de Sitter Spacetime: Static patch vs Global coordinates

Question

Two well-known coordinate charts on the dS spacetime are the global coordinates and the static patch coordinates. In the global coordinates, the D-dimensional dS metric takes the following form $$ ds^2 = -dt_g^2 + l^2 \cosh^2(t_g/l) d\Omega_{D-1}^2$$ where $t_g$ is the global time and $l$ is the radius of the dS space. Clearly, the metric components are time-dependent and hence non-static. In fact, this metric shares the form with the FRW metric which describes an expanding universe in the future.

With the help of the static patch coordinates, we can write the dS metric in the form $$ds^2 = -(1-r^2/l^2)dt_s^2 + \frac{1}{1-r^2/l^2}dr^2 + r^2 d\Omega^2_{D-2}$$ where $t_s$ is the time in the static patch. Clearly, this is a static metric with a timelike Killing vector $\partial_{t_s}$. My question is, how can the same manifold express both an FRW expanding universe and a static Schwarzschild-like spacetime just because we are using different coordinates? For example, we know that the energy is conserved in a static spacetime whereas it is not conserved in an FRW metric. Wouldn't this give rise to contradicting statements on the same manifold? Clearly I am missing something here, so any help is appreciated.

Answer 1

The properties you mention are coordinate-dependent but actually they are (conformal-)Killing vector dependent. The complete dS spacetime has several such vector fields so that all those properties may co-exist. Furthermore, since they refer to intrinsic geometric structures, they may have a physical meaning.

The global coordinates you mention actually define intrinsic geometric global structures proper of solutions of FLRW spacetimes:

(a) a global foliation made of spacelike 3D conformally-isometric manifolds with intrisic positive constant curvature, whose metric is $d\Omega_{D-1}$ up to a scale factor,

(b) a conformal Killing time globally defined $$\frac{\partial}{\partial \eta}:=\ell \cosh(t_g/\ell) \frac{\partial}{\partial t_g}\:.$$ With respect to that notion of conformal time $\eta$ only massless fields (photons) admit conserved (conformal) energy described in terms of a density in the rest spaces of the foliation.

However, referring to subregions of the spacetime, further (infinitely many) geometric structures take place. In those regions, further conservation laws are admitted. In every static region, the corresponding Killing vector $\partial_{t_s}$ permits a local conservation law referred to its flow and densities assigned on the orthogonal notion of rest space.

The situation is quite similar to that of Minkowski spacetime. There, we have a pair of timelike Killing vectors (actually infinitely many such pairs). One is the standard global Minkowski Killing vector time $\partial_t$, the other is the so-called Rindler Killing vector $\partial_\tau$ which is timelike in the Killing wedges $|x|> t$ giving rise to further conservation laws of corresponding notions of energy.

Coming back to dS spacetime and its physics, the crucial physical fact that makes relevant $\partial_{t_g}$ shows up when assuming that this spacetime is a solution of the Einstein equations for a suitable type of gravitational sources.

According to FLRW models, the integral curves of $\partial_{t_g}$ are the worldlines of the sources of gravitation for the Einstein equations (the stories of the clusters of galaxies in general, but in the case of the pure dS spacetime, the four-velocity field of the perfect fluid made of dark energy).

Furthermore, the above-mentioned 3-surfaces orthogonal to those curves define the common rest spaces where the expansion of the universe is measured and where the background cosmic radiation appears to be isotropic with the measured temperature $\simeq 3 K$ (in the pure dS spacetime there is no radiation however).

In this cosmological model the static regions have no direct meaning, because the integral lines of the static time are not the worldlines of the source of the gravitation.

Answer 2

The de Sitter spacetime possesses scale invariance. That is to say, the only matter content allowed in a consistent theory of dS spacetime has conformal symmetry, and therefore does not feel the expansion of spacetime. In this sense, the dilution characteristic of FLRW spacetimes is not felt by the matter in a consistent dS cosmology, and hence it possesses a time-like killing vector.

In theories where the universe has matter content, the universe is only approximately de Sitter, as we observe in the slightly negative CMB power spectrum tilt. This slight tilt represents the slightly broken scale invariance of de Sitter during inflation: the size of the horizon, $l$, in your metric, is a function of time in any realistic theory, and therefore these new coordinates do not possess a timelike killing vector.