I'm looking at general relativity and particularly considering what happens at the Big Bang. I think the Friedman equation is [itex]H^2=\frac{8\pi G}{3}\rho[/itex] so I see that as the matter density goes to infinity, [itex]H[/itex] goes to infinity. According to this video (around 10:10), this is where the problem lies with GR's description of the Big Bang. The thing is I don't really get what the Hubble Parameter actually is, so I don't understand what the problem is with [itex]H[/itex] going to infinity. I know [itex]H[/itex] is [itex]\frac{\dot{a}}{a}[/itex], but I can't really find a description of what the scale factor is either. All I know is that it has something to do with the rate of expansion of the universe, although I don;t know how exactly they are linked.

The distances whose growth it governs are ones large enough not to be affected by local gravity within clusters of matter, that tends to hold things together. They are so called "proper distances" which you would measure by conventional means if you could pause expansion long enough, at some given moment, to give you time to do it. We can go into more detail about the definition of distances later.

I like to describe H in terms of a percentage growth rate per million years. Right now distances are growing at the rate of about 1/144 % per million years. Have to go help in kitchen, back soon.

Back now. Company tonight! Lot to do.

a(t) is a number that grows with time showing size of a generic distance relative to its size now.
a(now) = 1, So a(t) = 1/2 back when distances were all half their present size.

a-dot or a'(t) is the derivative or slope of a(t). How much it increases per unit time.

so a'(t) divided by a(t) is the FRACTIONAL GROWTH. By what fraction of itself does it increase per unit time.

So imagine a time t back a long time ago when distances were HALF present, so a(t) = 0.5
And I don't mean distances within our solar system or our galaxy or even our little group of galaxies which holds together by its own gravity. I mean cosmological distances, between clusters of galaxies even, BIG distances not between things bound together in groups.

And suppose you plot that curve a(t) which shows how distances have been growing and you find that the slope is a'(t) = 0.005 per million years.
You find that a(t) is growing by an amount 0.005 per million years.
So then what is a'/a

Will divide 0.005 by 0.5

It is 0.0001 which is 1/100 of one percent. So that means that at that time in universe history, the scale factor a(t) was growing 1/100 % per million years.

That means that all the large scale intergalactic distances were growing at that percentage rate, then.

The Hubble growth rate, H(t) is DEFINED to be a'(t)/a(t)
which is the fractional growth rate of distances at some given time t.

BTW Joanna, thanks for posting that YouTube. I think the woman in it is a very good interviewer: smart and simpatico, also she probes a little deeper than most media interviewers. She isn't content with oversimplifications, she wants to understand in a some depth.

I'd suggest that you keep patiently asking questions here at PF until you feel the Hubble growth rate H(t) has been adequately explained and you have a sense of understanding it. People don't want to load readers with too much explanation, you have to indicate what makes sense to you and what doesn't and when you are ready for the next installment.

Thank you Marcus! I get the Hubble parameter now, but with regards to the Big Bang, is this why general relativity fails when it comes to explaining the Big Bang? I have seen the Schwarzschild metric and from this I can see why you get into trouble with black holes (dividing by zero and, one you get inside, it seems you have imagery time), but have not seen anything similar with the Big Bang. Is $H$ going to infinity the problem in the GR equations when it comes to the Big Bang?

For a matter or radiation dominated universe, the Hubble parameter varies as 1/t, where t is the time since the singularity a(0) = 0. The breakdown of GR is rather related to the high density of the universe at very early times (which in turn is implied by the universe being very very small) and that quantum effects have to be considered.

When it comes to the Schwarzschild solution, time does not become imaginary inside the horizon - the coordinate t simply describes a spatial direction instead of the time direction. The coordinate r is instead the time direction. The coordinates are simply badly chosen and results in an apparent singularity at the horizon. This singularity can be removed by an appropriate change of coordinates. The metric keeps the same signature and there is one time direction and 3 spatial directions everywhere.

Is there an equation that shows that general relativity breaks down at the Big Bang, or is it just that general relativity is known to break down at singularities because it does not take into account quantum effects?

Joanna, attend Stanford University's Leonard Susskind's YouTube lecture series, 150+ episodes that includes cosmology and the Hubble parameter in complete context.