Time dilation on Earth's surface due to revolution

TIME DILATION ON EARTH’S SURFACE DUE TO REVOLUTION

INTRODUCTION

Earth is the 3^rd planet in our Solar System. It’s 150 million km [93 million miles] from the Sun. It completes one revolution around the Sun in approximately 365 Earth days. Consider our planet Earth which revolves at a certain velocity and another analogous Earth which doesn’t revolve. According to the Special Theory of Relativity, a clock on our Earth would run slower than that on a non-revolving Earth. We’ll find the time gained by our earth relative to the stationary one.

ASSUMPTIONS

• The revolution path is exactly circular.

• Earth 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 ≈ 365 days = 31536000 s]

ω = 2*3.14/31536000

ω = 1.9923*10^-7 rad/s

The tangential velocity is given by,

v = R*ω [m/s]

R – Average distance between Sun and Earth [R = 150 million km]

v = 150*10⁶*1000*1.9923*10^-7

v = 29884.5 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*√ [1-9.9231*10^-9]

t' = t*√ [0.9999999900769]

t' = t* 0.99999999503845

CONCLUSION

We can observe that proper and actual time isn’t the same which proves that time dilates on revolving earth 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 Earth will gain time over the stationary one. In fact in one year it gains 0.15 seconds just due to revolution and 37microseconds due to rotation. The time gained by revolution is more than rotation because Earth revolves at 29.8 km/s as compared to rotation which is 0.463 km/s.

Time	t’ [Stationary Earth] (s)	t [Revolving Earth] (s)	Difference (s)
1 minute	60	59.999999702307	0.000000297693
1 hour	3600	3599.99998213842	0.00001786158
1 day	86400	86399.9995713221	0.0004286779
1 month	2592000	2591999.98713966	0.01286034
1 year	31536000	31535999.8435326	0.1564674