Using the standard model of cosmology we calculate the Hubble time to obtain an estimate of the age of the universe.
This model assumes a beginning of time in the past. But that point is a true singularity in the sense that even if we switch to another coordinate system it will still be a singularity.
Thus that event in space-time cannot be reached through any path. Geodesics simply end there. One can even note that the beginning of time is not a point of the manifold and does not belong to it.
Now the question is that how is it possible to estimate a finite age of the universe even though the beginning of time is a singularity at infinity and not even included in the topology of space-time?
That point is at infinity and traveling to the past we approach it asymptomatically yet we had assumed a beginning and a finite age. Isn't this a contradiction?
Answer
Strictly speaking the FLRW metric doesn't specify that time starts at the Big Bang. It specifies only that the Big Bang is a singular point so it is impossible to analytically continue a geodesic back in time past the Big Bang.
If it helps to make things clearer, exactly the same happens with an object falling into a black hole. A geodesic that crosses the horizon must reach the singularity and it cannot be analytically continued forward in time past the singularity. Hence the claim common in the popular science press that time stops at the centre of a black hole, just as it's claimed that time started at the Big Bang. Neither statement is especially helpful.
In both cases the proper length of the geodesic is finite. Since the elapsed time for an observer on the geodesic is just the length divided by $c^2$ this means the time from the Big Bang until now is finite, just as the proper time to hit the singularity in a black hole is finite.
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