Time dilation in equator versus pole of Earth

TIME DILATION IN EQUATOR VERSUS POLE OF EARTH

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 three locations on our planet Earth, the center, equator and the North Pole. All points rotate with the same angular velocity but the center doesn’t have a tangential velocity. According to the Special Theory of Relativity, a clock on Earth’s equator would run slower than that on the pole with respect to the center. We’ll find the time gained by equator and pole relative to the stationary center.

ASSUMPTIONS

1. Earth is not revolving around the Sun.

2. 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 at the center is given by,

v = R*ω [m/s]

R – Average radius of earth [R = 0m][center]

v = 0*7.2722*10^-5

v = 0 m/s

Since tangential velocity is zero, time won’t dilate hence is absolute. We’ll consider time at the center for reference.

The tangential velocity at the equator is given by,

v = R*ω [m/s]

R – Equatorial radius of earth [R = 6378.13 km]

v = 6378.13*1000*7.2722*10^-5

v = 463.8303 m/s

The tangential velocity at the North Pole is given by,

v = R*ω [m/s]

R – Polar radius of circular motion
R = 0 since the axis of pole passes through the center of Earth

v = 0*1000*7.2722*10^-5

v = 0 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]

The time dilation at Equator is,

t' = t*√ [1-(v/c) ²]

t' = t*√ [1-2.3904*10^-12]

t' = t*√ [0.99999999999761]

t' = t * 0.999999999998805

The time dilation at Pole is,

t' = t*√ [1-(v/c) ²]

t' = t*√ [1-0]

t' = t*√ [1]

t' = t* 

CONCLUSION

We can observe that pole and equatorial time isn’t the same which proves that time dilates only on equator and not on pole or center. We’ll consider 5 different t’ values and calculate the t value. The larger the t’ the more is the difference between t and t’. Thus equatorial region will gain around 37 microsecond than the polar region or center in one year.

Time	t’ [Center of Earth] (s)	t [Pole] (s)	t [Equator] (s)
1 minute	60	60	59.9999999999283
1 hour	3600	3600	3599.9999999957
1 day	86400	86400	86399.9999998968
1 month	2592000	2592000	2591999.9999969
1 year	31536000	31536000	31535999.9999623