ASTR 3740 Spring 2007 Homepage
Back to ASTR 3740 Problem Sets
The Reissner-Nordström 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 free-faller
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 free-faller by tff, so that tff = s along the worldline of the free-faller. The free-faller changes their radial position r in a proper time tff at free-fall velocity
| (1.5) |
What is the value of the free-fall 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 r0, the turnaround radius, does the free-fall velocity v go to zero, besides r ® ¥?
Plot the free-fall 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 free-faller has through the BH.
No credit: Integrate to obtain an explicit expression for the free-fall time tff 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 free-fallers with E = 1 and E = 0 respectively.