Time dilation on Alpha arae's surface due to rotation

TIME DILATION ON ALPHA ARAE’S SURFACE DUE TO ROTATION

INTRODUCTION

Alpha Arae is a distant star in our Solar System with a distance of 270 light years from the Earth. It rotates as fast as 375km/s so it looks like an ellipsoid. Consider the star Alpha arae which rotates at a certain velocity and another analogous Alpha arae which doesn’t rotate. According to the Special Theory of Relativity, a clock on the rotating Alpha arae would run slower than that on a non-rotating Alpha arae. We’ll find the time gained by rotating Alpha arae relative to the stationary one.

ASSUMPTIONS

• Alpha arae is a perfect homogeneous sphere.

• The effect of Gravitational time dilation is negligible.

• Alpha arae rotates properly like a planet and doesn’t exhibit differential rotation.

CALCULATION

The Angular velocity is related to the tangential velocity by,

v = R*ω [m/s]

R – Average radius of Alpha arae [R = 3130650 km]

v – Tangential velocity of Alpha arae [v = 375 km/s]

ω = (375*1000) / (3130650*1000)

ω = 1.1978*10^-4 rad/s

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

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*√ [1-1.5625*10^-6]

t' = t*√ [0.9999984375]

t' = t* 0.999999218749695

CONCLUSION

We can observe that proper and actual time isn’t the same which proves that time dilates on rotating Alpha arae 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 the rotating Alpha arae will gain time over the non-rotating one. It almost gains 25 seconds in one year.

Time	t’ [Non-rotating Alpha arae] (s)	t [Rotating Alpha arae] (s)	Difference (s)
1 minute	60	59.9999531249817	0.0000468750549
1 hour	3600	3599.9971874989	0.0028125033
1 day	86400	86399.9324999736	0.0675000791
1 month	2592000	2591997.97499921	0.02500237
1 year	31536000	31535975.3624904	24.6375289