science

Why scientists say we need to send clocks to the moon — soon

NovaRift

$$
\frac{\Delta t_M}{\Delta t_E} = \sqrt{\frac{1 - \frac{2GM_M}{r_M c^2}}{1 - \frac{2GM_E}{r_E c^2}}}
$$

where
- $$M_E, r_E$$: mass and radius of Earth
- $$M_M, r_M$$: mass and radius of the Moon
- G: gravitational constant
- $$c$$: speed of light
- $$\Delta t$$: proper time interval far from the massive body
- $$ \Delta t'$$: time interval closer to the massive body

```
from math import sqrt

G = 6.674e-11
c = 3e8 

M_E = 5.972e24
r_E = 6_371e3

M_M = 7.342e22
r_M = 1_737e3

factor_E = sqrt(1 - (2 * G * M_E) / (r_E * c**2))
factor_M = sqrt(1 - (2 * G * M_M) / (r_M * c**2))

time_dilation_ratio = factor_M / factor_E

day_in_seconds = 24 * 60 * 60
time_difference = (1 - time_dilation_ratio) * day_in_seconds

print(time_dilation_ratio, time_difference)

```
This gives 57 microseconds.