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

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 to arise 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/c 2. This defined the Schwarzschild radius, demonstrated to be the point at which the escape velocity equals the speed of light. At this point, we halted our look inwards and discussed what Bob would observe as Al plunges into the black hole. He would become infinitely redshifted and his clock would appear to stop (calculations show that the reddening of light occurs very quickly, hence she would not be observed after about 10-4 s for a 10 solar mass black hole). We only hinted at what would happen to Al. Since he is free falling he is in an IRF. However the limits of the inertial frame would rapidly shrink as he approached the hole and would soon observe non-inertial effects. Basically his 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 the topic in terms of the Schwarzschild coordinates and then discuss a set of different coordinates that helps describe the scenario near and within a black hole more clearly. Afterwards, we hope to touch very briefly on a cosmological model (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 Horizon The Schwarzschild metric describes spacetime outside of a spherically symmetric body, dr 2 ds 2 = (1 2GM )c 2 dt 2 1 1 2GM r 2 d 2, where d = ( d 2 + sin 2 d 2 ). The coordinates t, r,, are the ones which are used by a far away observer. At the Schwarzschild radius, R s, it is observed that dt 0 and dr. The first limit corresponds to the far away observer viewing the in falling clock to stop at the horizon. The question remains, what does the in falling observer feel as the horizon is approached and passed? The answer 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 is approached. The Schwarzschild metric remains valid outside of the massive object at the center, now concentrated to a singular point at the origin. What about at R s and inside of the horizon? The spatial term, (1-2GM/ ) -1 dr 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 faraway coordinates as used by the far away observer. Before we discuss this region a serious limitation must be discussed first. These coordinates are used for comparing the distances measured out by the shell observers as compared to what the far away observer would measure if there were no source present. To extend this notion inside of the horizon becomes vague, for the far away observer can never observe anything inside of the horizon. What we will discuss next requires a significant qualifier. IF Bob could observe the trajectory of Al falling into the black hole he would use the Schwarzschild metric with the far-away coordinates. Inside of the horizon the curvature factor, (1-2GM/ ), changes sign. Hence the Schwarzschild metric becomes, ds 2 = ( 2GM 1)c 2 dt 2 + dr 2 2GM r 2 d 2. 1 Observe what has happened, the time and space coefficient have exchanged signs. What this implies is that the nature of time and radial distance have switched. If we are stubborn and insist on interpreting this scenario, we should switch the meaning of t and r, i.e. set r = t and t = r. Doing this we would have inside of the horizon, ds 2 = (1 2GM t 'c 2 dt ' 2 )c2 dr ' 2 1 2GM t ' 2 d 2. t 'c 2 The light cone can be identified as the region of spacetime demarcated by light rays. Note now that the light cone points (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. 2,

However, we switched the meaning of t and r. This means that inside of the horizon the future light cone points inward 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 once inside of the horizon, the only trajectories are those which end on the singularity. The forward light rays terminate on the singularity. There are no trajectories which can maintain a stable orbit inside of the horizon. Once inside, the only 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, and terminates on the singularity at a finite time. This is rather strange behavior. Do objects travel backwards in time once 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 black hole and dividing by 2. These shell observers then recorded events and reported to Bob at a far off distance. Inside of the horizon, it is not even possible to set up such observers as there are no stable orbits within. No matter how strong their rockets are to attempt to maintain their distance, they end up in the singularity. In addition, even if such observers could be set up, they could never communicate the results outside of the horizon. Again, the far away observer 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 the horizon but these coordinates are only really suited to the region outside of the horizon. (Of course, they are valid mathematically inside, but it is difficult to relate them to actual observations). To proceed, we need to change view points. Clearly using the far away coordinates will not give us a meaningful description, the shell observer coordinates would fail in the same way, the natural way to proceed is to attempt to describe this scenario from the point of view of the in falling object. Eddington-Finkelstein Coordinates To 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 flat spacetime metric in its local vicinity. However, we want to extend this to a region going beyond his local IRF which requires the Schwarzschild metric. As the object is free falling, it is in an IRF. If this object were to periodically fire light pulses radially inwards and outwards, the inward ray would start off at c. That is, the slope of the line would initially be 45 o. This construction, setting up coordinates which describe worldlines of light falling in at 45 o, leads to new coordinates known as Eddington-Finkelstein coordinates. The way to determine these coordinates, and the subsequent Schwarzschild metric obtained, is shown in the appendix (in addition a separate set of coordinates, with a more natural physical basis, are described in appendix 2). Here we will simply state the result, 2

dr dt = dt + c 1 2GM ds 2 = c 2 (1 2GM c 2 r )d t 2 2c 2GM c 2 r d tdr 1+ 2GM c 2 r. The Schwarzschild metric then becomes, dr 2 r 2 d 2. This complicated looking metric (the first time we ve seen off-diagonal terms) is simply the Schwarzschild metric expressed in coordinates more suited to describe the motion from the object s view. Although it is not expected that you do any calculations involving this metric, notice one thing; this metric no longer diverges at R s. 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 these new coordinates. It is also illuminating to find the worldlines of light rays emitted from this object from time to time. 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 same conclusions can be ascertained from this plot as were gotten from the previous, far-away, coordinates. First, at the Schwarzschild radius the worldline of the light ray emitted radially away from the black hole is vertical. This means that the light ray never reaches a region of r > R s. 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 the center. The highlighted green region shows the possible future points of travel within the horizon. There are no stable orbits within the horizon. Spinning Black Holes, Spinning Electrically Charged Black Holes It 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 the electrical charge of the object. This was paraphrased by the quote that black holes have no hair. It is interesting to note 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 be simple, say a billiard ball. A billiard ball can be described by its total mass, electrical charge, and angular momentum. However, on the microscopic scale the imperfections in the ball and its mass distribution come into play. Thus to describe exactly what a billiard ball will do requires a vast amount of information about its detailed structure: 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 3

momentum, J. No other information can be ascertained (or is necessary). Hence black holes are extremely simple objects to describe. With this in mind we will present the metric in the presence of a black hole of angular momentum J. This metric is called the Kerr metric. There is not enough time to describe how to get the solution of Einstein s equation and hence the form will simply be displayed, ds 2 = 1 2GM c 2 dt 2 + 4Ma dr 2 dtd c 2 r r 1 2GM 1+ a2 r + 2 Ma2 r 2 d 2, 2 r 3 where a = J/M the angular momentum per unit mass. Again, you are not expected to use this metric or completely understand its meaning but this is the second of the three metrics used to describe black holes. In fact, since most macroscopic objects have no net electrical charge, this metric is actually the most significant. As most stars have a net angular momentum, most black holes will be described by this metric. The most general metric is known as the Kerr-Newman metric and describes an electrically charged, spinning black hole. We will not display this metric to keep things simple. Kruskal Extension We now want to develop another map that describes the nature of spacetime for the free falling observer. We begin with the Schrodinger coordinate view developed earlier and picture the worldline of a free falling observer and a shell observer at some distance r. The new coordinates, 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 (in fact, 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 lines of simultaneity, or constant time T. As before we begin again with the fact that for the free 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 this must be seen as a patchwork of small local frames. We will examine a series of points and 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 the same as the far away observers. Basically, the faraway radial coordinate, r, is a vertical line just as X is. 4

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 it out to an infinite (far-away) distance at an infinite (far-away) time. The far away observer views the light ray 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 the horizon. The field of black hole research is an active and deep one. What we have not touched upon here is how exactly black holes form, what type of stars, the nature of the matter contained, etc. We only represented what the form of the metric is after a black hole forms. A full discussion of the formation of black holes and more speculative discussions of the singularity, white holes, and worm holes, would require several weeks to develop. As has been the theme through out this camp, it is merely the geometry that describes these exotic objects. 5

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 spherical symmetry, 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 we describe the inbound light rays to travel at c in these coordinates (that is at 45 degrees). The other condition is the familiar one that light rays travel with interval equal to zero. We use these two facts to determine the function f, and then to find the form in these new coordinates. 1) t = t + f r 2) r ct = 1 3) 0 = (1 K)c 2 t 2 r 2 1 K r 2 2, where K = 2GM Condition 3 is merely the fact that light rays travel along geodesics of null interval. The first step is to square out condition 1 for the difference t and insert it into the metric 3. The angular term is neglected for simplicity, t 2 = t 2 + 2 f t r + f 2 r 2 Next, divide through by the new time coordinate, 0 = (1 K )c 2 ( t 2 + 2 f t r + f 2 r 2 ) r 2 (1 K ) 2 r 0 = 1+ 2cf + c 2 f 2 2, where = 2 (1 K) ct. To represent the in falling ray of light to travel at c, we set = -1. This gives for our equation, expressed as a quadratic equation in f as, The solution of this equation, via the quadratic equation, is, 0 = f 2 2 c f + 1 1 1. c 2 (1 K) 2 f = 1 c 1± 1 2 4 4 + 4 1 K ( ) 2 Choosing the positive solution we have, t = t + (1+ 1 1 K = 1 c (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), s 2 = c 2 (1 2GM c 2 r ) t 2 2c 2GM c 2 r t r 1+ 2GM c 2 r r 2 r 2 2, Appendix 2 : Alternative Coordinates to Eddington-Finkelstein. [Note: This appendix is adapted from Project B, page B-13 of Exploring Black Holes, Introduction to General Relativity, by Edwin Taylor and John Wheeler.] 6

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, Gravity Wave 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 other questions have been answered). The structure and geometry of the universe is still an open question. It is becoming clear, however, that to answer these questions requires probing further and further back in time. This means probing closer to the initial singularity which was the big bang. In this regime the foundations upon which general relativity rest begin to exhibit quantum properties. 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. A full theory incorporating both of these theories is yet to be developed, and may not be in our lifetime. To a certain extent certain aspects of quantum theory can be incorporated into a general relativistic description of cosmology at early times. However, this can only be pushed back so far, to about 10-42 seconds after the big bang. This may not seem to be a problem since it is such a short time, however what happens before this time dictates what happens after and a complete picture 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 question which could alleviate the whole problem. There are a small number of researchers who propose this view. There are technical problems with this viewpoint and, thus, does not really make the problem go away. As was mentioned above, in order to explore regimes near singularities both of these theories come into play. We can get an idea of the scale of the regime by making some basic arguments. One result of quantum theory is that energy conservation can be violated for very short times. The limit of this violation imposed by quantum mechanics is related to the famous Heisenberg Uncertainty Principle, here between time and energy. The statement is that a quantum fluctuation of energy can occur for an interval of time such that the product of the change in energy and the time interval 341.05410EtJs= is less than the value of Planck s constant, i.e.. Thus the larger the energy created (particles pop out of the vacuum ) the shorter the time it has before they must annihilate back into the vacuum. The range over which this fluctuation can effect 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 M p c 2 =. The last term is known as the M p c 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 R p is less than the Schwarzschild radius, 2GM/c 2. This implies that the fluctuation is a black hole. To describe such an object requires both general relativity and quantum mechanics. The scale at which this occurs is generally viewed as the regime where quantum gravity must come into play. We can easily find the scale at which this occurs, the Planck scale, by equating the two distances (Compton wavelength and Schwarzschild radius), R Sc = R Compton GM p c 2 = M p c M p = c G = 2.18 108 kg R p = G =1.6210 35 m c 3 7

To explore fluctuations greater than the Planck mass, or equivalently to measure distances shorter than the Planck length R p, neither general relativity nor quantum field theory can be used alone. We can also get a sense of the energy of the fluctuation and the time limit by going back to the uncertainty relation. E p = M p c 2 = 1.22 10 19 GeV E p t p < t p = 8 M p c 2 = 51044 s Again, to describe fundamental particles with an energy of the scale of the Planck energy or time intervals less than the Planck time requires a theory of quantum gravity. We can take this time limit as the approximate limit in exploring the initial conditions of the big bang. What this argument suggests is that we can push the separate theories of general relativity and quantum field theory (quantum mechanics united with special relativity) back to a time of about 10-42 s or so. 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 initial conditions of the universe and answer all questions about its development. Hence, such a theory would be enormously powerful. Over the past 50 years it has become clear that such a theory is not going to be easily developed. What are the difficulties in uniting these two powerful theories? There are several different ways to point out the conflicts. First, it is clear that general relativity needs to have some modification on the extremely small, high energy scale. At the center of black holes and the beginning of the universe, the theory calls for a singularity. This singularity is a point of infinite spacetime curvature and energy density. Such singularities are mathematically unacceptable. However in low curvature, low energy, regions the general theory is an accurate theory. Quantum field theory (the unification of quantum mechanics and special relativity) on the other hand is an accurate theory on short distance, moderately high energy, scales. On the large scale, low energy scale, quantum mechanics transitions to classical mechanics. A transition which is not entirely well understood. There is much work today on this transition regime between quantum mechanics and classical physics. Another problem in bringing together these two theories is the question of what exactly is being quantized. To discuss quantization, first consider classical electromagnetic theory and its quantized form quantum electrodynamics. The process of quantizing the electromagnetic theory replaces the notion of an electromagnetic wave with particles (quanta) which mediate the electric and magnetic forces. The photon is the quanta of EM radiation. Classically it is electromagnetic waves (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 no space to discuss the details). For general relativity what is to be quantized? Recall that the Einstein equation gives the metric solution for a particular distribution of energy. The metric is the field which mediates the gravitational force (we are drawing an analogy to electricity and magnetism, again, there is no gravitational force but the curvature of spacetime). So the quantity to quantize is the metric itself. Or, put another way, spacetime itself must be quantized! This is surely 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 is no longer a continuum of spacetime? Tying this together with quantum mechanics which employs time and space as parameters to describe the wavefunctions of the quanta here being space and time itself. It is somewhat self-referential and causes difficulties in even beginning to construct a theory. Quantum theory relies on a spacetime background, however here we are doing away with such a concept and replacing it with discrete particles.

There are several different programs actively pursuing the goal of quantum gravity. Some come from the side of quantum field theory, beginning by treating gravity as any other field they would quantize, modifying general relativity in the process (string theory falls under this category). Others come from the relativist side, preferring to alter the quantization program to fit in with general relativity. Still others think that such a theory is so radical that we should start from scratch and hopefully general relativity and quantum mechanics will fall out in the appropriate limits. Each tactic has benefits and sheds light on the problem as a whole but no one program 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 and the different approaches. 9