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Homework Help: Combined time dilation due to SR and GR

  1. Nov 21, 2008 #1
    1. The problem statement, all variables and given/known data

    A cosmonaut spends a few years in an orbit above the Earth. We would like to estimate how his age will differ from his age if he had stayed on Earth. We will consider two seperate effects.

    (a) First calculate the effect due to time dilation from Special Relativity. Let the cosmonaut be orbiting in a circular orbit at a height 200 km above the Earth's surface. Assume that the velocity at the Earth's surface is negligible. What is the ratio of the cosmonaut's time interval compared to the time interval at the Earth's surface?

    (b) The second effect is due to gravitational redshift. Write down an expression for the ratio between the time intervals at the surface of the Earth and in the cosmonaut's spaceship. What is the value of this ratio for the values given in the previous part of the question?

    (c) In part (a) we assumed that the velocity at the Earth's surface was negligible. Explain why this is a reasonable assumption.

    [You may take the radius of the Earth to be 6380km.]

    2. Relevant equations

    Doppler shift equation:

    [tex]\lambda[/tex]1 / [tex]\lambda[/tex]2 = 1 + z = sqrt[(1 +v/c)/(1-v/c)]

    Gravitational redshift:

    [tex]\lambda[/tex] / [tex]\lambda[/tex]0 = 1 + z = [1 - 2GM/(c[tex]^{}2[/tex]R)]^(-1/2)

    3. The attempt at a solution

    I know time is proportional to 1/frequency, so I'm going to need to use the Doppler shift equation in part (a) and the Gravitational redshift equation for part (b).

    The fact I am given a [tex]\Delta[/tex]R (200km) as the distance above the Earth makes me think I'm going to need to apply calculus to these equations. But I'm not exactly sure how to approach this

    For (a) I think I might need to use the radial velocity equation to determine v, then perhaps differentiate this. But I am not sure whether this is the right approach, and even if it is how to go about it.

    I am not sure at all about (c).
  2. jcsd
  3. Nov 22, 2008 #2
    I think I'm getting somewhere with this, is the right approach to use the Newton error approximation:

    For Z(x,y,...)

    (∆Z)² = (∂Z/∂x)²(∆x)² +(∂Z/∂y)²(∆y)² + ...

    Then to find v for Doppler Shift I can use: v=SQRT(MG/R), and I can use the given equation directly for red shift from GR.

    My concern is that when I use the Newton method, what value do I use for r, is it at the point of the cosmonaut, on the surface of the Earth, or in between?

    Also I still have no idea about (c) :S.

  4. Nov 22, 2008 #3
    Would I be best of posting the error approximation in the maths section?
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