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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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# B Integral in Hubble Flow

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