The force of
gravity also causes time dilation to occur. In the aforementioned atomic-clock-jet-plane experiment, which was conducted on November 22, 1975 by a U.S. Navy
P3C antisubmarine patrol plane, only about 10% of the time dilation experienced was due to
velocity. In fact, a clock on the second floor of a building runs slower to an observer in a
free-floating frame than one on the first floor. The equation for gravitational time dilation:
tf = to * √(1 - 2M/r)
Where:
tf = time for the observer in the free-float frame
to = time for the object undergoing time dilation
M = mass of the gravitational body expressed in meters (G/c2 * Masskg
r = the reduced circumference (C/2π)
The time dilation due to the
Earth or even the
Sun's gravity is quite small - the
curvature of
spacetime only differs from flat space by one part in a million in the Sun's case - but gravitational time dilation is quite measureable if there are any
black holes nearby.
Time dilation due to a black hole's gravity can be determined by the simplified equation:
to = tf / √(1 - Ch/Co)
Where:
tf = time for the observer in the free-float frame
to = time for the object undergoing time dilation
Ch = circumference about the event horizon
Co = circumference of the orbiting object
Consider a black hole with a circumference of 10,000 kilometers. An hovering object with a orbital circumference of 20,000 kilometers will appear to have been orbiting for about 1.4 days for every day that passes in the life of the outside observer. Should that object drop to 10,001 kilometers, more than 100 days will pass in the free-float frame while only a day passes for the orbiter!
Sneaking even closer... should the circumferences differ by just a meter... spend just a day orbiting at that distance... and almost ten years will pass.
Perceptive folks will notice that when the object's circumference equals that of the event horizon, division by zero occurs. It's at that point that infinite time dilation occurs. I think that's why no one can observe an object crossing the event horizon, although I think it might have something to do with the infinite red shift of light at the event horizon. Maybe Pseudomancer has the answer...