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Lecture 10

In the previous lecture we described the Schwarzschild exterior solution for spherically symmetric objects.The effects of time dilation and gravitational redshift of light emanating from a massive object was demonstrated toarise from this metric.The Schwarzschild radius is an exterior solution, meaning it is only valid on the outside of the massive object.However, we observed that the metric diverges when r = 2GM/c2.This defined the Schwarzschild radius, demonstrated to be the point at which the escape velocity equals the speed oflight. At this point, we halted our look inwards and discussed what Bob would observe as Al plunges into the blackhole. He would become infinitely redshifted and his clock would appear to stop (calculations show that thereddening of light occurs very quickly, hence she would not be observed after about 10-4s for a 10 solar mass blackhole). We only hinted at what would happen to Al. Since he is free falling he is in an IRF. However the limits of theinertial frame would rapidly shrink as he approached the hole and would soon observe non-inertial effects. Basicallyhis head and feet would experience a net ‘tidal’ force which would eventually rip him to shreds.

In this final lecture we want to discuss what occurs inside of the event horizon. We will first approach thetopic in terms of the Schwarzschild coordinates and then discuss a set of different coordinates that helps describe thescenario near and within a black hole more clearly. Afterwards, we hope to touch very briefly on a cosmologicalmodel (history of the universe) and discuss how general relativity has influenced our view of the universe. Lastly,we will discuss on a very qualitative level the current status and unanswered questions.

Inside the Event HorizonThe Schwarzschild metric describes spacetime outside of a spherically symmetric body,

ds2 = (12GM

rc2)c2dt2

dr 2

1 2GM

rc2

r 2d2 ,

where d = ( d2+ sin2 d2).The coordinates t, r, , are the ones which are used by a far away observer. At the Schwarzschild radius,

Rs, it is observed that dt 0 and dr . The first limit corresponds to the far away observer viewing the in fallingclock to stop at the horizon. The question remains, what does the in falling observer feel as the horizon is approached and passed? Theanswer is nothing (outside of tidal forces), as the in falling observer is in a free falling reference frame (IRF). Again,the region over which this free falling frame remains an inertial reference frame shrinks as the black hole isapproached. The Schwarzschild metric remains valid outside of the massive object at the center, now concentratedto a singular point at the origin. What about at Rs and inside of the horizon? The spatial term, (1-2GM/rc2)-1 dr 2,diverges at this point. The coordinates used are not valid at this one radius.

Now we examine the Schwarzschild radius inside of the horizon. The coordinates, (t,r,,), are the far-away coordinates as used by the far away observer. Before we discuss this region a serious limitation must bediscussed first. These coordinates are used for comparing the distances measured out by the shell observers ascompared to what the far away observer would measure if there were no source present. To extend this notion insideof the horizon becomes vague, for the far away observer can never observe anything inside of the horizon. What wewill discuss next requires a significant qualifier. IF Bob could observe the trajectory of Al falling into the black holehe would use the Schwarzschild metric with the far-away coordinates. Inside of the horizon the curvature factor, (1-2GM/rc2), changes sign. Hence the Schwarzschild metric becomes,

ds2 = (2GM

rc21)c2dt2 +

dr 2

2GM

rc21

r 2d2 .

Observe what has happened, the time and space coefficient have exchanged signs. What this implies is that thenature of time and radial distance have switched. If we are stubborn and insist on interpreting this scenario, weshould switch the meaning of t and r, i.e. set r’ = t and t’ = r. Doing this we would have inside of the horizon,

ds2 = (12GM

t 'c2)c2dr '2

dt '2

1 2GM

t 'c2

t '2 d2 .

The light cone can be identified as the region of spacetime demarcated by light rays. Note now that the light conepoints (at a point) in the forward cone of the t’-r’ plane, exactly as in the t-r plane a ways outside of the horizon.

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However, we switched the meaning of t and r. This means that inside of the horizon the future light cone pointsinward along the –r direction. This is shown in the following plot.

The future light cone is the region in which the in falling object is allowed to travel. This plot demonstrates that onceinside of the horizon, the only trajectories are those which end on the singularity. The forward light rays terminateon the singularity. There are no trajectories which can maintain a stable orbit inside of the horizon. Once inside, theonly future is to travel to the singularity.

Notice also that the trajectory of the in falling object travels from an infinite time, backwards in time, andterminates on the singularity at a finite time. This is rather strange behavior. Do objects travel backwards in timeonce inside? Of course not. Recall our qualification. This construction was used with the far-away coordinates.Recall that the shell observer’s radii were set up by finding the circumference of a trajectory encircling the blackhole and dividing by 2. These shell observers then recorded events and reported to Bob at a far off distance. Insideof the horizon, it is not even possible to set up such observers as there are no stable orbits within. No matter howstrong their rockets are to attempt to maintain their distance, they end up in the singularity. In addition, even if suchobservers could be set up, they could never communicate the results outside of the horizon. Again, the far awayobserver can gather no information of what occurs inside of the horizon, recall the big IF.

Ok, where does that leave us? The Schwarzschild coordinates gave us a hint at what happens inside of thehorizon but these coordinates are only really suited to the region outside of the horizon. (Of course, they are validmathematically inside, but it is difficult to relate them to actual observations). To proceed, we need to change viewpoints. Clearly using the far away coordinates will not give us a meaningful description, the shell observercoordinates would fail in the same way, the natural way to proceed is to attempt to describe this scenario from thepoint of view of the in falling object.

Eddington-Finkelstein CoordinatesTo describe the motion of the object falling in from the object’s perspective we will need to change the

coordinates to ones more suited to the object. As the object remains in an (ever shrinking) IRF, it can use the flatspacetime metric in its local vicinity. However, we want to extend this to a region going beyond his local IRF whichrequires the Schwarzschild metric.

As the object is free falling, it is in an IRF. If this object were to periodically fire light pulses radiallyinwards and outwards, the inward ray would start off at c. That is, the slope of the line would initially be 45o. Thisconstruction, setting up coordinates which describe worldlines of light falling in at 45o, leads to new coordinatesknown as Eddington-Finkelstein coordinates. The way to determine these coordinates, and the subsequentSchwarzschild metric obtained, is shown in the appendix (in addition a separate set of coordinates, with a morenatural physical basis, are described in appendix 2). Here we will simply state the result,

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dt = dt +dr

c 1 2GM

rc2

. The Schwarzschild metric then becomes,

ds2 = c2(1

2GM

c2r)dt 2 2c

2GM

c2rdtdr 1+

2GM

c2r

dr 2 r 2d2 .

This complicated looking metric (the first time we’ve seen off-diagonal terms) is simply theSchwarzschild metric expressed in coordinates more suited to describe the motion from the object’s view. Althoughit is not expected that you do any calculations involving this metric, notice one thing; this metric no longer divergesat Rs. This reflects the fact that for the in falling object, it is always in a freely falling frame, an ever shrinking IRF.As the object passes the horizon, nothing significant happens. It doesn’t even notice that it has passed.

In order to get an idea of the nature inside of the horizon we plot the worldline of this object in thesenew coordinates. It is also illuminating to find the worldlines of light rays emitted from this object from time totime. These trace out the lightcones of possible future travel.

This plot is the best view we have to describe what happens inside of the horizon. Many of the sameconclusions can be ascertained from this plot as were gotten from the previous, far-away, coordinates. First, at theSchwarzschild radius the worldline of the light ray emitted radially away from the black hole is vertical. This meansthat the light ray never reaches a region of r > Rs. All worldlines within the horizon slope inwards. This tells us,again, that all worldlines within the horizon terminate on the singularity. There is no future but motion towards thecenter. The highlighted green region shows the possible future points of travel within the horizon. There are nostable orbits within the horizon.

Spinning Black Holes, Spinning Electrically Charged Black HolesIt was stated earlier that, due to conservation laws always held to be true, that the only information

detectable from outside of the horizon is the mass of the object, the angular momentum of the object, and theelectrical charge of the object. This was paraphrased by the quote that black holes have no hair. It is interesting tonote that black holes, often thought to be difficult to understand, are the simplest objects to describe in the universe.The reason is that there are only three parameters that describe such a beast. Contrast this to something thought to besimple, say a billiard ball. A billiard ball can be described by its total mass, electrical charge, and angularmomentum. However, on the microscopic scale the imperfections in the ball and its mass distribution come intoplay. Thus to describe exactly what a billiard ball will do requires a vast amount of information about its detailedstructure: mass asymmetries, electric dipole, quadropole and higher moments, its magnetic moments as well, etc.There is nothing else to describe a black hole except its total mass M, its net electrical charge, Q, and its angular

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momentum, J. No other information can be ascertained (or is necessary). Hence black holes are extremely simpleobjects to describe.

With this in mind we will present the metric in the presence of a black hole of angular momentum J. Thismetric is called the Kerr metric. There is not enough time to describe how to get the solution of Einstein’s equationand hence the form will simply be displayed,

ds2 = 12GM

c2r

c2dt2 +

4Ma

rdtd dr 2

1 2GM

rc2

1+a2

r 2+

2 Ma2

r3

r 2d 2 ,

where a = J/M the angular momentum per unit mass. Again, you are not expected to use this metric or completelyunderstand its meaning but this is the second of the three metrics used to describe black holes. In fact, since mostmacroscopic objects have no net electrical charge, this metric is actually the most significant. As most stars have anet angular momentum, most black holes will be described by this metric. The most general metric is known as theKerr-Newman metric and describes an electrically charged, spinning black hole. We will not display this metric tokeep things simple.

Kruskal Extension

We now want to develop another map that describes the nature of spacetime for the freefalling observer.

We begin with the Schrodinger coordinate view developed earlier and picturethe worldline of a free falling observer and a shell observer at some distance r. The newcoordinates, X and T, will represent those of the free falling observer.

This map is a patchwork of local inertial frames of the free falling observer (infact, of all free falling observers). For this observer, who is at rest in their own frame,vertical lines represent worldlines of constant position X. Horizontal lines represent linesof simultaneity, or constant time T. As before we begin again with the fact that for thefree falling observer light rays travel at c locally and are represented by 45 degree lines.

So our map, so far, looks like any other flat spacetime IRF however here thismust be seen as a patchwork of small local frames. We will examine a series of pointsand rays to find how to represent the shell observer and Schrodinger (far away) coordinates within this map.1) Far away. If we travel far away from the black hole the

coordinates of the free falling observer are thesame as the far away observers. Basically, the far-away radial coordinate, r, is a vertical line just as Xis.

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2) Shell observer. A shell observer at constant r will appear as an accelerating observer to the free falling observer.

3) Light ray emitted at horizon. When the free falling observer just passes the horizon, a light ray will just make itout to an infinite (far-away) distance at an infinite (far-away) time. The far away observer views the lightray to be still at the horizon. Thus, this light ray maps out the horizon as seen by the far away observer.

We have described the necessary geometry around black holes and have discussed what occurs upon entering thehorizon. The field of black hole research is an active and deep one. What we have not touched upon here is howexactly black holes form, what type of stars, the nature of the matter contained, etc. We only represented what theform of the metric is after a black hole forms. A full discussion of the formation of black holes and more speculativediscussions of the singularity, white holes, and worm holes, would require several weeks to develop. As has been thetheme through out this camp, it is merely the geometry that describes these exotic objects.

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Appendix 1: The Form of the Eddington-Finkelstein coordinates.

To obtain the form of the coordinates describing we perform a coordinate transformation of the time coordinate.The most general transformation would involve all four coordinates, however due to the restriction of sphericalsymmetry, the transformation can not involve the two angles, and . To bypass some messy arithmetic and

calculus, we will express the general most general form of the differential t. t = gt + f r . However, we

can simplify this by absorbing one of the coefficients (we’ll choose g), into the definition of the new differential

coordinate. Hence the general transformation is of the form, t = t + f r . The question is now, what is the

form of the function f? To find the form, we note the two conditions upon the coordinates we desire. Mainly, that wedescribe the inbound light rays to travel at c in these coordinates (that is at 45 degrees). The other condition is thefamiliar one that light rays travel with interval equal to zero. We use these two facts to determine the function f, andthen to find the form in these new coordinates.

1) t = t + f r

2)

r

ct = 1

3) 0 = (1 K )c2t2

r 2

1 K r 22 , where K =

2GM

rc2

Condition 3 is merely the fact that light rays travel along geodesics of null interval. The first step is to square outcondition 1 for the difference t and insert it into the metric 3. The angular term is neglected for simplicity,

t2 = t 2 + 2 f tr + f 2r2

0 = (1 K )c2 t 2 + 2 f tr + f 2r2( ) r2

(1 K )

,

Next, divide through by the new time coordinate,

0 = 1+ 2cf + c2 f 2 2

2

(1 K )2, where =

r

ct.

To represent the in falling ray of light to travel at c, we set = -1. This gives for our equation, expressed as aquadratic equation in f as,

0 = f 2

2

cf +

1

c21

1

(1 K )2

.

The solution of this equation, via the quadratic equation, is,

f =1

c1±

1

24 4+

4

1 K( )2

=

1

c(1±

1

1 K) .

Choosing the positive solution we have, t = t + (1+

1

1 K)r and inserting this into the form of the

Schwarzschild metric gives us the form of this metric discussed above (reinserting the form of K),

s2 = c2(1

2GM

c2r)t 2 2c

2GM

c2rtr 1+

2GM

c2r

r2 r 22

Appendix 2 : Alternative Coordinates to Eddington-Finkelstein.

[Note: This appendix is adapted from Project B, page B-13 of “Exploring Black Holes, Introduction toGeneral Relativity”, by Edwin Taylor and John Wheeler.]

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Future Directions

To this date, general relativity has passed all experimental tests with flying colors.Further more accurate tests will be conducted in the coming years, (Gravity Probe B, GravityWave detectors). The areas of active research, where there are still some unanswered questions,concerning general relativity are cosmology and black holes. (This is not to say all otherquestions have been answered). The structure and geometry of the universe is still an openquestion. It is becoming clear, however, that to answer these questions requires probing furtherand further back in time. This means probing closer to the initial singularity which was the bigbang. In this regime the foundations upon which general relativity rest begin to exhibit quantumproperties. The other area where this comes into play is at the singularities inside of black holes.The interplay of general relativity and quantum mechanics is a very tricky one to understand. Afull theory incorporating both of these theories is yet to be developed, and may not be in ourlifetime.

To a certain extent certain aspects of quantum theory can be incorporated into a generalrelativistic description of cosmology at early times. However, this can only be pushed back sofar, to about 10-42 seconds after the big bang. This may not seem to be a problem since it is sucha short time, however what happens before this time dictates what happens after and a completepicture can not be developed until a theory of quantum gravity is developed.

Why do we need to bring these two theories together anyways? This is a good questionwhich could alleviate the whole problem. There are a small number of researchers who proposethis view. There are technical problems with this viewpoint and, thus, does not really make theproblem go away. As was mentioned above, in order to explore regimes near singularities bothof these theories come into play. We can get an idea of the scale of the regime by making somebasic arguments.

One result of quantum theory is that energy conservation can be violated for very shorttimes. The limit of this violation imposed by quantum mechanics is related to the famousHeisenberg Uncertainty Principle, here between time and energy. The statement is that aquantum fluctuation of energy can occur for an interval of time such that the product of thechange in energy and the time interval is less than the value of Planck’s constant, i.e.341.05410EtJs=

.Thus the larger the energy created (particles ‘pop out of the vacuum’) the shorter the time it hasbefore they must annihilate back into the vacuum. The range over which this fluctuation caneffect nearby objects is limited by the speed of light, R < ct. From the uncertainty relation

above we see that the range is

R ct = c

E= c

Mpc2=

Mpc

. The last term is known as the

Compton wavelength for a particle of mass M. This is a measure of the ‘fuzziness’ of a particle.Consider now a quantum fluctuation which occurs whose ‘radius’ Rp is less than the

Schwarzschild radius, 2GM/c2. This implies that the fluctuation is a black hole. To describe suchan object requires both general relativity and quantum mechanics. The scale at which this occursis generally viewed as the regime where quantum gravity must come into play. We can easilyfind the scale at which this occurs, the Planck scale, by equating the two distances (Comptonwavelength and Schwarzschild radius),

RSc= R

Compton

GMp

c2=

Mpc M

p=c

G= 2.18108 kg

Rp=G

c3=1.621035 m

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To explore fluctuations greater than the Planck mass, or equivalently to measure distancesshorter than the Planck length Rp, neither general relativity nor quantum field theory can be usedalone. We can also get a sense of the energy of the fluctuation and the time limit by going backto the uncertainty relation.

Ep= M

pc2 =1.221019GeV

Ept

p< t

p=

Mpc2= 51044 s

Again, to describe fundamental particles with an energy of the scale of the Planck energy or timeintervals less than the Planck time requires a theory of quantum gravity. We can take this timelimit as the approximate limit in exploring the initial conditions of the big bang. What thisargument suggests is that we can push the separate theories of general relativity and quantumfield theory (quantum mechanics united with special relativity) back to a time of about 10-42 s orso. Prior to this time physics is governed by the unknown theory of quantum gravity.

If a theory of quantum gravity is developed, the hope is that it will describe the initialconditions of the universe and answer all questions about its development. Hence, such a theorywould be enormously powerful. Over the past 50 years it has become clear that such a theory isnot going to be easily developed.

What are the difficulties in uniting these two powerful theories? There are severaldifferent ways to point out the conflicts. First, it is clear that general relativity needs to havesome modification on the extremely small, high energy scale. At the center of black holes andthe beginning of the universe, the theory calls for a singularity. This singularity is a point ofinfinite spacetime curvature and energy density. Such singularities are mathematicallyunacceptable. However in low curvature, low energy, regions the general theory is an accuratetheory. Quantum field theory (the unification of quantum mechanics and special relativity) on theother hand is an accurate theory on short distance, moderately high energy, scales. On the largescale, low energy scale, quantum mechanics transitions to classical mechanics. A transitionwhich is not entirely well understood. There is much work today on this transition regimebetween quantum mechanics and classical physics. Another problem in bringing together these two theories is the question of what exactly isbeing quantized. To discuss quantization, first consider classical electromagnetic theory and itsquantized form quantum electrodynamics. The process of quantizing the electromagnetic theoryreplaces the notion of an electromagnetic wave with particles (quanta) which mediate the electricand magnetic forces. The photon is the quanta of EM radiation. Classically it is electromagneticwaves (or electric and magnetic fields) which mediate the forces. This is what is being quantized.(The process is more complicated then simply replacing waves with particles but there is nospace to discuss the details). For general relativity what is to be quantized? Recall that theEinstein equation gives the metric solution for a particular distribution of energy. The metric isthe ‘field’ which mediates the gravitational force (we are drawing an analogy to electricity andmagnetism, again, there is no gravitational force but the curvature of spacetime). So the quantityto quantize is the metric itself. Or, put another way, spacetime itself must be quantized! This issurely a strange requirement. What does it mean to replace spacetime with quanta (gravitons)which mediate the gravitational force? What do these particles propagate through, since there isno longer a continuum of spacetime? Tying this together with quantum mechanics whichemploys time and space as parameters to describe the wavefunctions of the quanta – here beingspace and time itself. It is somewhat self-referential and causes difficulties in even beginning toconstruct a theory. Quantum theory relies on a spacetime background, however here we aredoing away with such a concept and replacing it with discrete particles.

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There are several different programs actively pursuing the goal of quantum gravity. Somecome from the side of quantum field theory, beginning by treating gravity as any other field theywould quantize, modifying general relativity in the process (string theory falls under thiscategory). Others come from the relativist side, preferring to alter the quantization program to fitin with general relativity. Still others think that such a theory is so radical that we should startfrom scratch and hopefully general relativity and quantum mechanics will fall out in theappropriate limits. Each tactic has benefits and sheds light on the problem as a whole but no oneprogram has been successful yet.

To go into the details of each would send us to far out of our path. On the CD are a few,somewhat non-technical, review articles by prominent researchers discussing the problem andthe different approaches.