Integral in Hubble Flow

• B
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?]

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Orodruin
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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.

@PeterDonis Thanks PeterDonis. You haven't forgot my previous question, and neither do I.
My point is that you shouldn't ask them at all; you should first spend some time looking up references--...
Sean Carroll's lecture notes on GR ...
Thanks again
Carroll's article gives a good overview of the issues involved here:

http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

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.
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..

You are making a large number of unjustified assumptions...
The fact that your final expression contains a logarithm of a parameter with physical dimension should be a huge warning sign.
Okay..., perhaps (and it's true) my math is very poor. My question is beyond my understanding. But, I read this...
Redshift is largely the only yardstick we have for the distant universe. While it has served us well and made many confirmed predictions that does not 'prove' it is invincible. But, without it we are lost in illusions of reality. We must either place faith in observational evidence or our ancient religious beliefs. I believe the former offers us a better future.