Reissner-Nordstrom spacetime-diagram in Eddington-Finkelstein coordinates

honeytrap

I am currently trying to understand the causal structure of the Reissner-Nordstrom (RN) spacetime.

My goal is to get this visual description of the RN spacetime Eddington-Finkelstein coordinates (derived from the metric):



But during my calculation several problems occured (see below)!

I am starting with the Reissner-Nordstrom Metric in Eddington-Finkelstein coordinates:


Then I set [itex]\varphi[/itex]=constant and [itex]\theta[/itex]= constant (from dΩ), which means d[itex]\varphi[/itex]=d[itex]\theta[/itex]=0

and I consider therefore the 2-dim term

.

By integration of (regarding "r") I get:



Furthermore because of du=0 => u=constant.

Now I should be able to draw (to plot) the radial light trajectories and the above spacetime diagram.

BUT: It is a well-known fact that you have to premise that m>q (in order to avoid imaginary event horizons r_+= m+√(m^2-q^2) and r_-= m-√(m^2-q^2)). But with this requirement I get an imaginary square root (in the integrated term)?! And this cannot lead to the above sketch!

My question is now:

- What is wrong with my approach? Do draw the spacetime sketch there cannot be any imaginary components

- Can someone show me the detailed steps how to get the desired spacetime diagram derived from the metric (e.g. I have the metric, how can I get the above sketch and understand the causal structure)?

- Is there an easy way to plot the spacetime diagram online?

THANK YOU!

PS: I my RN-diagram I will directly use the u-axis (instead the t-axis) and the r-axis.