Internal-external-dynamics decoupling in canonical general relativity

Gerhard Schäfer is a retired professor at the University of Jena. His main scientific interests are equations of motion in general
relativity and their applications in astronomy and astrophysics.

Research on general-relativistic equations of motion based on Hamiltonian or canonical frameworks is not quite a main-stream doing; likely because of the all-over covariance of the theory and canonical is just not covariant but rather quite the opposite. Covariance under spacetime coordinate transformations makes the theory a spacetime-local one with its local scalars, vectors and tensors, the canonical picture on the other side is at home in the phase space of the dynamics which combines position and momentum variables. Crucial object-changing operations in spacetime are covariant derivatives, crucial ones in phase space are Poisson brackets.

What is the benefit of performing research in general relativity within a canonical framework? Let us concentrate on gravitating systems living in asymptotically flat spacetimes. Then there exist global quantities — energy, linear momentum, angular momentum, Lorentz-boost vector — which are nicely conserved. If those quantities are calculated within a canonical framework even more is achieved because the energy expression is a Hamiltonian too which is known to be the generator of the evolution in time of the whole system. Additionally, the total linear momentum and the total angular momentum have simple universal structures independent from the specific dynamics. Only the Hamiltonian and the Lorentz-boost vector depend on the specific dynamics. Furthermore, the mentioned conserved quantities fulfil the Poincaré algebra which guarantees global Lorentz invariance. Clearly, in regard to global quantities, a canonical formulation is very appealing. But is the local spacetime aspect of the dynamics or the spacetime picture now completely lost? Not at all! Imagine a gravitationally interacting n-body system in asymptotically flat spacetime. Take its Hamiltonian and let in a limiting process become one of the bodies a test body or a “spectator object”. Then one obtains the Hamiltonian ( $\rm H$ ) of a particle (rest mass $m$ , linear momentum $p_i$ ) in the spacetime generated by the remaining (n-1) heavy particles (bodies) and a whole local spacetime picture arises with lapse ( $\rm N$ ) and shift ( ${\rm N}^i$ ) functions and metric ( $g_{ij}$ , inverse $\gamma^{ij}$ ) of three-dimensional slices, reading:

To simplify equations as much as possible, the speed of light $c$ therein is always put equal to one.

Ira Georg is a PhD fellow at the University of Cape Town in South Africa.

Let us go back to global aspects. The globally valid Lorentz group induces the famous relation between total energy $E$ or Hamiltonian $\rm H$ and total linear momentum ${\bf P}$ in the form ${\rm H} = \sqrt{M^2 + {\bf P}^2}$ . Here $M$ is the rest mass of the total system which is not changing if the whole system is getting boosted, i.e. $M$ and ${\bf P}$ do commute with each other in terms of the canonical Poisson brackets. The interesting question arises for the internal canonical variables which commute with both the total linear momentum and its conjugate position variable. In our CQG paper, this goal is achieved for a self-gravitating binary system within general relativity up to the second post-Newtonian order of approximation, i.e. to the order $1/c^4$ beyond Newtonian gravity. It is trivial to calculate the internal canonical coordinates in terms of the single particle position and momentum variables in the center-of-mass frame where ${\bf p}_1 + {\bf p}_2 =0$ holds and where the relative position vector is simply given by ${\bf x} = {\bf x}_1 - {\bf x}_2$ and its canonical conjugate by ${\bf p} = {\bf p}_1$ , and also the inversion which represents the single particle variables in terms of internal relative ones is not difficult to achieve and goes just by using the Lorentz-boost vector. Here an interesting remark is worth to be made. A phase space knows nothing about the dynamics, it is kinematics only. So the Newtonian phase space is identical in structure (same Poisson brackets) with the second post-Newtonian one, only their Hamiltonians (and boost vectors) are different. So there must exist a canonical transformation which connects the two phase spaces determined by the different Hamiltonians. The generators of those canonical tranformations up to even the third post-Newtonian approximation were calculated in a previous paper, again an offspring of one of the authors’ research based on Hamiltonian general relativity. A very useful application hereof is the topic of our CQG paper which itself should be nicely applicable to hierarchical triple systems or binary systems undergoing gravitational recoil. Those topics will be part of future research. Clearly, the successful decoupling is not the least argument for choosing a canonical framework.

Read the full article in Classical and Quantum Gravity or on arXiv:
Canonical center and relative coordinates for compact binary systems through second post-Newtonian order
Ira Georg and Gerhard Schäfer 2015 Class. Quantum Grav. 32 145001

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