Age of the Universe:
(Here are my notes from a very long discussion in these forums)
Do all observers agree on the age of our Universe?
Crowell:
No, they don't all agree.
But in an FRW cosmological model, there are preferred observers, who are essentially observers who detect no dipole asymmetry in the CMB. Such observers agree with one another on the amount of clock time since the Big Bang, and this is what we mean when we speak of the age of the universe in such a model.
In the real universe, a clock on the Earth's surface is not a bad approximation to such a clock. The solar system isn't moving at any large fraction of c relative to the CMB, and there is not a huge amount of gravitational time dilation between the Earth's surface and a point that is, say, outside the local group of galaxies.
There is not just one such frame for the whole cosmos. There is one such frame for every point in the cosmos. Global frames of reference don't exist in GR.
The existence of these preferred frames is also not a general characteristic of GR. It's just a characteristic of this particular solution of the GR field equations.
The age of the universe as usually discussed is for an observer who is at rest relative to the average motion of the matter and radiation in the universe (the "Hubble flow"), and is in the context of homogeneous models, which wouldn't include any structure such as black holes, etc. Yes, you're right, different observers can measure different ages of the universe on their clocks. You can't be "on" a black hole, but an observer hovering just outside a black hole's event horizon would say that according to her clock, the universe is very young. There is no limit on how young the universe could be according to such an observer. The same applies to an observer moving at nearly the speed of light relative to the Hubble flow.