Radial fall in the Scwarzschild field

Lemaitre coordinates

In the Schwarzschild metric
ds2 = (1 − rg
r
 ) dt2− dr2
1 − rg
r
 − r2(dθ2 +sin2θdφ2),
(1)
the gravitational radius rg=2M is a singular point and inside the gravitational radius r becomes time-like and t becomes space-like. This singularity however is not a physical divergence but rather an artifact of the (false) assumption, that a static Schwarzschild coordinates can be realized under the gravitational radius with material bodies.
A transformation to the Lemaitre coordinates τ, ρ
dτ
=
dt +   
√
rg
r
 
 dr
1− rg
r
 , 
(2)
dρ
=
dt +   
√
r
rg
 
 dr
1− rg
r
(3)
leads to the Lemaitre metric, where the singularity at rg is removed,1
ds2 = dτ2 − rg
r
 dρ2− r2(dθ2 +sin2θdφ2) ,
(4)
where r=[[3/2](ρ−τ)]2/3rg1/3, which is obtained by integrating dρ−dτ = √{[(r)/(rg)]}dr.
The Lemaitre coordinates are synchronous2 and are thus realized by a system of clocks in a free radial fall from infinity towards the origin.

Radiall fall towards the origin

For a free falling body, dρ = 0, equation (3) gives
dt = −   
√
r
rg
 
 1
(1− rg
r
 )
 dr   .
(5)
In the region r >~rg we have to the lowest order
dt = − rg
r−rg
 dr  ,   ⇒   r−rg
r0−rg
 = e−[(t−t0)/(rg)].
(6)
Apparently, it takes a free falling body infinitely long t-time3 to reach the Schwarzschild radius.
On the contrary, a free falling Lemaitre clock moves from some given r0 to the gravitational radius (and also to the origin) within finite τ-time ∆τ,

∆τ = −⌠
⌡ 		rg

r0 
   
√
r
rg
 
 dr = 2
3
r03/2−rg3/2
rg1/2

 .
(7)

Event horizon and black holes

Along the trajectory of a radial light ray ds2=0=dτ2−√{[(rg)/(r)]}dρ2, which gives
dρ = ±   
√
r
rg
 
 dτ  ,
(8)
where plus and minus describe the rays of light sent correspondingly up and down. Inserting dρ = dτ+√{[(r)/(rg)]}dr into (8) leads to
dr=

 ±1−   
√
rg
r
 


 dτ.
(9)
Apparently if r < rg then there is always dr < 0 and thus the light rays emitted radially inwards and outwards both end up at the origin. In other words no signal can escape from inside the gravitational radius - a phenomenon called an event horizon.
Thus a massive object with a size less than the gravitational radius, called a black hole, is completely under the event horizon and its interior is totally invisible. The black holes can be detected if they interact with the matter outside the event horizon.

Excercises

  1. What is the escape velocity4 at a coordinate r in the Schwarzschild field? Investigate the limits r << rg (should be Newtonian) and r→ rg (should become impossible to escape, right?).
    1. show that the equation of motion of a free radially moving body in the Schwazschild field can be written as

      		1− rg
      r

      		 dt
      ds
      		 =E ,  E2−
      		 dr
      ds

      		 2

       
      		 =1− rg
      r
      		 .
    2. show that if the body has reached r=∞ the integration constant E is its energy divided by mass;
    3. show that for a body, starting a free radial fall from infinity with zero velocity, E=1.
  2. What is the g-force5 experienced by an observer at rest at a coordinate r > rg in the Schwarzschild field? Investigate the limits r << rg (should be Newtonian) and r→ rg (should be something horrible, I guess).

Footnotes:

1 there remains a genuine singularity at the origin
2 the metric has the form ds2=dτ2−...
3 the time used by the outer observer
4 a body starting a free radial fall at infinity with zero velocity will reach a given r with [(dr)/(ds)] equal escape velocity;
5 the g-force is equal the proper-acceleration [(d2r)/(ds2)] of a free radially falling body at the moment when the body is at rest

File translated from TEX by TTH, version 3.77.
On 10 Oct 2007, 14:12.