Gravitational Time Dilation

Employees on the top floor of a 10 story office building age slightly slower than all other employees working on the floors below theirs. This time dilation is caused by gravity itself. Gravity also distorts the flow of time, and can be quantified with the equation(a simplified version of the Schwarzschild solution):

Δt' = The change in time in the gravitationally influenced reference frame
Δt = The change in time in a reference frame an infinite distance from any mass (a "standard" hour)
c = The speed of light
G = The gravitational constant
M = The mass of the object being approached (such as Earth)
r = The distance from the object being approached

For example:
Let's say a spaceship is approaching the black hole in Messier 87. How close to this black hole must the spaceship get to experience time twice as slowly? This black hole weighs around 6.6 billion times as much as our sun, which weighs 1.989 x 10^30 kg.

The black hole weights: M = 1.989 x 10^30 x 6,600,000,000 = 1.3127 x 10^40 kg.
c =  299,792,458 m/s
G = 6.673 x 10^-11 N*(m/kg)^2
Δt = 1
Δt' = .5

r = ?

Rewriting the equation gives : r = -2GM/c^2((Δt'/Δt)^2-1)
                                                 r = -2(6.673x10^-11)(1.3127x10^40) / (-.75)(299,792,458)^2
                                                 r = 2.599x10^13 m

Which is roughly 6 times the distance between Earth and Neptune. And that is only for time to be slowed by a factor of 2. Assuming an adequately small radius of the actual space hole, on the surface of the black hole time itself would stand still, since the square root term would become 0 or even imaginary.

In simple terms, time slows down closer to the center of a massive object's gravitation field. A real world example would be a nuclear submarine tour. "Participants on a  6 month nuclear submarine tour at 300 meters below sea level would emerge .5 seconds younger than people who remained at sea level for those 6 months" (Davies). Similarly, astronauts come back from space missions having aged slightly less than people earth, but this is caused by a combination of relative velocity time dilation and gravitation time dilation. The increase in time dilation caused by their relative velocity is greater than the decrease in time dilation caused by being further from the earth, and thus they experience a net increase in time dilation.

Next: Time Travel via. Maximized Velocity
Previous: Relative Velocity Time Dilation
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