8.6 Penrose Diagrams
8.6.2 Schwarzschild Spacetime
Penrose diagrams become a powerful tool to explore the global properties and the causal structure of more complicated spacetimes.
Let us now consider the Schwarzschild spacetime in Kruskal–Szekeres coordi- nates. The line element is given in Eq. (8.78). With the following coordinate trans- formation
5The symbolI is usually pronounced “scri”.
Fig. 8.2 Penrose diagram for the Minkowski spacetime
V = 1
2tanT +R
2 +1
2tanT −R
2 ,
U = 1
2tanT +R
2 −1
2tanT −R
2 , (8.85)
the line element becomes ds2= 32G3NM3
r e−r/(2GNM)
4 cos2 T +R
2 cos2 T−R 2
−1
−d T2+d R2
+r2dθ2+r2sin2θdφ2. (8.86)
Figure8.3shows the Penrose diagram for the maximal extension of the Schwarzschild spacetime with its asymptotic regionsi+,i−,i0,I+, andI−. We can distinguish four regions, indicated, respectively, by I, II, III, and IV in the figure.
Region I corresponds to our universe, namely the exterior region of the Schwarzschild spacetime in Schwarzschild coordinates. Region II is the black hole, so the Schwarzschild spacetime in Schwarzschild coordinates has only regions I and II. The central singularity of the black hole atr=0 is represented by the line with wiggles above region II. The event horizon of the black hole atr=rSis the red line at 45◦separating regions I and II. Any ingoing light ray in region I is captured by the black hole, while any outgoing light ray in region I reaches future null infinity I+. Null and time-like geodesics in region II cannot exit the black hole and they necessarily fall to the singularity atr =0.
Regions III and IV emerge from the extension of the Schwarzschild spacetime.
Region III corresponds to another universe. The red line at 45◦separating regions II and III is the event horizon of the black hole atr =rS. Like in region I, any light ray in
160 8 Schwarzschild Spacetime
Fig. 8.3 Penrose diagram for the maximal extension of the Schwarzschild spacetime
region III can either cross the event horizon or escape to infinity. No future-oriented null or time-like geodesics can escape from region II. Our universe in region I and the other universe in region III cannot communicate: no null or time-like geodesic can go from one region to another.
Region IV is a white hole. If a black hole is a region of the spacetime where null and time-like geodesics can only enter and never exit, a white hole is a region where null and time-like geodesics can only exit and never enter. The red lines at 45◦ separating region IV from regions I and III are the horizons atr =rSof the white hole.
Problems
8.1 Let us consider a massive particle orbiting a geodesic circular orbit in the Schwarzschild spacetime. Calculate the relation between the particle proper time and the coordinate timetof the Schwarzschild metric.
8.2 Let us consider the Penrose diagram for the Minkowski spacetime, Fig.8.2.
We have a massive particle that emits an electromagnetic pulse att=0. Show the trajectories of the massive particle and of the electromagnetic pulse in the Penrose diagram.
8.3 Let us consider the Penrose diagram for the maximal extension of the Schwarzschild spacetime, Fig. 8.3. Show the future light-cone of an event in re- gion I, of an event inside the black hole, and of an event inside the white hole.
References
1. C.W. Misner, K.S. Thorne, J.A. Wheeler,Gravitation(W. H. Freeman and Company, San Fran- cisco, 1973)
2. P.K. Townsend, gr-qc/9707012
Chapter 9
Classical Tests of General Relativity
In 1916, Albert Einstein proposed three tests to verify his theory [2]:
1. The gravitational redshift of light.
2. The perihelion precession of Mercury.
3. The deflection of light by the Sun.
These three tests are today referred to as the classical tests of general relativity, even if, strictly speaking, the gravitational redshift of light is a test of the Einstein Equivalence Principle [6], while the other two are tests of the Schwarzschild solution in the weak field limit.1 In 1964, Irwin Shapiro proposed another test [4], which is often called the fourth classical test of general relativity. Like the perihelion precession and the light deflection, even the test put forward by Shapiro is actually a test of the Schwarzschild solution in the weak field limit.
In the case of the perihelion precession of Mercury and of the deflection of light by the Sun, we can test, respectively, the trajectories of massive and massless test- particles in the Schwarzschild background. As discussed in Sect.8.4, we have two differential equations, Eqs. (8.64) and (8.65), which we rewrite here for convenience
u=0, (9.1)
u−kGNM
L2z +u−3GNM
c2 u2 =0, (9.2)
where k=1 for massive particles and k=0 for massless particles. These two equations are the counterpart of Eqs. (1.100) and (1.101) of Newton’s gravity. Equa- tions (1.100) and (9.1) are identical and are simply the equation for a circle, so there
1Note, for example, that the Schwarzschild metric is a solution even in some alternative theories of gravity. Since the Mercury perihelion precession and the light deflection are only sensitive to the trajectories of particles, these tests can only verify the Schwarzschild metric (assuming geodesic motion), they cannot distinguish Einstein’s gravity from those alternative theories of gravity in which the Schwarzschild metric is a solution of their field equations.
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are no interesting implications. Equation (1.101) has the solution (1.103), and the orbits are ellipses, parabolas, or hyperbolas according to the value of the constant A. Equation (9.2) has the last term proportional tou2, which is absent in Newton’s gravity and introduces relativistic corrections.