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The ReissnerNordström metric describes the geometry of empty space in and around a spherically symmetric black hole of mass M and charge Q. In units c = G = 1, the metric is
 (1.1) 
 (1.2) 

 (1.3) 

 (1.4) 
(a) Horizons
Horizons in the RN geometry occur where a worldline that is at rest in the geometry, dr = dq = df = 0, is also a null geodesic, ds = 0. What is the condition on the metric coefficient B for a horizon to occur?
For the RN geometry, what are the radii of the horizons in terms of the mass M and charge Q? Evaluate these radii, in units of the BH mass M, for the case where Q/M = 0.8.
What condition on the charge to mass ratio Q/M of the BH is necessary for horizons to exist? FYI, the critical case is called an extremal black hole, which proves to be a case of special interest  for example, the innermost circular orbit of a charged particle with the same charge to mass as the BH is at the horizon, for an extremal BH.
(b) Radial freefaller
A person who falls radially from zero velocity at infinity has unit energy per unit mass, E = 1, and zero angular momentum per unit mass, L = 0. Why? [Hint: Impose the condition of zero velocity on the equations of motion (1.3) in the limit r ® ¥.]
Denote the proper time experienced by such a radial freefaller by t_{ff}, so that t_{ff} = s along the worldline of the freefaller. The freefaller changes their radial position r in a proper time t_{ff} at freefall velocity
 (1.5) 
What is the value of the freefall velocity at a horizon? There are two possible signs to this value, one corresponding to a black hole, the other to a white hole. Which is which?
In the RN geometry, at what radius r_{0}, the turnaround radius, does the freefall velocity v go to zero, besides r ® ¥?
Plot the freefall velocity v as a function of radius r for the case Q/M = 0.8. Don't forget the two possible signs of the square root.
Using your plot of the velocity v as a guide, describe in words the trip that the radial freefaller has through the BH.
No credit: Integrate to obtain an explicit expression for the freefall time t_{ff} as a function of radius r.
(c) River model
Show that the coordinate transformation
 (1.6) 
 (1.7) 
(d) Zero energy geodesic
Return to the equations of motion (1.3) and consider the case of a geodesic with zero energy and angular momentum, E = 0 and L = 0. What is the radial velocity dr/ds on this orbit?
What are the minimum and maximum radii of the geodesic, where the velocity goes to zero?
Plot the radial velocity dr/ds on a diagram.
No credit: Integrate to obtain an explicit expression for the proper time s as a function of radius r on this orbit.
(e) Penrose diagram
Sketch a Penrose diagram of the RN geometry, and on it sketch the trajectories of the two cases you have considered, radial freefallers with E = 1 and E = 0 respectively.