From: John K Clark (johnkc@well.com)
Date: Thu Jan 23 1997 - 22:00:16 MST
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On Wed, 22 Jan 1997 Eliezer Yudkowsky <sentience@pobox.com> Wrote:
>Special Relativity says that lost time is equal to the 'tau'
> factor, equalling:
>
> __________
> / v^2
> / 1 - ---
> \/ c^2
>[...]
>Somebody else says that you actually gain time due to General
>Relativity, which says that time slows down close to a large
>mass [i.e. speeds up in an airplane], but I can't calculate
>that.
- -
On Wed, 22 Jan 1997 Hal Finney <hal@rain.org> Wrote:
>Photons redshift as they climb out of a gravitational well
>because they lose energy. This means the higher observer
>actually sees things running slower lower down, which is
>interpreted as time running faster for the higher observer.
>A photon of energy E has an effective gravitational mass of
>E/c^2, and gravitational potential energy over height h in
>constant gravity field g is m*g*h. So, the photon climbing
>in the Earth's surface gravitational field of g = 9.8 m/s^2
>will lose energy equal to (E/c^2) * g * h, or E * (g/c^2) * h.
>The fractional time change will be equal to the fractional
>energy loss, and that is just (g/c^2) * h. g/c^2 is about
>10^-14 using units of meters. A commercial jet flying at
>10000 m will therefore have a speedup due to GR of about
>10^-10.
The trouble is, g is not constant, it changes as the square of the distance
from the Earth's center, it is only equal to 9.8 m/s^2 at the surface, just
4000 miles up it is a quarter of that value. The easiest way to figure
out how much slower a clock on the surface of the earth is running compared
to your clock that is far from the earth and not in a gravitational field is
to pretend that the surface clock is moving at the escape velocity of the
earth away from you, and then plug that speed into the formula that Eliezer
gave. The reason this works is because of Einstein's principle of equivalence,
you can't tell if the Doppler shift (time dilation) of the object you are
observing is because it's moving away from you or because it's in a
gravitational field.
If you figure Earth's escape velocity is about 7 miles per second and c is
186000 and put that into Eliezer's formula, you find that the clock on the
Earth's surface is only running 99.99999993% as fast as your clock in deep
space. Certainly this is not important for an object as light as the Earth
but there are places in the Universe where it becomes overwhelming. At the
event horizon of a Black Hole the escape velocity is c, the speed of light,
so you can see from the formula that time must grind to a complete halt there.
The gravitational field on the second story of a house is a little weaker
than it is on the first story 10 feet below, so time will run slower down
there, about 3 parts in 10^16 slower. Incredibly this tiny difference has
actually been detected experimentally, and it was done more than 30 years ago,
using the Mossbauer effect.
John K Clark johnkc@well.com
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