Proper Time : Constant Velocity Clock vs Constant Acceleration Clock

In summary: Doing the math, I get 2.598E6 s = 30.07 days. So my answer is a little closer to 30 days than your answer was.
  • #1
morrobay
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Homework Statement


3 clocks at to
A clock on Earth
B clock above Earth 4.66 * 1014 meters
C clock above Earth 3.30 * 1013 meters
B accelerates to .6 c and arrives at Earth when Earth clock reads 2.6 *106 sec
= 30 days With velocity 1.8 *10^8 m/sec with gamma = .8 B clock reads 2.07 * 10^6 sec = 24 days
C clock travels to Earth with constant acceleration in respect to Earth frame of 9.81 m/sec2
and arrives at the same time as B . C clock velocity 2.5 * 10^7 m/sec
What is the proper time on Clock C ?

Homework Equations


Integral to to 2.6 * 106 sec. sqrt [ 1-v(t)2/c2] dt
So Int to to t1 sqrt [ 1-1.01*10-15t2] dt

The Attempt at a Solution

I put Sqrt{1-ax2] in The Integrator
and plugged in values and got 15 days proper time on C clock ?
 
Last edited:
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  • #2
morrobay said:

Homework Statement


3 clocks at to
A clock on Earth
B clock above Earth 4.66 * 1014 meters
C clock above Earth 3.30 * 1013 meters
B accelerates to .6 c and reaches Earth when Earth clock reads 2.6 *106 sec
= 30 days . with gamma = .8 B clock reads 2.07 * 10^6 sec = 24 days
C clock travels to Earth with constant acceleration in respect to Earth frame of 9.81 m/sec2
and arrives at the same time as B
What is the proper time on Clock C ?

Homework Equations


Integral to to 2.6 * 106 sec. sqrt [ 1-v(t)2/c2] dt
So Int to to t1 sqrt [ 1-1.01*10-15t2] dt

The Attempt at a Solution

I put Sqrt{1-ax2] in The Integrator
and plugged in values and got 15 days proper time on C clock ?

The evaluation of the above integral =
1/2x sqrt [ 1-ax2] + sin-1 ( sqrt a) x/2 sqrt a

Note: I am posting this problem because there are endless discussions on proper time
but not many numerical answers
 
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  • #3
morrobay said:
The evaluation of the above integral =
1/2x sqrt [ 1-ax2] + sin-1 ( sqrt a) x/2 sqrt a

For the above problem the values for proper time :
x = 2.6 * 106 sec
a= 1.01 * 10 -15
1/2 x = 1.3 *106
x2 = 6.76 * 1012 sec2
a1/2= 3.17 *10-7
2(a)1/2 = 6.35 *10-7
With these values in the above evaluation:
1/2x [1-ax2]1/2 = 1.29 *106 sec
sin-1 (a)1/2 x / 2 a1/2
= sin-1 .824 = 55
so 55/6.35*10-7 = 8.66*107 added to term on left
1.29*106 sec = 87.8*10^6 sec and is not correct since it is more than A clock ?
 
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  • #4
morrobay said:

The Attempt at a Solution

I put Sqrt{1-ax2] in The Integrator
and plugged in values and got 15 days proper time on C clock ?
All your work is good. But it seems like you are getting bad values out of your numerical integration routine. Perhaps it is a numerical precision problem.

I plug the same integral into Mathematica and get 2.588E6 s = 29.95 days.
 
  • #5
Would you expect that the acellerating clock C in the original problem would have
essentially the same proper time as clock A ( 30 days proper time ) ?
Also clock B had 24 days proper time with .6c

note correction : a = 1.07 * 10 ^-15 from v(t^2)/c^2 = (9.81)^2 m/s^2 / 9*10^16
but does not change values too much.
Yes I am having a few numerical precision problems.
 
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  • #6
morrobay said:
Would you expect that the acellerating clock C in the original problem would have
essentially the same proper time as clock A ( 30 days proper time ) ?
Yes. 3.3E13 m / 2.6E6 s is only an average speed of .04 c which corresponds to an average time dilation factor less than 1.001, so I would expect the clock to not be significantly time dilated overall.
 

1. What is proper time?

Proper time is the time that is experienced by an observer who is moving along with a clock or a system. It is the time measured by a clock that is at rest relative to the observer.

2. What is a constant velocity clock?

A constant velocity clock is a clock that is moving at a constant speed and in a straight line. This type of clock experiences uniform motion and its time measurement is not affected by its velocity.

3. What is a constant acceleration clock?

A constant acceleration clock is a clock that is undergoing acceleration, which means its speed is changing. This type of clock measures time differently than a constant velocity clock and its time measurement is affected by its acceleration.

4. How does the concept of proper time relate to the theory of relativity?

The concept of proper time is a fundamental principle in the theory of relativity. It states that the laws of physics should be the same for all observers who are moving at a constant velocity. This means that the measurement of time should also be the same for all observers in this scenario.

5. Which clock, constant velocity or constant acceleration, measures proper time?

Both clocks measure proper time, but they do so in different scenarios. A constant velocity clock measures proper time in scenarios where the observer is moving at a constant velocity. A constant acceleration clock measures proper time in scenarios where the observer is undergoing acceleration.

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