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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 Gullstrand-Painleve 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 freely-falling 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
- Theory Of Relativity, Black hole