Answer 1:
Sorry for all the math below, but it's probably
best to point directly to the equations: Time_dilation
(for deltat) 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(1v^{2}/c^{2})
is the
equation for time dilation. Divide both sides by
delta_t0 to get:
delta_t / delta_t0 = 1 /
sqrt(1v^{2}/c^{2})
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*c^{2}))
...then divide
both sides by t_f... t_0 / t_f = sqrt(1 
2*G*M/(r*c^{2}))
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. Gravitational_constant
Now set the two equations equal:
1
/ sqrt(1v^{2}/c^{2}) = sqrt(1 
2*G*M/(r*c^{2}))
Square both sides
to get rid of the square root: 1 /
(1v^{2}/c^{2}) = 1 
2*G*M/(r*c^{2})
Move things around
by algebra:
2*G*M/(r*c^{2}) = 1 
1 / (1v^{2}/c^{2})
r = M /
((1  1 / (1v^{2}/c^{2})) *
c^{2} / (2*G)) or
equivalently,
M = (1  1 /
(1v^{2}/c^{2})) * c^{2} /
(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.
