On Oct 23, 2009, at 10:11 AM, Roarty, Francis X wrote:
What about the acceleration at the very center of the earth, I know
It cancels but is time dilated even slower than on the surface or
does dilation cancel too?
Regards
Fran
If an elevator existed to the center of the earth and you took it to
the center, time dilation, (1 - (2 G M)/(r c^2))^0.5, would reduce
to zero when you reached the center. This is due to the fact that the
*effective* M is given by M = k * r^3, because the effect of the
spherical shell mass outside that radius sums to zero. The time
dilation factor f is thus:
f = (1 - (2 G (k * r^3))/(r c^2))^0.5
f = (1 - (2 G (k * r^2))/(c^2))^0.5
and:
(limit r ->0) f = (limit r ->0) (1 - (2 G (k * r^2))/(c^2))^0.5
(limit r ->0) f = (1 - 0)^0.5 = 1
and there is no time dilation. Not that you would notice.
Given the mass of the earth is 5.9736 × 10^24 kg and radius r is
6,378.1 km, and G = 6.67259x10^-11 m^3/(kg s^2), we have:
(2 G M)/(r c^2) = ( 2 (6.67259x10^-11 m^3/(kg s^2)) (5.9736×10^24
kg)) / ( (6,378.1 km) / (2.9979x10^8 m/s)^2)
(2 G M)/(r c^2) = 1.3878x10^-9
and
(1 - (2 G M)/(r c^2))^0.5 = 0.999999999326
so time is slowed by about 1.4828 parts in billion, or about 0.02128
seconds per year at the earth's surface, if I did all that right
(which is doubtful given my track record!)
Best regards,
Horace Heffner
http://www.mtaonline.net/~hheffner/
