Null Coordinates: Understanding & Exploring

[stevebd1]

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-

$$v=t + r$$

$$u=t - r$$

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-

$$c^2d\tau^2=ds^2=c^2dt^2-dx^2-dy^2-dz^2$$

and consider just $t, x$ and set the spacetime to $s=0$, we get-

$$ds=0=cdt-dx$$

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 1-2The 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..'-

$$v=t + r^\star$$

$$u= t - r ^\star$$

For the Schwarzschild metric, r* is-

$$r^\star=r+2M\,\ln\left|\frac{r}{2M}-1\right|$$

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?

 

[stevebd1]

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-

$$dv=dt+\frac{r^2+a^2}{\Delta}\,dr$$

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

 

[Entropia]

Thank you for sharing your thoughts on null coordinates and their application in Minkowski space and curved spacetime. It seems like you have a good understanding of the concept and its relation to the Eddington-Finkelstein coordinates.

As for your question about the spurious c and its introduction in the metric, it is important to note that the c in the equation represents the speed of light and is a constant in all frames of reference. Therefore, it does not need to be introduced separately in the metric.

Regarding the behavior of V and U in Kruskal-Szekeres coordinates, it is normal for them to tend to infinity as r gets larger. This is because these coordinates are used to describe the entire spacetime and not just a specific region. As r gets larger, the spacetime curvature also increases, leading to the coordinates approaching infinity. This is a common feature in many coordinate systems used in general relativity.

Overall, your understanding and exploration of null coordinates is commendable. Keep up the good work!