Lecture Notes on General Relativity (2024)

7. THE SCHWARZSCHILD SOLUTION AND BLACK HOLES

We now move from the domain of the weak-field limit to solutions ofthe full nonlinear Einstein's equations. With the possible exceptionof Minkowski space, by far the most important such solution is thatdiscovered by Schwarzschild, which describes spherically symmetricvacuum spacetimes. Since we are in vacuum, Einstein's equationsbecome R = 0. Of course, if we have a proposed solution to a set of differential equations such as this, it would suffice toplug in the proposed solution in order to verify it; we would liketo do better, however. In fact, we will sketch a proof of Birkhoff'stheorem, which states that the Schwarzschild solution is the uniquespherically symmetric solution to Einstein's equations in vacuum.The procedure will be to first present somenon-rigorous arguments that any spherically symmetric metric (whetheror not it solves Einstein's equations) must take on a certain form,and then work from there to more carefully derive the actual solutionin such a case.

"Spherically symmetric" means "having the same symmetries as a sphere." (In this section the word "sphere" means S², notspheres of higher dimension.)Since the object of interest to us is the metric ona differentiable manifold, we are concerned with those metrics thathave such symmetries. We know how to characterize symmetries ofthe metric - they are given by the existence of Killing vectors.Furthermore, we know what the Killing vectors of S²are, and that there are three of them. Therefore, a spherically symmetric manifold is one that has three Killing vector fields which are just like thoseon S². By "just like" we mean that the commutator of theKilling vectors is the same in either case - in fancier language,that the algebra generated by the vectors is the same. Something thatwe didn't show, but is true, is that we can choose our three Killingvectors on S² to be (V⁽¹⁾, V⁽²⁾,V⁽³⁾), such that

(7.1)

The commutation relations are exactly those of SO(3), the group of rotations in three dimensions. This is no coincidence, of course,but we won't pursue this here. All we need is that a sphericallysymmetric manifold is one which possesses three Killing vector fieldswith the above commutation relations.

Back in section three we mentioned Frobenius's Theorem, which statesthat if you have a set of commuting vector fields then there exists aset of coordinatefunctions such that the vector fields are the partial derivativeswith respect to these functions. In fact the theorem does not stop there, but goes on to say that if we have some vector fields which donot commute, but whose commutator closes - the commutator ofany two fields in the set is a linear combination of other fields inthe set - then the integral curves of these vector fields "fittogether" to describe submanifolds of the manifold on which they areall defined. The dimensionality of the submanifold may be smaller than the number of vectors, or it could be equal, but obviously notlarger. Vector fields which obey (7.1) will of course form 2-spheres.Since the vector fields stretch throughout the space, every point willbe on exactly one of these spheres. (Actually, it's almost every point- we will show below how it can fail to be absolutely every point.)Thus, we say that a spherically symmetric manifold can be foliatedinto spheres.

Let's consider some examples to bring this down to earth. The simplestexample is flat three-dimensional Euclidean space. If we pick an origin,then is clearly spherically symmetric with respect to rotations around this origin. Under such rotations (i.e., under the flow of the Killing vector fields) points move into each other, buteach point stays on an S² at a fixed distance from theorigin.

It is these spheres which foliate .Of course, they don't really foliate all of the space, sincethe origin itself just stays put under rotations - it doesn't movearound on some two-sphere. But it should be clear that almost all ofthe space is properly foliated, and this will turn out to be enough forus.

We can also have spherical symmetry without an "origin" to rotatethings around. An example is provided by a "wormhole", with topology × S². If we suppress a dimension and draw our two-spheres as circles, such a space might look like this:

In this case the entire manifold can be foliated by two-spheres.

This foliated structure suggests that we put coordinates on our manifoldin a way which is adapted to the foliation. By this we mean that, ifwe have an n-dimensional manifold foliated by m-dimensional submanifolds, we can use a set of m coordinate functionsuⁱ on the submanifolds and a set of n - m coordinate functionsv^I to tell us which submanifold we are on. (So i runs from 1 to m, whileI runs from 1 to n - m.) Then the collection ofv's and u's coordinatize the entire space. If the submanifolds are maximallysymmetric spaces (as two-spheres are), then there is the followingpowerful theorem: it is always possible to choose the u-coordinatessuch that the metric on the entire manifold is of the form

(7.2)

Here (u) is the metric on the submanifold.This theorem is saying two things at once: that there are no crossterms dv^Idu^j, and that both g_IJ(v) and f (v) are functions of the v^I alone, independent of the uⁱ. Proving the theorem is a mess, but you are encouraged to look in chapter 13 of Weinberg.Nevertheless, it is a perfectly sensible result. Roughly speaking,if g_IJ or f depended on the uⁱ then the metric would changeas we moved in a single submanifold, which violates the assumption ofsymmetry. The unwanted cross terms, meanwhile, can be eliminated bymaking sure that the tangent vectors /v^I areorthogonal to the submanifolds - in other words, that we line up our submanifolds in the same way throughout the space.

We are now through with handwaving, and can commence some honestcalculation.For the case at hand, our submanifolds are two-spheres, on which wetypically choose coordinates (,) in which the metric takesthe form

(7.3)

Since we are interested in a four-dimensional spacetime, we need twomore coordinates, which we can call a and b. The theorem (7.2)is then telling us that the metric on a spherically symmetric spacetimecan be put in the form

(7.4)

Here r(a, b) is some as-yet-undetermined function,to which we have merely given a suggestive label. There is nothing to stop us, however, from changing coordinates from (a, b) to(a, r), by inverting r(a, b). (The one thing that couldpossibly stop us would be if r were a function of a alone; in this case we could justas easily switch to (b, r), so we will not consider thissituation separately.) The metric is then

(7.5)

Our next step is to find a function t(a, r) suchthat, in the (t, r) coordinate system, there are no cross terms dtdr + drdt in the metric. Notice that

(7.6)

so

(7.7)

We would like to replace the first three terms in the metric (7.5) by

(7.8)

for some functions m and n. This is equivalent to therequirements

	(7.9)
	(7.10)

and

(7.11)

We therefore have three equations for the three unknownst(a, r), m(a, r), and n(a, r), justenough to determine them precisely (up to initial conditions for t). (Of course, they are "determined"in terms of the unknown functions g_aa,g_ar, and g_rr, so in this sense they are still undetermined.)We can therefore put our metric in the form

(7.12)

To this point the only difference between the two coordinates t andr is that we have chosen r to be the one which multiplies themetric for the two-sphere. This choice was motivated by what we knowabout the metric for flat Minkowski space, which can be writtends² = - dt² + dr²+ r²d. We know that the spacetimeunder consideration is Lorentzian, so either m or n willhave to be negative. Let us choose m, the coefficient of dt², to benegative. This is not a choice we are simply allowed to make, and infact we will see later that it can go wrong, but we will assume it fornow. The assumption is not completely unreasonable, since we know that Minkowski space is itself spherically symmetric, and will therefore bedescribed by (7.12). With this choice we can trade in the functionsm and n for new functions and , such that

(7.13)

This is the best we can do for a general metric in a sphericallysymmetric spacetime. The next step is to actually solve Einstein'sequations, which will allow us to determine explicitly the functions(t, r) and (t, r). It is unfortunately necessary tocompute the Christoffel symbols for (7.13), from which we can getthe curvature tensor and thus the Ricci tensor. If we use labels(0, 1, 2, 3) for (t, r,,) in the usual way, the Christoffelsymbols are given by

(7.14)

(Anything not written down explicitly is meant to be zero, or relatedto what is written by symmetries.) Fromthese we get the following nonvanishing components of the Riemanntensor:

(7.15)

Taking the contraction as usual yields the Ricci tensor:

(7.16)

Our job is to set R = 0. From R₀₁ = 0 we get

(7.17)

If we consider taking the time derivative of R₂₂ = 0and using = 0, we get

(7.18)

We can therefore write

(7.19)

The first term in the metric (7.13) is therefore - e^2f(r)e^2g(t)dt².But we could always simply redefine our time coordinate by replacing dt e^-g(t)dt; in other words, we are free to choose t such that g(t) = 0, whence (t, r) = f (r). We therefore have

(7.20)

All of the metric components are independent of the coordinate t.We have therefore proven a crucial result: any spherically symmetricvacuum metric possesses a timelike Killing vector.

This property is so interesting that it gets its own name: a metricwhich possesses a timelike Killing vector is called stationary.There is also a more restrictive property: a metric is calledstatic if it possesses a timelike Killing vector which is orthogonal to a family of hypersurfaces. (A hypersurface in ann-dimensional manifold is simply an (n - 1)-dimensionalsubmanifold.) The metric (7.20) is notonly stationary, but also static; the Killing vector field isorthogonal to the surfaces t = const (since there are nocross terms such as dtdr and so on). Roughly speaking, a static metric isone in which nothing is moving, while a stationary metric allows thingsto move but only in a symmetric way. For example, the static sphericallysymmetric metric (7.20) will describe non-rotating stars or black holes,while rotating systems (which keep rotating in the same way at all times)will be described by stationary metrics. It's hard to remember whichword goes with which concept, but the distinction between the twoconcepts should be understandable.

Let's keep going with finding the solution. Since bothR₀₀ and R₁₁ vanish, we can write

(7.21)

which implies = - + constant. Once again, we canget rid of the constant by scaling our coordinates, so we have

(7.22)

Next let us turn to R₂₂ = 0, which now reads

(7.23)

This is completely equivalent to

(7.24)

We can solve this to obtain

(7.25)

where is some undetermined constant. With (7.22) and (7.25), our metric becomes

(7.26)

We now have no freedom left except for the single constant , sothis form better solve the remaining equations R₀₀ = 0 andR₁₁ = 0; it is straightforward to check that it does, for any value of .

The only thing left to do is to interpret the constant interms of some physical parameter. The most important use of aspherically symmetric vacuum solution is to represent the spacetimeoutside a star or planet or whatnot. In that case we would expectto recover the weak field limit as r . In this limit, (7.26) implies

(7.27)

The weak field limit, on the other hand, has

(7.28)

with the potential = - GM/r. Therefore the metrics do agree inthis limit, if we set = - 2GM.

Our final result is the celebrated Schwarzschild metric,

(7.29)

This is true for any spherically symmetric vacuum solution to Einstein's equations; M functions as a parameter, which we happento know can be interpreted as the conventional Newtonian mass that wewould measure by studying orbits at large distances from thegravitating source. Note that as M 0 we recoverMinkowski space, which is to be expected. Note also that the metricbecomes progressively Minkowskian as we go to r ;this property is known as asymptotic flatness.

The fact that the Schwarzschild metric is not just a good solution,but is the unique spherically symmetric vacuum solution, is known asBirkhoff's theorem. It is interesting to note that the resultis a static metric. We did not say anything about the sourceexcept that it be spherically symmetric. Specifically, we did notdemand that the source itself be static; it could be a collapsingstar, as long as the collapse were symmetric. Therefore a processsuch as a supernova explosion, which is basically spherical, would beexpected to generate very little gravitational radiation (in comparisonto the amount of energy released through other channels). This isthe same result we would have obtained in electromagnetism, where theelectromagnetic fields around a spherical charge distribution do notdepend on the radial distribution of the charges.

Before exploring the behavior of test particles in the Schwarzschildgeometry, we should say something about singularities. From the formof (7.29), the metric coefficients become infinite at r = 0 andr = 2GM - an apparent sign that something is goingwrong. The metric coefficients, of course, are coordinate-dependent quantities, and as such we should not make toomuch of their values; it is certainly possible to have a "coordinatesingularity" which results from a breakdown of a specific coordinatesystem rather than the underlying manifold. An example occurs atthe origin of polar coordinates in the plane, where the metric ds² = dr² +r²d becomes degenerate and the component g = r^-2 of the inverse metric blows up, even though that point of the manifold is no different from any other.

What kind of coordinate-independent signal shouldwe look for as a warning that something about the geometry is out ofcontrol? This turns out to be a difficult question to answer, andentire books have been written about the nature of singularities ingeneral relativity. We won't go into this issue in detail, butrather turn to one simple criterion for when something has gone wrong -when the curvature becomes infinite. The curvature is measured bythe Riemann tensor, and it is hard to say when a tensor becomes infinite,since its components are coordinate-dependent. But from the curvaturewe can construct various scalar quantities, and since scalars arecoordinate-independent it will be meaningful to say that they becomeinfinite. This simplest such scalar is the Ricci scalar R = gR,but we can also construct higher-order scalars such as RR,RR, RRR, and so on. If anyof these scalars (not necessarily all of them) go to infinity as weapproach some point, we will regard that point as a singularity of thecurvature. We should also check that the point is not "infinitelyfar away"; that is, that it can be reached by travelling a finite distance along a curve.

We therefore have a sufficient condition for a point to be considereda singularity. It is not a necessary condition, however, and it isgenerally harder to show that a given point is nonsingular; for our purposes we will simply test to see if geodesics are well-behaved at the point inquestion, and if so then we will consider the point nonsingular.In the case of the Schwarzschild metric (7.29), direct calculation reveals that

(7.30)

This is enough to convince us that r = 0 represents an honest singularity. At the other trouble spot, r = 2GM, you couldcheck and see that none of the curvature invariants blows up. We thereforebegin to think that it is actually not singular, and we have simplychosen a bad coordinate system. The best thing to do is to transformto more appropriate coordinates if possible. We will soon see thatin this case it is in fact possible, and the surface r =2GM is very well-behaved (although interesting) in the Schwarzschild metric.

Having worried a little about singularities, we should point out thatthe behavior of Schwarzschild at r 2GM is of little day-to-dayconsequence. The solution we derived is valid only in vacuum, andwe expect it to hold outside a spherical body such as a star. However,in the case of the Sun we are dealing with a body which extends to aradius of

(7.31)

Thus, r = 2GM is far inside the solar interior, where we do not expect the Schwarzschild metric to imply. In fact, realistic stellarinterior solutions are of the form

(7.32)

See Schutz for details. Here m(r) is a function ofr which goes to zero faster than r itself, so there are no singularities to deal with at all. Nevertheless, there are objects for which thefull Schwarzschild metric is required - black holes - and thereforewe will let our imaginations roam far outside the solar system in thissection.

The first step we will take to understand this metric more fully is to consider the behavior of geodesics. We need the nonzeroChristoffel symbols for Schwarzschild:

(7.33)

The geodesic equation therefore turns into the following fourequations, where is an affine parameter:

	(7.34)
	(7.35)
	(7.36)

and

(7.37)

There does not seem to be much hope for simply solving this set ofcoupled equations by inspection. Fortunately our task is greatlysimplified by the high degree of symmetry of the Schwarzschild metric.We know that there are four Killing vectors: three for the sphericalsymmetry, and one for time translations. Each of these will lead toa constant of the motion for a free particle; if K is a Killingvector, we know that

(7.38)

In addition, there is another constant of the motion that we alwayshave for geodesics; metric compatibility implies that along the paththe quantity

(7.39)

is constant.Of course, for a massive particle we typically choose = ,and this relation simply becomes = - gUU = + 1. For a massless particle we always have = 0. We will also be concerned with spacelike geodesics (even though they do not correspondto paths of particles), for which we will choose = - 1.

Rather than immediately writing out explicit expressions for thefour conserved quantities associated with Killing vectors, let's thinkabout what they are telling us. Notice that the symmetries theyrepresent are also present in flat spacetime, where the conservedquantities they lead to are very familiar. Invariance under timetranslations leads to conservation of energy, while invariance underspatial rotations leads to conservation of the three components ofangular momentum. Essentially the same applies to the Schwarzschildmetric. We can think of the angular momentum as a three-vector witha magnitude (one component) and direction (two components). Conservationof the direction of angular momentum means that the particle will movein a plane. We can choose this to be the equatorial plane of our coordinate system; if the particle is not in this plane, we can rotatecoordinates until it is. Thus, the two Killing vectors which lead toconservation of the direction of angular momentum imply

(7.40)

The two remaining Killing vectors correspond to energy and themagnitude of angular momentum. The energy arises from the timelikeKilling vector K = , or

(7.41)

The Killing vector whose conserved quantity is the magnitude of theangular momentum is L = , or

(7.42)

Since (7.40) implies that sin = 1 along the geodesics ofinterest to us, the two conserved quantities are

(7.43)

and

(7.44)

For massless particles these can be thought of as the energy and angular momentum; for massive particles they are the energy andangular momentum per unit mass of the particle. Note that the constancy of (7.44) is the GR equivalent of Kepler's second law(equal areas are swept out in equal times).

Together these conserved quantities provide a convenient way tounderstand the orbits of particles in the Schwarzschild geometry.Let us expand the expression (7.39) for to obtain

(7.45)

If we multiply this by (1 - 2GM/r) and use our expressionsfor E and L, we obtain

(7.46)

This is certainly progress, since we have taken a messy system ofcoupled equations and obtained a single equation for r().It looks even nicer if we rewrite it as

(7.47)

where

(7.48)

In (7.47) we have precisely the equation for a classical particle of unit mass and "energy" E² moving in a one-dimensional potential given by V(r). (The true energy per unit mass is E,but the effective potential for the coordinate r responds to E².)

Of course, our physical situation is quite different from a classicalparticle moving in one dimension. The trajectories under considerationare orbits around a star or other object:

The quantities of interest to us are not only r(),but also t() and (). Nevertheless, we can go along way toward understanding all of the orbits by understanding theirradial behavior, and it is a great help to reduce this behavior to a problem we know how to solve.

A similar analysis of orbits in Newtonian gravity would have produceda similar result; the general equation (7.47) would have been thesame, but the effective potential (7.48) would not have had the lastterm. (Note that this equation is not a power series in 1/r, it isexact.) In the potential (7.48) the first term is just a constant, thesecond term corresponds exactly to the Newtonian gravitational potential,and the third term is a contribution from angular momentum which takesthe same form in Newtonian gravity and general relativity. The last term,the GR contribution, will turn out to make a great deal of difference,especially at small r.

Let us examine the kinds of possible orbits, as illustrated in thefigures. There are different curves V(r) for different valuesof L; for any one of these curves, the behavior of the orbit can bejudged by comparing the E² to V(r). The general behavior of the particle will be to move in the potential until it reaches a"turning point" where V(r) = E², where it will begin moving in the other direction. Sometimes there may be no turning point to hit,in which case the particle just keeps going. In other cases theparticle may simply move in a circular orbit at radius r_c = const; this can happen if the potential is flat, dV/dr =0. Differentiating (7.48), we find that the circular orbits occur when

(7.49)

where = 0 in Newtonian gravity and = 1 in generalrelativity. Circular orbits will be stable if they correspond toa minimum of the potential, and unstable if they correspond to amaximum. Bound orbits which are not circular will oscillatearound the radius of the stable circular orbit.

Turning to Newtonian gravity, we find that circular orbits appear at

(7.50)

For massless particles = 0, and there are no circular orbits;this is consistent with the figure, which illustrates that there are nobound orbits of any sort. Although it is somewhat obscured in thiscoordinate system, massless particles actually move in a straightline, since the Newtonian gravitational force on a massless particle iszero. (Of course the standing of massless particles in Newtonian theoryis somewhat problematic, but we will ignore that for now.) In terms ofthe effective potential, a photon with a given energy E will come in from r = and gradually "slow down" (actually dr/dwill decrease, but the speed of light isn't changing) until it reachesthe turning point, when it will start moving away back to r = . The lower values of L, for which the photon will come closer before it starts moving away, are simply those trajectories whichare initially aimed closer to the gravitating body.For massive particles there will be stable circular orbits at theradius (7.50), as well as bound orbits which oscillate around thisradius. If the energy is greater than the asymptotic value E = 1,the orbits will be unbound, describing a particle that approaches thestar and then recedes. We know that the orbits in Newton's theory areconic sections - bound orbits are either circles or ellipses, whileunbound ones are either parabolas or hyperbolas - although we won'tshow that here.

In general relativity the situation is different, but only for r sufficiently small. Since the difference resides in the term - GML²/r³, as r the behaviors are identical in the two theories. But as r 0 the potential goes to - rather than + as in the Newtonian case. At r = 2GM the potential is always zero; inside this radius is the black hole, whichwe will discuss more thoroughly later. For massless particlesthere is always a barrier (except for L = 0, for which the potentialvanishes identically), but a sufficiently energetic photon willnevertheless go over the barrier and be dragged inexorably down tothe center. (Note that "sufficiently energetic" means "in comparisonto its angular momentum" - in fact the frequency of the photon isimmaterial, only the direction in which it is pointing.) At the topof the barrier there are unstable circular orbits.For = 0, = 1, we can easily solve (7.49) to obtain

(7.51)

This is borne out by the figure, which shows a maximum ofV(r) at r = 3GM for every L. This means that a photon canorbit forever in a circle at this radius, but any perturbation will cause it to fly away either to r = 0 orr = .

For massive particles there are once again different regimes dependingon the angular momentum. The circular orbits are at

(7.52)

For large L there will be two circular orbits, one stable and one unstable. In the L limit their radii are given by

(7.53)

In this limit the stable circular orbit becomes farther and farther away, while the unstable one approaches 3GM, behavior whichparallels the massless case. As we decrease L the two circularorbits come closer together; they coincide when the discriminant in(7.52) vanishes, at

(7.54)

for which

(7.55)

and disappear entirely for smaller L. Thus 6GM is thesmallest possible radius of a stable circular orbit in the Schwarzschild metric. There are also unbound orbits, which come in from infinity and turn around, and bound but noncircular ones, whichoscillate around the stable circular radius. Note that suchorbits, which would describe exact conic sections inNewtonian gravity, will not do so in GR, although we would have tosolve the equation for d/dt to demonstrate it. Finally, there areorbits which come in from infinity and continue all the way in to r = 0; this can happen either if the energy is higher than thebarrier, or for L < GM, when the barrier goes away entirely.

We have therefore found that the Schwarzschild solution possessesstable circular orbits for r > 6GM and unstablecircular orbits for 3GM < r < 6GM. It's important to rememberthat these are only the geodesics; there is nothing to stop an accelerating particle fromdipping below r = 3GM and emerging, as long as it stays beyondr = 2GM.

Most experimental tests of general relativity involve the motion oftest particles in the solar system, and hence geodesicsof the Schwarzschild metric; this is therefore a good place topause and consider these tests. Einstein suggested three tests:the deflection of light, the precession of perihelia, and gravitationalredshift. The deflection of light is observable in the weak-fieldlimit, and therefore is not really a good test of the exactform of the Schwarzschild geometry. Observations of this deflectionhave been performed during eclipses of the Sun, with results whichagree with the GR prediction (although it's not an especially cleanexperiment). The precession of perihelia reflectsthe fact that noncircular orbits are not closed ellipses; to a goodapproximation they are ellipses which precess, describing a flower pattern.

Using our geodesic equations, we could solve for d/d as a power series in the eccentricity e of theorbit, and from that obtain the apsidal frequency ,defined as 2 divided by the time it takes for the ellipse toprecess once around.For details you can look in Weinberg; the answer is

(7.56)

where we have restored the c to make it easier to compare withobservation.(It is a good exercise to derive this yourself to lowest nonvanishingorder, in which case the e² is missing.) Historically theprecession of Mercury was the first test of GR. For Mercury the relevant numbers are

(7.57)

and of course c = 3.00 × 10¹⁰ cm/sec. This gives = 2.35 × 10^-14 sec^-1. In other words, the major axis of Mercury's orbit precesses at a rate of 42.9 arcsecs every 100years. The observed value is 5601 arcsecs/100 yrs. However,much of that is due to the precession of equinoxes in our geocentriccoordinate system; 5025 arcsecs/100 yrs, to be precise. Thegravitational perturbations of the other planets contribute anadditional 532 arcsecs/100 yrs, leaving 43 arcsecs/100 yrsto be explained by GR, which it does quite well.

The gravitational redshift, as we have seen, is another effectwhich is present in the weak field limit, and in fact will be predictedby any theory of gravity which obeys the Principle of Equivalence.However, this only applies to small enough regions of spacetime; overlarger distances, the exact amount of redshift will depend on themetric, and thus on the theory under question. It is thereforeworth computing the redshift in the Schwarzschild geometry. We consider two observers who are not moving on geodesics, but are stuckat fixed spatial coordinate values (r₁,,) and(r₂,,). According to (7.45), the proper time ofobserver i will be related to the coordinate time t by

(7.58)

Suppose that the observer ₁ emits a light pulse which travels to the observer ₂, such that ₁ measures the timebetween two successive crests of the light wave to be .Each crest follows the same path to ₂, except that theyare separated by a coordinate time

(7.59)

This separation in coordinate time does not change along the photontrajectories, but the second observer measures a time between successivecrests given by

(7.60)

Since these intervals measure the proper time betweentwo crests of an electromagnetic wave, the observed frequencies will berelated by

(7.61)

This is an exact result for the frequency shift; in the limit r>> 2GM we have

(7.62)

This tells us that the frequency goes down as increases, whichhappens as we climb out of a gravitational field; thus, a redshift.You can check that it agrees with our previous calculation based onthe equivalence principle.

Since Einstein's proposal of the three classic tests, further testsof GR have been proposed. The most famous is of course the binarypulsar, discussed in the previous section. Another is the gravitationaltime delay, discovered by (and observed by) Shapiro. This is justthe fact that the time elapsed along two different trajectories betweentwo events need not be the same. It has been measured by reflectingradar signals off of Venus and Mars, and once again is consistent withthe GR prediction. One effect which has not yet been observed isthe Lense-Thirring, or frame-dragging effect. There has been a long-term effort devoted to a proposedsatellite, dubbed Gravity Probe B, which would involve extraordinarilyprecise gyroscopes whose precession could be measured and the contribution from GR sorted out. It has a ways to go before beinglaunched, however, and the survival of such projects is alwaysyear-to-year.

We now know something about the behavior of geodesics outside the troublesome radius r = 2GM, which is the regime ofinterest for the solar system and most other astrophysical situations. We will next turn to the study of objects which are described by the Schwarzschild solutioneven at radii smaller than 2GM - black holes. (We'll use theterm "black hole" for the moment, even though we haven't introduceda precise meaning for such an object.)

One way of understanding a geometry is to explore its causal structure,as defined by the light cones. We therefore consider radial null curves, those for which and are constant and ds² = 0:

(7.63)

from which we see that

(7.64)

This of course measures the slope of the light cones on a spacetime diagramof the t-r plane. For large r the slope is ±1,as it would be in flat space, while as we approach r = 2GM we get dt/dr ±, and the light cones "close up":

Thus a light ray which approaches r = 2GM never seems toget there, at least in this coordinate system; instead it seems toasymptote to this radius.

As we will see, this is an illusion, and the light ray (or a massiveparticle) actually has no trouble reaching r = 2GM. But anobserver far away would never be able to tell. If we stayed outside whilean intrepid observational general relativist dove into the blackhole, sending back signals all the time, we would simply see thesignals reach us more and more slowly.

This should be clear from the pictures, and is confirmedby our computation of / when we discussedthe gravitational redshift (7.61). As infalling astronautsapproach r = 2GM, any fixed interval of their propertime corresponds to a longer and longer interval from our point of view. This continues forever; we would never seethe astronaut cross r = 2GM, we would just see them movemore and more slowly (and become redder and redder, almost as if they wereembarrassed to have done something as stupid as diving into a black hole).

The fact that we never see the infalling astronauts reach r =2GM is a meaningful statement, but the fact that their trajectory inthe t-r plane never reaches there is not. It is highlydependent on our coordinate system, and we would like to ask a more coordinate-independent question (such as, do the astronauts reachthis radius in a finite amount of their proper time?). The bestway to do this is to change coordinates to a system which is betterbehaved at r = 2GM.There does exist a set of such coordinates, which we now set out tofind. There is no way to "derive" a coordinate transformation, ofcourse, we just say what the new coordinates are and plug in theformulas. But we will develop these coordinates in several steps,in hopes of making the choices seem somewhat motivated.

The problem with our current coordinates is thatdt/dr along radial null geodesics which approachr = 2GM; progress in the r direction becomes slowerand slower with respect to the coordinate time t. We can try to fix this problemby replacing t with a coordinate which "moves more slowly" alongnull geodesics. First notice that we can explicitly solve thecondition (7.64) characterizing radial null curves to obtain

(7.65)

where the tortoise coordinate r^* is defined by

(7.66)

(The tortoise coordinate is only sensibly related to r whenr 2GM, but beyond there our coordinates aren't very good anyway.) In terms of the tortoise coordinate the Schwarzschild metric becomes

(7.67)

where r is thought of as a function of r^*.This represents some progress, since the light cones now don'tseem to close up; furthermore, none of the metric coefficients becomesinfinite at r = 2GM (although both g_tt and g_r^*r^* becomezero). The price we pay, however, is that the surfaceof interest at r = 2GM has just been pushed to infinity.

Our next move is to define coordinates which are naturally adaptedto the null geodesics. If we let

(7.67)

then infalling radial null geodesics are characterized by = constant, while the outgoing ones satisfy = constant.Now consider going back to the original radial coordinate r,but replacing the timelike coordinate t with the new coordinate. These are known as Eddington-Finkelstein coordinates. In terms of them the metric is

(7.69)

Here we see our first sign of real progress. Even though the metriccoefficient g vanishes at r = 2GM, there is no real degeneracy; the determinant of the metric is

(7.70)

which is perfectly regular at r = 2GM. Therefore the metric isinvertible, and we see once and for all that r = 2GM issimply a coordinate singularity in our original (t, r,,) system.In the Eddington-Finkelstein coordinates the condition for radialnull curves is solved by

(7.71)

We can therefore see what has happened: in this coordinate system the light cones remain well-behaved at r = 2GM, and thissurface is at a finite coordinate value. There is no problem in tracingthe paths of null or timelike particles past the surface.On the other hand, something interesting is certainly going on.Although the light cones don't close up, they do tilt over, suchthat for r < 2GM all future-directed paths are in thedirection of decreasing r.

The surface r = 2GM, while being locally perfectlyregular, globally functions as a point of no return - once a test particle dipsbelow it, it can never come back. For this reason r = 2GMis known as the event horizon; no event at r 2GM can influence anyother event at r > 2GM. Notice that the event horizonis a null surface, not a timelike one. Notice also that since nothing can escape theevent horizon, it is impossible for us to "see inside" - thus thename black hole.

Let's consider what we have done. Acting under the suspicion thatour coordinates may not have been good for the entire manifold, wehave changed from our original coordinate t to the new one ,which has the nice property that if we decrease r along a radialcurve null curve = constant, we go right through theevent horizon without any problems. (Indeed, a local observer actuallymaking the trip would not necessarily know when the event horizon hadbeen crossed - the local geometry is no different than anywhere else.) We thereforeconclude that our suspicion was correct and our initial coordinatesystem didn't do a good job of covering the entire manifold. The regionr 2GM should certainly be included in our spacetime, since physical particles can easily reach there and pass through. However,there is no guarantee that we are finished; perhaps there are otherdirections in which we can extend our manifold.

In fact there are. Notice that in the (, r) coordinatesystem we can cross the event horizon on future-directed paths, butnot on past-directed ones. This seems unreasonable, since we startedwith a time-independent solution. But we could have chosen instead of , in which case the metric would have been

(7.72)

Now we can once again pass through the event horizon, but this timeonly along past-directed curves.

This is perhaps a surprise: we can consistently follow either future-directed or past-directed curves through r = 2GM,but we arrive at different places. It was actually to be expected, sincefrom the definitions (7.68), if we keep constant and decreaser we must have t + , while if we keep constant and decrease r we must have t - .(The tortoise coordinate r^* goes to - as r 2GM.)So we have extended spacetime in two different directions, one tothe future and one to the past.

The next step would be to follow spacelike geodesics to see if we would uncover still more regions. The answer is yes, we would reachyet another piece of the spacetime, but let's shortcut the process bydefining coordinates that are good all over. A first guess might be touse both and at once (in place of t and r),which leads to

(7.73)

with r defined implicitly in terms of and by

(7.74)

We have actually re-introduced the degeneracy with which we startedout; in these coordinates r = 2GM is "infinitely far away" (ateither = - or = + ). The thing to dois to change to coordinates which pull these points into finitecoordinate values; a good choice is

(7.75)

which in terms of our original (t, r) system is

(7.76)

In the (u', v',,) system the Schwarzschild metric is

(7.77)

Finally the nonsingular nature of r = 2GM becomescompletely manifest; in this form none of the metric coefficients behave in any special wayat the event horizon.

Both u' and v' are null coordinates, in the sense that their partial derivatives /u' and /v'are null vectors. There is nothing wrong with this, since the collection of four partial derivative vectors (two null and twospacelike) in this system serve as a perfectly good basis for thetangent space. Nevertheless, we are somewhat more comfortable workingin a system where one coordinate is timelike and the rest are spacelike. We therefore define

(7.78)

and

(7.79)

in terms of which the metric becomes

(7.80)

where r is defined implicitly from

(7.81)

The coordinates (v, u,,) are known as Kruskalcoordinates, or sometimes Kruskal-Szekres coordinates. Note thatv is the timelike coordinate.

The Kruskal coordinates have a number of miraculous properties.Like the (t, r^*) coordinates, the radial nullcurves look like they do in flat space:

(7.82)

Unlike the (t, r^*) coordinates, however, theevent horizon r = 2GM is not infinitely far away; in fact it is defined by

(7.83)

consistent with it being a null surface.More generally, we can consider the surfaces r = constant. From(7.81) these satisfy

(7.84)

Thus, they appear as hyperbolae in the u-v plane. Furthermore,the surfaces of constant t are given by

(7.85)

which defines straight lines through the origin with slopetanh(t/4GM). Note that as t ± thisbecomes the same as (7.83); therefore these surfaces are thesame as r = 2GM.

Now, our coordinates (v, u) should be allowed torange over every value they can take without hitting the realsingularity at r = 2GM; the allowed region is therefore- u and v² < u² + 1. We can now drawa spacetime diagram in the v-u plane (with and suppressed), known as a "Kruskal diagram", which represents theentire spacetime corresponding to the Schwarzschild metric.

Each point on the diagram is a two-sphere.

Our original coordinates (t, r) were only good forr > 2GM, which is only a part of the manifold portrayed on the Kruskal diagram. It isconvenient to divide the diagram into four regions:

The region in which we started was region I; by followingfuture-directed null rays we reached region II, and by followingpast-directed null rays we reached region III. If we had exploredspacelike geodesics, we would have been led to region IV.The definitions (7.78) and (7.79) which relate (u, v) to(t, r) are really only good in region I; in the other regions it is necessary to introduce appropriate minus signs to prevent thecoordinates from becoming imaginary.

Having extended the Schwarzschild geometry as far as it will go,we have described a remarkable spacetime. Region II, of course,is what we think of as the black hole. Once anything travels fromregion I into II, it can never return. In fact, every future-directedpath in region II ends up hitting the singularity at r = 0; once youenter the event horizon, you are utterly doomed. This is worthstressing; not only can you not escape back to region I, you cannoteven stop yourself from moving in the direction of decreasing r,since this is simply the timelike direction. (This couldhave been seen in our original coordinate system; for r <2GM, t becomes spacelike and r becomes timelike.) Thus you can no morestop moving toward the singularity than you can stop getting older.Since proper time is maximized along a geodesic, you will live thelongest if you don't struggle, but just relax as you approach thesingularity. Not that you will have long to relax. (Nor that thevoyage will be very relaxing; as you approach the singularity thetidal forces become infinite. As you fall toward the singularityyour feet and head will be pulled apart from each other, while your torso is squeezed to infinitesimal thinness. The grislydemise of an astrophysicist falling into a black hole is detailedin Misner, Thorne, and Wheeler, section 32.6. Note that they useorthonormal frames [not that it makes the trip any more enjoyable].)

Regions III and IV might be somewhat unexpected. Region III is simplythe time-reverse of region II, a part of spacetime from which thingscan escape to us, while we can never get there. It can be thoughtof as a "white hole." There is a singularity in the past, out of which the universe appears to spring. The boundary of region III is sometimescalled the past event horizon, while the boundary of region II is calledthe future event horizon. Region IV, meanwhile, cannot be reachedfrom our region I either forward or backward in time (nor can anybody from over therereach us). It is another asymptotically flat region of spacetime,a mirror image of ours. It can be thought of as being connected toregion I by a "wormhole," a neck-like configuration joining twodistinct regions. Consider slicing up the Kruskal diagram into spacelikesurfaces of constant v:

Now we can draw pictures of each slice, restoring one of the angular coordinates for clarity:

So the Schwarzschild geometry really describes two asymptoticallyflat regions which reach toward each other, join together via awormhole for a while, and then disconnect. But the wormholecloses up too quickly for any timelike observer to cross it from oneregion into the next.

It might seem somewhat implausible, this story about two separatespacetimes reaching toward each other for a while and then lettinggo. In fact, it is not expected to happen in the real world, sincethe Schwarzschild metric does not accurately model the entire universe. Remember that it is only valid in vacuum, for exampleoutside a star. If the star has a radius larger than 2GM, weneed never worry about any event horizons at all. But we believethat there are stars which collapse under their own gravitationalpull, shrinking down to below r = 2GM and further into asingularity, resulting in a black hole. There is no need for a white hole, however,because the past of such a spacetime looks nothing like that of thefull Schwarzschild solution. Roughly, a Kruskal-like diagram forstellar collapse would look like the following:

The shaded region is not described by Schwarzschild, sothere is no need to fret about white holes and wormholes.

While we are on the subject, we can say something about the formationof astrophysical black holes from massive stars. The life of a staris a constant struggle between the inward pull of gravity and theoutward push of pressure. When the star is burning nuclear fuel atit* core, the pressure comes from the heat produced by this burning.(We should put "burning" in quotes, since nuclear fusion is unrelatedto oxidation.) When the fuel is used up, the temperature declines andthe star begins to shrink as gravity starts winning the struggle.Eventually this process is stopped when the electrons are pushed soclose together that they resist further compression simply on thebasis of the Pauli exclusion principle (no two fermions can be in thesame state). The resulting object is called a white dwarf.If the mass is sufficiently high, however, even the electron degeneracy pressure is not enough, and the electrons will combinewith the protons in a dramatic phase transition. The result is aneutron star, which consists of almost entirely neutrons (althoughthe insides of neutron stars are not understood terribly well).Since the conditions at the center of a neutron star are very differentfrom those on earth, we do not have a perfect understanding of theequation of state. Nevertheless, we believe that asufficiently massive neutron star will itself be unable to resist thepull of gravity, and will continue to collapse. Since a fluid ofneutrons is the densest material of which we can presently conceive,it is believed that the inevitable outcome of such a collapse isa black hole.

The process is summarized in the following diagram of radius vs.mass:

The point of the diagram is that, for any given mass M,the star will decrease in radius until it hits the line. Whitedwarfs are found between points A and B, and neutron stars between points C and D. Point B is at a height ofsomewhat less than 1.4 solar masses; the height of D is less certain, but probablyless than 2 solar masses. The process of collapse is complicated, andduring the evolution the star can lose or gain mass, so the endpointof any given star is hard to predict. Nevertheless white dwarfs areall over the place, neutron stars are not uncommon, and there are anumber of systems which are strongly believed to contain black holes.(Of course, you can't directly see the black hole. What you can seeis radiation from matter accreting onto the hole, which heats up asit gets closer and emits radiation.)

We have seen that the Kruskal coordinate system provides a veryuseful representation of the Schwarzschild geometry. Before movingon to other types of black holes, we will introduce one moreway of thinking about this spacetime, the Penrose (or Carter-Penrose,or conformal) diagram. The idea is to do a conformal transformationwhich brings the entire manifold onto a compact region such that wecan fit the spacetime on a piece of paper.

Let's begin with Minkowski space, to see how the technique works.The metric in polar coordinates is

(7.86)

Nothing unusual will happen to the , coordinates, butwe will want to keep careful track of the ranges of the other twocoordinates. In this case of course we have

(7.87)

Technically the worldline r = 0 represents a coordinate singularityand should be covered by a different patch, but we all know what isgoing on so we'll just act like r = 0 is well-behaved.

Our task is made somewhat easier if we switch to null coordinates:

(7.88)

with corresponding ranges given by

(7.89)

These ranges are as portrayed in the figure, on which eachpoint represents a 2-sphere of radius r = u -v. The metric in these coordinates is given by

(7.90)

We now want to change to coordinates in which "infinity" takeson a finite coordinate value. A good choice is

(7.91)

The ranges are now

(7.92)

To get the metric, use

(7.93)

and

(7.94)

and likewise for v. We are led to

(7.95)

Meanwhile,

(7.96)

Therefore, the Minkowski metric in these coordinates is

(7.97)

This has a certain appeal, since the metric appears as a fairlysimple expression multiplied by an overall factor. We can make iteven better by transforming back to a timelike coordinate and a spacelike (radial) coordinate , via

(7.98)

with ranges

(7.99)

Now the metric is

(7.100)

where

(7.101)

The Minkowski metric may therefore be thought of as related by a conformal transformation to the "unphysical" metric

(7.102)

This describes the manifold × S³, where the 3-sphere ismaximally symmetric and static. There is curvature in this metric,and it is not a solution to the vacuum Einstein's equations. This shouldn't bother us, sinceit is unphysical; the true physical metric, obtained by a conformaltransformation, is simply flat spacetime. In fact this metric isthat of the "Einstein static universe," a static (but unstable)solution to Einstein's equations with a perfect fluid and a cosmologicalconstant. Of course, the full range of coordinates on × S³would usually be - < < + , 0 ,while Minkowski space is mapped into the subspace defined by (7.99).The entire × S³ can be drawn as a cylinder, in which each circle is a three-sphere, as shown on the next page.

The shaded region represents Minkowski space. Note that each point (,) on this cylinder is half of a two-sphere, where the other half is the point (, - ). We can unroll the shaded region to portray Minkowski space as a triangle, as shownin the figure.

The is the Penrose diagram. Each point represents a two-sphere.

In fact Minkowski space is only the interior of the abovediagram (including = 0); the boundaries are not part of theoriginal spacetime. Together they are referred to as conformalinfinity. The structure of the Penrose diagram allows us to subdivide conformal infinity intoa few different regions:

(⁺ and ^- are pronounced as "scri-plus" and"scri-minus", respectively.) Note that i⁺, i⁰, and i^- are actually points, since = 0 and = are the north andsouth poles of S³. Meanwhile ⁺ and ^- areactually null surfaces, with the topology of × S².

There are a number of important features of the Penrose diagram forMinkowski spacetime. The points i⁺, andi^- can be thought of as the limits of spacelike surfaces whose normals are timelike;conversely, i⁰ can be thought of as the limit oftimelike surfaces whose normals are spacelike. Radial null geodesics are at ±45° in the diagram.All timelike geodesics begin at i^-and end at i⁺; all null geodesics begin at ^- and end at ⁺; all spacelike geodesics both begin and end at i⁰. On the other hand, there can be non-geodesic timelike curves that end at null infinity (if they become "asymptotically null").

It is nice to be able to fit all of Minkowski space on a small pieceof paper, but we don't really learn much that we didn't alreadyknow. Penrose diagrams are more useful when we want to representslightly more interesting spacetimes, such as those for black holes.The original use of Penrose diagrams was to compare spacetimes toMinkowski space "at infinity" - the rigorous definition of"asymptotically flat" is basically that a spacetime has a conformalinfinity just like Minkowski space. We will not pursue these issuesin detail, but instead turn directly to analysis of the Penrosediagram for a Schwarzschild black hole.

We will not go through the necessary manipulationsin detail, since they parallel the Minkowski case withconsiderable additional algebraic complexity. We wouldstart with the null version of the Kruskal coordinates, inwhich the metric takes the form

(7.103)

where r is defined implicitly via

(7.104)

Then essentially the same transformation as was used inflat spacetime suffices to bring infinity into finitecoordinate values:

(7.105)

with ranges

The (u", v") part of the metric (that is, at constantangular coordinates) is now conformally related to Minkowski space.In the new coordinates the singularitiesat r = 0 are straight lines that stretch from timelikeinfinity in one asymptotic region to timelike infinity in the other. The Penrose diagram for the maximally extended Schwarzschild solution thus looks like this:

The only real subtlety about this diagram isthe necessity to understand that i⁺ andi^- are distinct from r = 0 (there are plenty of timelike paths that do nothit the singularity). Notice also that the structure ofconformal infinity is just like that of Minkowski space,consistent with the claim that Schwarzschild is asymptoticallyflat. Also, the Penrose diagram for a collapsing star thatforms a black hole is what you might expect, as shown on thenext page.

Once again the Penrose diagrams for these spacetimes don'treally tell us anything we didn't already know; their usefulnesswill become evident when we consider moregeneral black holes. In principle therecould be a wide variety of types of black holes, depending onthe process by which they were formed. Surprisingly, however,this turns out not to be the case; no matter how a blackhole is formed, it settles down (fairly quickly) into astate which is characterized only by the mass, charge, andangular momentum. This property, which must be demonstratedindividually for the various types of fields which one mightimagine go into the construction of the hole, is oftenstated as "black holes have no hair." You can demonstrate, for example, that a hole which is formed froman initially inhom*ogeneous collapse "shakes off" anylumpiness by emitting gravitational radiation. This is anexample of a "no-hair theorem." If we are interested inthe form of the black hole after it has settled down, we thusneed only to concern ourselves with charged and rotatingholes. In both cases there exist exactsolutions for the metric, which we can examine closely.

But first let's take a brief detour to the world of blackhole evaporation.It is strange to think of a black hole "evaporating," but inthe real world black holes aren't truly black - they radiateenergy as if they were a blackbody of temperature T = /8kGM, where M is the mass of the hole and k is Boltzmann's constant. The derivation of this effect, known as Hawkingradiation, involves the use of quantum field theory in curvedspacetime and is way outside our scope right now. The informalidea is nevertheless understandable.

In quantum field theorythere are "vacuum fluctuations" - the spontaneous creation andannihilation of particle/antiparticle pairs in empty space. These fluctuations are precisely analogous to the zero-pointfluctuations of a simple harmonic oscillator. Normally suchfluctuations are impossible to detect, since they average outto give zero total energy (although nobody knows why; that'sthe cosmological constant problem). In the presence of an eventhorizon, though, occasionally one member of a virtual pair willfall into the black hole while its partner escapes to infinity.The particle that reaches infinity will have to have a positiveenergy, but the total energy is conserved; therefore the blackhole has to lose mass. (If you like you can think of the particlethat falls in as having a negative mass.) We see the escaping particles as Hawking radiation.It's not a very big effect, and the temperature goes down as themass goes up, so for black holes of mass comparable to the sun itis completely negligible. Still, in principle the black hole couldlose all of its mass to Hawking radiation, and shrink to nothing inthe process. The relevant Penrose diagram might look like this:

On the other hand, it might not. The problem with this diagram isthat "information is lost" - if we draw a spacelike surfaceto the past of the singularity and evolve it into the future, partof it ends up crashing into the singularity and being destroyed.As a result the radiation itself contains less information thanthe information that was originally in the spacetime. (This isthe worse than a lack of hair on the black hole. It's one thingto think that information has been trapped inside the event horizon,but it is more worrisome to think that it has disappeared entirely.) But such a processviolates the conservation of information that is implicit in bothgeneral relativity and quantum field theory, the two theories thatled to the prediction. This paradox is considered a big deal thesedays, and there are a number of efforts to understand how the information can somehow be retrieved. A currently popular explanationrelies on string theory, and basically says that black holes have a lotof hair, in the form of virtual stringy states living near the eventhorizon. I hope you will not be disappointed to hear that we won'tlook at this very closely; but you should know what the problem isand that it is an area of active research these days.

With that out of our system, we now turnto electrically charged black holes. Theseseem at first like reasonable enough objects, since thereis certainly nothing to stop us from throwing some netcharge into a previously uncharged black hole. In anastrophysical situation, however, the total amount of chargeis expected to be very small, especially when compared withthe mass (in terms of the relative gravitational effects).Nevertheless, charged black holes provide a useful testingground for various thought experiments, so they are worthour consideration.

In thiscase the full spherical symmetry of the problem is stillpresent; we know therefore that we can write the metric as

(7.106)

Now, however, we are no longer in vacuum, since the holewill have a nonzero electromagnetic field, which in turnacts as a source of energy-momentum. The energy-momentumtensor for electromagnetism is given by

(7.107)

where F is the electromagnetic field strength tensor. Since we have spherical symmetry, the most general fieldstrength tensor will have components

(7.108)

where f (r, t) and g(r, t) aresome functions to be determined by the field equations, and components not written are zero. F_tr corresponds to a radial electric field, while Fcorresponds to a radial magnetic field. (For those of you wonderingabout the sin, recall that the thing which should be independentof and is the radial component of the magnetic field,B^r = F. For a spherically symmetric metric, = is proportional to (sin)^-1, so we want a factor of sinin F.) The field equations in this case areboth Einstein's equations and Maxwell's equations:

(7.109)

The two sets are coupled together, since the electromagnetic fieldstrength tensor enters Einstein's equations through theenergy-momentum tensor, while the metric enters explicitly intoMaxwell's equations.

The difficulties are not insurmountable, however, and aprocedure similar to the one we followed for the vacuumcase leads to a solution for the charged case as well. Wewill not go through the steps explicitly, but merely quotethe final answer. The solution is known as the Reissner-Nordstrøm metric, and is given by

(7.110)

where

(7.111)

In this expression, M is once again interpreted as the mass of the hole; q is the total electric charge, andp is the total magnetic charge. Isolated magnetic charges (monopoles)have never been observed in nature, but that doesn't stopus from writing down the metric that they would produce ifthey did exist. There are good theoretical reasons to thinkthat monopoles exist, but are extremely rare. (Of course, there isalso the possibility that a black hole could have magnetic chargeeven if there aren't any monopoles.) In fact theelectric and magnetic charges enter the metric in the sameway, so we are not introducing any additional complications by keepingp in our expressions.The electromagnetic fields associated with this solution are given by

(7.112)

Conservatives are welcome to set p = 0 if they like.

The structure of singularities and event horizons is morecomplicated in this metric than it was in Schwarzschild,due to the extra term in the function (r) (which canbe thought of as measuring "how much the light cones tip over"). One thing remains the same: at r = 0 there is a true curvature singularity (as could be checked by computing the curvaturescalar RR). Meanwhile,the equivalent of r = 2GM will be the radius where vanishes. This will occur at

(7.113)

This might constitute two, one, or zero solutions,depending on the relative values of GM² andp² + q². We therefore consider eachcase separately.

Case One - GM² < p² + q²

In this case the coefficient is always positive (never zero),and the metric is completely regular in the (t, r, ,)coordinates all the way down to r = 0. The coordinate t isalways timelike, and r is always spacelike. But there stillis the singularity at r = 0, which is now a timelike line.Since there is no event horizon, there is no obstructionto an observer travelling to the singularity and returningto report on what was observed. This is known as a naked singularity, one which is not shielded by anhorizon. A careful analysis of the geodesics reveals, however,that the singularity is "repulsive" - timelike geodesicsnever intersect r = 0, instead they approach and then reverse course and move away. (Null geodesics can reach the singularity,as can non-geodesic timelike curves.)

As r the solution approaches flat spacetime,and as we have just seen the causal structure is "normal" everywhere.The Penrose diagram will therefore be just like that of Minkowskispace, except that now r = 0 is a singularity.

The nakedness of the singularity offends our sense ofdecency, as well as the cosmic censorship conjecture, whichroughly states that the gravitational collapse of physicalmatter configurations will never produce a naked singularity.(Of course, it's just a conjecture, and it may not be right; thereare some claims from numerical simulations that collapse ofspindle-like configurations can lead to naked singularities.)In fact, we should not ever expect to find a black hole withGM² < p² + q²as the result of gravitational collapse. Roughly speaking, this condition states that the total energy of the holeis less than the contribution to the energy from the electromagneticfields alone - that is, the mass of the matter which carried thecharge would have had to be negative. This solution is thereforegenerally considered to be unphysical. Notice also that thereare not good Cauchy surfaces (spacelike slices for which everyinextendible timelike line intersects them) in this spacetime, sincetimelike lines can begin and end at the singularity.

Case Two - GM² > p² + q²

This is the situation which we expect to apply in real gravitationalcollapse; the energy in the electromagnetic field is less than thetotal energy. In this case the metric coefficient (r) ispositive at large r and small r, and negative inside the twovanishing points r_± = GM±. The metric has coordinate singularities at both r₊ andr_-; in both cases these could be removed by a change of coordinates as wedid with Schwarzschild.

The surfaces defined by r = r_± are bothnull, and in fact they are event horizons (in a sense we will make precise in a moment).The singularity at r = 0 is a timelike line (not a spacelikesurface as in Schwarzschild). If you are an observer falling into the black hole from far away, r₊ is just like 2GM in the Schwarzschild metric; at this radius r switches from being a spacelike coordinateto a timelike coordinate, and you necessarily move in thedirection of decreasing r. Witnesses outside the black hole alsosee the same phenomena that they would outside an uncharged hole -the infalling observer is seen to move more and more slowly, andis increasingly redshifted.

But the inevitable fall from r₊ to ever-decreasingradii only lasts until you reach the null surface r = r_-,where r switches back to being a spacelike coordinate and the motion inthe direction of decreasing r can be arrested. Therefore youdo not have to hit the singularity at r = 0; this is to be expected,since r = 0 is a timelike line (and therefore not necessarily in yourfuture). In fact you can choose either to continue on to r = 0, orbegin to move in the direction of increasing r back through thenull surface at r = r_-. Then r will onceagain be a timelike coordinate, but with reversed orientation; you are forced to movein the direction of increasing r. You will eventually bespit out past r = r₊ once more, which is likeemerging from a white hole into the rest of the universe. From here you can chooseto go back into the black hole - this time, a different hole thanthe one you entered in the first place - and repeat the voyageas many times as you like. This little story corresponds to theaccompanying Penrose diagram, which of course can be derived morerigorously by choosing appropriate coordinates and analyticallyextending the Reissner-Nordstrøm metric as far as it will go.

How much of this is science, as opposed to science fiction?Probably not much. If you think about the world as seen froman observer inside the black hole who is about to cross the eventhorizon at r_-, you will notice that they can look backin time to see the entire history of the external (asymptotically flat)universe, at least as seen from the black hole. But they see this(infinitely long) history in a finite amount of their proper time -thus, any signal that gets to them as they approach r_- isinfinitely blueshifted. Therefore it is reasonable to believe(although I know of no proof) that any non-spherically symmetricperturbation that comes into a Reissner-Nordstrøm black holewill violently disturb the geometry we have described. It's hard to say what the actual geometry will look like, but there is no verygood reason to believe that it must contain an infinite number ofasymptotically flat regions connecting to each other viavarious wormholes.

Case Three - GM² = p² + q²

This case is known as the extreme Reissner-Nordstrømsolution (or simply "extremal black hole"). The mass is exactly balanced in some sense by the charge -you can construct exact solutions consisting of several extremalblack holes which remain stationary with respect to each otherfor all time. On the one hand the extremal hole is an amusing theoretical toy; these solutions are often examined in studies of the informationloss paradox, and the role of black holes in quantum gravity.On the other hand it appears very unstable, since adding just alittle bit of matter will bring it to Case Two.

The extremal black holes have (r) = 0 at a single radius,r = GM. This does represent an event horizon, but the rcoordinate is never timelike; it becomes null at r = GM,but is spacelike on either side. The singularity at r = 0 is a timelike line, as in the other cases. So for this black holeyou can again avoid the singularity and continue to move to thefuture to extra copies of the asymptotically flat region, butthe singularity is always "to the left." The Penrose diagramis as shown.

We could of course go into a good deal more detail about thecharged solutions, but let's instead move on to spinning black holes. It is much more difficult to findthe exact solution for the metric in this case, since we havegiven up on spherical symmetry. To begin with all that ispresent is axial symmetry (around the axis of rotation), but we canalso ask for stationary solutions (a timelike Killing vector).Although the Schwarzschild and Reissner-Nordstrøm solutions werediscovered soon after general relativity was invented, the solutionfor a rotating black hole was found by Kerr only in 1963. Hisresult, the Kerr metric, is given by the following mess:

(7.114)

where

(7.115)

and

(7.116)

Here a measures the rotation of the hole and M is the mass. It is straightforward to include electric and magnetic chargesq and p, simply by replacing 2GMr with 2GMr - (q² + p²) / G;the result is the Kerr-Newman metric. All of the interesting phenomenapersist in the absence of charges, so we will set q = p =0 from now on.

The coordinates (t, r,,) are known as Boyer-Lindquist coordinates. It is straightforward to check thatas a 0 they reduce to Schwarzschild coordinates. Ifwe keep a fixed and let M 0, however, we recoverflat spacetime but not in ordinary polar coordinates. The metricbecomes

(7.117)

and we recognize the spatial part of this as flat space inellipsoidal coordinates.

They are related to Cartesian coordinates in Euclidean3-space by

(7.118)

There are two Killing vectors of the metric (7.114), both of which are manifest; since the metric coefficients are independentof t and , both = and = are Killing vectors. Of course expresses the axial symmetry of the solution.The vector is not orthogonal to t = constant hypersurfaces, and in fact is not orthogonal to any hypersurfaces at all; hence this metric is stationary, but not static. (It's not changingwith time, but it is spinning.)

What is more, the Kerr metric also possesses something called a Killing tensor. This isany symmetric (0, n) tensor which satisfies

(7.119)

Simple examples of Killing tensors are the metric itself, andsymmetrized tensor products of Killing vectors. Just as a Killingvector implies a constant of geodesic motion, if there exists aKilling tensor then along a geodesic we will have

(7.120)

(Unlike Killing vectors, higher-rank Killing tensors do notcorrespond to symmetries of the metric.)In the Kerr geometry we can define the (0, 2) tensor

(7.121)

In this expression the two vectors l and n are given (withindices raised) by

(7.122)

Both vectors are null and satisfy

(7.123)

(For what it is worth, they are the "special null vectors" of thePetrov classification for this spacetime.) With these definitions,you can check for yourself that is a Killing tensor.

Let's think about the structure of the full Kerr solution. Singularitiesseem to appear at both = 0 and = 0; let's turn ourattention first to = 0. As in the Reissner-Nordstrøm solution there are three possibilities: G²M² > a², G²M² = a², and G²M² < a². Thelast case features a naked singularity, and the extremal case G²M² = a² isunstable, just as in Reissner-Nordstrøm. Since these cases are of less physical interest, and time is short,we will concentrate on G²M² > a². Thenthere are two radii at which vanishes, given by

(7.124)

Both radii are null surfaces which will turn out to be event horizons. The analysis of these surfaces proceeds in close analogywith the Reissner-Nordstrøm case; it is straightforward to findcoordinates which extend through the horizons.

Besides the event horizons at r_±, the Kerrsolution also features an additional surface of interest. Recall that in thespherically symmetric solutions, the "timelike" Killing vector = actually became null on the (outer) event horizon, and spacelike inside. Checking to see where the analogous thing happens for Kerr, we compute

(7.125)

This does not vanish at the outer event horizon; in fact, at r =r₊ (where = 0), we have

(7.126)

So the Killing vector is already spacelike at the outer horizon,except at the north and south poles ( = 0) where it is null.The locus of points where = 0 is known as theKilling horizon, and is given by

(7.127)

while the outer event horizon is given by

(7.128)

There is thus a region in between these two surfaces, known asthe ergosphere. Inside the ergosphere, you must move inthe direction of the rotation of the black hole (the direction); however, you can still towards or away from the event horizon (and there is no trouble exiting the ergosphere).It is evidently a place where interestingthings can happen even before you cross the horizon; more detailson this later.

Before rushing to draw Penrose diagrams, we need to understand thenature of the true curvature singularity; this does not occur atr = 0 in this spacetime, but rather at = 0. Since = r² +a²cos² is the sum of two manifestly nonnegativequantities, it can only vanish when both quantities are zero, or

(7.129)

This seems like a funny result, but remember that r = 0 is nota point in space, but a disk; the set of points r = 0, = /2is actually the ring at the edge of this disk. The rotationhas "softened" the Schwarzschild singularity, spreading it outover a ring.

What happens if you go inside the ring? A careful analyticcontinuation (which we will not perform) would reveal that youexit to another asymptotically flat spacetime, but not an identicalcopy of the one you came from. The new spacetime is describedby the Kerr metric with r < 0. As a result, never vanishesand there are no horizons. The Penrose diagram is much like that forReissner-Nordstrøm, except now you can pass through the singularity.

Not only do we have the usual strangeness of these distinct asymptotically flat regions connected to ours through the blackhole, but the region near the ring singularity has additionalpathologies: closed timelike curves. If you consider trajectorieswhich wind around in while keeping and tconstant and r a small negative value, the line element along sucha path is

(7.130)

which is negative for small negative r. Since these paths areclosed, they are obviously CTC's. You can therefore meet yourselfin the past, with all that entails.

Of course, everything we say about the analytic extension of Kerris subject to the same caveats we mentioned for Schwarzschild andReissner-Nordstrøm; it is unlikely that realistic gravitationalcollapse leads to these bizarre spacetimes. It is nevertheless alwaysuseful to have exact solutions. Furthermore, for the Kerr metricthere are strange things happening even if we stay outside theevent horizon, to which we now turn.

We begin by considering more carefully the angular velocity of thehole. Obviously the conventional definition of angular velocitywill have to be modified somewhat before we can apply it to somethingas abstract as the metric of spacetime. Let us consider the fateof a photon which is emitted in the direction at someradius r in the equatorial plane ( = /2) of a Kerr blackhole. The instant it is emitted its momentum has no components inthe r or direction, and therefore the condition that itbe null is

(7.131)

This can be immediately solved to obtain

(7.132)

If we evaluate this quantity on the Killing horizon of the Kerrmetric, we have g_tt = 0, and the two solutions are

(7.133)

The nonzero solution has the same sign as a; we interpret thisas the photon moving around the hole in the same direction as thehole's rotation. The zero solution means that the photon directedagainst the hole's rotation doesn't move at all in this coordinatesystem. (This isn't a full solution to the photon's trajectory, justthe statement that its instantaneous velocity is zero.) This is an example of the "dragging of inertial frames" mentioned earlier. The point of this exercise is to note thatmassive particles, which must move more slowly than photons, arenecessarily dragged along with the hole's rotation once they are inside the Killing horizon. This dragging continues as we approachthe outer event horizon at r₊; we can define theangular velocity of the event horizon itself, , to be the minimum angularvelocity of a particle at the horizon. Directly from (7.132) wefind that

(7.134)

Now let's turn to geodesic motion, which we know will besimplified by considering the conserved quantities associated withthe Killing vectors = and = .For the purposes at hand we can restrict our attention to massiveparticles, for which we can work with the four-momentum

(7.135)

where m is the rest mass of the particle. Then we can take asour two conserved quantities the actual energy and angular momentumof the particle,

(7.136)

and

(7.137)

(These differ from our previous definitions for the conservedquantities, where E and L were taken to be the energy andangular momentum per unit mass. They are conserved either way, of course.)

The minus sign in the definition of E is there because atinfinity both and p are timelike, so their inner product is negative, but we want the energy to be positive. Inside the ergosphere, however, becomes spacelike; we can therefore imagine particles for which

(7.138)

The extent to which this bothers us is ameliorated somewhat by therealization that all particles outside the Killing horizonmust have positive energies; therefore a particle inside theergosphere with negative energy must either remain on a geodesicinside the Killing horizon, or be accelerated until its energy ispositive if it is to escape.

Still, this realization leads to a way to extract energy from a rotating black hole; the method is known as the Penrose process.The idea is simple; starting from outside the ergosphere, you armyourself with a large rock and leap toward the black hole. If wecall the four-momentum of the (you + rock) system p⁽⁰⁾, thenthe energy E⁽⁰⁾ = - p⁽⁰⁾ is certainly positive,and conserved as you move along your geodesic. Once you enter theergosphere, you hurl the rock with all your might, in a veryspecific way. If we call your momentum p⁽¹⁾ and that ofthe rock p⁽²⁾, then at the instant you throw it we haveconservation of momentum just as in special relativity:

(7.139)

Contracting with the Killing vector gives

(7.140)

But, if we imagine that you are arbitrarily strong (and accurate),you can arrange your throw such that E⁽²⁾ < 0, asper (7.158).

Furthermore, Penrose was able to show that you can arrange theinitial trajectory and the throw such that afterwards you follow ageodesic trajectory back outside the Killing horizon into theexternal universe. Since your energy is conserved along the way,at the end we will have

(7.141)

Thus, you have emerged with more energy than you enteredwith.

There is no such thing as a free lunch; the energy you gained camefrom somewhere, and that somewhere is the black hole.In fact, the Penrose process extracts energy from therotating black hole by decreasing its angular momentum;you have to throw the rock against the hole's rotationto get the trick to work. To see this more precisely,define a new Killing vector

(7.142)

On the outer horizon is null and tangent tothe horizon. (This can be seen from = , = , and the definition (7.134) of .)The statement that the particle with momentum p⁽²⁾crosses the event horizon "moving forwards in time" is simply

(7.143)

Plugging in the definitions of E and L, we see thatthis condition is equivalent to

(7.144)

Since we have arranged E⁽²⁾ to be negative, and is positive, we see that the particle must have a negativeangular momentum - it is moving against the hole's rotation.Once you have escaped the ergosphere and the rock hasfallen inside the event horizon, the mass and angular momentumof the hole are what they used to be plus the negativecontributions of the rock:

(7.145)

Here we have introduced the notation J for the angularmomentum of the black hole; it is given by

(7.146)

We won't justify this, but you can look in Wald for anexplanation. Then (7.144) becomes a limit on how much youcan decrease the angular momentum:

(7.147)

If we exactly reach this limit, as the rock we throw inbecomes more and more null, we have the "ideal" process,in which J = M/.

We will now use these ideas to prove a powerful result:although you can use the Penrose process to extract energyfrom the black hole, you can never decrease the area of theevent horizon. For a Kerr metric, one can go through astraightforward computation (projecting the metric andvolume element and so on) to compute the area of the eventhorizon:

(7.148)

To show that this doesn't decrease, it is most convenientto work instead in terms of the irreducible mass ofthe black hole, defined by

(7.149)

We can differentiate to obtain, after a bit of work,

(7.150)

(I think I have the factors of G right, but it wouldn'thurt to check.) Then our limit (7.147) becomes

(7.151)

The irreducible mass can never be reduced; hence the name.It follows that the maximum amount of energy we can extractfrom a black hole before we slow its rotation to zero is

(7.152)

The result of this complete extraction is a Schwarzschildblack hole of mass M_irr.It turns out that the best we can do is to start with anextreme Kerr black hole; then we can get out approximately29% of its total energy.

The irreducibility of M_irr leads immediately tothe fact that the area A can never decrease. From (7.149)and (7.150) we have

(7.153)

which can be recast as

(7.154)

where we have introduced

(7.155)

The quantity is known as the surface gravityof the black hole.

It was equations like (7.154) that first started people thinking about the relationship between black holes andthermodynamics. Consider the first law of thermodynamics,

(7.156)

It is natural to think of the term J as"work" that we do on the black hole by throwing rocks intoit. Then the thermodynamic analogy begins to take shapeif we think of identifying the area A as the entropyS, and the surface gravity as 8G times thetemperature T. In fact, in the context of classical general relativity the analogy is essentially perfect.The "zeroth" law of thermodynamics states that in thermal equilibrium the temperature is constant throughoutthe system; the analogous statement for black holes is thatstationary black holes have constant surface gravity onthe entire horizon (true). As we have seen, the first law (7.156) is equivalent to (7.154). The second law,that entropy never decreases, is simply the statement thatthe area of the horizon never decreases. Finally, the thirdlaw is that it is impossible to achieve T = 0 in anyphysical process, which should imply that it is impossibleto achieve = 0 in any physical process. It turnsout that = 0 corresponds to the extremal blackholes (either in Kerr or Reissner-Nordstrøm) - wherethe naked singularities would appear. Somehow, then,the third law is related to cosmic censorship.

The missing piece is that real thermodynamic bodiesdon't just sit there; they give off blackbody radiationwith a spectrum that depends on their temperature. Blackholes, it was thought before Hawking discovered his radiation, don't do that, since they're truly black. Historically,Bekenstein came up with the idea that black holes shouldreally be honest black bodies, including the radiation atthe appropriate temperature. This annoyed Hawking, who setout to prove him wrong, and ended up proving that therewould be radiation after all. So the thermodynamic analogyis even better than we had any right to expect - althoughit is safe to say that nobody really knows why.