Time dilation on Earth due to Rotation

TIME DILATION ON EARTH’S SURFACE DUE TO ROTATION

INTRODUCTION

Earth is the 3^rd planet in our Solar System. It completes one rotation about its axis in 23 hours 56 minutes and 4 second. Consider our planet Earth which rotates at a certain velocity and another analogous earth which doesn’t rotate. According to the Special Theory of Relativity, a clock on our Earth would run slower than that on a non-rotating Earth. We’ll find the time gained by our earth relative to the stationary one.

ASSUMPTIONS

• Earth is not revolving around the Sun.

• 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 – Rotational time period [T ≈ 24 hours = 86400s]

ω = 2*3.14/86400

ω = 7.2722*10^-5 rad/s

The angular velocity is same at all points on earth since it doesn’t exhibit differential rotation. But the tangential velocity on surface varies with the distance from the center.

The tangential velocity is given by,

v = R*ω [m/s]

R – Average radius of earth [R = 6371000m]

v = 6371*1000*7.2722*10^-5

v = 463.311m/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*√ [1-2.39*10^-12]

t' = t*√ [0.999999999997615]

t' = t* 0.999999999998808

CONCLUSION

We can observe that proper and actual time isn’t the same which proves that time dilates on rotating earth relative to the non-rotating 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 non-rotating one. In fact it’ll gain 37 microseconds in one year.

Time	t’ [Non-rotating Earth] (s)	t [Rotating Earth] (s)	Difference (s)
1 minute	60	59.9999999999285	0.0000000000715
1 hour	3600	3599.99999999571	0.00000000429
1 day	86400	86399.999999897	0.000000103
1 month	2592000	2591999.99999691	0.00000309
1 year	31536000	31535999.9999624	0.0000376