Stefan van der Sterren Supervisor Dr. Sebastian de Haro Amsterdam university College

Transcription

1 Black Hole Entropy Stefan van der Sterren Supervisor Dr. Sebastian de Haro Amsterdam university College

2 ABSTRACT Since Karl Schwarzschild in 1915 discovered that the distance from the center of an object, where all the mass of the object were compressed within a symmetrically sphere, the escape speed from the surface would equal the speed of light. This distance is known as the Schwarzschild radius. Such an object further called black hole would be formed from the collapse of at least 3 solar masses, and would not allow anything to come out of it, not even light. In the seventies Jacob Bekenstein and Stephen Hawking calculated the entropy of such a black hole. Hawking also proposed that black holes actually radiate, and eventually evaporate. In the nineties string theory could also be used to calculate black hole entropy. The entropy of a black hole is a gauge for the number of arrangements (states or also called information) of matter that was trapped in the black hole. It turns out that entropy is related to surface area of the black hole. So, all information can be found on this boundary. Initially a formula for entropy was established based on parallels between thermodynamics, information and black holes by Bekenstein. It is therefore that we first visit thermodynamic entropy and information entropy, followed by some metrics of a Schwarzschild black hole, as it is important to know how a particle behaves around the event horizon. This event horizon will also be looked at in some more detail and the penrose energy extraction process will also be treated. After the basics for thermodynamic, information entropy and black holes are established the links with black hole entropy will become apparent. These relations will be used to derive black hole entropy like Bekenstein did, but also via Hawking radiation and in short via string theory. Even though several methods to derive black hole entropy have been established, not all questions surrounding the topic have been answered. For example a black hole can potentially evaporate till only an elementary particle remains or can reach an immense size, like the suspected black hole in the center of galaxies, calculations become more cumbersome in these extreme situations and have raised questions for further research. 1

3 Contents Introduction Entropy Multiplicity tends to increase Entropy change Information entropy Entropy of an information source Entropy of a tossing event Information loss Black hole Origination of a black hole Escape velocity Black hole size Black hole classification Penrose-Hawking singularity theorem(s) Schwarzschild solutions Schwarzschild metric Other coordinates to enhance/extend the Schwarzschild solution Horizons and the Penrose process Event horizons Killing horizons Kerr horizons Penrose process Entropy of a black hole Four laws of black holes Calculating entropy Hawking radiation Application of string entropy to black holes Conclusion Acknowledgements References

4 Introduction Already in 1783 John Michell realized that gravity could be so overwhelming strong that nothing could escape: not even light. It has been investigated further by Laplace in Later many scientists were inspired by this phenomenon, but it took some time before the existence of a black hole could be proved. In the early 20 th century Schwarzschild made much progress by calculating the gravitational field of stars, using Einstein s field equation. Originally black holes were assumed to be the remainder of gravitational collapsed stars and they were characterized by the fact that you can enter them, but nothing at all comes out of it. The most important feature of the black hole is therefore the event. The event horizon seems to demarcate a point of no return. Schwarzschild calculated the radius of the event horizon, what is today known as the Schwarzschild radius. In 1972 Bekenstein noted that there are a number of similarities between black hole physics and thermodynamics. He was the first to suggest that black holes have a well-defined entropy based on the surface area of the black hole and generalized the second law of thermodynamics to apply to black holes. In 1974 Hawking proposed the existence of Hawking radiation and affirmed Bekenstein s work. The result of this radiation is that it seems black holes do release information after all. Over the last decades many scientists entered the subject of black hole entropy to prove, enhance, add, or counter theorems and postulations. Many arguments were about the event horizon. As entropy is related to this horizon it is important to learn more about the metrics of the event horizon and the various ways to calculate to entropy. This paper will discuss black hole entropy based on a study of the literature. The first section explains both thermodynamic entropy and information entropy. The second section is about black holes; the metrics of a Schwarzschild black hole, the event horizon and the 3

5 Penrose energy extraction process. In the last section we will discuss various ways of deriving black hole entropy; Bekenstein s method, via the Hawking radiation and in short via string theory. 1. Entropy Everyone will understand that if you separate the atoms a BMW and throw them together in the air it is highly unlikely that the atoms will fall back such that your car looks brand new. Perhaps if you would do it over and over again, you would eventually have something like an automobile, but would it work? There are so many possible ways of assembling the atoms that nearly all arrangements look like a pile of something [1]. The term entropy is often thought about as disorder. Most people would agree that a shuffled deck of cards is more disorderly than a sorted deck, and indeed, shuffling increases the entropy because it increases the number of possible arrangements. However, many people would say that a glass of crushed ice appears more disorderly than a glass of an equal amount of water. In this case, though, the water has much more entropy, since there are so many more ways of arranging the molecules, and so many more ways of arranging the larger amount or energy among them [2, p75]. 1.1 Multiplicity tends to increase Suppose you have 3 true coins each with a head and tail side. A true coin is defined as when tossing there is a percent chance coming up with head (H) or tail (T). With 3 coins there are 8 possible outcomes, microstates : TTT, TTH, THT, THH, HTT, HTH, HHT and HHH. There are four possible macrostates: 4

6 3 x H, has 1 microstate (HHH) 2 x H, has 3 microstates (THH, HTH, HHT) 1 x H, has 3 microstates (TTH, THT, HTT) 0 x H, has 1 microstate (TTT) The number of microstates of a certain macrostate is called multiplicity. The chance of being in macrostate x is Ω(x) where Ω (x) is the multiplicity of x. Being in a Ω (all) certain macrostate means being in one of the microstates associated with that macrostate. Often only macrostates can be observed. Thus the more microstates associated with one macrostate (=multiplicity) the greater the uncertainty there is about being in which microstate. Figure 1: gas molecules in a container (no inside membranes or walls) Suppose the gray area depicts gas molecules in a container. What is the probability of finding the configuration shown in figure 1? With N gas molecules, out of all the allowed microstates, only 1 in 2 N microstates has all the molecules in the left half. Thus, the probability of this arrangement is 2 N [2]. Since multiplicity can be a very large number the Boltzmann constant k is used. This also gives it units J, which helps it relate to temperature as a higher K temperature generally means a higher entropy. Entropy is denoted as: S = k ln Ω In words: entropy is the logarithm of the number of ways of arranging things in a system. Generally the more particles in a system and the more energy it contains, the greater its multiplicity and thus S. Other possibilities to increase entropy are for example: 5

7 Increasing the space the system is in Breaking large particles into smaller ones Interaction of substances, which were initially separated (mixing) In general particles and energy tend to rearrange themselves until the multiplicity is at (or very near) its maximum value. This seems to be true for any system, provided that it contains enough particles and units of energy for the statistics of very large numbers to apply. In this respect the second law of thermodynamics is defined as follows: Any large system in equilibrium will be found in the macrostate with the greatest multiplicity, aside from fluctuations that are normally too small to measure. [2, p74] 1.2 Entropy change Entropy change can be calculated as follows, provided the process is quasi static: Q = T ds If two different monoatomic ideal gases (A and B), each with the same energy, volume and number of particles are in a container, separated by a partition, are being mixed by removing the partition the entropy increases [2]. A B Figure 2: Two different gases separated by a partition. When, in figure 2. the partition is removed each gas expands to fill the whole container, mixing the two gasses and as a result creating entropy. This increase in entropy is called entropy of mixing. The entropy increase of gas A can be calculated as follows: 6

8 ΔS A = Nk ln V f V i = Nk ln 2 As the entropy of gas B increases by the same amount, the total entropy increase is: ΔS total = ΔS A + ΔS B = 2Nk ln 2 There is only an increase in entropy if the gases are different. If they are the same, the entropy doesn t increase when the partition is removed. The term 2Nk ln 2 is called the entropy of mixing. The Sackur-Tetrode equation explains the entropy of mixing monatomic ideal gas in more detail: S = Nk ln V 3 N 4πmU 3Nh Information entropy The word information can mean many things, such as I think your information is wrong, For your information. or You can find the information.. [1]. Suppose a small piece of paper contains a text of 10 characters and someone s keyboard holds approximately 100 symbols (including upper and lower case, numbers and punctuation marks), the number of possibilities to organize the message is The piece of paper is said to contain rounded 70 bits of information, as an information bit is defined as [1]: log 2 2 n = n bits Shannon [3] defined information entropy as a quantity that measures how much information and at what rate information is produced. 7

9 Shannon says further: Suppose we have a set of possible events whose probabilities of occurrence are p 1, p 2,, p n. Then all we know about these events is their probability of taking place, which does not seem much. The question that then occurs is can we find a measure, H, indicating how much choice there is or how uncertain we are of the outcome? Further, if there is such measure H = (p 1, p 2,, p n ) than it follows based on three additional requirements that H should be: n H = K p i log p i i=1 K is a positive constant. These three requirements are the following 1. H is to be continuous in the p i. As Log (x) is defined for every possible value p i, where 0 < p i 1, this requirement holds 2. If all p i are equal to p i = 1, then H should be a monotonically increasing function of n. n As p i = 1 n for i = 1, 2,, n and since p 1 = p 2 = = p n It follows n p i = n. p i = n. 1 n = 1 i=1 And so (leaving K out of the equation) This leads to n H = p i log p i i=1 H = log p i = log 1 = log n n Thus log n < log (n + 1) 8

10 From this it follows that this requirement holds as being a monotonically increasing function means f(x + 1) > f(x), where f is some function. 3. If a choice be broken down into two successive choices, the original H should be the weighted sum of the individual values of H [3, p10]. The meaning of this will become clear in the following example. Figure 3: Decomposition of a choice of 3 possibilities. The graph on the left hand side shows three possibilities p 1 = 1, p 2 2 = 1, p 3 3 = 1. On the 6 right hand side we need to choose first between 2 possibilities, each with probability 1 2. Following the lower branch we can make another choice with probabilities 2, 1. The final 3 3 results of the two graphs are the same. We thus require that H 1 2, 1 3, 1 6 = H 1 2, H 2 3, 1 3. The coefficient 1 is because this second choice only occurs half of the time. 2 Generalized, this gives (see figure 4): H(1 x, ax, bx, cx) = H(1 x, x) + x H(a, b, c) = (1 x) log (1 x) + ax log (ax) + bx log (bx) + cx log(cx) 9

11 = (1 x) log(1 x) + ax log(a) + ax log(x) + bx log(b) + bx log(x) + cx log (c) + cx log(x) = (1 x) log (1 x) + x(a log(a) + b log(b) + c log(c)) + (a + b + c) x log (x) Figure 4: Decomposition in more general As a + b + c = 1 it follows = (1 x) log (1 x) + x log x + x(a log(a) + b log (b) + c log (c)) = H(1 x) + H(x) + x H(a, b, c) Entropy of an information source Some stochastic processes are mathematically known as discrete Markoff processes and will be briefly discussed next, using two examples by Shannon. 10

12 For each possible state i there will be a set of probabilities p i (j) of producing the various possible (text) symbols j. There is an entropy H i for each state. The entropy of the source is the weighted average of H i. This is the entropy of the source per symbol of text. H = P i H i If the Markoff process is performed at a definite time rate there is also an entropy per second, where f i is the average number of symbols produced per second. i H t = f i H i H t measures the amount of information generated by the source per symbol or per second. If the logarithmic base is 2, H t will represent bits per symbol or per second. Figure 5 shows a graphical presentation of a typical Markoff process. If. A happens, there is a chance of 50% that A happens again and 20% chance for B to occur and 30% for C. A possible outcome of the process would be: ABAACACBAA. The process would have H = 1.49 bits. i Figure 5: Typical Markov process 11

13 Figure 6: Markov process with more choices. Figure 6 shows an example of a Markov process with more choice. The following probability table can be set up. A B C H A = 1.49 P A = 0.4 A H B = 1.16 P B = 0.2 B H C = 1.30 P C = 0.4 C H = Entropy of a tossing event Let s look at the event table below where we have 3 true coins. Macrostate Number of Occurrence Probability head mircostates 0 1 (T,T,T) 1/8 1 3 (T,T,H), (T,H,T), (H,T,T) 3/8 2 3 (T,H,H), (H,T,H), (H,H,T) 3/8 3 1 (H,H,H) 1/8 We can calculate entropy in two ways: according the Statistical Mechanics [2] or by information entropy [3]. S = k log Ω (1) 12

14 n H = K p i log p i i=1 (2) The calculated entropy (leaving the constants k/k out) is (applying 1) while the second is (applying 2). Some remarks: H is basically a measure of uncertainty Similarity H/S is zero when Ω/p = 1 Both S and H have an extreme (maximum) S when the probability of that macrostate is highest and H when the uncertainty is highest For both H and S this could be seen as a measure of equality as the state furthest from both extremes gives the highest entropy This can be further illustrated by the following. Taking a case of two possibilities with probabilities p and q = 1 p. This gives H = (p log(p) + q log(q)) Which can be plotted as shown in figure 7. 13

15 Figure 7: Entropy in the case of two possibilities with probabilities p and (1 p) The difference between Thermodynamic Entropy and Information Entropy is that the value of S doesn t depend on the multiplicity of the other macrostates, while H depends on the multiplicities of (all) other marcostates Information loss Wheeler [4] suggested that information is fundamental to the physics of the universe. Wheeler believed that all material objects are composed of bits of information. A bit is as small as the smallest possible size; the fundamental quantum of distance discovered by Max Planck. According to this "it from bit" doctrine, all things physical are information-theoretic in origin. If you could read the code at any point in time you could understand what was going on in that particular space of the universe. As entropy tends to increase, meaning patterns change with time and so does the code ; the information flows. There are some comments to be made to this doctrine. The first law of Thermodynamics says that energy is conserved [2], but what about information conservation? Suppose you know the present with perfect precision, you can predict the future for all time. Reversely you could also be absolutely sure about the past. However conservation implies in this respect that the process, transferring information from one state to the other, is fully understood and doesn t hide any uncertainty. The Second law of thermodynamics says that the total entropy of the world always increases. The change of potential, kinetic, chemical, and other forms of energy into heat always favors more heat and less of those organized, non-chaotic forms of energy. Thus organized energy degrades to heat, not the other way around [2]. Heisenberg s Uncertainty 14

16 Principle is interesting in this respect. He says that you cannot measure the exact position of a particle and at the same time its exact velocity. So if information flows how can you be absolutely sure about the present, past and future arrangements of information? Initially black holes were considered reservoirs of hidden information as nothing came out of it. Indeed they are most densely packed information storage containers in nature, but nowadays it seems black holes can decay and radiate away. So they release information after all. 2. Black hole 2.1 Origination of a black hole Already in the late 18th century it was suggested by John Michell and Pierre-Simon Laplace that black holes exist and that their gravity would be so strong such that no light can escape. But how are black holes born? Black holes are thought to form from stars or other massive objects. If and when these objects collapse under their own gravity, a black hole is formed. As the centre of such a black hole is approached the density goes to infinity: in other words, a singularity. During most of a star's lifetime, nuclear fusion in the core generates electromagnetic radiation, which includes photons, the particles of light. This radiation exerts an outward pressure that exactly balances the inward pull of gravity caused by the star's mass. 15

17 As the nuclear fuel is exhausted, the outward forces due to radiation diminish, allowing the gravitation to compress the star. The contraction of the core causes its temperature to rise and allows the remaining nuclear material to be used as fuel. The star is saved from further collapse, but only for a while. Eventually, all possible nuclear fuel is used up and the core collapses. How far it collapses, into what kind of object, and at what rate, is determined by the star's final mass and the remaining outward pressure that the burnt-up nuclear residue (largely iron) can muster. If the star is sufficiently massive or compressible, it may collapse to a black hole. If it is less massive or made of stiffer material, its fate is different: it may become a white dwarf or a neutron star. [5]: 16

18 2.2 Escape velocity The escape velocity is defined as the minimal velocity that one (with mass M 2 ) needs, without adding energy/propulsion, to escape a spherical object with mass M 1, while being at a distance R from the centre of that object. To escape to infinite distance we can use energy conservation. At infinite separation, the gravitational potential energy is zero, and the minimum kinetic energy is also zero. Thus the total energy with which a projectile can barely escape to infinity from an object s gravitational pull is zero. Energy conservation then implies: 1 2 M 2 2v esc = M 2G M 1 R v esc = 2 G M 1 R If v esc = c, where c is light speed, then R = R s, where R s is the so-called Schwarzschild Radius, which we will come back to later. Note that v esc exceeds the speed of light if R < R s [6]. 2.3 Black hole size Black holes are being ranked as follows, where AU is defined as Astronomical Unit (Earth- Sun distance, being km): Class Mass Size Supermassive black hole ~ MSun ~ AU Intermediate-mass black hole ~103 MSun ~103 km = REarth Stellar black hole ~10 MSun ~30 km Micro black hole (also called Primordial black hole) up to ~MMoon up to ~0.1 mm 17

19 Supermassive black holes seem to evolve from stellar black holes and appear to exist in the centre of galaxies. Stellar black holes are being formed from the collapse of at least 3 solar masses and their density is only a little above of nuclear density (10 18 kg/m2). The micro black holes have an extremely small radius (the mass of a big mountain would have a radius of a nanometre). So what about an elementary particle? According Jacobson [6] an elementary particle cannot be a black hole. Its Compton wavelength λ c is much greater than its Schwarzschild radius (note for a proton λ R s ). The Planck mass is special because the reduced Compton wavelength (λ c = ћ Mc ), for this mass, is equal to half of the Schwarzschild radius. This special distance is called the Planck length. The Schwarzschild radius is proportional to the mass, whereas the Compton wavelength is proportional to the inverse of the mass. Planck mass: M p = ћc G 1/2, kg Planck energy: E p = ( ћc5 G ) 1/2, GeV Planck length: L p = ( ћg c 3) 1/2, m From v = 2GM R it follows that R = 2GM v 2, while R s = 2GM c 2 and λ c = ћ Mc To find M p it follows: λ c = 1 2 R s = L p ћ M p c = GM p c 2 M p 2 = ћ c G M p = ћc /G When filling in it gives: R s = 2Gm c 2 = 2G c 2 ћc G = 2 ћg c 3 = 2L p and so L p = 1 2 R s 18

20 2.4 Black hole classification The no-hair theorem states that, once it achieves a stable condition/equilibrium, a black hole has only three independent physical properties: mass, charge, and angular momentum (spin). No-hair just expresses that anything that falls into the black hole disappears behind the black event horizon and is inaccessible to external observers [1]. The simplest black holes have mass but neither electric charge nor angular momentum. Caroll [7 p.238]: Stationary solutions are of special interest because we expect them to be the end states of gravitational collapse. The alternative might be some sort of oscillating configuration, but oscillations will ultimately be damped as energy is lost through the emission of gravitational radiation; in fact, typical evolutions will evolve quite rapidly to stationary configuration. A static spacetime is a stationary spacetime with a time reflection symmetry. Thus a spinning black hole is stationary but not static. The following grouping can be made: Non-rotating ( J = 0 ) Rotating ( J 0 ) Uncharged ( Q = 0) Schwarzschild (of EF) Kerr Charged ( Q 0) Reissner-Nordström Kerr-Newman Explanation EF-metric: Unique static vacuum solution with event horizon Kerr metric: Stationary vacuum solution with event horizon, parameterised by M and J. RN (Reissner-Nordström) metric: Stationary vacuum solution with event horizon, parameterised by M, Q e and Q m (magnetic charge). Kerr-Newman metric: Unique stationary vacuum solution with J and Electromagnetic field. 19

21 Stable black holes can be completely described at any moment in time by eleven parameters: mass-energy M, linear momentum P (three components), angular momentum J (three components), position X (three components), electric charge Q. 2.5 Penrose-Hawking singularity theorem(s) A singularity, a point of almost infinite density, in solutions of the Einstein field equations is: 1. Situation where matter is forced to be compressed to a point (a space-like singularity; feature of uncharged non-rotating solutions) or 2. Situation where certain light rays come from a region with infinite curvature (time-like singularity; feature of charged or rotating solutions) Both have the property of geodesic incompleteness: Some light-paths or particle-paths cannot be extended beyond a certain proper-time or affine-parameter (affine parameter is the null analog of proper time). It is still an open question whether time-like singularities ever occur in the interior of real charged or rotating black holes, or whether they are artifacts of high symmetry and turn into spacelike singularities when realistic perturbations are added. 2.6 Schwarzschild solutions In General Relativity (GR), the unique static spherically symmetric vacuum solution is the Schwarzschild metric. Birkhoff s theorem says basically the same and in particular that there are no time-dependent solutions in this form Schwarzschild metric 20

22 The Schwarzschild metric, in spherical coordinates ( t, r, θ, ), is given by: ds 2 = 1 2GM r dt GM 1 dr 2 + r 2 dω 2 (3) r Where dω 2 is the metric on a unit two-sphere. dω 2 = dθ 2 + sin 2 θ dφ 2 The constant M is interpreted as the mass of the gravitating object. Some notes: if M 0, you get Minkowski space if r 0 (in the term with dt 2 ) the metric coefficient becomes infinite r 2GM (= r s ), in the term with dr 2, the metric coefficient also becomes infinite For both r = 2GM and r = 0 there are singularities or there is something wrong with the metric. To find out whether these are real singularities we need a way to check whether the curvature becomes infinite. The curvature can be measured with the Riemann tensor, but both tensors and the metric are coordinate dependent. This means it cannot show if the curvature really goes to infinity or that it is due to the choice of coordinates. Therefore we need some scalar as scalars are coordinate independent. By constructing scalars from the curvature we can figure out whether it is due to the choice of coordinates or whether we have a real singularity. The simplest such scalar is the Ricci scalar R = g μν R μν. We take the following scalar R μνρσ R μνρσ = 48G2 M 2 If this scalar goes to infinity, the curvature, and it is a singularity. So it seems that r 0 represents a singularity. However r 2GM does not seem to be a singularity. When 21 r 6

23 we change to more appropriate coordinates the surface r 2GM is actually very wellbehaved in the Schwarzschild metric and in fact demarcates the event horizon of a black hole [7]. Let us have a closer look at the Schwarzschild metric. Consider radial null curves, those for which φ and θ are constant and ds 2 = 0. So, which can be written as ds 2 = 0 = 1 2GM r dt GM 1 dr 2 r 1 dt 2GM = ± 1 dr r This measures the slope of the light cones on a spacetime diagram in the t r plane. For large r the slope is ±1, as it would be in flat space, while as r 2GM it give dt dr ±, and the light cones close up as shown in figure 8 [7]. It means that when a light ray approaches r = 2GM it never seems to get there (at least in this coordinate system); instead it seems to asymptote to this radius. Figure 8: In Schwarzschild coordinates, light cones appear to close up as we approach r 2GM. 22

24 It is obviously an illusion as a particle can reach the radius without issue. However an observer far away could never tell. An observer at fixed r and a beacon falling freely into a black hole, sending signals at constant proper time intervals τ 1. These signals will take more and more time to reach the observer, see Figure 9 [7]. Thus we will never observe the beacon vanish into the black hole. Figure 9: An observer at fixed r receives the signals at successively long time intervals τ Other coordinates to enhance/extend the Schwarzschild solution To solve the problem of the coefficient going to infinity we change to tortoise coordinate r, which is only useful if r 2GM. r = r + 2GM log ( r 2GM 1) dr = 1 + 2GM r 1 dr 2GM GM = 1 + r 1 2GM 1 = 1 2GM 1 r Writing dr 2 (= 1/dr 2 ) leads to: dr 2 = 1 2GM r 2 dr 2 (4) Inserting this result in the Schwarzschild metric (3) gives: 23

25 ds 2 = 1 2GM r dt GM 1 1 2GM 2 r r dr 2 + r 2 dω 2 ds 2 = 1 2GM r dt GM 1 dr 2 + r 2 dω 2 r ds 2 = 1 2GM r dt2 + dr 2 + r 2 dω 2 (5) The light cones do not close up anymore and the metric coefficient does not become infinite at r = 2GM, but the downside is that surface of interest, the event horizon, has been put to infinity. Eddington-Finkelstein built on the tortoise coordinates and came out with coordinates that are naturally adapted to the null geodesic. ν = t + r u = t r Infalling radial null geodesics are characterized by ν = constant and the outgoing ones by u = constant. From ν = t + r it follows t = v r and t 2 = v 2 2vr + r 2 From equation (4) it follows: dr 2 = 1 2GM 2 r dr 2 and to dr = 1 2GM 1 r dr (6) When using the Schwarzschild equation (3) and replacing t 2 by (v 2 2vr + r 2 ), and r 2 by r 2, and for ease we use A = 1 2GM, the result is: r ds 2 = (A) dv 2 2dvdr + dr 2 + (A ) 1 dr 2 + r 2 dω 2 ds 2 = (A)dv 2 + (A)2dv(A) 1 dr (A)(A) 2 dr 2 + (A ) 1 dr 2 + r 2 dω 2 ds 2 = (A)dv 2 + 2dvdr + r 2 dω 2 24

26 Replacing A leads to: ds 2 = 1 2GM r dv2 + 2dvdr + r 2 dω 2 (7) Though the metric coefficient vanishes at r = 2GM there is no problem here as the determinant of the metric is: g = r 4 sin 2 θ In the Eddington-Finkelstein coordinates the condition for radial null curves is solved by: dv 0, (infalling) dr = 2 (1 2GM r ) 1 (outgoing) In this coordinate the system light cones remain well behaved at r = 2GM. Although the light cones do not close up, like the Schwarzschild ones they do till over. This means that for r < 2GM all future-directed paths are in the direction of decreasing r: see figure 10 [7]. Figuur 10: Schwarzschild light comes in the (v, r) coordinates of (equation 7). In these coordinates we can follow future-directed timeline paths past r = 2GM. In the (v, r) coordinate system we can cross the event horizon on future-directed paths, but not on past-directed ones. If we would choose u instead of v a particle would be able to pass through the event horizon, but this time only along past-directed curves. Therefore, a combination of both u and v might solve this. We will now try to get to Kruskal coordinates. 25

27 Continuing with equation (5) and further leaving out dω 2, which does not change under coordinate transformation. ds 2 = 1 2GM r dt2 + dr 2 (8) v = t + r u = t r t = 1 (v + u) and 2 r = 1 (v u) (9) 2 Thus, t 2 + r 2 = 1 4 (v2 + u 2 + 2vu) (v2 + u 2 2vu) = ( 1 4 ( v2 u 2 2vu + v 2 + u 2 2vu) = 1 4 ( 4vu) t 2 + r 2 = vu dt 2 + dr 2 = dvdu (10) Plugging (10) in (8) gives: ds 2 = 1 2GM (dvdu) (11) r Where r is defined implicitly in terms of u and v by: 1 r (v u) = r + 2GM ln 2 2GM 1 For convenience we write: r s = 2GM. Using this gives: Dividing by r s and rearranging: 1 2 (v u) = r + r s ln r r s 1 ln r r s 1 = 1 2r s (v u) r r s r r s 1 = e 1 (v u) r 2r s r s 26

28 r s r. e 1 2r s (v u). e r r s Plugging the left part of (12) in (11) gives: r s = r r 1 = 1 r s (12) r s r ds 2 = r v u r s r. e 2r s. e r s dudv (13) Changing coordinates and rearranging: u u = e2r s and v = e v 2r s ( u v v u = e 2r s ) du = 1 u e2r s du and dv = 2r s 1 e 2r s v 2r s dv Plugging this result in (13) gives: So, du dv = 1 4r s 2 e v u 2r s dudv 2 4r s du dv v u = e 2r s dudv ds 2 = 4r s 3 r r e r s du dv (14) T = 1 2 (u + v ) and R = 1 2 (u v ) Plugging this in (14) gives then: T 2 R 2 = vu and dt 2 dr 2 = du dv ds 2 = 4r s 3 r r e r s (dt 2 dr 2 ) (15) Like the (t, r ) coordinates, the radial null curves look like they do in flat space: T = ± R + constant The event horizon is however not infinitely far away; it is defined by: T = ± R 27

29 If r = constant, than T 2 R 2 = constant. Thus they appear as hyperbolae in the R-T plane. The surfaces of constant t are given by: T R = tanh ( t 4GM ) which defines straight lines through the origin with slope tanh ( t 4GM ). If t ± than T = ± R and therefore t ± represents the same surface as r = 2GM. Range of T, R is such that singularity at r = 0 cannot be reached. The allowed region is therefore: R T 2 < R A spacetime diagram in the T-R plane is known as the Kruskal diagram. See figure 11. [7] Figure 11: Kruskal diagram the Schwarzschild solution in Kruskal coordinates, where all light comes in at ±45. Kruskal divided the diagram in 4 regions. See figure

30 Figure 12: Regions of the Kruskal diagram. 1. Region I: r > 2GM, 2. Following future directed null lines => Region II 3. Following past-direct null lines => Region III 4. Follow space-like geodesics => Region IV Region II is seen as the black hole. Everything going from region I to region II cannot go back. It is even such that when inside the black hole, the particle will keep on moving in the direction of reducing r until the singularity at r = 0 is reached. The region III is actually the reverse of region II. It is named a white hole. Everything leaves it, but nothing can reach it. The boundary of region III is the past event horizon, while the boundary of region II is the future event horizon. Region IV cannot be reached either. It is thought of as being connected to region I by a wormhole (Einstein-Rosen bridge). 2.7 Horizons and the Penrose process In most physical theories we hope to have a well-defined initial value problem, so that information about a state at any one moment of time can be used to predict (or retrodict) the state at any other moment of time. As a consequence, any two states that are connected by a solution to the equations of motion should require the same amount of information to be specified. But in GR, it seems, we take a very complicated collection of matter, collapse it 29

31 into a black hole, and end up with a configuration described completely by mass, charge and angular momentum. In classical GR this might not bother us so much, since the information can be thought of as hidden behind the event horizon rather than truly being lost. But when quantum field theory is taken into account, we find that holes evaporate and eventually disappear, and the information seems to be truly lost. Conceivably, the outgoing Hawking radiation responsible for the evaporation somehow encodes information about what state was originally used to make a black hole, but how that could happen is completely unclear. Understanding this information loss paradox is considered by many to be a crucial step in building a sensible theory of quantum gravity. [7, p239] Event horizons The most important feature of a black hole is the event horizon. An event horizon is a hypersurface separating those spacetime points that are connected to infinity by a timelike path from those that are not [connected to infinity] [7 p239]. Therefore, we first define being at infinity. Infinity could be defined as the spacetime sufficiently far away from the black hole such that it is no longer affected by the black hole. When infinity (past, future null infinity φ ± and spatial infinity i 0 ) can be approximated by Minkowski space, we say that the whole space is asymptotically flat. [7]. See figure 13 and

32 Figure 13: future null infinity φ +, spacelike infinity i 0, and past null infinity φ. [7] Figure 14: A normal Minkowski diagram. A hypersurface can be defined by f(x) = constant for some function f(x). A null hypersurface, such as the event horizon, can be seen as a collection of null geodesics. If we take a piece of paper lying on a flat table, draw a straight line on it, then curve the paper around a sphere for example, the drawn line is a geodesic. It can be defined by the following equation dx μ dx ν ε = g μν dλ dλ Where ε is some constant, if ε = 0 it is a null geodesic and λ will not be fixed. Simply said, a geodesic is a null geodesic if its tangent vector has norm null. For massive particles λ = τ and ε = 1; a timelike geodesic. The gradient of f(x), μ f, is normal to, as in 3d the gradient is a set of points perpendicular to the surface. Another way of establishing whether it is a null hypersurface is checking if the normal vector is null. It also happens to be that the tangent vectors ξ μ to these geodesics are proportional to the normal vectors. 31

33 ξ μ = dxμ dλ = h(x)gμν ν f h(x) can be chosen in such a way that geodesics are affinely parameterized, so the tangent vectors will obey: ξ μ ξ μ = 0, ξ μ μ ξ ν = 0 Cosmic censorship conjecture is defined as: Naked singularity cannot form in gravitational collapse from generic, initially nonsingular states in an asymptotically flat spacetime obeying the dominant energy condition. [7, p243] In other words, a naked singularity is a singularity that is not hidden behind an event horizon. It can also not be formed by the collapse of a star for example. Hawking s area theorem Assuming the weak energy condition and cosmic censorship, the area of future event horizon in an asymptotically flat spacetime is non-decreasing. [7, p243] For Schwarzschild black holes the area depends monotonically on the mass, so this Hawking s area theorem implies that Schwarzschild black holes can only increase in mass. But for rotating black holes this is no longer the case; the area depends on a combination of mass and angular momentum, and we can actually extract energy from a black hole by decreasing its rotation (see Penrose process section). It is also possible to decrease mass of a black hole through quantum-mechanical Hawking radiation as quantum field theory in curved spacetime can violate the weak energy condition Killing horizons 32

34 A killing vector implies that the one of the coordinates is independent of the other coordinates of the metric. If X is the killing vector field then it obeys the killing equation μ X ν + ν X μ = 0 The Schwarzschild metric has 4 killing vectors due to its rotational invariance and time independence. R = φ = y x + x y S = z x x z T = z y + y z K = t A vector can be classified in three groups depending on its tangent vector. If X is the tangent vector and g the metric with signature (-,+,+,+) then g(x, X) < 0 g(x, X) = 0 g(x, X) > 0 Timelike Null Spacelike It can also be written as g ij v i v j > 0 (for timelike). In the Schwarzschild metric, the Killing vector goes from being timelike to spacelike at the event horizon. In general, if a Killing vector field χ μ is null along some null hypersurface, than is a Killing horizon of χ μ. To every Killing horizon we associate a quantity called surface gravity. A Killing vector field χ μ with Killing horizon (where χ μ is a normal vector to ) obeys along the Killing horizon the geodesic equation: χ μ μ χ v = κ χ v The parameter κ is the surface gravity and is constant over the horizon except for a bifurcation two-sphere (e.g. in the center of the Kruskal diagram (figure 11). 33

35 In words the surface gravity could be defined as: In a static, asymptotically flat spacetime, the surface gravity is the acceleration of a static observer near the horizon, as measured by a static observer at infinity [7, p245]. To calculate the surface gravity for a Schwarzschild black hole (with mass M) is, we have κ 2 = 1 2 ( μχ ν )( μ χ ν ) For a static observer it holds that its four-velocity U μ is proportional to the time translation killing field [7]. K μ = V(x)U μ As the four-velocity is normalized to 1, the function V is the magnitude of the killing vector V = K μ K μ This V relates to the frequency of a photon as seen by the static observer, therefore it is also called the redshift factor. Expressing the four-acceleration a μ = U σ σ U μ in terms of the redshift factor gives a μ = μ ln V The magnitude of this acceleration is a = V 1 μ V μ V This acceleration goes to infinity close the event horizon of the black hole, which is an killing horizon, as it takes an infinite acceleration to keep a particle out of the black hole. However, an observer at infinity is never able to see a particle enter a black hole; he only sees the acceleration redshifting by a factor V. Therefore, the surface gravity is κ = Va = μ V μ V evaluated at the killing horizon Σ. Applying this to the Schwarzschild metric gives 34

36 From this it results that K μ = (1,0,0,0) and U μ = [ 1 2GM r 1 2, 0,0,0] V = 1 2GM r and a = GM r 2 1 2GM r 1/2 This then results in the surface gravity being κ = 1 4GM Thus it appears that the surface gravity decreases as the mass increases. In other words, the surface gravity of a big black hole is actually weaker than that of a small black hole Kerr horizons For rotating black holes (stationary) we have to look for a axial symmetry rather than spherical symmetry (static) solution of the Einstein field equations. A solution was found by Kerr with the following metric: Where, ds 2 = 1 2GMr ρ 2 dt2 2GMar sin2 θ ρ 2 (dt dφ + dφdt) + ρ2 dr2 + ρ 2 dθ 2 + sin2 θ ρ 2 [(r 2 + a 2 ) 2 a 2 sin 2 θ]dφ 2 (16) (r) = r 2 2GMr + a 2 (17a) And ρ 2 (r, θ) = r 2 + a 2 cos 2 θ (17b) 35

37 As the Kerr-solution is stationary but not static, the event horizons at r ± are not Killing horizons. Figure 15 shows the horizon structure of the Kerr metric. [7] Figure 15: Horizon structure of the Kerr metric (side view). The event horizons are null surfaces that demarcate points past which it becomes impossible to return to a certain region of space. The stationary limit surface, in contrast, is timelike except where it is tangent to the event horizon (at the poles); it represents the place past which it is impossible to be a stationary observer. This stationary limit surface is the place where K μ K μ = 0. The ergosphere is the area between the stationary limit surface and the outer event horizon. It is a region is where it is possible to enter and leave again, but not to remain stationary Penrose process Energy can be extracted from a black hole itself, if the hole is spinning or charged, by classical processes. [6] Note: If quantum effects are included, then it turns out that one can even extract energy from a non-rotating, neutral black hole, either by letting it evaporate via Hawking radiation or by mining it [6]. 36

38 The Penrose process is related to the fact that if a particle with negative energy is inside the ergosphere it must stay inside the ergosphere or be accelerated until its energy is positive in order to escape. The reason is that in the outside of the ergosphere a particle must have positive energy, while this is not necessary inside the ergosphere. In regular space both the Killing vector K μ and the four momentum P μ are timelike. However, K μ becomes spacelike inside the ergosphere. It is therefore possible for particles to obey: E = K μ P μ < 0 (18) As P μ = mu μ and U = (γc, γv x x, γv y, γv z ) it follows that P 0 = E I will illustrate this with an example. If there is a person with rock moving along a geodesic their momentum is P (0)μ, which is positive and conserved as you move along your geodesic. When you enter the ergosphere and then throw the rock such that you (yourself) will start to move on a geodesic leaving the ergosphere. When the rock is thrown very hard and against the direction of rotation, the energy of the rock becomes negative. The fact that the rock has to be thrown against the direction of rotation will be shown later on. As momentum is conserved it follows that P (0)μ = P (1)μ + P (2)μ with P (1)μ your own momentum and P (2)μ the momentum of the rock after the throw. Using E = K μ P (0)μ gives: E (0) = E (1) + E (2). If E (2) < 0 than E (1) > E (0), which means that you have gained energy. This gained energy is extracted from the rotational energy of the black hole (thus decreasing the angular momentum of the black hole). For Kerr killing horizon is a linear combination of the timetranslation and rotational Killing vectors and is represented by: χ μ = K μ + Ω H R μ (19) 37

39 Ω H is the angular velocity of the horizon. The statement that a particle with momentum P (2)μ crossing the horizon is: P (2)μ χ μ < 0 (20) The angular momentum is given by: L = R μ P μ (21) When plugging (19) in (20) and using (18) and (21) the result is: P (2)μ (K μ + Ω H R μ ) < 0 P (2)μ K μ + P (2)μ Ω H R μ < 0 E (2) + L (2) Ω H ) < 0 Thus, L (2) < E(2) Ω H (22) As E (2) was arranged to be negative and Ω H positive the particle must have a negative angular momentum and is thus thrown against the rotation direction. Also note that L (2) becomes a limit on how much you can decrease the angular momentum. Once you have escaped the ergosphere and the rock has fallen inside the event horizon, the mass and angular momentum of the black hole are as before plus the negative contribution of the rock: δm = E (2) δj = L (2) (23a) (23b) So, both M and J are limited. Rewriting (23a) and using (22) and (23b) gives: 38

40 δj < δm Ω H When we reach the limit it gives δj = δm/ω H. Taking into account the area theorem (the area of the event horizon is non-decreasing) we can calculate the outer event horizon at, where a = J/M: r + = GM + G 2 M 2 a 2 Using (r) = r 2 2GMr + a 2 and r = r +, this gives that (r + ) = G 2 M 2 + G 2 M 2 a 2 + 2GM G 2 M 2 a 2 2GM GM + G 2 M 2 a 2 + a 2 = 2G 2 M 2 2G 2 M 2 a 2 + a 2 + 2GM + 2GM G 2 M 2 a 2 2GM GM + G 2 M 2 a 2 = 0 If we use the Kerr metric (16) and apply = 0 (cause r = r + ) and dt = 0, dr = 0 we get: ds 2 = ρ 2 dθ 2 + sin2 θ ρ 2 [(r 2 + a 2 ) 2 ]dφ 2 (24) As ρ 2 (r, θ) = r 2 + a 2 cos 2 θ it follows that ρ 2 (r +, θ) = r a 2 cos 2 θ. When plugging in ρ 2 (r +, θ) in (24) the results is: ds 2 = (r a 2 cos 2 θ)dθ 2 + sin2 θ (r a 2 ) 2 r a 2 cos 2 θ dφ2 (25) The determinant is: γ = (r a 2 ) 2 sin 2 θ and can be proved as follows: ds 2 = Edθ + 2Fdφdθ + Gdφ ds 2 = [dφdθ] E F F dθ G dφ The determinant is EG F 2. Since F = 0 it leaves EG. So in (25) the terms for E and G are: 39

41 As γ = E. G it follows: E = (r a 2 cos 2 θ) and G = sin2 θ (r a 2 ) 2 r a 2 cos 2 θ γ = (r a 2 ) 2 sin 2 θ γ = (r a 2 ) sin θ (26a) (26b) To calculate the horizon area we apply: 2π π A = γ 0 0 dθdφ (27) Plugging in (26b) in (27) and solving the first integral first leads to: π (r a 2 ) sin θ 0 Solving the second integral leads to the horizon area A: = (r a 2 ) ( (r a 2 )) = 2(r a 2 ) 2π A = 2 (r a 2 ) dφ 0 A = 4π (r a 2 ) (28) To show that the area does not decrease, it is convenient to work instead in terms of the irreducible mass of the black hole, defined by: 2 M irr = A/16πG 2 A = 16πG 2 2 M irr δm irr = δa = 32πG 2 M irr δm irr δa 32πG 2 M irr (29) Proceeding with (28) and using a = J/M and r + = GM + G 2 M 2 a 2, gives: A = 4π (G 2 M 2 + G 2 M 2 a 2 + 2GM G 2 M 2 a 2 + a 2 ) 40

42 = 4π 2G 2 M 2 + 2GM G 2 M 2 a 2 = 4π 2G 2 M 2 + 2G G 2 M 4 M 2 J2 M 2 = 8πG GM 2 + G G 2 M 4 J 2 Differentiating leads to: 1 δa = 8πG 2GMδM + 2 G 2 M 4 J. 2 (4G2 M 3 δm 2JδJ) 1 = 8πG 2GM + 2 G 2 M 4 a 2 M. 2 4G2 M 3 1 δm 8πG 2J. δj 2 G 2 M 4 a 2 M2 = 8πG 2GM + 2G2 M 2 G 2 M 2 a. δm 8πG J 2 M. 1 δj G 2 M 2 a2 = 8πG 2GM + 2G2 M 2 a. δm δj (30) G 2 M 2 a2 G 2 M 2 a2 Ω H has been defined as: Ω H = dφ dt (r +) = So, plugging in r + = GM + G 2 M 2 a 2, leads to: Ω H = a r a 2 a and Ω 1 2G 2 M 2 + 2GM G 2 M 2 a 2 H = 2G2 M 2 + 2GM G 2 M 2 a 2 a Continuing with (30) and applying a Ω G 2 M 2 a 2 H 1 = 2GM + 2G2 M 2 gives: G 2 M 2 a 2 δa = 8πG Ω 1 a a H. δm δj G 2 M 2 a2 G 2 M 2 a2 41

43 δa = 8πG a G 2 M 2 a (Ω 2 H 1 δm δj) (31) Plug (31) in (29): δm irr = 1 32πG 2 M irr 8πG δm irr = a G 2 M 2 a 2 (Ω H 1 δm δj) a 4GM irr G 2 M 2 a 2 (Ω H 1 δm δj) Use X = a 4GM irr G 2 M 2 a 2 to simplify: 1 X δm irr = Ω H 1 δm δj Ω H 1 δm = δj + 1 X δm irr We earlier found: δj < δm Ω H and thus: δj < δj + 1 X δm irr This leads to: δm irr > 0 As the area is related to the irreducible mass, also the area has to increase under all processes. 42

44 3. Entropy of a black hole 3.1 Four laws of black holes The four laws of Black Hole thermodynamics are subsequently: Zeroth law: For stationary black holes, the surface gravity is constant on the horizon. This law is not just true for spherically symmetric non-rotating black holes, but also for rotating black holes that are not spherically symmetric [9] First law: For a rotating charged black hole, the First Law takes the form [6]: dm = κ da + ΩdJ + ΦdQ 8πG Second law: The area of the event horizon always increases [10]. Third law: The surface gravity of the horizon cannot be reduced to zero in a finite number of steps, but it does not always hold for extremal black holes [6]. In thermodynamics, the zeroth law states that temperature is constant in a body in thermal equilibrium. As stationary is comparable to equilibrium and if κ is like a temperature both laws are nearly identical. The same holds for the first law. In thermodynamics this is de = TdS PdV. We know that E = mc 2, thus saying that m is similar to E makes perfect sense. As we said κ is like a temperature, which means that A should be like entropy. That A is like entropy is strengthened by the second law as both are to always increase. 43

45 However, saying that a black hole would have something as a temperature, meaning that it would radiate, was initially considered strange as nothing can come out of a black hole [9]. In 1975 Hawking showed that a black hole does radiate black body radiation with a certain temperature. We will come back on that later. There is more overlap. Most processes increase the irreducible mass of the black hole, however only a few processes leave the irreducible mass unchanged. These are the reversible processes. This is the same in thermodynamics, where only processes that leave the entropy unchanged are reversible. As we have seen A = 16πM ir,which again shows the connection between thermodynamic entropy and the area of a black hole [10]. Entropy is also related to degradation of energy, as degradation of the quality of energy leads to an increase of entropy. Irreducible mass can be viewed in the same light, as it represents that mass from which no more work can be extracted, in other words the lowest quality of energy. The generalised version of the second law was defined by Bekenstein [10, p7] and says: When common entropy goes down a black hole, the common entropy in the black hole exterior plus the black hole entropy never decreases. This statement means that we must regard black-hole entropy as a genuine contribution to the entropy content of the universe. 3.2 Calculating entropy From an information point of view, the entropy of a black hole can be seen as the inaccessibility of information about the internal configuration of the black hole. For instance, there could be three black holes, which can be described with only three factors (M,Q,J) according to the no-hair theorem. Those values could be the same for all three black holes, but one could be born form a star collapsing, another from the collapse of a neutron star and 44