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Schwarzschild radial light rays

  1. Apr 24, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider Schwarzschild spacetime.
    A) Show that the equation for ingoing/outgoing radial light rays is dt/dr = +-r/(r-2m) in t,r coordinates and dt*/dr = -1, dt*/dr=(r+2m)/(r-2m) in t*,r coordinates
    B) Sketch the local light cones in t,r and t*,r coordinates
    C) Explain which coordinates give a truer physical description of the local light cones.
    D) Explain the motion of a radial light ray emitted near r=2m.


    2. Relevant equations
    schwarzschild metric 0=g[itex]_{\mu\upsilon}[/itex](x(t))[itex]\dot{x}[/itex][itex]^{\mu}[/itex](t)[itex]\dot{x}[/itex][itex]^{\upsilon}[/itex](t)= -[itex]\frac{r-2m}{r}[/itex] +[itex]\frac{r}{r-2m}[/itex][itex]\dot{r}[/itex][itex]^{2}[/itex]+r[itex]^{2}[/itex]sin[itex]^{2}[/itex][itex]\theta\phi[/itex][itex]^{2}[/itex]+r[itex]^{2}[/itex][itex]\dot{\theta}[/itex][itex]^{2}[/itex]

    t[itex]_{*}[/itex]=t+2mln([itex]\frac{r}{2m}[/itex]-1)

    3. The attempt at a solution
    My main struggle here is with part A) ... C) I also am not 100% sure of.


    A) I have managed to show dr/dt*=[itex]\frac{r-2m}{r+2m}[/itex] for the outgoing t*,r coordinates, by using null coordinates, but I am clearly missing something here.

    B) *i have this one completed also*

    C)I believe that the t coordinates give a truer physical description (for us), as these are introduced in order to adapt to the light rays. i.e. t* is the time coordinate adapted to the light rays, thus, this coordinate system should give the physical description for the light rays... but t should be physically truer for us (?)

    D)I have said that all light rays near the r=2m will eventually hit the singularity, however the outward directed ray from outside the r=2m line will always remain outside r=2m, 'escaping' the blackhole.

    Any hints/help are greatly appreciated!
     
  2. jcsd
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