# Constraints in general relativity

In this review on inflation, on Pg. 135, Baumann talks about the energy and the momentum constraints for gravity. Are these equations the $$G_{00} = T_{00}$$ and $$G_{0i} = T_{0i}$$ components of the Einstein equations respectively? If so, how does one know that they are constraints?

Also what is the nature of these constraints, are they primary/secondary or first class/second class?

## 1 Answer

The standard (ADM) Hamiltonian formalism of general relativity involves the choice of a 3-foliation of the underlying spacetime by the choice of a time coordinate $$\tau$$. With this specified, one can define the time-evolution vector by the choice of a three-vector $$\beta^{i}$$ and scalar $$\alpha$$, which gives a time evolution vector $$t^{a} = \alpha n^{a} + \beta^{a}$$, where $$n^{a}$$ is the unit timelike normal to the surfaces of constant $$\tau$$.

Then, one turns the crank of the standard Lagrangian-to-Hamiltonian procedure. If this is done correctly, $$\alpha$$ and the $$\beta^{i}$$ can be treated like Lagrange multipliers, and therefore, their variations will be constraints. It will also be the case that none of these constraint equations will involve time derivatives of the conjugate momenta.

It can be shown, however, that their Poisson brackets are not closed, and instead, you will get terms proportional to the constraints for the poisson brackets of the constraints. So, if you take variations of the Hamiltonian in the directions of the constraint violoations, you can get growing constraint violations. I believe I've seen different arguments about how this counts as "first class/second class" (I may be wrong here), but as a practical concern, when you're numerically solving Hamiltonian GR, you have to be worried about growth in constraint-violating terms, and it is useful to modify your EOM to damp constraint-violating terms.