T constant lines in Minkowski Conformal Diagram

[LAHLH]

Hi,

I've been trying to work out exactly why the t=const and r=const lines look like they do in the Minkowski conformal diagram.

I started with the usual Minkowski metric in polar coords (t,r) then go into null coords, then pull in the infinities by using arctan transformations, finally I then go back to non-null coords (T,R).

I find that the coords must be related as: $$t=\frac{1}{2}\left(\tan{\left(\frac{T-R}{2}\right)}+\tan{\left(\frac{T+R}{2}\right)}\right)=\frac{1}{2}\frac{\sin{(T)}}{\cos{(T)}+\cos{(R)}}$$

The last equality following from a few sum product trig idents etc. Thus it would seem to me that t=constant lines satisfy:

$$\cos{(T)}+\cos{(R)}=c\sin{(T)}$$ and there is also the condition $$0\leq R < \pi$$ and $$|T|+R < \pi$$

Having Maple solve these, they kind of look correct if I choose to piece together the correct +/- solutions it spits out, but I'm not sure, and I thought finding these t=const lines would be simpler some how. Is there another way?

 

[bcrowell]

This book http://homepages.physik.uni-muenchen.de/~winitzki/T7/ has a good explanation of the extent to which the standard transformation for Minkowski space is or is not arbitrary, and what criteria we'd like it to obey. In general, geodesics look very complicated when you put them on a conformal diagram (strange asymmetric S shapes), so it doesn't surprise me too much if the t=const lines are complicated.

 

[LAHLH]

Thanks for the reply. Also the equation for lines of constant r seem to satify $$\cos{(T)}+\sin{(R)}=a\sin{(R)}$$ where the constant $$a \geq 0$$. These reproduce the r constant lines perfectly in the upper half of the triangle but cut off at the lower half, very strange, there must be some branch of the solution Maple is missing when it solves this equation I guess.

I just really thought it would easier to get the equations than it seems, but I guess not if they really are arc trig type functions of other trig functions and there is no simpler form? seems like no book on GR I can find actually solves and shows explicitley the equations for these lines in the conformal diagram but just shows the diagram

 

[bcrowell]

The whole point of conformal diagrams is that they throw away all geodesic structure and only preserve conformal structure. Therefore the question of what geodesics look like on a conformal diagram isn't one that most authors would probably care about. The equations used for mapping also have a high degree of arbitrariness built into them (as discussed in the Winitzki book), so an equation for a geodesic is not anything fundamental about physics.