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Hyperphysics: Hafel-Keating experiment

  1. Jul 21, 2010 #1
    http://hyperphysics.phy-astr.gsu.edu/HBASE/relativ/airtim.html#c5

    I don't understand the approximation T0 = -TS that they make in the final step of the section "Kinematic Time Shift Calculation". From this, and the other equations in this section, I get

    [tex]-T_0=T_0\left ( 1+\frac{R^2\omega^2}{2c^2} \right )[/tex]

    [tex]-1=1+\frac{R^2\omega^2}{2c^2}[/tex]

    [tex]c^2=\frac{R^2\omega^2}{-4}[/tex]

    [tex]c=\pm \frac{R\omega}{2i}[/tex]

    but how can this be when c is a constant positive real number, not dependent on the product of the rotation of the earth with its radius? And

    [tex]T_A=T_S-T_S\left ( \frac{2R\omega v+v^2}{2c^2} \right )[/tex]

    [tex]T_0\left ( 1+\frac{R^2\omega^2}{2c^2} \right )=-T_S\left ( -1+ \frac{2R\omega v+v^2}{2c^2} \right )[/tex]

    [tex]T_0\left ( 1+\frac{R^2\omega^2}{2c^2} \right )=T_0\left ( -1+ \frac{2R\omega v+v^2}{2c^2} \right )[/tex]

    [tex]4c^2= 2R\omega v+v^2 - R^2\omega^2[/tex]

    [tex]c=\frac{\sqrt{(R\omega+v)^2-2R^2\omega}}{2}[/tex]

    which can't be right, since c doesn't depend on these arbitrary variables: radius of the earth, etc.
     
  2. jcsd
  3. Jul 21, 2010 #2
    HK is poorly explained using SR, a correct explanation requires GR. I am quite sure that I gave a GR-based explanation for HK somewhere in this forum. It is simply calculating the proper time [tex]\tau[/tex] by integrating the expression in coordinate time t. The expression can be derived straight from the general Schwarzschild metric setting:

    [tex]dr=d\theta=0[/tex],
    [tex]\frac{d\phi}{dt}=\omega +\frac{v_1}{R}[/tex]
    [tex]\frac{d\phi}{dt}=\omega -\frac{v_2}{R}[/tex]

    depending on the direction of plane motion
     
    Last edited: Jul 21, 2010
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