Killing tensor/vector very basic context question about consevation

As Peter noted, given a matter distribution ##T^{\mu\nu}## the statement ##\nabla_{\mu}T^{\mu\nu} = 0## is local so it doesn't manifestly take into account the conservation laws for the gravitational field that the matter distribution interacts with i.e. it does not manifestly include the dynamics of the background space-time; this can be seen for example by transforming to local inertial coordinates wherein the above statement becomes ##\partial_{\mu}T^{\mu\nu} = 0## just as in flat space-time. This is of course because the dynamics of the background space-time is only gauge invariant at the level of curvature ##R_{\mu\nu\gamma\delta}##, not at the level of the connection ##\nabla_{\mu}##, due to the equivalence principle. In ##\nabla_{\mu}T^{\mu\nu} = 0## the dynamics of the gravitational field is incorporated in a complicated and non-gauge invariant way into ##\nabla_{\mu}##.

Thus it isn't a surprise that ##\nabla_{\mu}T^{\mu\nu} = 0## holds in any space-time irrespective of the space-time symmetries, or lack thereof. Because a local conservation law cannot be construct from gauge invariant quantities that manifestly takes into account the dynamics of the gravitational field, again due to the equivalence principle, one must resort to quasi-local conservation laws. That is, one finds a non-gauge invariant quantity ("pseudotensor") ##t^{\mu\nu}## which is constructed only from ##g_{\mu\nu}## and ##\partial_{\delta}g_{\mu\nu}## while satisfying ##\partial_{\mu}(T^{\mu\nu} + t^{\mu\nu}) = 0##, and as such can always be made to vanish in a local inertial coordinate system but which upon integration over a finite region of space-time ("quasi-local") yields gauge invariant conserved charges.

One such choice is the Landau-Lifshitz pseudotensor ##t^{\mu\nu}_{LL}##, whose exact expression can be found here: http://en.wikipedia.org/wiki/Stress–energy–momentum_pseudotensor#Metric_and_affine_connection_versions. So for example if we take any coordinate system ##x^{\mu}## on some region of this space-time and consider a space-like hypersurface ##\Sigma## in this coordinate system then we can define a conserved total 4-momentum ##P^{\mu} = \int_{\Sigma}d^3 x(-g)(T^{\mu0}+ t^{\mu0}_{LL})## that clearly includes the dynamics of both the matter distribution and the gravitational field it interacts with. It is important to note that as long as the Einstein equations are valid, ##\nabla_{\mu}T^{\mu\nu} = 0## is equivalent to ##\partial_{\mu}(T^{\mu\nu} + t^{\mu\nu}_{LL}) = 0##; all the latter conservation law does is manifestly include the dynamics of the gravitational field by shifting it implicitly from ##\nabla_{\mu}## into ##t^{\mu\nu}_{LL}##.

Clearly ##\frac{d P^{\mu}}{dx^0} = 0## holds irrespective of whether or not the space-time possesses a Killing field so what does this have to do with them? Consider a space-time with a time-like Killing field ##\xi^{\mu}## and for simplicity assume it is asymptotically flat. One can always find an asymptotic Lorentz frame ##(t,\vec{x})## in which the space-time has no 3-momentum e.g. if one has a static black hole then this will be the asymptotic rest frame of the black hole. In this frame the space-time simply has a mass ##M## (called the ADM mass which in this case is equivalent to the Komar mass) that is constant, ##\frac{dM}{dt} = 0## i.e. the space-time has no dynamics; of course ##\xi^{\mu} = \delta^{\mu}_t## in these coordinates.

All this means is, instead of having to go through the formalism described above in order to make manifest the conservation laws due to the combined dynamics of the matter fields and the gravitational field they interact with, one can simply use ##\xi^{\mu}## to describe the dynamics of the matter fields alone since the space-time itself has no dynamics on account of ##\xi^{\mu}##; essentially what one is doing is defining a static gravitational potential ##\phi = \log(-\xi_{\mu}\xi^{\mu})^{1/2}## to encode the dynamics of matter interacting with the static gravitational field. For example for a freely falling test particle one uses the conserved energy ##E = -\xi_{\mu}p^{\mu}## which, in a non-trivial way, includes both the gravitational potential energy and the kinetic energy of the particle (the split becomes manifest in the non-relativistic limit).

This is not unlike what one does in EM. In flat space-time we still have the local conservation law ##\partial_{\mu}(T^{\mu\nu}_{\text{Matter}} + T^{\mu\nu}_{\text{EM}}) = 0## for charged matter fields interacting with an EM field propagating in the flat space-time but if the EM field is static then it has no dynamics, just a constraint equation ##\vec{\nabla}\cdot \vec{E} = 4\pi \rho##, and we simply describe the dynamics of the charged particle through an electrostatic potential ##\phi##.