The observable and non-observable parts of the Universe

[PeterDonis]

But $a(0) = 0$?

Yes, that's why the thingie I wrote is not a valid Riemannian metric for $t = 0$. But it's still a perfectly well-defined mathematical formula.

 

[PAllen]

Most FLRW cosmologies have an irremovable singularity in that not only is there geodesic incompleteness, but curvature invariants become infinite, which prevents an extension of the manifold to remove geodesic incompleteness. However, the special case of a(t)=t, the Milne cosmology is interesting in relation to what @PeterDonis is discussing. Here, geodesic incompleteness is removable because curvature is identically zero everywhere, and a coordinate transform takes you to Minkowski coordinates. Here, the limiting t=0 surface (not considered part of the Milne model) is simply the light cone from t=0, spatial origin. This is topologically R3, because it is S2XR plus the central point. However, metrically, any way of computing its volume is zero because one component of the volume element must be light like. Here we are using the Minkowski metric which is perfectly well defined, and is a true (pseudoriemannian) metric.

 

[PainterGuy]

Thank you all of you!

I see that post #13, #14, #15 were more direct replies to my post. I appreciate and value your advice, and could see your point. There were many points about the big bang which had bothered me for years but I'm glad that I have the basic answers and can picture it better.

There is no doubt that a book should be the primary source of learning, and it gives you a detailed and coherent view of a subject. At the same you need to devote more time to go through a book which some of us find it hard for several reasons.

I have another question cosmic microwave background radiation, CMBR, which is closely related to the observable universe. Could I ask it here or start a new thread?

Thank you!

 

[PAllen]

Thank you all of you!

I see that post #13, #14, #15 were more direct replies to my post. I appreciate and value your advice, and could see your point. There were many points about the big bang which had bothered me for years but I'm glad that I have the basic answers and can picture it better.

There is no doubt that a book should be the primary source of learning, and it gives you a detailed and coherent view of a subject. At the same you need to devote more time to go through a book which some of us find it hard for several reasons.

I have another question cosmic microwave background radiation, CMBR, which is closely related to the observable universe. Could I ask it here or start a new thread?

Thank you!

Start a new thread.