When is the charge generating isometries given by a symplectic product?

Question

I am reading the following paper https://arxiv.org/abs/2309.15897 and have the following confusion: The authors look at the covariant phase space of linearised general relativity after one expands around some background spacetime with a bifurcate killing horizon. The authors first define the following function

$F_x = W(\gamma, \mathcal{L}_x\gamma)$, where $W$ is the canonical symplectic product on phase space of linearized metric perturbations, $\gamma$ are the first-order perturbations and $\mathcal{L}_x$ is the lie-derivative w.r.t. the killing vector field $x$ associated to the bifurcate killing horizon.

They claim in eq. C. 18 that

$\{F_x,\gamma(f)\} = \gamma(\mathcal{L}_x f)$.

holds, which looks to me as if $F_x$ generates the diffeomorphism associated to x.

What I find confusing is that it appears that one can write the charge simply as a symplectic product $F_x = W(\gamma, \mathcal{L}_x\gamma)$. However the charge was defined in eq. 8 of https://arxiv.org/abs/gr-qc/9911095 to have an variation that is given by the symplectic product.

It looks a little as if they think the symplectic product is already the integral of the variation which might be true but I don't know why. Maybe it is because one has restricts to a linear theory?

Answer 1

It turns out that for isometries, the charge defined via the symplectic product always satisfies the equation $\delta H = \Omega(\delta \phi, \mathcal{L}_x\phi)$, because if $H = \frac{1}{2}\Omega(\phi,\mathcal{L}_x\phi)$, then

$\delta H = \frac{1}{2} (\Omega(\delta\phi,\mathcal{L}_x\phi) + \Omega(\phi,\mathcal{L}_x\delta \phi)) = \frac{1}{2} (\Omega(\delta\phi,\mathcal{L}_x\phi) - \Omega(\mathcal{L}_x\phi,\delta \phi)) = \frac{1}{2} (\Omega(\delta\phi,\mathcal{L}_x\phi) + \Omega(\delta\phi,\mathcal{L}_x\phi)) = \Omega(\delta\phi,\mathcal{L}_x\phi)$

Here it was important in the second equality that the Lie derivative acts as an antisymmetric operator if one performs a partial integration to shift the derivative to the first argument. This is true for a general metric if the vector field is an isometry, because one then does not get extra terms arising from derivatives of the metric when performing the partial integration.