Null coordinates

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Main Question or Discussion Point

There’s one thing regarding Eddington-Finkelstein coordinates I’m still not entirely sure about. According to most sources, in Minkowski space, the ingoing and outgoing null coordinates are expressed-

[tex]v=t + r[/tex]

[tex]u=t - r[/tex]

Where v is the ingoing null coordinate and u is the outgoing null coordinate.

A null coordinate is when spacetime=zero (i.e. time=0 for light) so if we take Minkowski spacetime-

[tex]c^2d\tau^2=ds^2=c^2dt^2-dx^2-dy^2-dz^2[/tex]

and consider just [itex]t, x[/itex] and set the spacetime to [itex]s=0[/itex], we get-

[tex]ds=0=cdt-dx[/tex]

which in some way resembles the outgoing null coordinate in Eddington-Finklestein coordinates.

Source- http://www.phys.ufl.edu/~det/6607/public_html/grNotesMetrics.pdf pages 1-2


The results also apply in curved spacetime and the null coordinates get a bit more sophisticated when introducing the tortoise coordinate, which in some way relates the local behaviour of light relative to the observer at infinity or as "www2.ufpa.br/ppgf/ASQTA/2008_arquivos/C4.pdf"[/URL] puts it, '..In some sense, the tortoise coordinate reflects the fact that geodesics take an infinite coordinate time to reach the horizon..'-

[tex]v=t + r^\star [/tex]

[tex]u= t - r ^\star[/tex]

For the Schwarzschild metric, r* is-

[tex]r^\star=r+2M\,\ln\left|\frac{r}{2M}-1\right|[/tex]

And slightly more sophisticated for Kerr metric (see [URL]https://www.physicsforums.com/showpost.php?p=2098839&postcount=4"[/URL])

I’m still not entirely sure of what to make of t. Is there a spurious c that needs to be introduced or is this introduced later in the metric? Is t simply a countdown to zero at the centre of mass, matching r at large distances? When calculating the v and u coordinates based on t simply equalling r, the coordinates v and u do seem to make sense.


When transferring over to Kruskal-Szekeres coordinates-

[tex]V=e^{(v/4M)[/itex]

[tex]U=e^{(u/4M)[/itex]

Which works fine with both tending to zero at the event horizon of a black hole, the only query I have is that when r gets larger, V and U tend to infinity fairly quickly before r really gets too large. Is this the norm?
 
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Answers and Replies

  • #2
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While the quantity for the tortoise coordinate (r*) is relatively easy to establish, can anyone confirm what quantity is used for t? Does it match the radius as a quantity ranging from infinity to zero as you approach the object or is it assumed to simply be zero in respect of null coordinates and v and u are based on the tortoise coordinate only? The equation for the change in ingoing null coordinates is-

[tex]dv=dt+\frac{r^2+a^2}{\Delta}\,dr[/tex]

which implies that t does have a quantity but is it counting up or down as you approach the object?
 
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