I'd like to know your interpretation.

[EhsanZ]

With Hobson's notation:

H=(da/dt)/a

dH/dt = ((d2a/dt2)/a) - ((da/dt)/a)2

If ((d2a/dt2)/a) is negative, (dH/dt) is certainly negative.
But if ((d2a/dt2)/a) is positive, (dH/dt) can be positive or negative.

What is your interpretation?

 

[mathman]

dH/dt = ((d2a/dt2)/a) - ((da/dt)/a)2

Above is incorrect. Should be:

dH/dt = (d²a/dt²)/a - (da/dt)/a²

Your second term is wrong.

 

[I like Serena]

No, it is correct.
I think you're forgetting the application of the chain rule.

I also think EhsanZ's interpretation is correct, although I do not know what "Hobson's notation" is.
I couldn't find it with google, except for in this thread. ;)

 

[BillSaltLake]

Actually I think the only thing wrong with second eq. is a lack of superscripting of the 2s:

dH/dt = ((d²a/dt²)/a) - ((da/dt)/a)²

 

[George Jones]

With Hobson's notation:

H=(da/dt)/a

dH/dt = ((d2a/dt2)/a) - ((da/dt)/a)2

If ((d2a/dt2)/a) is negative, (dH/dt) is certainly negative.
But if ((d2a/dt2)/a) is positive, (dH/dt) can be positive or negative.

What is your interpretation?

d^2 a/dt^2 > 0 and dH/dt < 0 means that the expansion of the universe is accelerating while the Hubble constant is decreasing. We think that this is happening today.

although I do not know what "Hobson's notation" is.
I couldn't find it with google, except for in this thread. ;)

This (I presume) refers to the notation used in the book General Relativity: An Introduction for Physicists by Hobson, Efstathiou, and Lasenby.

EhsanZ, play around with the spatially flat, matter-only Lemaitre model given on page 406 and in problem 15.23. This a good analytic approximation to our universe.

 

[Chronos]

George is correct. If you look at the equations dispassionately, it is obvious the result can never be negative.

 

[mathman]

No, it is correct.
I think you're forgetting the application of the chain rule.

I also think EhsanZ's interpretation is correct, although I do not know what "Hobson's notation" is.
I couldn't find it with google, except for in this thread. ;)

You're right. My bad.

 

[EhsanZ]

dH/dt = ((d2a/dt2)/a) - ((da/dt)/a)2

Above is incorrect. Should be:

dH/dt = (d²a/dt²)/a - (da/dt)/a²

Your second term is wrong.

No my friend! U made a mistake.

 

[EhsanZ]

You're right. My bad.

Doesn't matter! Everybody makes mistakes.

 

[EhsanZ]

No, it is correct.
I think you're forgetting the application of the chain rule.

I also think EhsanZ's interpretation is correct, although I do not know what "Hobson's notation" is.
I couldn't find it with google, except for in this thread. ;)

I meant the notation that Hobson had used in his book named "General Relativity: An Introduction for Physicists ".

 

[EhsanZ]

Actually I think the only thing wrong with second eq. is a lack of superscripting of the 2s:

dH/dt = ((d²a/dt²)/a) - ((da/dt)/a)²

Yes, you're right my friend. I should've written it more carefully.
Thanks