464 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
y
x
r 1
y ̃
Figure 7.13: Due to the lensing effect, one atxcan also see the star atyas if it is another star
aty ̃.
customary to characterize redshift by a dimensionless quantityzin the formula
1 +z=
λobserv
λemit,
whereλobservandλemitrepresent the observed and emitted wavelenths. There are three red-
shift types:
Doppler effect, cosmological redshift, gravitational redshift.The gravitational redshift in a black hole are caused by boththe gravitational fields of the
emitting object and the black hole.
The first type of redshift, due to the gravitational field, is formulated as
(7.5.36) 1 +z=
√
1 −^2 cmG (^2) r
√
1 −^2 cmG (^2) r
0
,
wheremis the mass of theemitting object,r 0 is its radius, andris the distance between the
object and the observer.
The second type of redshifts, due to the cosmological effector black hole effect, is
(7.5.37) 1 +z=
√
−g 00 (r 0 )
√
−g 00 (r 1 ),
whereg 00 is the time-component of the black hole gravitational metric,r 0 andr 1 are the
positions of the observer and the emitting object (including virtual images).
If a universe is not considered as a black hole, then the gravitational redshift is simply
given by (7.5.36) and is very small for remote objects. Likewise, the cosmological redshift
is also too small to be significant. Hence, astronomers have to think the main portion of the
redshift is due to the Doppler effect:
(7.5.38) 1 +z=
√
1 +v/c
√
1 −v/c