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Gravitational time dilation at 1000 km and 100 solar masses

Gravitational time dilation is a phenomenon predicted by Einstein's theory of general relativity, where time passes more slowly in regions of stronger gravitational fields. The gravitational time dilation factor between two points with a gravitational potential difference of ΔΦ can be calculated using the following formula:

t_0 / t_f = sqrt(1 - 2 * ΔΦ / (c^2 * Δr))

where t_0 and t_f are the proper times measured at the points, c is the speed of light, and Δr is the distance between the points.

In this case, we want to calculate the gravitational time dilation at a distance of 1000 km from a massive object with a mass of 100 solar masses. Assuming that the object is spherically symmetric, the gravitational potential at a distance r from the center of the object can be calculated as:

Φ = - G * M / r

where G is the gravitational constant and M is the mass of the object. Substituting r = 1000 km and M = 100 * M_sun (where M_sun is the mass of the sun), we get:

Φ = - G * (100 * M_sun) / (1000 km) = - 2.963 * 10^11 J/kg

Using the formula above, we can calculate the gravitational time dilation factor between a point at this distance and a point infinitely far away (where ΔΦ = 0):

t_0 / t_f = sqrt(1 - 2 * (-2.963 * 10^11 J/kg) / (c^2 * 1000 km)) = 0.9999999999997617

This means that time passes about 1.0000000000002383 times faster at a point infinitely far away compared to a point at a distance of 1000 km from the massive object.

gravitational time dilation | 

radius | 1000 km (kilometers)

mass | 100 M_☉ (solar masses)

time in rest frame | 1 second

time seen by stationary observer | 1.191 seconds

t = t_0/sqrt(1 - (2 G M)/(r c^2)) | 

t | time is seen by a stationary observer

r | radius

M | mass

t_0 | time in the rest frame

G | Newtonian gravitational constant (≈ 6.674×10^-11 m^3/(kg s^2))

c | speed of light (≈ 2.998×10^8 m/s)

(assuming a nonrotating spherical body)