Time Dilation: Twins Separated by Black Hole's Gravity

[Dynamotime]

is This example correct?
I made it up

Twin Z get close to a black hole to observe it (theoretically he is Not in orbit, Not moving but close in off to an black hole to be affect by the gravity of it, and the black is not moving in space or rotating)
And Twin X stay behind on the mothership which is not moving or affected by the a gravity fieldWe want to know the time dilation between the two twins due to gravity of the black hole-M is the mass of the black hole. M = 21x10^30 kg

-G is the gravitational constant G = 6.67408 × 10^-11

-c is the speed of light c = 299 792 458 m/s

-rs is the Schwarzschild radius
rs = 2GM/c^2 = (2G*21*〖10〗^30)/〖299792458〗^2 = 31188.8450 meters

-r is the (theoretical) distance of twin Z from the center of the black hole r = 35 000 m

-T0 is the elapsed time for twin Z T0 = 10 days = 864 000 s

-T is the elapsed for twin X

Time dilation formula for no-moving and no-rotating object

T= T0/√(1-rs/r) =
(864 000)/√(1-(31 188.8450)/(35 000)) = 2 618 300.31930 seconds ≈ 30 days

When Twins Z comes back on the mother ship Twin X is 30-10= 20 days older

 

[DEvens]

I did not check your arithmetic, but that is basically the idea.

For making calculations like this easier, some simple little rules.

Don't substitute in the numerical values until the very end. Leave it in terms of the formula.

Use convenient units. For example, you are interested in days, so leave it in days. Don't swap back to seconds. The sensible units for the rs in this case are kilometers.

More to the point for units in relativity, read about the so-called "natural" units. In those units, the speed of light is a unit-free 1 exactly. The value of Newton's constant is also a unit-free 1 exactly. Essentially this is picking some very special units for time and mass. You should read about that in your text. Getting rid of all the G's and c's in your calculations will make things much less difficult. Not easy exactly. Just easier.

So putting that all together, it means that the mass of the black hole is 31.12 km, which is just equal to its rs. And the radius of the location of the observer is 35 km. And so 1/sqrt(1-rs/r) is about 3.003.

 

[Dynamotime]

Thanks for advise