# B Integral in Hubble Flow

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1. Jul 2, 2016

### Stephanus

Dear PF Forum,
While learning why the net energy of the universe is zero. I've been reading about the expansion of the universe, and of course in it, Hubble Flow.
https://en.wikipedia.org/wiki/Hubble's_law = 73km/s per Mega parsec
https://en.wikipedia.org/wiki/Parsec = 3.26 light year.
In the end I come up with this number.
Hubble flow is 2.37E-18 m/s per meter. Of course as often said, it only works in space where there's very little gravity. Not here on earth or in interstellar medium, perhaps in intergalactic space?

Let's say H = 2.37E-18 m/s per meter or H = 2.37E-18/t.
Now the next question is, how long does it take to reach say... 100,000 trillion meters?
So t = distance/velocity or t = m/v
But this velocity changes over distance, so I have to use integral.
t = m/v in dm
while v = m {distance}*H
so
$t = \int\frac{1}{Hx}dx$
$t = \frac{1}{H}\ln{(x)}$
I try to include 100,000 trillion metres in this equation, so
$t = 4.22 * 10^{17} * \ln{(10^{17})} = 4.22 * 10^{17} * 39.14 = 1.65 * 10^{19}$
So, it takes ... 5.24 * 10 11 years to reach 100,000 trillion metres. Okay...

Now, this. How long does it take to reach 1 meter?
Since ln(1) = 0, so it takes 0 seconds to reach 1 meter?
Did I make a mistake in my calculation?
This is my integral
$t = \int\frac{1}{Hx}dx$
Thank you very much.

[Add: I post this in General physics forum, not in Math forum, because I think even if it is mathematically correct, it's physically impossible?
What if there's an alien civilization where their meter and their seconds is not the same as ours. And our 1 meter is zero second?]

Last edited: Jul 2, 2016
2. Jul 2, 2016

### Orodruin

Staff Emeritus
You are making a large number of unjustified assumptions among them:
• The expansion of the universe occurs with a constant Hubble parameter.
• The Hubble flow starts out at a distance of 1 m. (This sets the lower limit of your integral.)
Obviously, the latter of these assumptions directly implies zero time to reach a distance of 1 m. The fact that your final expression contains a logarithm of a parameter with physical dimension should be a huge warning sign.

3. Jul 2, 2016

4. Jul 3, 2016

### Stephanus

@PeterDonis Thanks PeterDonis. You haven't forgot my previous question, and neither do I.
Thanks again

It seems that I've been drifting a light year away from my original curiosity. The total energy of the (this?) universe is not zero. This has no well defined answer. Still I want to know why it doesn't. Thank you very much for your helps staffs/mentors. I'll do my own reading..

5. Jul 10, 2016

### Stephanus

Okay..., perhaps (and it's true) my math is very poor. My question is beyond my understanding. But, I read this...
Wiki states that: Hubble flow is 73km/sec per Mega persec
= 73km/sec per 3.26 million ly.
So, if the red shift indicates close to 300,000 km/sec then the distance would be... 13.397 billions light year.
Is this simply math concept correct?
So the galaxy that is 13.397 billions light year (nearly the age of the universe) run away from us at close to the speed of light?
Thank you very much.

6. Jul 10, 2016

### Chronos

Irrelevant. A galaxy that distant has left our observable universe and will redshift into oblivion long before any light it emits 'now' can ever reach us.