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ASTR 3740 Relativity & Cosmology Spring 2007. Problem Set 5. Due Wed 17 Mar

1. Geodesics in the Reissner-Nordström geometry

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

ds2 = B  dt2 - dr2
B
- r2 do2
(1.1)
where do2  dq2+sin2q df2 is the metric on the surface of a unit 3-sphere, and
B = 1 - 2 M
r
+ Q2
r2
 .
(1.2)
Similarly to Problem Set 4, the equations of motion of a (neutral) particle freely-falling in the Reissner-Nordström geometry are
B dt
ds
= E
 
r2 df
ds
= L
(1.3)


dr
ds


2

 
+ Veff2 = E2
 
where s is the proper time of the particle, and E and L are constants, the particle's energy and angular momentum per unit mass. The quantity Veff is the effective potential given by
Veff2 = B

1 + L2
r2


 .
(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

v - dr
dtff
 .
(1.5)
What is this velocity v in terms of the metric coefficient B?

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

dt = dtff - v
1 - v2
dr
(1.6)
transforms the metric (1.1) into the river metric
ds2 = dtff2 - ( dr+v dtff )2 - r2 do2  .
(1.7)
[Hint: It is easiest to derive this by expressing the metric coefficient B in terms of v.

(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.