TIME
DILATION ON NEUTRON STAR DUE TO ROTATION
INTRODUCTION
A Neutron Star is
formed after super nova explosion when the remnant mass is 3 to 5 solar masses.
They’re primarily composed of neutrons and have the same density as that of an
atom. They have a strong surface gravity and escape velocity which is one third
the speed of light. The closest Neutron star is about 600 light years away. It rotates
extremely fast with an angular velocity of 716 rad/s. Consider a Neutron Star
which rotates at a certain velocity and another analogous Neutron Star which
doesn’t rotate. According to the Special Theory of Relativity, a clock on
rotating Neutron Star would run slower than that on a non-rotating Neutron Star.
We’ll find the time gained by rotating Neutron Star relative to the stationary one.
ASSUMPTIONS
- Neutron Star is a perfect homogeneous sphere.
- The effect of Gravitational time dilation is negligible.
- Neutron Star 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
Neutron Star [R = 11 km]
v – Tangential velocity
of Neutron Star
ω – Angular velocity of
Neutron Star [716 rad/s or 43000rpm]
v = 11*1000*716
v = 7876000 m/s
The angular velocity is
same at all points on Neutron Star 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*√ [1-6.8923*10-4]
t' = t*√ [0.99931077]
t' = t* 0.999655325599779
CONCLUSION
We can observe that proper and actual time isn’t
the same which proves that time dilates on rotating Neutron Star 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
Neutron Star will gain a lot of time over the non-rotating one due to its
extremely fast rotation. In fact it gains 10873 seconds [3 hours approximately]
in one year.
Time
|
t’
[Non-rotating Neutron Star] (s)
|
t
[Rotating Neutron Star] (s)
|
Difference
(s)
|
||
1
minute
|
60
|
|
|
||
1
hour
|
3600
|
|
|
||
1
day
|
86400
|
|
|
||
1
month
|
2592000
|
|
|
||
1
year
|
31536000
|
|
|
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