Gravity and Strings

7.1 Schwarzschild’s solution 193
metric in Kruskal–Szekeres coordinates takes the form
ds 2 = 4R3 −r
R
Se S 2 2
(dcT) − dX
r
− r 2 d 2 (2) , (7.18)
where r is a function of T and X that is implicitly given by the coordinate transformations
between the pairs t and r and T and X:
r
− 1 e
RS
r
RS = X 2 − c 2 T 2 ,
(7.19)
ct X + cT
= ln
= 2arctanh(cT/ X),
X − cT
RS
so the Schwarzschild time t is an angular coordinate and the constant-r lines are
similar to hyperbolas that asymptotically approach the X =±cT lines.
A convenient feature of the Kruskal–Szekeres coordinates is that the T, X part is conformally
flat and at each point in the T, X plane the light cones have the same form
as in Minkowski spacetime and no particle can have a worldline forming an angle
smaller than π/4 with the X axis. The r = RS lightlike hypersurface that separates
these two quadrants (I, the exterior, and II, the interior) iscalled the event horizon.
It is then clear that particles or signals can go from the exterior to the interior but
no signal or particle (including light signals) can go from the interior to the exterior.
For this reason, the object described by the full Schwarzschild metric (with no star at
rE > RS) iscalled a black hole (BH).
The existence of an event horizon has very important consequences. First, the freefalling
observer can never come back from the BH and cannot send any information
that contradicts the Schwarzschild observer’s experiences. In this way, the two different
observations are made compatible, completely against our classical intuition.
Second, it is impossible for the Schwarzschild observer to have any experience of
the physical singularity at r = 0. This is pictorially expressed by saying that “the
singularity is covered by the event horizon.”
8. There is another kind of diagram that can be useful for studying the causal structure
of the spacetime: Penrose diagrams (see, for instance, [508]). They are obtained by
performing a conformal transformation of the metric (that preserves the light-cone
structure) such that the infinity is brought to a finite distance in the new metric. A
Penrose diagram of Schwarzschild’s spacetime is drawn in Figure 7.2.
Apart from the existence of an event horizon, we also see clearly in this diagram
that the fate of the free-falling observer will always be to reach the singularity r = 0,
which is now a spacelike hypersurface in which he/she will be crushed by infinite
tidal forces.
9. We know that there are many objects in the Universe whose gravitational fields are
very well described by a region r > rE > RS of the Schwarzschild spacetime, but
what kind of object gives rise to the r < RS region, that is, to the BH metric?