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Lets say that I know how much time dilation will occur at a certian speed, using that, how will I be able to calculate the amount of gravity needed to cause the same amount of time dilation?
Question Date: 2007-09-01
Answer 1:

Sorry for all the math below, but it's probably best to point directly to the equations:
Time_dilation (for delta-t)
Gravitational_time_dilat (for t0)

In the equations used above, you're looking for ratios. For example, for time dilation due to velocity (special relativity):

delta_t = delta_t0 / sqrt(1-v2/c2)

is the equation for time dilation. Divide both sides by delta_t0 to get:

delta_t / delta_t0 = 1 / sqrt(1-v2/c2)

This is the proportion that time will slow down (say, 1/10th of normal time etc.) for a given velocity (v). "c" is the speed of light.

Similarly, for time dilation due to gravity (general relativity):

t_0 = t_f * sqrt(1 - 2*G*M/(r*c2))

...then divide both sides by t_f... t_0 / t_f = sqrt(1 - 2*G*M/(r*c2))

This is the proportion that time will slow down if you are a distance "r" from a given mass "M". "G" is the universal gravitational constant.

Now set the two equations equal:

1 / sqrt(1-v2/c2) = sqrt(1 - 2*G*M/(r*c2))

Square both sides to get rid of the square root: 1 / (1-v2/c2) = 1 - 2*G*M/(r*c2)

Move things around by algebra:

2*G*M/(r*c2) = 1 - 1 / (1-v2/c2)

r = M / ((1 - 1 / (1-v2/c2)) * c2 / (2*G))
or equivalently,

M = (1 - 1 / (1-v2/c2)) * c2 / (2*G) * r

From this we can calculate how close (r) you'll need to be to a planet or star with mass M in order to have the same time dilation as traveling at velocity "v". Or equivalently, how big a mass M you'll need to have if you're a distance "r" from it.

If you start plugging in numbers to the equation above, be careful that you always use consistent units. In other words, don't put in km/hour for one part if you're using meters and seconds elsewhere, for example. It's best to convert all your numbers into meters, kilograms, and seconds before you use them.

Answer 2:

You can't use the same reasoning for both of these scenarios. Time dilation at a fraction of the speed of light according to Special Relativity is based on the viewpoint of constant speed in flat space,without considering gravity. The equation for time dilation that you are thinking of is only good for inertial reference frames, without considering gravity at all. You have to use General Relativity which accounts for curvature of space-time due to momentum and energy to calculate how much time will be slowed by a gravitational potential.Basically, clocks run slower the closer to a large mass (more gravity)they are.

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