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Phy 442/542 — HW#11

11.1 Static spherically symmetric space-time: the Ricci tensor

The metric of the static spherically symmetric space-time is given by the line element

g dxdx  = A(r)dt2 + B(r)dr2 + r2d2 + r2 sin2 d2 :

Calculate the components of the Ricci tensor,

R  = @   @  

 +

 

 

  :

Hint: Use the Christo¤el coe¢cients calculated in a previous problem.

Answer: the non-vanishing components are

Rtt   =   A00

2B  A

0

4B

A0

A  + B

0

B

+   A

0

rB  ;

Rrr   =   A00

2A +

  A0

4A

A0

A  +

 B 0

B

+

  B0

rB  ;

R   = 1 1

B 

r

2B

A0

A 

B0

B

  ;

R   = sin2 R   :

11.2 Visualizing Schwarzschild’s space-time: Flamm’s paraboloid

Visualizing 4d spaces is not easy. In cases of high symmetry one can get a good ideaof how the space is curved by studying cross sections in which one or more coordinatesare held …xed. For Schwarzschild’s static space-time take a slice at   t  = const :  Thisyields a curved 3d space which is still too di¢cult to visualize. Take an additional sliceand consider the equatorial plane at    =  =2.

(a)  Show that the metric for this 2d equatorial slice is

d`2

=  dr2

1 2m=r + r2

d2

:

(b)  To “see” how this surface is curved we imagine that it as a surface embedded ina …ctitious ‡at 3d space. Calculate the equation of this surface of revolution andshow that it is a paraboloid.

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Hint: In cylindrical coordinates  (r;;z) the …ctitious ‡at 3d space has metric

d`2 = dr2 + r2d2 + dz2 :

The equation   z   =   h(r)   de…nes a surface of revolution . Find the metric  d`2 of thesurface for a generic height function  h(r) and then determine the  h(r) that reproducesthe metric in part (a) for  r > 2m.

11.3 Gravitational redshift in a Schwarzschild black hole

Photons of energy h!e  are emitted by a source (say, hydrogen atoms) at rest at radiusre. They travel radially out to an observer at rest at  ro.

(a)  Show that an observer at rest at radius r  has four-velocity

U  = (U 0; 0; 0; 0)   where   U 0 = (1 2M 

r  )1=2 :

(b)  Show that if the photon energy as measured by an observer at rest at the source

is p 0̂ = h!e  then the energy detected by the observer is

h!o = h!e

1 2M=re1 2M=ro

1=2:

Hint: the conserved photon “energy” is    p0, and the observed photon energy ish!o  =  p  U .

(c)   Show that as the source approaches the horizon,   re   !   rS    = 2M , it becomese¤ectively unobservable by a distant observer.

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