4.5.4. Eddington-Finkelstein Coordinates

Consider a particle free falling from some point r> r S  towards the origin.  As shown in §4.5.3, it takes a duration Δt  for it to reach r= r S .  Also shown in §4.5.2 is that the corresponding amount of proper time Δτ  required is finite.  Thus, the singularity at r= r S  is an artifact of the coordinate system ( ct,r,θ,φ ) .  It can be removed with a suitable choice of coordinates.  A well-known example is the Eddington- Finkelstein coordinates obtained by demanding the null radial geodesics to be straight lines when expressed in terms of them [see §16.6, D’Inverno].

 

To be more specific, consider 1st the incoming null geodesics [see §4.5.3]

        ct=( r+ r S ln|   r r S   |  +α )

For r> r S , we can set

        c t ¯ =ct+ r S ln( r r S )                                (a)

to get

        c t ¯ =r+α                                              (b)

which is a straight line in the graph c t ¯  vs r.  Substituting

        cd t ¯ =cdt+ r S dr r r S   =cdt+ r S r ( 1 r S r ) 1 dr

into the line element (4.34) gives

        c 2 d τ 2 =( 1 r S r )d t ¯ 2 2 r S r d t ¯ dr( 1+ r S r )d r 2 r 2 ( d θ 2 + sin 2 θ  d φ 2 )           (c)

which is regular for all r0 .  Thus, the transformation (a) extends the coordinate range from r S <r<  to 0<r< Calling r S <r<  region I and 0<r< r S  region II, we say that (c) is an analytic extension of (4.34) from region I into region II as t .  Note that even though (a) is not defined for r< r S , the extension is valid as long as (a) applies to a finite overlap of the domains of (c) and (4.34).  Further simplification is obtained by introducing the advanced time parameter

        v=c t ¯ +r   =ct+r+ r S ln( r r S )                                                      (4.48)

so that (c) becomes

        c 2 d τ 2 =( 1 r S r )d v 2 2dvdr r 2 ( d θ 2 + sin 2 θ  d φ 2 )                       (4.49)

Thus, the incoming null geodesic (b) becomes v=α=const , which represents straight lines of slope -1 in the graph c t ¯  vs r [see Fig.4.3].

 

For outgoing particles, we introduce the time-reversed coordinate

        ct*=ct r S ln( r r S )

together with the retarded time parameter

        w=ct*r=ctr r S ln( r r S )                                                    (4.50)

so that (4.34) is analytically extended from region I into region II* ( 0<r< r S  ):

        c 2 d τ 2 =( 1 r S r )d w 2 +2dwdr r 2 ( d θ 2 + sin 2 θ  d φ 2 )                      (4.50a)

Thus, the outgoing null geodesics are w=const , which represent straight lines of slope +1 in the graph ct*  vs r [see Fig.16.12, D’Inverno].  Note that the forward light cones in region II point to the right because we are dealing with a time-reversed solution.