### 4.5.4. Eddington-Finkelstein Coordinates

Consider a particle free falling from some
point
r>
r
S
towards the origin. As shown in §4.5.3, it takes a duration
Δt→∞
for it to reach
r=
r
S
. Also shown in §4.5.2 is that the
corresponding amount of proper time
Δτ
required is finite. Thus, the singularity at
r=
r
S
is an artifact of the coordinate system
(
ct,r,θ,φ
)
. It can be removed with a suitable choice of
coordinates. A well-known example is the** Eddington- Finkelstein coordinates**
obtained by demanding the null radial geodesics to be straight lines when
expressed in terms of them [see §16.6, D’Inverno].

To be more specific, consider 1^{st}
the incoming null geodesics [see §4.5.3]

ct=−(
r+
r
S
ln|
r−
r
S
| +α
)

For
r>
r
S
,
we can set

c
t
¯
=ct+
r
S
ln(
r−
r
S
)
(a)

to get

c
t
¯
=−r+α
(b)

which is a straight line in the graph
c
t
¯
vs *r*. Substituting

cd
t
¯
=cdt+
r
S
dr
r−
r
S
=cdt+
r
S
r
(
1−
r
S
r
)
−1
dr

into the line element (4.34) gives

c
2
d
τ
2
=(
1−
r
S
r
)d
t
¯
2
−2
r
S
r
d
t
¯
dr−(
1+
r
S
r
)d
r
2
−
r
2
(
d
θ
2
+
sin
2
θ d
φ
2
)
(c)

which is regular for all
r≠0
. Thus, the transformation (a) extends the coordinate
range from
r
S
<r<∞
to
0<r<∞
. Calling
r
S
<r<∞
region I and
0<r<
r
S
region II, we say that (c) is an **analytic extension** of (4.34) from
region I into region II as
t→∞
.
Note that even though (a) is not defined
for
r<
r
S
,
the extension is valid as long as (a) applies to a finite overlap of the
domains of (c) and (4.34). Further
simplification is obtained by introducing the **advanced time parameter**

v=c
t
¯
+r
=ct+r+
r
S
ln(
r−
r
S
)
(4.48)

so that (c) becomes

c
2
d
τ
2
=(
1−
r
S
r
)d
v
2
−2dvdr−
r
2
(
d
θ
2
+
sin
2
θ d
φ
2
)
(4.49)

Thus, the incoming null geodesic (b)
becomes
v=α=const
,
which represents straight lines of slope -1 in the graph
c
t
¯
vs *r*
[see Fig.4.3].

For outgoing particles, we introduce the **time-reversed** coordinate

ct*=ct−
r
S
ln(
r−
r
S
)

together with the **retarded time parameter**

w=ct*−r=ct−r−
r
S
ln(
r−
r
S
)
(4.50)

so that (4.34) is analytically extended
from region I into region II* (
0<r<
r
S
):

c
2
d
τ
2
=(
1−
r
S
r
)d
w
2
+2dwdr−
r
2
(
d
θ
2
+
sin
2
θ d
φ
2
)
(4.50a)

Thus, the outgoing null geodesics are
w=const
,
which represent straight lines of slope +1 in the graph
ct*
vs *r*
[see Fig.16.12, D’Inverno]. Note that
the forward light cones in region II point to the right because we are dealing
with a time-reversed solution.