Time dilation on Moon due to revolution

TIME DILATION ON MOON DUE TO REVOLUTION

INTRODUCTION

Moon is Earth’s only Natural satellite which was formed billions of years ago when it separated from the Earth.  It’s 400,000 km [250,000 miles] from the Earth. It completes one revolution around the Earth in approximately 27 Earth days. Consider our satellite Moon which revolves at a certain velocity and another analogous Moon which doesn’t revolve. According to the Special Theory of Relativity, a clock on our Moon would run slower than that on a non-revolving Moon. We’ll find the time gained by our Moon relative to the stationary one.

ASSUMPTIONS

• The revolution path is exactly circular.

• Moon is a perfect homogeneous sphere.

• The effect of Gravitational time dilation is negligible.

CALCULATION

The Angular velocity is given by,

ω = 2π/T [rad/s]

T – Revolution time period [T ≈ 27 days 7 hours 43.19 minutes 11.5 second]

T = 27*86400 + 7*3600 + 43.19*60 + 11.5 s

T = 2360602.9 s

ω = 2*3.14/2360602.9

ω = 2.6616*10^-6 rad/s

The tangential velocity is given by,

v = R*ω [m/s]

R – Average distance between Earth and Moon [R = 400,000 km]

v = 400000*1000*2.6616*10^-6

v = 1064.64 m/s

According to the Special Theory of Relativity, the time dilation equation is,

t' = t/γ [s]

t’ – Actual time or Moving observer’s time. [s]

t - Proper time or Stationary observer’s time. [s]

γ – Relativistic gamma factor, γ = 1/√ [1-(v/c) ²]

c - Velocity of light [c = 3*10⁸ m/s]

t' = t*√ [1-(v/c) ²]

t' = t*√ [1-1.2593*10^-11]

t' = t*√ [0.999999999987407]

t' = t* 0.999999999993703

CONCLUSION

We can observe that proper and actual time isn’t the same which proves that time dilates on revolving Moon relative to the stationary one. We’ll consider 5 different t’ values and calculate t value. The larger the t’ the more is the difference between t and t’. Thus our Moon will gain time over the stationary one. In fact in one year it gains 0.1986 milliseconds due to revolution.

Time	t’ [Stationary Moon] (s)	t [Revolving Moon] (s)	Difference (s)
1 minute	60	59.9999999996222	0.0000000003778
1 hour	3600	3599.99999997733	0.00000002267
1 day	86400	86399.9999994559	0.0000005441
1 month	2592000	2591999.99998368	0.00001632
1 year	31536000	31535999.9998014	0.0001986