question; maybe I should just read on to get it but I`m curious about this now ... :-)

Answer

Well, perhaps one should consider reading The Hamiltonian formulation of General Relativity: myths and reality for further mathematical details. But I would like to remind to you with most constrained Hamiltonian systems, the Poisson bracket of the constraint generates gauge transformations.

For General Relativity, foliating spacetime $\mathcal{M}$ as $\mathbb{R}\times\Sigma$ ends up producing diffeomorphism constraints $\mathcal{H}^{i}\approx0$ and a Hamiltonian constraint $\mathcal{H}\approx 0$. Note I denote weak equalities as $\approx$.

This is first considered in Peter G. Bergmann and Arthur Komar's "The coordinate group symmetries of general relativity" Inter. J. The. Phys. 5 no 1 (1972) pp 15-28.

Since you asked, I'll give you a few exercises to consider!

Exercise 1: Lie Derivative of the Metric

The Lie derivative of the metric along a vector $\xi^{a}$ is $$ \mathcal{L}_{\xi}g_{ab}=g_{ac}\partial_{b}\xi^{c}+g_{bc}\partial_{a}\xi^{c}+\xi^{c}\partial_{c}g_{ab} $$ Show that this may be rewritten as $$ \mathcal{L}_{\xi}g_{ab}=\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a} $$ where $\nabla$ is the standard covariant derivative.

Exercise 2: Constraints generate diffeomorphisms

Recall that the Hamiltonian and momentum constraints are $$\mathcal{H} = \frac{16\pi G}{\sqrt{q}}\left(\pi_{ij}\pi^{ij}-\frac{1}{2}\pi^{2}\right)-\frac{\sqrt{q}}{16\pi G}{}^{(3)}\!R,\quad\mathcal{H}^{i} = -2D_{j}\pi^{ij}$$ and $\pi^{ij}=\displaystyle\frac{1}{16\pi G}\sqrt{q}(K^{ij}-q^{ij}K)$ with $K_{ij}=\displaystyle\frac{1}{2N}(\partial_{t}q_{ij}-D_{i}N_{j}-D_{j}N_{i})$. Let $$H[\widehat{\xi}] = \int d^{3}x\left[\hat{\xi}^{\bot}\mathcal{H}+\widehat{\xi}^{i}\mathcal{H}_{i} \right]$$ Show that $\mathcal{H}[\widehat{\xi}]$ generates (spacetime) diffeomorphisms of $q_{ij}$, that is, $$\left\{H[\widehat{\xi}],q_{ij}\right\}=(\mathcal{L}_{\xi}g)_{ij}$$ where $\mathcal{L}_{\xi}$ is the full spacetime Lie derivative and the spacetime vector field $\xi^{\mu}$ is given by $$\widehat{\xi}^{\bot}=N\xi^{0}, \quad \widehat{\xi}^{i}=\xi^{i}+N^{i}\xi^{0}$$ The parameters $\{\widehat{\xi}^{\bot},\widehat{\xi}^{i}\}$ are known as "surface deformation" parameters.

(Hint: use problem 1 and express the Lie derivative of the spacetime metric in terms of the ADM decomposition.)

Addendum: I'd like to give a few more references on the relation between the diffeomorphism group and the Bergmann-Komar group.

From the Hamiltonian formalism, there are a few references:

1. C.J. Isham, K.V. Kuchar "Representations of spacetime diffeomorphisms. I. Canonical parametrized field theories". Annals of Physics 164 2 (1985) pp 288–315


2. C.J. Isham, K.V. Kuchar "Representations of spacetime diffeomorphisms. II. Canonical geometrodynamics" Ann. Phys. 164 2 (1985) pp 316–333

The Lagrangian analysis of the symmetries are presented in:

1. Josep M Pons, "Generally covariant theories: the Noether obstruction for realizing certain space-time diffeomorphisms in phase space." Classical and Quantum Gravity 20 (2003) 3279-3294; arXiv:gr-qc/0306035


2. J.M. Pons, D.C. Salisbury, L.C. Shepley, "Gauge transformations in the Lagrangian and Hamiltonian formalisms of generally covariant theories". Phys. Rev. D 55 (1997) pp 658–668; arXiv:gr-qc/9612037


3. J. Antonio García, J. M. Pons "Lagrangian Noether symmetries as canonical transformations." Int.J.Mod.Phys. A 16 (2001) pp. 3897-3914; arXiv:hep-th/0012094

For more on the hypersurface deformation algebra, it was first really investigated in Hojman, Kuchar, and Teitelboim's "Geometrodynamics Regained" (Annals of Physics 96 1 (1976) pp.88-135).