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Frequency shift of light between observers in Schwarzschild space-time

  1. Jul 11, 2008 #1

    Mentz114

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    Considering the simple case of two observers O1 and O2 lying on the same radius at positions r=r1 and r=r2 respectively.

    Using a result from Stephani(1) I work out that the ratio of frequencies of light sent radially between these observers is given by this ratio, numerator and denominator evaluated at the points r=r1 and r=r2 respectively,

    [tex]\frac{\nu_1}{\nu_2} = \frac{(g_{mn}u^m k^n)_1}{(g_{mn}u^m k^n)_2}[/tex] -------- (1)

    where u(i)n is the 4-velocity of O1 and O2 and kn is a null vector defined by the photon trajectory ( up to a constant which will cancel out) so that

    [tex]k_{m;n}k^n = 0[/tex] --------- (2)

    which is the geodesic condition for a (transverse ?) plane wave.

    We need to find the null vector kn. Because only k0 and k1 are non-zero for a radial photon it is not difficult to solve for k from equation (2), up to a factor, which is all we need. I got the components of kn,

    [tex]\left(1 - \frac{2M}{r}\right)^{-\frac{1}{2}}, \left(1 - \frac{2M}{r}\right)^{\frac{1}{2}}, 0, 0[/tex]

    which satisfy equ (1) and also gmnkmkn = 0.

    Now we can calculate [itex](g_{mn}u^m k^n)_i[/itex] which gives,

    [tex](g_{mn}u^m k^n)_i = u_i^1\left(1-\frac{2M}{r_i}\right)^{-\frac{1}{2}} - u_i^0\left(1-\frac{2M}{r_i}\right)^{\frac{1}{2}}[/tex].

    Which looks as if it could be right. If the observers are at rest wrt to each other, then we can write ui0 = 1 and ui1 = 0, which reduces equation (1) to the usual gravitational frequency shift.

    For u to be a proper 4-vector gmnumun = 1 ( speed of light), so it should be possible to generalise this result a bit more.

    If it's correct. This must have been worked out somewhere.



    (1) 'General Relativity', Stephani, (Cambridge, 1993).
     
  2. jcsd
  3. Jul 15, 2008 #2

    Mentz114

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    I haven't had much time for this and resources on the web for geometric optics seem limited. I did notice that my formula derived above,

    [tex](g_{mn}u^m k^n)_i = u_i^1\left(1-\frac{2M}{r_i}\right)^{-\frac{1}{2}} - u_i^0\left(1-\frac{2M}{r_i}\right)^{\frac{1}{2}}[/tex]

    gives the correct SR result for M=0, namely

    [tex]g_{mn}u^m k^n = u^1 - u^0 = \gamma\beta - \gamma = \gamma(\beta - 1)[/tex].

    The sign is different, but that disappears when we take the ratio.
     
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