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Unformatted text preview: Q. 1 a) The simplest way to find the radial velocity of an observer falling freely from infinity is to note that he begins his fall from rest at infinity and thus l = 0 and dt/dτ = 1 which means e = 1. Since e and l are constants of the motion we can simply plug these values into the general expression for an orbit in Schwarzschild (eq. 9.26) and find that dr/dτ = radicalBig 2 M r . We wish to have the coordinate speed dr/dt ′ which is obtained simply by taking our value for e , g rr and g tr and plugging them into the timelike Killing vector conservation law, eq. 9.21 to get dt/dτ . In this case we find that dr/dt ′ = dr/dτ , since dt/dτ = 1. To find the speed of a radial photon simply take the line element and set ds = dθ = dφ = 0 and then solve the quadratic to show that dr/dt ′ = ± 1 radicalBig 2 M r . This is shown, for instance, in the Wikipedia entry for GullstrandPainleve coordinates. Thus a freely falling observer, if he subtracts his own coordinate speed from that for the radial photons finds that they have speed of ± 1. b) None of the downward moving photons will be seen by anyone above, so neither observer will see these (even the freelyfalling observer agrees that these downward photons are moving away from him at speed 1). The upward photons are all invisible to the observer at infinity and all visible to a falling observer who falls all the way to the singularity. It might appear that the first emitted photon from outside the horizon willall the way to the singularity....
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 Fall '11
 Kennefick
 Theory Of Relativity, Black hole

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