Bremen notes on black hole physics

Transcription

1 Bremen notes on black hole physics Martin Scholtz and Norman Gürlebeck May 16, 2017

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3 Contents 1 Introduction Classical notion: Escape velocity Evidence for their existence Supermassive black holes Special relativity Events Spacetime interval Minkowski spacetime Causal structure Differential geometry Topological spaces Manifolds Curves Vector fields Covector fields Direct product and direct sum Tensor fields Metric tensor Connection Geodesics Curvature Lie derivative Pull-backs and push-forwards Killing equations Unit sphere General relativity 51 5 Schwarzschild spacetime Derivation of the Schwarzschild metric Thermodynamics of black holes and singularities Causal structure Hawking area theorem

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5 Chapter 1 Introduction In the end of the life of stars, when the fuel which sustains the pressure resisting the gravitational pull is depleted, stars collapse according to their mass to either White dwarfs: M < 1.4M Neutron stars: 1.4M M 3M Black Holes: M > 3M The mass of the Sun is M kg White dwarfs are sustained by the pressure by a totally degenerate electron gas. Its radius is R 10 4 km and its mass is M M. Neutron stars are supported by the pressure of a totally degenerate neutron gas with a radius of 10 km and a mass of (1 2)M. Black holes can have many different masses: stellar black holes: 1 M < M < 100 M intermediate mass black holes: 100 M < M < 10 6 M supermassive black holes: 10 6 M < M Theoretically, the mass is not restricted. In fact, mini-black holes might exist too: primordial black holes, big bang, particle accelerators. Many of this lectures results also applicable to these. All three were treated in the lecture Relativistic Astrophysics last semerster. Here we study the physical implications and mathematical results regarding black holes in detail. 1.1 Classical notion: Escape velocity The escape velocity for a spherically symmetric compact object (mass M, radius R) is the minimal velocity necessary for a particle (mass m) initially located at the surface to reach infinity. picture E kin (R) = E pot (r = ) E pot (R) 1 ( 2 m v esc = m 0 + M G ) R 2 M G v esc = R 5

6 Newtonian gravitational constant G = m 3 kg 1 s 2 and speed of light c = m s 1. The condition v esc c now implies 1 2 M G R c 2. Equality is reached for the Schwarzschild radius R S = 2 M G c 2. First discovered by J. Mitchell: 1783 (1.1) (1.2) P.S Laplace: 1796 Schwarzschild radius for the Sun is R S (M ) 3 km. After Einsteins discovery of General Relativity (1915), a solution to the Einstein equations was found describing such a black hole (Schwarzschild: 1915). That the radius comes out coincidentally exactly as in this classical argument. The term black hole was coined by J.A. Wheeler, as the tale goes, in Evidence for their existence Starting out as a mathematical oddity only completely understood in the end of the 1950s to the end of the There were found to be a general feature of general relativity if the matter behaves sufficiently nice (energy conditions). Stellar black holes Stellar black holes, which originate via the collapse of stars are observed frequently. Gravitational waves picture The most recent observation is via gravitational waves last year. The first event GW less than a second and is explained by two merging black holes of M 36M and M 29M with an energy output of 3M ; the event happened ly away 1. We use most times in the lecture geometric units where the time is measured in the terms of the distance light travels in that time, i.e. in m(eters), which yields c = 1. The mass is measured via its Schwarzschildradius, i.e. in m(eters) as well, which implies G = 1. We restore these physical constants only in cases where we take limits to classical situations. X-rays A bright X-ray source Cygnus X-1 was observed in The radiation varies on a bout a 1 s timescale over the entire spectrum. This is only possible if they are causally related which limits the size of the emission region to less than 200, 000 km. Only in 1970s the angular resolution improved to localize it on the sky and find another source in the radio band associated with an blue giant O-star. The spectral lines vary with a period of 5.5 days (in frequency-doppler effect, in intensity - deformation), which led to the expectation that the two form a binary system, where the X-ray emission is produced by the infall of matter from the star into the black hole. If this accreted matter would hit a solid surface, the X-ray spectrum should have a sharp cut-off. However, it dies slowly in accordance with the interpretation of an event horizon. Altogether masses of 15M of the black hole and 18M for the blue star are implied. Still other explanations are possible but more hypothetical. There are other stellar black holes observed ( 20). 1 One light year is approximately 1ly m. 6

7 1.3 Supermassive black holes Motion of stars picture The motion of stars around our galactic center implies a central mass near that center in the order of M. It is also visible in the X-ray and infrared spectrum. These stars are about 1000 R S (Sgr A*) away from the mass, which implies that Newtonian theory is already a very good theory to explain it. Such super-massive black holes are expected in many other galactic centers if not all. Shadows Another possibilty coming up this month is the observation of the shadow of Sgr A*. picture interstellar picture drawing of the idea Imagine there are light sources distributed every where but not between us and the black hole. We as observers revceive light from many different directions, thou if we trace the light races back there will be a dark region in the sky the shadow. The angular resolution of the Event Horizon Telescope, a very long baseline ground based observatory in the far infrared, which should provide the first pictures this month should suffice to see the shadow of Sgr A*. 7

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9 Chapter 2 Special relativity Einstein s special theory of relativity (SR) is based on the two famous postulates: 1. Principle of relativity. Laws of physics must acquire the same mathematical form in all inertial frames of reference. 2. Invariance of the speed of light. The speed of the light propagation (in vacuum) does not depend on the choice of the frame of reference. To be more precise, the first postulate guarantees that there exists an inertial frame S, that is, a frame in which the law of inertia holds. Any other frame S which is moving with constant speed with respect to S is inertial too. For the description of the physical phenomena, both frames are perfectly equivalent, no one is preferred and the physical laws are the same for both. The second postulate asserts there exists a speed c which is invariant: even tough the frames S and S are in mutual motion, they will agree on the speed of any light signal. The value of this invariant speed c, of course, depends on the choice of the units. In SI system of units, the speed of light is by definition c = m s 1. (2.1) For practical purposes of theoretical physicist, though, it is often convenient to set c = 1, as explained in the introduction. 2.1 Events. Einstein s postulates can be reformulated in a geometrical way which turned out to be crucial for the formulation of general relativity. By an event we mean a specific place in the space at some specific time. Hence, when we talk about the events, we do not mean what happened, just where and when it happened. In order to specify the event, we have to specify i) the observer who witnesses the event and ii) coordinates of the event, i.e. one temporal coordinate t and three spatial coordinates x, y and z, as measured by the specified observer. We will abbreviate the four-tuple (t, x, y, z) by the symbol x µ, where index µ = 0, 1, 2, 3 and, by definition, x 0 = t, x 1 = x, x 2 = y, x 3 = z. (2.2) In order to emphasize that the four coordinates x µ are assigned to a specific event A, we will sometimes write x µ (A) = (t A, x A, y A, z A ). Analogously, another event B will have coordinates x µ (B). In general, different observers will assign different coordinates to the same event. For example, the event A will have coordinates x µ (A) in the frame of reference of the observer S, while the same event A will have coordinates x µ (A) in the frame S. Of course, the question is, how are coordinates x µ and x µ related, provided that we know the relative state of motion of the two observers. 9

10 Event A is, in the given frame S, uniquely characterized by the four-tuple of coordinates x µ (A) = (t A, x A, y A, z A ), µ = 0, 1, 2, 3. (2.3) The same event will have different coordinates x µ (A) in a different frame S. Spatial part of the 4-tuple x µ will be denoted by x i (A) = (x A, y A, z A ), i = 1, 2, 3. (2.4) When no confusion arises, we omit the argument A and write simply x µ for the coordinates of an event in one frame and x µ for the coordinates of the same event in a different frame. 2.2 Spacetime interval. Consider two infinitesimally close events A and B with the coordinates x µ (A) = (t, x, y, z), x µ (B) = x µ (A) + dx µ = (t + dt, x + dx, y + dy, z + dz). (2.5) Let A be an emission of a light signal at time t at the position with coordinates x i. Infinitesimally later, at time t + dt, the signal propagates to the space position with the coordinates x i + dx i. The spatial distance between the two points is dr = dx 2 + dy 2 + dz 2. (2.6) By the second postulate, speed of light is c in every frame and hence dr = c dt. (2.7) These two relations imply c 2 dt 2 + dx 2 + dy 2 + dz 2 = 0. (2.8) However, by the first postulate, any other inertial frame is equally good for the description of the same to events A and B. An observer in such a frame will use different set of coordinates x µ, but will derive the same relation c 2 dt 2 + dx 2 + dy 2 + dz 2 = 0; (2.9) notice that the speed of light c is the same by the second postulate. Remarkable feature of equations (2.8) and (2.9) is that, although the different observers assign different coordinates to the same pair of events, there is a specific combination of the coordinates (or the differentials of the coordinates) which has the same value for both observers: it vanishes in both frames. More precisely, we have shown that if the events A and B are connected by the light propagation, the quantity (2.8) vanishes for any observer. The key observation by Minkowski is that quantity (2.8) is non-zero in general, but it has always the same value for each observer. Now, let A and B be arbitrary, but infinitesimally close events, so that their coordinates differ by dx µ as before. Let us define the quantity called spacetime interval by ds 2 = c 2 dt 2 + dx 2 + dy 2 + dz 2. (2.10) In general, different observer will assign a different spacetime interval to the same pair of the events, ds 2 = c 2 dt 2 + dx 2 + dy 2 + dz 2. (2.11) 10

11 We have already shown that if A and B are connected by a light signal, ds 2 = ds 2 = 0. If they are not connected by a light signal, the spacetime interval will be non-zero. Using the fact that ds 2 and ds 2 are both infinitesimal and they always vanish simultaneously, we can write ds 2 = f ds 2, (2.12) where f is a function to be found. First, the space and time are assumed to be homogeneous, so that all spatial points and all instants of time are equivalent. Then, function f can depend neither on the spatial position nor on the time; it can depend on the relative velocity of the two frames only. Second, the space is assumed to be isotropic, so that all spatial directions are equivalent. Then, function f cannot depend on the direction of the relative motion o the two frames, it can depend only on the magnitude of the velocity, i.e. on the speed. With these considerations we can write the relation between the spacetime intervals in the two frames in the form ds 2 = f( v ) ds 2, (2.13) where v is the relative velocity of the frames. Now, consider three frames S 1, S 2 and S 3, and let v ij be the relative velocity of frames S i and S j. Then we have (ds 2 ) 2 = f( v 12 ) (ds 2 ) 1, (ds 2 ) 3 = f( v 23 ) (ds 2 ) 2, (ds 2 ) 3 = f( v 13 ) (ds 2 ) 1, (2.14) which implies f( v 12 ) f( v 23 ) = f( v 13 ). (2.15) This relation must hold for any relative motion of the respective frames. In particular, suppose that S 3 is in fact at rest with respect to S 1. Then v 12 = v 23 =: v and v 13 = 0. (2.16) We also have f(0) = 1, because if the frames coincide, also the associated spacetime intervals must be the same. Hence, condition (2.15) reduces to f(v) 2 = 1 (2.17) which implies 1 f(v) = 1. (2.18) We have shown that function f(v) which appears in (2.13) is in fact a constant function 1, which means that for arbitrary pair of (infinitesimally close) events, the spacetime interval acquires the same numerical value in all inertial frames. Although the coordinates of the events are different (which corresponds to different spatial and temporal intervals between the two events), the specific combination of the coordinates is the same for each observer. This observation is at the hearth of both special and general relativity. The spacetime interval between the two infinitesimally close events A (with coordinates x µ ) and B (with coordinates x µ + dx µ ) is defined by ds 2 = c 2 dt 2 + dx 2 + dy 2 + dz 2 (2.19) and it is independent of the choice of the coordinates (i.e. the choice of the observer). 1 What about the case f(v) = 1? 11

12 Exercise 1 Consider two frames S[x µ ] and S [x µ ]. Let S move uniformly with the velocity v along the x-axis of the frame S; suppose that at time t = 0 both frames had the same origin. Show that coordinates of an arbitrary event A with respect to S and S are related by the Lorentz transformation t = t v x c 2, x = x v t, y = y, z = z. (2.20) 1 v2 c 2 1 v2 c 2 Hint. i) Write down the general linear transformation between dt, dx and dt, dx with undetermined coefficients. ii) Express the spacetime interval (2.19)in both coordinate systems and by comparing the two expressions, find the conditions on the coefficients. iii) Use the identity cosh 2 α sinh 2 α = 1 to parametrize the coefficients which identically satisfy the conditions from ii). iv) Consider a point which is at rest in the frame S, i.e. point with dx = 0, dx = v dt. Relate the parameter α from iii) to the velocity v. v) Express the transformation (dt, dx) (dt, dx ) in terms of the velocity, and finally integrate to get (2.20). Exercise 2 Check that transformation inverse to (2.20) is t = t + v x c 2 1 v2 c 2, x = x + v t, (2.21) 1 v2 c 2 which corresponds to replacement t t, x x, v v. Exercise 3 Derive three elementary special-relativistic effects: dilation of time, t = contraction of length, 1 v2 t; (2.22) c2 x x = ; (2.23) 1 v2 c 2 addition theorem for velocities, v 13 = v 12 + v v 12 v 23, c 2 which replaces the classical law v 13 = v 12 + v 23. (2.24) 2.3 Minkowski spacetime. The invariance of the spacetime interval (2.19) is in many respects similar to the invariance of the line element in the Euclidean space. For example, in two dimensions, distance between two infinitesimally close points is given by the Pythagorean theorem dr 2 = dx 2 + dy 2. (2.25) 12

13 Exercise 4 Define the coordinates x and y by x = x cos θ + y sin θ, y = x sin θ + y cos θ, (2.26) where ( ) cos θ sin θ R = sin θ cos θ is the rotation matrix in (x, y) plane by angle θ. Check the invariance of (2.25), i.e. check that dx 2 + dy 2 = dx 2 + dy 2. (2.27) (2.28) The Cartesian coordinates (x, y) in the Euclidean plane are not unique. In particular, we can rotate the Cartesian coordinate system, and the rotated system is perfectly equivalent to the previous one. Hence, the values of the coordinates are not so important: a given point will have different coordinates in different coordinate systems. However, there are invariant geometrical properties, like the distance between the points, which do not depend on the (arbitrary) choice of the coordinates. Now it is obvious that the description of points in the Euclidean space is in many respects similar to description of events by different inertial observers. Events are described by the spatial and temporal coordinates. Observers see the same events, but they describe them by different coordinates. Nevertheless, there is some specific combination of the coordinates called spacetime interval given by (2.19), which is independent of particular coordinate system and thus has the same value for all observers. This shows that there is some underlying geometrical structure. Spatial distances and time intervals depend on the observer, but the spacetime interval does not. For an analogy in the Euclidean plane, consider the point P with the coordinates (0, 1) (i.e. x = 0, y = 1). The origin O has coordinates (0, 0). So, the x-interval between O and P is x = 0, because both points have the same x coordinate. On the other hand, the y-interval between the two points is y = 1 0 = 1. Of course, the distance is r 2 = x 2 + y 2 = 1. Now, rotate the coordinate system by angle π/2 in the positive sense, so that in the new (x, y ) coordinates point P has coordinates x = 1, y = 0. Now, the x interval between O and P is x = 1 and similarly y = 0. We see that the coordinate intervals depend on the choice of the coordinate system and have no geometrical meaning. But also in the transformed coordinate system, the distance between the two points is r 2 = x 2 + y 2 = 1. It is the distance between the two points, which is invariant, i.e. coordinate independent. In the same spirit, we interpret all possible events as points of some abstract set called spacetime or Minkowski spacetime; we denote it by M. Description of the events in the reference frames of different observers is equivalent to choosing different coordinates on M. The distance between two points in M is given by the spacetime interval and it is independent of the choice of the coordinates. Minkowski spacetime M is an abstract set of events. An inertial observer describes these events in terms of the (spacetime) coordinates x µ = (t, x, y, z). Distance between two infinitesimally close events is given by the spacetime interval ds 2 = c 2 dt 2 + dx 2 + dy 2 + dz 2. Imagine a point particle moving in an arbitrary way, i.e. with arbitrary velocity (not necessarily uniformly). Let us consider arbitrary inertial observer whose clocks measure time t and who describes the position of such particle by the three spatial coordinates x = x(t), y = y(t), z = z(t), or, more concisely, x i = x i (t) (recall Eq. (2.4)). In the space, the position vector defines the spatial trajectory, which is a mapping t (x(t), y(t), z(t)). (2.29) Similarly, in the spacetime, the motion of the particle can be described by the 4-dimensional curve called worldline, which is a mapping t x µ (t). (2.30) 13

14 For example, the (spatial) trajectory of a particle which is at rest in a given frame, is simply a point, because t x i (t) is a constant function. The worldline of such particle is, on the other hand, a line in the spacetime M; its spatial coordinates remain constant, but the time coordinate varies. 2.4 Causal structure. The spacetime interval resembles the Pythagorean theorem, but the presence of negative term has important physical consequences 2. First, consider an observer moving uniformly with arbitrary speed. This observer is at rest with respect to his/her own frame. As the time passes, the origin O of the frame does not move, so that dx = dy = dz = 0, or, in the notation (2.4), dx i = 0. The only varying coordinate of the origin O is time and hence the spacetime interval reduces to ds 2 = c 2 dt 2. (2.31) In this situation, dt is equal to the time elapsed on the clocks of the observer we are talking about. In other words, dt is time measured by the clocks which are not moving with respect to the observer; for this reason we say that such clocks measure the proper time of the observer and usually we denote the proper time by dτ, so that the spacetime interval ds 2 = c 2 dτ 2. (2.32) In this way we have found the physical interpretation of the spacetime interval. Up to factor c 2, the spacetime interval ds 2 is equal to the proper time elapsed on the clocks which travel from event A to (infinitesimally close) event B. 2 Sometimes it is useful to define the spacetime interval as ds 2 = c 2 dt 2 dx 2 dy 2 dz 2. In that case, temporal term is positive and spatial terms are negative. This is not important, tough. Physically important thing is that time has opposite sign than remaining spatial terms and the overall sign is just a matter of convention. 14

15 Chapter 3 Differential geometry In general relativity we need to study spacetimes more general than the Minkowski spacetime introduced in section 2. We want to take into account that different observers can describe the events in different coordinate systems, but all such coordinate systems are equivalent and physical laws must have the same form in all coordinate systems; this is called principle of general covariance. Second, we need to consider spacetimes with more general metrics than the Minkowski metric; such spacetimes are called curved, because their geometry resembles the geometry of curved surfaces in Euclidean space. For us, the most basic mathematical structure behind these considerations is a manifold. Step by step, we will equip a general manifold with other geometrical structures: metric tensor and connection, which give rise to the notion of a curvature of the manifold. After we build the mathematical formalism, we will postulate the equations which relate the geometry of a spacetime to its matter content. 3.1 Topological spaces. Roughly speaking, manifold is set of points on which we can introduce coordinate systems. The definition of the manifold ensures that these coordinate systems are, in a specific sense, compatible, and therefore it does not matter which particular system we use for the description of the properties of the manifold. Once we define the manifold, we will use the convenient coordinate systems to study the properties of a manifold, but at the end we will be mainly interested in the properties which are coordinate independent. First we have to introduce the notion of topological space. This concept generalizes the notion of the open set familiar from elementary mathematics. Consider, for example, an open ball of radius r around a point P in the space R n, which is a set O P (r) = {Q R n P Q < r}, (3.1) i.e. it is a set of such points Q whose distance from P is (strictly) less than r. Clearly, the boundary of O P (r) is the sphere of radius r centered at P, but no point of the ball actually lies on this boundary. In this sense, the set O P (r) is open and intuitively it is very clear what it means for a set to be open. There is a problem, however, that in order to define the open set we already need the notion of a distance P Q. That is fine in the Euclidean space, where we have a well-defined notion of distance given by the Pythagorean theorem, but on a general set there is a priori no such notion. Thus, we have to reformulate the definition of an open set in such a way which does not rely on the distance, nor on any other more advanced notion. Instead of constructive definition, we have to analyze which properties of Euclidean open sets are important (compared to non-open sets). These properties will then define the open set in general. In other words, we have to consider the openness as a separate structure. 15

16 Topological space is a set X equipped with the topology τ, where a τ 2 X is a collection of subsets (called open sets) of X satisfying the following axioms: 1. both empty set and entire set X are open sets, τ, X τ; (3.2a) 2. any countable union of open sets is an open set, U n τ, if n N : U n τ, (3.2b) n where the labeling set N is either finite or coincides with the set of natural numbers; 3. any finite intersection of open sets is an open set, U n τ, if n N : U n τ, (3.2c) n where the labeling set N is finite. a By symbol 2 X we mean the set of all subsets of X. The reader is invited to check that usual open sets in R n satisfy all these axioms. The reason why we need only finite number of sets in the point (3.2c) is the following: imagine a sequence of open sets U n = ( 1 n, 1 n ) for n N. The union of all these sets is an interval ( 1, 1) which is an open set, but the intersection is a single point {0}. We do not want to consider an isolated point as an open set. For a finite number of sets U n, their intersection will remain an open set. We can see that if we are interested in the properties of an abstract set X, we cannot a priori say which subsets of X are open. We have to provide an additional structure, in this case the topology τ, which is simply a collection of open sets. In other words, we have to enumerate which sets are open and be careful that these sets satisfy the required axioms. Hence, the topological space, i.e. space in which the notion of open set exists, is not simply the set X, but a pair (X, τ), that is the underlying set together with the topology. In differential geometry, we use various mappings all the time, but we require them to be sufficiently well-behaved. For example, a continuity is again a term which, in standard calculus, relies on the notion of distance between the real numbers (because the absolute value enters the definition). In topology we have to reformulate the notion of continuity which could apply to topological spaces. A mapping f : X Y (3.3) between the two topological spaces (X, τ) and (Y, σ) is called continuous, if V σ : f 1 (V ) τ, (3.4) i.e. if the preimage f 1 (V ) of any set V which is open in Y is an open set in X. One can check again that this definition of continuity agrees with the usual definition for functions of real variables. At this stage, however, we cannot define the notion of differentiable mapping. In topology we usually study properties of the objects which are not related to the metric (i.e. lengths, angles, areas) but properties which do not change under continuous deformations. For example, sphere can be continuously deformed to a cube and therefore the sphere and the cube are considered as the same in the 16

17 topological sense. On the other hand, sphere is not topologically equivalent to a torus, because the creation of the hole (sphere + hole = torus) requires a discontinuous transformation. Mathematically, mappings which realize such continuous deformations are called homeomorphisms 1. Mapping f : X Y between the two topological spaces is called homeomorphism if it is continuous, bijective, and has a continuous inverse. Topological spaces are said to be homeomorphic, if there exists a homeomorphism between them. By a neighborhood of point P X of a topological space (X, τ) we mean any open set U τ containing P, i.e. P U. Topological space is called Hausdorff, if any two points P, Q X have non-intersecting neighborhoods, i.e. if for any such P and Q there exist neighborhoods U P and V Q such that U V =. In these lectures we consider only the Hausdorff spaces. Let us see an artificial example how a non- Hausdorff space can emerge. Consider two copies of real line and denote them by R 1 and R 2. Both sets contain exactly the same elements. Now, consider they disjoint union L = R 1 R2. (3.5) Disjoint union means that the elements of R 1 will be labeled by 1, i.e. (x, 1) R 1, while (x, 2) R 2 ; elements (x, 1) and (x, 2) are taken as distinct entities in the disjoint union. So, the space L contains 2 copies of the real line. Now we define an equivalence relation by (x, 1) (x, 2) for any x 0. (3.6) With this equivalence relation we can form a quotient space L = L / (3.7) That is, any nonzero real number x in R 1 will be identified with the same number in R 2, but we still have two copies of zero, (0, 1) and (0, 2). Now it is clear that L is a non-hausdorff space: any sequence converging to (0, 1) will converge to (0, 2) as well. Any neighborhood of (0, 1) is at the same time also a neighborhood of (0, 2) and there do not exist two disjoint neighborhoods of (0, 1) and (0, 2). We will not consider non-hausdorff spaces anymore. 3.2 Manifolds. Having introduced the notion of topological space, we can say whether a given subset is an open set or not. This gives us the topological notion of neighborhoods and continuity. In the next step we would like to introduce the coordinate systems on topological spaces. There are essentially two issues to be addressed. First, we have to specify what we mean by a coordinate system and what criteria it should satisfy. We will recognize that sometimes (by sometimes we mean usually) the coordinate system cannot cover entire topological space. Consequently, in order to cover entire space we have to introduce several coordinate systems which will work on different but overlapping regions. This raises the problem of compatibility of different coordinates on the overlaps. It is worth to emphasize that these are not just wild mathematical considerations. We will see that in order to understand the physics of a black hole horizon we have to carefully analyze different coordinate systems and the regions of their validity. Let us briefly discuss the obstacles in defining a single coordinate system for the entire space. Cartesian coordinates in the Euclidean space are global coordinates, because they cover the full space. Moreover, they satisfy the two conditions, 1 More precisely, for a continuous deformation one has to consider a family of homeomorphisms. 17

18 1. continuity, 2. bijectivity. The continuity means that two close points have similar coordinates, while the bijectivity means that each point of the space has a unique n tuple of coordinates and vice versa. In general case, all the three requirements (global character, continuity and bijectivity) cannot be simultaneously satisfied. As a trivial example, consider the sphere S 1, i.e. the unit circle, in the Euclidean plane. S 1 P O θ O Natural coordinate of the point P on S 1 is the angle θ between lines OO and OP, where O is some fixed point. The angle θ is certainly a global coordinate. There is an ambiguity in the angle, however, for the angle θ+2π describes the same point P. We can either say that angle θ acquires arbitrary real values, θ (, ), or restrict θ to the interval [0, 2π). Clearly, the first choice violates the bijectivity requirement, because one point P can correspond to infinitely many values of θ differing by an integer multiple of 2π. The second choice, on the other hand, violates the continuity requirement. Indeed, when point P approaches and crosses the point O from above, coordinate θ suddenly changes the value from 0 to a number close to 2π. No choice of the coordinate on S 1 will satisfy all the three requirements simultaneously. If we insist that the well-behaved coordinates should be bijective and continuous, we have to relax the globality condition. Instead of a single angle θ, let us define three angles α, β and γ like in the following figure. α = 5 6 π β = 0 U 1 α = 0 β = 5 6 π U 2 γ = 0 U 3 γ = 5 6 π In terms of the original angle θ (regarded as arbitrary real number), we define the three regions, { U 1 = 5 12 π < θ < 5 } { 1 12 π, U 2 = 3 π < θ < 7 } { 13 6 π, U 3 = 12 π < θ < 23 } 12 π. (3.8) Clearly, these regions together cover entire S 1 and each U i has nonzero overlap with the other two. Exercise 5 Express angles α, β and γ in terms of θ. Consider mapping ψ α : U 1 α which assigns a value of coordinate α to any point of S 1. What is the image of ψ α, i.e. what is the set V 1 = ψ α (U 1 )?. Show that ψ α is homeomorphism (and similarly ψ β, ψ γ ). What is the meaning of the mapping ψ αβ = ψ α ψ 1 β? (3.9) What is its domain? Is it a homeomorphism? 18

19 A coordinate chart on a topological space (X, τ) is a pair (U, ψ), where U τ is an open set and ψ is a homeomorphism ψ : U D R n. The components of R n valued mapping ψ will be typically denoted by ψ(p ) = (x 1 (P ),..., x n (P )), P U. (3.10) When no confusion arises, we also write simply ψ : P x i, i = 1, 2,... n. In other words, if a coordinate chart exists, it means that there exists an open set U on X and the points of this set can be described by n coordinates. Here, ψ is a coordinate mapping which assigns to any point P U an n tuple of numbers (coordinates). For simplicity, we represent the n tuple by symbol x i, where i = 1, 2,... n. The existence of a coordinate chart can be interpreted as that the space X is locally, in the domain U, topologically equivalent to R n. In standard calculus of n variables, we have the well-defined notion of a differentiable function. Mapping f : R m R n is in fact a collection of n mappings of m variables, f(x 1,... x m ) = (f 1 (x 1,..., x m ),..., f n (x 1,..., x m )). (3.11) Roughly speaking, mapping f is differentiable (class C 1 ), if all partial derivatives f i x p for i = 1,..., n, p = 1,..., m, (3.12) exist and are continuous. More generally, mapping is of smoothness class C k if all partial derivatives of order k (including mixed derivatives with respect to different variables) exist and are continuous. Class C 0 means simply continuous functions. If f is of class C k for all k = 0, 1, 2,..., it is said to be of class C (infinitely differentiable or smooth). Smooth mapping which is at the same time also (real) analytic, is said to be of class C ω. Since a coordinate mapping ψ acts on a topological space, rather than on R m, we cannot directly define the notions of differentiability and smoothness. The idea behind the definition of the manifold (coming soon) is to cover the topological space with coordinate charts. Since these chart will necessarily have overlaps, the description of geometrical objects (like curves) must be independent of the coordinates, i.e. results obtained in different coordinates must be consistent. For example, imagine a curve expressed in specific coordinates in which the curve is smooth. If we perform a coordinate transformation, the curve must still look smooth in the new coordinates. This cannot hold in general, however, because the differentiability can be destroyed by non-differentiable coordinate transformation. Therefore, in the definition of the manifold we will admit only sufficiently smooth coordinate transformations. A coordinate transformation between the two charts (U 1, ψ 1 ) and (U 2, ψ 2 ) with nonzero overlap, U 1 U 2, is the mapping ψ 12 = ψ 1 ψ 1 2 : D 2 D 1, (3.13) where D 1 and D 2 are the images of ψ 1 and ψ 2, respectively. If ψ 1 : P x i and ψ 2 : P y i, the coordinate transformation can be written as ψ 12 : y i x i (y). (3.14) 19

20 An atlas on a topological space (X, τ) is a collection of coordinate charts A = {(U α, ψ α )}, (3.15) such that the open sets U α cover the space X, X = α U α. (3.16) The atlas is said to be of class C k if every possible coordinate transformation is a C k mapping; that is, for any α, β such that U α U β, (3.17) the coordinate transformation ψ αβ = ψ α ψ β (3.18) is of class C k. A manifold is a triple (M, τ, A), where (M, τ) is a topological space and A is an atlas on this space. Manifold is of class C k if the atlas has the same class of smoothness. For simplicity, we will say just manifold M and do not specify the topology and the atlas. 3.3 Curves Curve on a manifold M is a mapping γ : R M : t γ(t). (3.19) The coordinate form of the curve on a given chart (U, ψ), ψ : P x i is the mapping γ = ψ γ : t x i (t). (3.20) We will not distinguish between the curve and its coordinate form, so that the curve is a mapping x i = x i (t). Let f : M R be a real function on the manifold. We can construct the coordinate form of f by defining f = f ψ 1 : x i f(x). (3.21) Again, we will not distinguish function and its coordinate form. 3.4 Vector fields The problem with an intuitive notion of a vector in the context of general manifolds rests in the fact that we cannot define a vector as an oriented line segment, as we usually do in the calculus in Euclidean space. Indeed, consider a vector tangent to the sphere as in the following figure. 20

21 Obviously, the endpoint of the vector does not belong to the sphere itself, instead, it lies in a plane tangent to the sphere. Here the idea of the arrow still works, because we can embed the sphere into the Euclidean space and in this ambient space the notion of oriented line still makes sense. However, in general we want to consider curved manifolds which are not embedded into a larger space. Although any manifold can be embedded into an Euclidean manifold of sufficient dimension 2, we want to describe intrinsic properties of the manifold, i.e. properties which have nothing to do with such embeddings. Therefore, we are forced to look for another definition of a vector which will still encapsulate the basic property of a vector: it is a quantity which has a direction and magnitude. An analogy with velocity is appropriate here. Velocity vector tells us where the particle will be after the infinitesimal time step. Geometrically, velocity is a vector tangent to the trajectory of the particle. So, if we have to give up the notion of a vector as an oriented line segment, we can still talk about curves (because they do belong to the manifold) and vector will be quantity characterizing the infinitesimally close point of the curve. Consider a curve γ : t x i (t) and a function f = f(x). We can compose these two mappings to obtain function f along the curve, f(t) = f(x(t)). The derivative of the function along the curve, i.e. with respect to the parameter t, is df dt = f dx i x i dt. (3.22) (3.23) Geometrically we interpret this as a derivative of f in the direction of vector X tangent to the curve. We define so that X i = dxi dt, df dt = Xi f x i. (3.24) (3.25) We can view this relation as the action of an object differential operator acting on the function f. Since it can act on arbitrary function, we define the operator itself as X = X i x i (3.26) and call this operator a vector, functions X i are called components of the vector in coordinates x i. Hence, vector in general is a mapping X which assigns to a function its derivative in a specific direction, X : f X(f) = X i f x i. (3.27) 2 Whitney s embedding theorem guarantees that any n dimensional Hausdorff, second-countable manifold (i.e. manifold whose topology has a countable basis) can be be embedded into the Euclidean space of dimension 2n. This is the upper bound, sometimes it is possible to embed a manifold into Euclidean space of smaller dimension, but for 2n it is always possible. 21

22 Exercise 6 Consider two overlapping charts ψ 1 : U 1 x i and ψ 2 : U 2 y i restricted to the nonzero overlap U 1 U 2, and corresponding coordinate transformations, ψ 12 = ψ 1 ψ 1 2 : y i x i (y), ψ 21 = ψ 2 ψ 1 1 : x i y i (x). (3.28) Consider a function f = f(x) and a curve γ : x i = x i (t). Express the derivative of f along the curve γ in coordinates y i and partial derivatives of f with respect to y i. Compare this expression with the derivative in coordinates x i. What can you deduce for the relation between components of the tangent vector in different coordinate systems? Vector X at point P of the manifold M is a differential operator X : F(M) R, (3.29) where F(M) is the algebra of the functions on the manifold (i.e. f F(M) is a function f : M R), which satisfies the Leibniz rule X(f g) = X(f) g + f X(g). (3.30) All vectors at point P form a linear space which is called tangent space of manifold M at point P and is denoted by T P M. In local coordinate chart with coordinates x i, any element of T P M can be written in the form X = X i i P, (3.31) where i = x i (3.32) and i P means partial derivative with respect to x i at point P. Hence, operators i P form a basis of T P M. Under the change of coordinates x i x i, the components X i and the operators i P transform as X i = X j x i x j, where all quantities are evaluated at point P. i = x j x i j, (3.33) That is, vector X at point P assigns to any function f its derivative along the direction of X at point P. Vector field X is an assignment of a vector X P to any point P M, X : P X P T P M. (3.34) Vector field can be decomposed as X = X i (x) i. (3.35) Equivalently, vector field X is a mapping X : F(M) F(M) : f X(f), (3.36) which satisfies the Leibniz rule (3.30). 22

23 Hence, the vector field X is a mapping which assigns a vector X P to any point P of the manifold. Such vector acts on the algebra of functions and returns the directional derivative of the function. So, when vector field acts on a function, X(f), it assigns a number (derivative) X P (f) to any point P. In other words, vector field acts on the function and produces a function, so it is a mapping F(M) F(M). Let γ : t x i (t) be a curve on the manifold in coordinates x i. In these coordinates, the tangent vector to the curve is X = X i i, We will denote this vector also by symbol where X i = dxi dt ẋi. (3.37) γ = X. (3.38) Conversely, consider X = X i i be a general vector field on the manifold. Curve γ is said to be the orbit of the vector field X if everywhere on the curve, X is tangent to the curve, i.e. γ = X. 3.5 Covector fields Vector spaces and bases. Let us start with some linear algebra. Consider a vector space V of dimension n. By defition this means there exists an n tuple of vectors called basis (e 1, e 2,... e n ), e i V, i = 1, 2,... n, (3.39) which are linearly independent but any vector v V can be written as a linear combination of the basis vectors, v = v 1 e v n e n v i e i. (3.40) The basis of a vector space is not unique, there are infinitely many bases. Indeed, from given basis we can form another basis simply by taking arbitrary linear combinations of the basis vectors, provided that the new set remains linearly independent. Formally, we form a new basis f i by the prescription f i = e j A j i, or, in the matrix form, f = e A (3.41) where A j i are components of nonsingular n n matrix. As we know from the linear algebra, nonsingular matrix means that its determinant is nonzero, which means there exists the inverse matrix (A 1 ) i j defined by 3 (A 1 ) i j A j k = δi k, or A 1 A = I n, (3.42) and the rank of the matrix is equal to n; this implies that vectors f i are linearly independent 4. The inverse transformation then reads e i = f j (A 1 ) j i, or e = f A 1. (3.43) Now we easily find how the components v i of the vector v transform. We have v = v i e i = v i (A 1 ) j i f j, (3.44) 3 I n is the identity matrix of dimension n. 4 Otherwise the image of the matrix A would only a subspace of the original vector V. 23

24 but at the same time, decomposition of v in basis f i in general reads v = v i f i, (3.45) so that the comparison of the two expressions yields v j = (A 1 ) j i vi. (3.46) If we work directly with the components of the vector v i, it is more usual to transform the components by matrix A i j and use the inverse matrix for the transformation of the basis and in what follows we shall do that. In a general vector space V, any two bases e i and f i are related by a nonsingular matrix A i j, f i = e j (A 1 ) j i, e i = f j A j i. (3.47) The components of vector v = v i e i = v i f i in different bases are related by v i = A i j v j, v i = (A 1 ) i j v j. (3.48) In the case that the vector space V = T P M is a tangent space of the manifold at point P, we can choose as a basis the coordinate basis e i = i P comprised of the differential operators representing directional derivatives in the direction of coordinate lines. Different coordinates x i induce a different basis f i = i P and the transformation matrix A i j and its inverse are given by A i j = x i x j, (A 1 ) i j = xi, (3.49) x j as relation (3.33) shows. At each point P we have a different vector space T P M and all these tangent space comprise the so-called tangent bundle T M. As vector fields, operators i are functions of the position, i.e. at each point P they represent the basis of corresponding tangent vector space T P M. Dual vectors. With any vector space V we can canonically (without additional structure) associate the so-called dual space V whose elements are called covectors. Covectors are linear functionals on V, i.e. mappings which take as an argument a single vector and return a real number. A dual vector or covector on n dimensional vector space V is a mapping α : V R such that for any v, w V and λ R, we have α(v + λ w) = α(v) + λ α(w). (3.50) All such mappings form an n dimensional vector space V called dual space. In order to check that the dual space V has the same dimension as the original space V, let us introduce a basis e i in V. Let α V be any dual vector, let v = v i e i V be an arbitrary vector, then by linearity we have α(v) = α(v i e i ) = v i α(e i ). (3.51) We see that the action of α on arbitrary given v is completely determined if we know the action of α on the basis vectors e i.. In other words, all possible covectors are parametrized by the n numbers α(e 1 ),..., α(e n ). 24

25 Exercise 7 Check that V is a vector space, i.e. check that i) it is closed under the addition α + β and under the multiplication with a real number λ α; ii) zero vector exists and is unique and each α has inverse α; iii) distributive properties λ(α + β) = λ α + λ β and (λ + µ)α = λ α + µ α hold. The dual basis to a basis e i of a vector space V is the n tuple of covectors θ 1,..., θ n, θ i V, satisfying θ i (e j ) = δ i j. (3.52) Exercise 8 Check that: the dual basis always exists (trivial); that θ i are linearly independent and therefore form a basis (hint: by contradiction, suppose they are linearly dependent, i.e. α i θ i = 0 for some nonzero sequence α i ; using the definition of the dual basis, show that in fact α i = 0); We have verified that the dual basis θ i can be taken as a basis of the dual space and therefore any covector α V can be written in the form α = α i θ i, where α i = α(e i ). (3.53) The action of a covector on a vector can be then written as α(v) = α i θ i (v) = α i θ i (v j e j ) = α i v j θ i (e j ) = α i v j δ i j = α i v i. (3.54) As an exercise, check the following result. Let e i, f i be two bases of a vector space V and θ i, θ i corresponding dual bases of V. Let e i and f i be related by e i = f j A j i. (3.55) Then, the dual bases are related by θ i = A i j θ j, (3.56) and the components of a covector α = α i θ i = α i θ i transform as α i = α i (A 1 ) j i, α i = α j A j i. (3.57) To summarize, we have seen objects of two types: with lower and upper indices. Lower indices are called covariant, the upper ones are called contravariant. Vectors live in a vector space V. In V we choose basis e i (with covariant index), the component of vectors have contravariant indices. Hence, the vector has expansion of the form v = v i e i. (3.58) The reason why we distinguish upper and lower indices is that they transform in a different way. If we transform the basis (covariant indices) by a matrix A 1, e i e j (A 1 ) j i (3.59) 25

26 the components of the vectors transform like v i A i j v j. (3.60) That is, transformation of covariant indices is done by a matrix which is inverse to the matrix performing the transformation of contravariant indices. As a result, vector v itself is invariant under the change of basis. This is a general rule: any summation through a pair of one covariant and one contravariant index is independent of the basis. Next, we have covectors living in the space V. They act on the vectors in V linearly, producing a real number. For a specific choice of basis e i in V, there is a preferred choice of the dual basis θ i in V. Then, any covector can be expanded in the dual basis as α = α i θ i. (3.61) Again, contravariant index on θ i transforms with the matrix A, while covariant index on α i transforms with the matrix A 1. A natural question arises here. Since V is itself a vector space, we can construct its dual, the space (V ). Will this produce a third type of index which will transform in yet another way? The answer is no, because space (V ) is canonically isomorphic to the original space V. That is, we can uniquely and independently of the basis assign a vector v V to any element X (V ). By definition, elements of (V ) are linear functionals on V, i.e. mapping which linearly act on covectors. For any X (V ), let us define a vector v V by the prescription X(α) = α(v) for any α V. (3.62) The reader is invited to check that this prescription correctly defines an isomorphism between V and (V ) and by construction this isomorphism is basis independent. Thus, we can identify spaces V = (V ) and we do not have to introduce the third type of index for the dual of the dual. This has a consequence that, as algebraic object, vector v V can be simultaneously regarded as element of (V ), i.e. it can act on covectors α V according to the prescription v(α) = α(v), (3.63) where the right hand side is well-defined. Notice that this is different from acting of a vector as a differential operator on function, v(f). Here we are talking about vectors as algebraic objects. Since vector as a differential operator cannot act on covector, from the specification of the argument in v( ) it is always clear whether we mean action of the algebraic vector on the algebraic covector as in v(α), or as the differential operator acting on the function on the manifold, as in v(f). Covector fields. Now we can return to the covector fields on a manifold. Each tangent space T P M has associated dual space TP M which is called the cotangent space. The collection of all cotangent spaces comprises the cotangent bundle T M. Let us find a convenient dual basis of the cotangent space. We introduce the gradient of a function f F(M) as a linear mapping df : T M R defined by df(x P ) = X P (f), where X P T P M T M. (3.64) That is, gradient df assigns to any vector X P at arbitrary point P a derivative of f in the direction of X. We can see that df acts on vectors and produces real numbers, and it is linear; hence, df is a covector. As an example, consider gradients of the coordinates, i.e. covectors of the form dx i. Acting on a vector basis i, we get dx i ( j ) = j x i = δ i j. (3.65) 26

27 Comparing this with (3.52) we infer that gradients dx i form a dual basis of the cotangent bundle. Then any covector α T M can be expanded as α = α i dx i. (3.66) Exercise 9 Using the chain rule for the derivatives, show that dx i transform as dx i = xi x j dx j. (3.67) The last exercise shows that quantities dx i transform as differentials. Indeed, that is the reason why we use the same symbol for the gradient. Usually we interpret the differential as an infinitesimal change, which is somewhat vague interpretation and especially mathematicians are not very happy with it. Here, dx i are well-defined objects (covectors), but we can work with them as with the differentials. We do not repeat the arguments about transformations of vectors and covectors, but summarize these transformation rules in a box. At this point, the reader should see the connections between general linear algebra setting and the situation on the manifold. Vectors can be expanded in the coordinate basis as v = v i i, covectors can be expanded in the dual basis as α = α i dx i. Under the change of coordinates, the bases i and dx i transform as i = xj x i j, dx i = x i x j dxj, (3.68) i.e. the transformation matrices for basis and dual basis are inverse to each other. Components of vectors and covectors transform as v i = x i x j vj, α i = xj x i α j. (3.69) 3.6 Direct product and direct sum Direct sum. Before we proceed to general tensor fields, it is useful to introduce two operations which allow us to construct new vector space from two vector spaces V and W. Let us start with the direct sum. Direct sum of the vector spaces V and W is a vector space V W with the underlying set V W (Cartesian product), i.e. the elements of V W are ordered pairs v w, where v V and w W. We define the structure of the vector space V W by v w + v w = (v + v ) (w + w ), (3.70a) λ(v w) = (λ v) (λ w), λ R. (3.70b) In human words, direct sum of two vector spaces is simply the set of all pairs (v, w), but we denote such pairs by v w. However, in order to get a vector space, we have to prescribe how to make linear combinations of such pairs. The definition in terms of ordered pairs can be simply written as (v, w) + λ(v, w ) = (v + λ v, w + λ w ), (3.71) which is a usual component-wise linear combination of order pairs. R 2 = R R. The familiar example is the plane 27

28 The definition extends in an obvious way to any finite number 5 of vector spaces. Having a finite family of vector spaces V i, their direct sum is V i = V 1 V 2 V n, (3.72) i and the elements of such space are v 1 v 2 v n, v i V i. (3.73) For any i = 1,..., n there is a canonical projection homomorphism P i : j V j V i defined by P i (v 1 v n ) = v i, (3.74) which selects the i th element of ordered n tuple of vectors. Conversely, there is a coprojection P i : V i j V j defined by P i (v i ) = 0 1 v i 0 n, (3.75) where 0 j is the zero vector of vector space V j ; usually we denote them all simply by 0. Image Ṽi = P i (V i ) is a subspace of j V j. Obviously, dim i V i = i dim V i. (3.76) Consider a direct sum U = V W and let e i be the basis of V, f a be the basis of W. Then E i = e i 0 = P 1 (e i ), F a = 0 P 2 (f a ), (3.77) form bases of subspaces Ṽ and W and a general element u U can be written as u = u i E i + u a F a. (3.78) Direct product. vector space V ; The definition of a direct product is a bit more abstract. Recall that so far we have its dual V, which is a space of linear functionals on V ; the double dual (V ) is canonically isomorphic to V, i.e. (V ) = V. That also means, that covector α V is a mapping α : V R, but at the same time, vector v V can be regarded as an element of (V ) and hence is a functional on V, i.e. also vector v V = (V ) is a mapping v : V R defined by v(α) = α(v). Consider two covectors α, β V. We can define their direct product as a mapping α β : V V R (3.79) by relation (α β)(u, v) = α(u) β(v). (3.80) That is, the direct product α β is a bilinear mapping which takes as arguments two vectors and assigns a real number to them. We can see that the object α β, in fact, acts on the space which is dual to the space of covectors. Similarly we could define the direct product of two vectors u, v as the mapping (u v) : V V R : (α, β) α(u) β(v). (3.81) Again, the object u v acts on the dual space to vectors. This observation leads us to the following definition. 5 We leave the infinite case to another book. 28

29 The direct product (or tensor product or Kronecker product) of two vector spaces V and W is the space V W of bilinear mappings x : V W R, for any x V W. (3.82) The direct product of vectors v V and w W is a bilinear mapping v w : V W R (3.83) defined by the prescription (v w)(α, β) = α(v) β(w), for any α V, β W. (3.84) Suppose we choose the bases e i and ẽ a in the spaces V and W, and that θ i and θ i are corresponding dual bases in V and W, so that any α V and β W can be expressed as α = α i θ i, β = β a θa. (3.85) Let x V W, then x(α, β) = x(α i θ i, β a θa ) = α i β a x ia, (3.86) where we have denoted x ia = x(θ i, θ a ). (3.87) These are the components of mapping x, but with respect to what basis? Consider the mapping x = x ia e i ẽ a V W, (3.88) then x (α, β) = x ia (e i ẽ a )(α j θ j, β b θb ) = x ia e i (α j θ j ) e a (β b θb ) = x ia α i β a. (3.89) This shows that x (α, β) = x(α, β) for any α, β, and therefore x = x. In other words, any mapping x V W can be expressed as x = x ia e i ẽ b. (3.90) If e i and ẽ a are basis for vector spaces V and W, and θ i and θ a are corresponding bases in the dual spaces V and W, the basis of V W can be written in the form E ia = e i ẽ a, (3.91) and any element x of V W can be written in the form x = x ia E ia = x ia e i ẽ a, (3.92) where the components of x are x ia = x(θ i, θ a ). (3.93) 29

30 3.7 Tensor fields Now we generalize the notion of vectors and covectors to tensors. Covector is a mapping which takes a vector and returns a number. Vector is a mapping which takes a covector and returns a number. Tensor is a mapping which takes some number of vectors and some number of covectors and returns a number. Tensor t of type (p, q) on the vector space V is a multilinear mapping t : V V }{{ V V R. (3.94) }}{{} p q That is, t is an element t V V V }{{} V T }{{} q p, (3.95) p q which takes p covectors α,... β, and q vectors u,... v, and returns a number t(α,..., β, u,..., v) R. (3.96) If e i is a basis of V and θ i is a corresponding dual basis in V, the components of the tensor t are p {}}{ i... j t = t(θ k }. {{.. } l i,..., θ j, e k,..., e l ). (3.97) q With respect to this basis, t can be expanded as t = t i...j k...l e i e j θ k θ l. (3.98) Tensor of type (p, q) has p contravariant (upper) indices and q covariant (lower) indices. In the case of the tensor fields on a manifold, at each point of the manifold we have a tangent space T P M, its dual T P M, and hence we can construct the space of tensors of type (p, q), (T P ) p qm = T P M T P M T P M T P M. (3.99) The collection of all spaces (T P ) p qm, i.e. for every point P M, will be called the tensor bundle Tq p of (p, q) tensors. Notice that product of a (p, q) tensor and a (p, q ) tensor is a (p + p, q + q ) tensor. Hence, space Tq p of (p, q) tensors is not closed under the tensor multiplication. Formally it is often convenient to introduce the tensor algebra by T = p,q T p q, (3.100) which is a direct sum of spaces of tensors of all possible types. This algebra is already closed under the tensor multiplication 6. Given the coordinates x i, vector fields i and covector fields dx i constitute a basis on the tangent bundle T M and cotangent bundle T M. Their appropriate tensor products therefore constitute a basis on the bundle Tq p M of (p, q) tensor fields. General tensor field is of the form t = t i...j k...l i j dx k dx l, (3.101) 6 Something similar happens in quantum mechanics when one constructs the Fock space, e.g. for the bosonic particles. There, tensors of type (p, 0) represents states of a system with p particles. The full Fock space is constructed as direct sum of all p particle Hilbert spaces. 30

31 where the rank of the tensor is not specified, but it will be always clear from the context. Using the chain rule we can immediately infer the transformation properties for the components of a general tensor field. Under the coordinate transformation x i x i, the components of a tensor field t Tq p transform as t i...j k...l = t m...n x i r...s x m x j x n x r x xs. (3.102) k x l 3.8 Metric tensor Metric tensor. We would like to measure the magnitudes of vectors and have an inner product (scalar product, dot product) on the space of vector fields. The role of the inner product will be played by the metric tensor. Motivation: length of the curve. In the Euclidean plane in the Cartesian coordinates, where l = B A dr = B A B dx2 + dy 2 ẋ2 B = + ẏ 2 dt = (X1 ) 2 + (X 2 ) 2 dt, (3.103) A A X i = dxi dt (3.104) is the tangent vector to a curve. We can also write dr 2 = δ ij dx i dx j. (3.105) In the polar coordinates x i = (r, θ) defined by x = r cos θ, y = r sin θ, (3.106) we have dr 2 = dr 2 + r 2 dθ 2 g ij dx i dx j, (3.107) where g ij = ( ) r 2. (3.108) So, in general, the line element is given by dr 2 = g ij dx i dx j. (3.109) At the same time, if X i is tangent vector to the curve, its length is given by X 2 = g ij X i X j, (3.110) which reduces to the Cartesian coordinates to X 2 = (X 1 ) 2 + (X 2 ) 2. (3.111) 31

32 The scalar product (dot product, inner product) is X Y = g ij X i Y j, (3.112) which reduces to the Cartesian product to X Y = X 1 Y 1 + X 2 Y 2. (3.113) Transformation of g ij : so that dr 2 = g ij dx i dx j = g ij g ij = g ij x i x k hence, g ij is a (0, 2) tensor. x i x k x j x l ddx k dx l, (3.114) x j ; (3.115) x l On a general manifold, metric tensor is a (0, 2) tensor g ij satisfying: Symmetry: g ij = g ji Non-degeneracy: det g 0 The line element is given by ds 2 = g ij dx i dx j. (3.116) The scalar product of two vectors is X Y = g ij X i Y j (3.117) Since g ij is non-degenerate, we can define the inverse g ij by g ij g jk = δ i k. (3.118) Using the metric tensor we can raise and lower the indices on vectors and covectors: X i = g ij X j, X i = g ij X j. (3.119) The inner product is then X Y = g ij X i Y j = X j Y j = X i Y i. (3.120) At each tangent space T P M, we can choose a basis in which g ij is a diagonal matrix (by Gram-Schmidt orthonormalization process). In such a basis, g ij has a canonical form g ij = diag( 1, 1,..., 1, 1, 1,..., 1), r + s = n = dim M; (3.121) }{{}}{{} r s in that case, we say that the metric has signature (r, s). Euclidean space of dimension n has signature (0, n) and the metric tensor in canonical form is δ ij. In general relativity, we use the signature (1, 3), because the canonical form of the metric tensor is g ij = ( 1, 1, 1, 1). (3.122) In general, canonical form of metric g ij will be denoted by η ij and appropriate signature will be clear from the context. 32

33 3.9 Connection Connection. We need to differentiate tensor fields, but partial derivative does not work. For example, the partial derivative of a vector does not transform as the (1, 1) tensor: ix j = x k x i x j x l X l x k + xk x i Xl 2 x j x k x l. (3.123) The first term looks like correct tensorial term, but the second does not. Notice that partial derivative of a function, i f, transforms correctly as (0, 1) tensor (it s the gradient, after all). Geometrical reason why the partial derivative does not produce a well-behaved tensor can be expressed also in the following ways: taking partial derivatives involves subtracting two vectors from nearby points; but vectors from different points on the manifold belong to different tangent spaces: we simply cannot linearly combine them; values of the components at different points are actually components with respect to different bases, it does not make sense to compare them. In order to define a well-behaved tensorial derivative, we have to introduce a new structure called connection. We define the connection or covariant derivative as a mapping m which acts on the components of any tensor field t i...j k...l of type (p, q) and produces a tensor field of type (p, q + 1), m : t i...j k...l mt i...j k...l. (3.124) The connection has to satisfy the following properties: on functions it reduces to the usual partial derivative, i f = i f; (3.125) on a product it acts according to the Leibniz rule, i (X j Y k ) = X j i Y k + Y k i X j (3.126) It can be shown (see the lecture notes) that connection is fully characterized by the Christoffel symbols Γ i jk. Covariant derivatives of vectors and covectors read i X j = i X j + Γ j ik Xk, i X j = i X j Γ k ij X k. (3.127a) When the manifold has no torsion, Christoffel symbols are symmetric, Γ i jk = Γ i kj. (3.128) and the connection itself is called symmetric connection, or torsion-free connection. We will consider only symmetric connections! Exercise: check, that for a symmetric connection we have i j f j i f = 0, (3.129) where f is arbitrary function (scalar). The operator on the left hand side is called commutator of covariant derivatives. It annihilates scalar quantities, but not general tensor fields. 33

34 In general relativity we will need connection which is compatible with the metric in the sense that metric tensor is covariantly constant: i g jk = 0. (3.130) In terms of partial derivatives, this condition reads i g jk Γ l ij g lk Γ l ik g jl = 0. (3.131) Inverting this relation we find (assuming symmetry of the Christoffel symbols) Γ i jk = 1 2 gil ( j g lk + k g jl l g jk ). (3.132) This is called metric connection or Riemann Levi-Civita (RLC) connection or metric compatible connection Geodesics Geodesics. Motivation. Consider the stationary flow of a fluid in classical mechanics. The volume element at the position vector r moves with the velocity v( r). The change of the velocity d v along the streamline is given by d v dt = v dx i x i dt = v vi x i, (3.133) which is the acceleration of the fluid element. If the fluid element moves with the constant velocity, its acceleration is zero and the streamlines are straight lines. Hence, the fact that the streamlines are straight lines can be expressed as the fact that the acceleration of the element is zero. In curved geometries we cannot define the straight lines, but we can define the curves with zero acceleration with respect to covariant derivatives. These are called geodesics and represent the curves which are, for given geometry, as straight as possible. Geodesic is a curve γ with the tangent vector X which satisfies X i i X j = 0, (3.134) or, more explicitly, X i i X j + Γ j ik Xi X k = 0. (3.135) If we introduce a parameter t along the curve γ : t x i, we can write X i = dxi dt, (3.136) and the geodesic condition acquires the form dx j dt + Γ j ik Xj X k = 0, (3.137) or directly in terms of the coordinates d 2 x i dt 2 + Γi jk dx j dt dx k dt = 0. (3.138) 34

35 Let us return to the analogy with the fluids. In fact, if the streamline is straight, the particles of the fluid are not necessarily moving with the constant speed. They can accelerate, but only along the streamline, their acceleration cannot have a normal part. That is, the acceleration of the fluid element can be proportional to the velocity itself. The condition for the straight streamlines is therefore, in general, d v dt v, where the proportionality factor can be arbitrary function. If a vector field X satisfies equation (3.134), we say that the geodesic is affinely parametrized. Non-affinely parametrized geodesic has in general form X i i X j = f X j, (3.139) (3.140) where f is arbitrary function. By appropriate rescaling of X we can always find affinely parametrized geodesic. Usually, it is easier and more natural to work with affinely parametrized geodesics. However, sometimes we have a natural non-affine parametrization of geodesics. For example, on the horizon of a black hole, the deviation from affine parametrization describes the surface gravity of a black hole Curvature Curvature. Standard motivating example: transport of the vector over the manifold can depend on path. In the flat space, it does not depend on path: B path 1 A path 2 Here we paralelly transport vector from point A to point B either along the path 1 or along the path 2. In both cases we obtain the same result. On the sphere, however: path 1 A path 2 A B C B 35

36 First, we transport the vector from the point A along the quarter of the meridian to the point B, then along the quarter of the equator to the point C, and back to point A. We see that the resulting vector is different from the initial one. On the other hand, we can transport the initial vector from the point A along the full meridian, and we end up with the same vector. Hence: the result of the parallel transport depends on the path. Infinitesimal version: X i ε C X i D ε ε X i X i X i A ε B Choose a 2d-surface and coordinates x and y. Choose a vector X i at point a. Transport X i along the coordinate line of x by ε to B. Then along y by ε to C, by ε along x to D, finally along y by ε back to A. Resulting vector is X i. In the flat space X i = X i and the vector is the same as the initial one. In curved space, the variation of the vector will be i δx i = ε 2 Rjkl ( x ) j ( y k ) X l. (3.141) Here, tensor R i jkl is the Riemann tensor defined by i j X k j i X k = R ijkl X l. (3.142) and it describes the curvature of the connection m. In coordinates, Symmetries: l Rijk = jγ l ik i Γ l jk + Γ m ik Γ l jm Γ m jk Γ l im (3.143) R ijkl = R [ij]kl = R ij[kl] = R klij (3.144) Bianchi identities 1st Bianchi identity (algebraic) R [ijk]l = 0, i.e. R ijkl + R jkil + R kijl = 0 (3.145) 2nd Bianchi identity (differential) [m R ij]kl = 0 (3.146) 36

37 Ricci tensor and scalar curvature (Ricci scalar) R ij = R k ikj, R = g ij R ij (3.147) Symmetry: R ij = R ji Contracted Bianchi identities i R ij = 1 2 jr (3.148) Riemann normal coordinates Metric in the diagonal form, g ij = η ij Christoffel symbols vanish at given point Then Γ i jk P = 0 (3.149) g ij = η ij 1 3 R ikjl x k x l + O ( x 3) (3.150) This shows that Γ i jk is a coordinate effect, but curvature R ijkl cannot be eliminated by the choice of the coordinates. The volume element is ( dv = R ik x i x k + O ( x 3)) dv 0, (3.151) where dv 0 = dx 1... dx n is the Euclidean volume element. That is: Ricci tensor describes the change of the volume, either contraction or the expansion V 0 V Example of volume contraction by the presence of the Ricci curvature. Interpretation of the Ricci scalar R. For 2D surfaces, scalar curvature is related to the Gauss curvature (up to factor 2) At given point P of surface S, erect a normal vector n Any plane containing n and P is called normal plane and its intersection with S is normal section Choose a specific normal plane and corresponding normal section it is a curve, say, γ = γ(s) Define the unit tangent vector t = γ(s), and unit vector N orthogonal to curve, lying in the normal plane Then, t is necessarily orthogonal to t (since t 2 = 1, we have t t = 0) t = κ(s) N, (3.152) the scalar κ(s) is called the curvature of the curve. 37

38 In this way we define the curvature of any normal section of a surface. For different normal plane, corresponding curvature will be, in general, different. The maximal and the minimal values of the curvature of a normal section at given point P are called principal curvatures κ 1 and κ 2. The Ricci scalar is then R = 2 κ 1 κ 2 2 R 1 R 2, (3.153) where R = 1/κ is the radius of the corresponding curvature. Weyl tensor is the trace-free part of the Riemann tensor so that R ijkl = C ijkl + 2 ( ) 2 gi[k R l]j g k[i R j]l n 2 (n 1)(n 2) R g i[k g l]j, (3.154) g ik C ijkl = 0 (3.155) Symmetries like the Riemann tensor C ijkl = C [ij]kl = C ij[kl] = C klij, C [ijk]l = 0. (3.156) Weyl tensor describes the deformations preserving the volume, the so-called shear (we will discuss the shear later) 3.12 Lie derivative Covariant derivative is not the only way how to differentiate the tensor fields. We explained that the reason why partial differentiation of the components tensor fields does not produce tensors: in order to define the partial derivative we have to subtract the tensors at different points of the manifold, which means tensor living in different vector spaces. Covariant derivative provides a prescription how to cancel the nontensorial terms in equation like (3.123). At the end, covariant derivative gives rise to the notion of parallel transport which is a way how to identify tensors at different points. If we want to compare tensors t P at point P and t Q at point Q, we have to specify a curve γ connecting the points P and Q, then find a tensor τ along the curve γ such that γ τ = 0 and τ P = t P. In principle we can imagine there are more ways how to identify different tangent spaces than by parallel transport. An indeed, this is the case. First we will study how a diffeomorphism on a manifold induces such an identification, then we will see how vector fields provide such diffeomorphisms Pull-backs and push-forwards Pull-back of a function. Suppose there is a mapping φ : M N, (3.157) 38

39 which to any point P of the manifold M assigns a different point Q = φ(p ) of the manifold N. In general, manifolds M and N can have different dimensions, say m and n. In coordinates, this means that to any point P with coordinates x i = x i (P ), mapping φ assigns a point Q with coordinates y a = y a (x): φ : x i y a = y a (x), i = 1, 2,... m, a = 1, 2,... n. (3.158) Next, suppose that on N we have a function f F(M), which is a mapping f : P R : y a f(y). Schematically, we have the following picture: M φ N Q[y i ] (3.159) P [x i ] f Using these two mappings, we can form another mapping called pull-back of the function f, denoted by φ f, which will be a function on M. Pull-back of a function f : N R with respect to mapping φ : M N, is a mapping φ f defined by f(y) R φ f = f φ, (3.160) so that φ f : M R : P f(φ(p )), (3.161) or in the coordinates, φ f : x i f(y(x)). (3.162) We have a mapping φ : M N. The term pull-back means that we are dragging some object, in this case the function f, against the direction of the mapping φ, i.e. form N to M. Push-forward of a vector. Suppose we have a vector field X on manifold M; such vector field is a differential operator X : F(M) F(M). Vector field on manifold N would be a differential operator Y : F(N) F(N) on N. Since we can pull the function f : N R back on M, any vector field X on M can, in fact, act on such a pull-back, effectively giving rise to the vector field on N. Since now we are dragging vector field in the direction of the mapping φ, corresponding operation is called push-forward. Some issues arise when φ is not an injective mapping. The reason is obvious: suppose there exist two point P, P M such that φ(p ) = φ(p ) = Q N. Then, vectors X P and X P on M will be dragged to vectors at Q on N, but there is no guarantee that these vectors will coincide. In order that the operation of the push-forward is well-defined, we need to require that φ is injective and we will assume this; as a consequence, mapping φ 1 is well-defined on the image of φ. Push-forward of the vector field X : F(M) F(M) with respect to mapping φ : M N is a vector field φ X : F(N) F(N) defined by (φ X)(f) = X(φ f) φ 1 for any f F(N). (3.163) 39

40 Let us see how this works. First, we take a point Q N and map it by φ 1 to point P = φ 1 (Q) M. At this point we have the pull-back φ f of f, which is a function on M, and hence the vector field X can act on this pull-back, yielding a function on M. In this way we obtained a function on N, so that the push-forward φ X is indeed a vector field on N. More explicitly, in the coordinates we have (φ X)(f) Q = X(φ f) φ 1 (Q) = X i (x(y)) f(y(x)) x i. (3.164) x(y) Using the chain rule we get (φ X)(f) Q = X i (x(y)) f(y) y a y a x i. (3.165) x(y) From the last relation we can easily infer the transformation of the components of the vector field under the push-forward. Under the push-forward (3.163), the components of the pushed-forward vector φ X are (φ X) a y = X i (x(y)) ya x i. (3.166) x(y) Thus, formally, the push-forward of a vector looks like the coordinate transformation, but here we do not transform the coordinates, rather we map the point P with coordinates x i (on M) to the point Q with coordinates y a (on N). Push-forward of a vector is a mapping φ : T P M T Q N, where Q = φ(p ). (3.167) M X P [x i ] T P M φ N T Q N Q[y i ] φ X Pull-back of a covector. Since we can now push vectors forward in the direction of the mapping φ : M N, we can also pull the covectors back against the direction of the mapping φ. Since a covector α TQ N on N acts on the vectors from the tangent space T Q N, we can let it act on the vectors which are push-forwards of vectors on M. In this way we effectively get a covector on M. Pull-back of a covector is a mapping φ : TQ N T P M, where Q = φ(p ), defined by (φ α)(x) P = α(φ X Q ) for any α T QN. (3.168) The components of a covector transform according to the formula (φ α) i (x) = xi y a α a (y(x)). (3.169) y(x) 40

41 Pull-back and push-forward of general tensor field. If we want to, for example, pull-back a general tensor field t from manifold N to manifold M, we have to specify how the pulled-back tensor acts on a corresponding combination of vectors and covectors on M. In order to do that, we have to push all these objects forward to manifold N, where the action of t is well-defined. That means we have to drag all vectors and covectors in the direction of the mapping φ. However, so far we have defined just the push-forward of a vector and we do not know how to push-forward a covector. It is clear, however, that the mapping φ 1 : N M, if it exists, can be used to pull covectors on M to N, so it is effectively the push-forward of a covector on M to a covector on N. In order to be able to pull tensors back and to push them forward, therefore, we need that the mapping φ is invertible. Moreover, since we need to differentiate this mapping in order to calculate the transformation of the tensor components, we will assume that the mapping φ is a diffeomorphism. Mapping φ : M N, where M and N are smooth manifolds of the same dimension, dim M = dim N = n, is called a diffeomorphism, if φ is bijective and smooth, φ 1 exists and is smooth as well, where we define a smooth mapping in the following way. Suppose that in local coordinates, φ is given by φ : x i y a. Then φ is smooth, if the mapping y a = y a (x) is smooth for all a = 1, 2,... n. Recall that in the usual theory of functions of many variables, we have a well defined notion of smoothness. Hence, the mapping between manifold φ : M N is smooth, if its coordinate representation y a = y a (x), which is already a mapping R n R n, is smooth in a usual sense. At the first glance, it seems to be a coordinate dependent statement. If φ looks smooth in one coordinate system, can it look non-smooth in a different system? In princple, yes. However, in the definition of the smooth manifold we allowed only such coordinate transformations which are themselves smooth and hence cannot break the smoothness of φ. Hence, since the atlas on a smooth manifold consists only of such coordinates, which are related by a smooth transformation, the smoothness of φ does not depend on particular coordinate system. Manifolds M and N of the same dimension are said to be diffeomorphic, we write M = N, if there exists a diffeomorphism φ : M N. Any manifold M is trivially diffeomorphic to itself, with one diffeomorphism being the identity id : M M : P P. The set of all diffeomorphisms on manifold M is denoted by Diff(M). The set Diff(M) is in fact a group with respect to the composition of the mappings. That is: 1. for any two diffeomorphisms φ, ψ Diff(M), their composition is a diffeomorphism, i.e. φ ψ Diff(M); 2. there exists an identity mapping id Diff(M) such that φ id = id φ = φ for any φ Diff(M); 3. for any φ Diff(M) there exists φ 1 Diff(M) such that φ φ 1 = id. Diffeomorphisms on a manifold can be regarded in an active or in a passive way. Active way means that the diffeomorphism moves the points of the manifold to another points. On the other hand, in coordinates, the diffeomorphism is a mapping φ : x i y a, which looks like the coordinate transformation. Hence, instead of regarding a diffeomorphism as an active transformation which deforms the manifold, we can say that the same point with coordinates x i is now described by new coordinates y a ; this is a passive way. Both approaches are equivalent. Since the coordinate transformations do not change the properties of the manifold, they merely change the labeling of the points, the same holds for the diffeomorphisms. In other words, two diffeomorphic manifolds are for all practical purposes same. In general relativity we usually say that the theory is diffeomorphism 41

42 invariant which means that the physical laws cannot change under the diffeomorphisms, which are merely coordinate transformations. In order to define push-forward/pull-back of general tensor field, we will consider only diffeomorphisms, because thanks to their smoothness and bijectivity we can freely drag tensor fields in the direction or against the direction of the diffeomorphism. Pull-back of arbitrary (p, q) tensor field τ on manifold N against the direction of a diffeomorphism φ : M N is a tensor field φ τ on manifold M defined by the relation (φ τ)(α,..., β, X,..., Y ) = τ((φ 1 ) α,..., (φ 1 ) β), φ X,... φ Y ), (3.170) where objects α,..., β, X,... Y are covectors/vectors on manifold M. If y a = y a (x) is the coordinate form of the diffeomorphism φ, the components of the pull-back are (φ τ) i...j k...l = y c ta...b c...d x k yd x l x i y a xj y b. (3.171) Push-forward of arbitrary tensor field τ on manifold M in the direction of a diffeomorphism φ : M N is a tensor field on manifold N defined by (φ τ)(α,..., β, X,..., Y ) = τ(φ α,..., φ β, (φ 1 ) X,..., (φ 1 Y )), (3.172) where objects α,..., β, X,... Y are covectors/vectors on manifold N. If y a = y a (x) is the coordinate form of the diffeomorphism φ, the components of the push-forward are (φ τ) a...b c...d = t i...j x k k...l y c xl y a y d x i yb x j. (3.173) Flow of the vector field. In the previous paragraph we have discussed how to drag (pull-back or pushforward) tensor field if there is a diffeomorphism φ. In this paragraph we will see how such diffeomorphisms arise in a natural way. Consider a vector field X on the manifold M which is non-vanishing in the region of interest. The orbits of this field are curves which do not intersect and through each point of a manifold there is exactly one orbit which meets this point. Families of curves with this property are called congruences. We define the flow of a vector field φ as a mapping which has two arguments, parameter t and point P M. As a result, the flow returns a point Q which lies on the orbit passing through P in parametric distance t, as the figure illustrates. φ s+t (P ) R P X φ t (P ) Q φ s (Q) 42

43 Flow of the vector field X is the mapping φ : R M M : (t, P ) φ t (P ), (3.174) such that for a fixed P, γ(t) = φ t (P ) is an orbit of vector field X passing through the point P : X P = d dt φ t (P ) = γ(0). t=0 (3.175) This mapping satisfies the properties φ 0 (P ) = P, φ s φ t = φ s+t. (3.176) For a fixed t, mapping φ t : P φ t (P ) is a diffeomorphism. Lie derivative. Lie derivative is a concept which allows us to differentiate arbitrary tensor fields in the direction of a given vector field X. One important difference compared to the covariant derivative: for the covariant derivative we need only the curve along which we differentiate, i.e. only one orbit and its tangent vector. For the Lie derivative, we need indeed full vector field, because the Lie derivative depends also on the vector field in the neighborhood of the curve along which we differentiate. For an illustration, consider two vector fields X and Y. We want to Lie differentiate vector field Y along the vector field X. Let us first have a look at the figure. (φ t ) Y Y Q Y Y P Q P X φ t (P ) The curves without arrows represent the orbits of vector field X which is everywhere tangent to these curves, for example at the point P. At the same time, we have the vector field Y which is represented by the magenta solid arrows at P and Q. Here, point Q lies on the same orbit as P and therefore we can get from P to Q by the flow of vector field φ, Q = φ t (P ). (3.177) Since φ t is a diffeomorphism, we can use it to drag tensor fields along it (or against it). In particular, we can pull the vector Y Q at Q back to the point P, obtaining the vector Y = φ t Y Q. (3.178) In this way we have obtained two vectors at the same point P and we can compare them! In particular, we can define the Lie derivative of vector field Y in the direction of field X by 1 lim t 0 t (φ t Y Q Y P ). 43 (3.179)

44 The Lie derivative of a (p, q) tensor field τ with respect to a vector field X is a (p, q) tensor field defined by X τ = d dt φ t τ. (3.180) t=0 Let us see what this abstract (but geometrically very intuitive) definition means in terms of the components of vector fields. Lie derivative of a vector field. Formulas for the Lie derivatives of vectors and covectors are different, so we start with the aforementioned example of vector field. Let us fix the point P M, so that we have a curve γ = φ(p ) : t φ t (P ). Let X = γ be the tangent vector to that curve, in particular X P = γ(0). We can introduce the coordinates on the curve yt i such that γ : t yt, i and y0 i = x i, where x i are coordinates of the point P. This implies X = ẏ i, and therefore one can write y i t = x i + X i t + O ( t 2). Let us now consider vector Y at point y i t, Y (y t ) = Y i (y t ) y i. We want to pull this vector back to the point x i, (φ t Y (y)) x = Y i (y t (x)) xj y i t Formula (3.182) now implies x j y i t = δ j i Xj y i t (3.181) (3.182) (3.183) x j. (3.184) + O ( t 2), (3.185) so that (φ t Y (y)) x = [Y j (x + t X + O ( t 2) ) t Y i (y t ) Xj y i t + O ( t 2)] x i. (3.186) Now, the left hand side is already vector Y dragged from point yt i back to the point x i. By definition, the Lie derivative is the derivative of the left hand side with respect to t, evaluated at t = 0: [ Y j (x + t X) X Y x = x i t X i t Y i (y t ) Xj yt i + O ( t 2)] t=0 x j. (3.187) Setting t = 0 and using the fact y0 i = x i we get the desired result: X Y x = [X i Y j ] x i Y i Xj x i x j. (3.188) Lie derivative of a vector field Y with respect to a vector field X is, in coordinates, given by so that X Y = [ X i i Y j Y i i X j] j, (3.189) ( X Y ) j = X i i Y j Y i i X j. (3.190) 44

45 Surprisingly, particular combination of vector components and derivatives which we now discovered plays an important role in several contexts and has a special name, commutator of vector fields. Commutator or the Lie bracket of vector fields X and Y is a vector field [X, Y ] = [ X i i Y j Y i i X j] j (3.191) with the components [X, Y ] j = X i i Y j Y i i X j. (3.192) In this notation, the Lie derivative of a vector field is given by the commutator, X Y = [X, Y ]. (3.193) The commutator is manifestly antisymmetric, [X, Y ] = [Y, X]. Vector Y is said to be Lie constant along X if X Y = 0, (3.194) i.e. if the two vector fields commute. By antisymmetry of the commutator, any vector field X is automatically Lie constant along itself, X X = 0. Notice that condition that vector field is covariantly constant along itself is the definition of geodesic. The condition that the vector field be Lie constant along itself, on the other hand, is always trivially satisfied. Lie derivative of a covector field. We will proceed analogously without detailed comments. The covector at point y i t is of the form α(y t ) = α i (y t ) dy i t, (3.195) and its pull-back to the point x i reads where now so that (φ t α(y t )) x = α i (x + t X) yi t x j dxj, (3.196) y i t x j = δi j + t Xi x j, (φ t α(y t )) x = (3.197) ] [α j (x + t X) + t α i (y t ) Xi x j dx j. (3.198) Differentiation with respect to t and putting t = 0 yields X α = [ X i i α j + α i j X i] dx j. (3.199) Lie derivative of a covector field α with respect to vector field X is a covector field with the components ( X α) j = X i i α j + α i j X i. (3.200) 45

46 One can now easily deduce the Lie derivative for arbitrary tensor field. Lie derivative of a tensor field t with respect to vector field X is given by ( X τ) i...j k...l = Xm m τ i...j k...l τ m...j k...l m X i τk...l i...m m X j + τ i...j m...l kx m + + τ i...j k...m lx m. (3.201) We ask the reader to verify the following relation between covariant derivative and the Lie derivative. If is a torsion-free connection compatible with the metric g ij, one can replace the partial derivative by the covariant derivative in all formulas for the Lie derivative of vectors, covectors and general tensors. Unless explicitly stated otherwise, we will by default consider manifolds equipped with the metric g ij and corresponding torsion-free connection compatible with that metric Killing equations The notion of Lie derivative is related to the important concept of symmetries of the metric. One cannot use the covariant derivative for the differentiation of the metric tensor because i g jk = 0 by definition. Simple partial differentiation does not work either because it produces nontensorial objects. Next natural candidate is the Lie derivative and indeed, the Lie derivative is exactly what measures the variations of the metric in an invariant way. In particular, the (generator of the) symmetry of the metric is a vector field along which the metric is Lie constant. Such vector fields are called Killing vector fields. Vector field X is called the Killing vector field (briefly: Killing vector), if X g = 0, i.e. X i i g jk + g ik j X i + g ji k X i = 0. (3.202) These equations are called Killing equations. In general relativity, metric is the primary object. It is the unknown variable in the Einstein equations, it gives rise to the topology of the manifold (hence to the manifold itself), to the connection and curvature. For this reason we prefer to express all formulas in terms of covariant derivative (which is derived from the metric) instead of partial derivative. The Lie derivative of the metric is particularly simple in terms of covariant derivative. Indeed, we can replace all partial derivatives entering the Lie derivative by covariant ones. Hence, the Lie derivative of the metric is X g ij = X k k g ij + g kj i X k + g ik j X k = i X j + j X i, (3.203) where we have employed the fact k g ij = 0, hence g ik j X k = j (g ik X k ) = j X i, and similarly for the third term. Killing equations are equivalent to the condition i X j + j X i = 0. (3.204) 46

47 Unit sphere Orthogonal group. We know that the sphere is invariant under rotations about its center. Such rotations form a group O(3), the so-called orthogonal group. Matrix R of type n n is called orthogonal, if it satisfies the condition R T R = id n, (3.205) where id n is the identity matrix of dimension n. Orthogonal matrices form the orthogonal group O(n). One can easily show 7 that det R = ±1, for any orthogonal matrix R. If we regard general matrix A as a linear transformation of a vector A : v = v 1. v n Av, (3.206) (3.207) then the orthogonal matrices are such linear transformations which preserve the Euclidean norm. Indeed, the norm is v 2 = v T v, (3.208) where the matrix multiplication is understood on the right hand side. The value of the norm does not change under the transformation v = Rv by an orthogonal matrix R, since we have v 2 = v T v = v T R T R v = v T v = v 2. (3.209) We know that the length of vectors does not change under rotations and reflections and general orthogonal matrix R is indeed a combination of some rotation and some reflection. In fact, any rotation can be realized as a sequence of two reflections. Matrix R which represents the sequence of even number of reflections is a pure rotation and det R = 1. For a matrix R comprised of odd number of reflections, det R = 1. Clearly, pure rotations form a subgroup of O(3), but reflections with det R = 1 do not form a subgroup, since the composition of two reflections is a pure rotation. Special orthogonal group SO(n) is a subgroup of orthogonal group O(n) which consists of orthogonal matrices R O(n) with the additional property det R = 1. (3.210) Special orthogonal transformations are also called pure rotations. Any rotation matrix R SO(3) can be parametrized by three angles α, β and γ representing angles of rotation about axes x, y and z: cos β 0 sin β cos γ sin γ 0 J x (α) = 0 cos α sin α, J y (β) = 0 1 0, J z (γ) = sin γ cos γ 0. 0 sin α cos α sin β 0 cos β (3.211) We can see that group SO(3) is in fact a manifold because we can introduce continuous coordinates x i = (α, β, γ) on SO(3). 7 General relations: det A = det A T, det(a B) = (det A)(det B), det id n = 1. 47

48 Lie group is manifold G equipped with the structure of a group, i.e. with the identity element e M continuous mappings m : G G G (multiplication) and i : G G (inverse element) such that for any x G m(e, x) = x; m(i(x), x) = m(i(x), x) = e. Roughly speaking, Lie group is a group which is simultaneously a manifold. Hence, SO(3) is an example of a Lie group. Lie groups have very interesting properties, because the group structure gives rise to many canonical structures which are not present on an empty manifold. Here we discuss one important property of Lie groups, and that is the possibility to reconstruct the elements of the group from the so-called generators. General theory is rather abstract and will be presented in a separate chapter in detail, but it is easy to see the meaning of a generator for matrix groups like SO(3). Generators of group SO(3) are matrices L i, i = 1, 2, 3 defined by L i = dj i(θ) dθ. (3.212) θ=0 That is, take one of the matrices (3.211), differentiate with respect to its parameter and set the value of the parameter equal to 0. Explicitly, we have L 1 = 0 0 1, L 2 = 0 0 0, L 3 = (3.213) Exercise 10 Exponential of the matrix can be defined by its Taylor expansion, exp A = k=0 1 k! Ak = id +A A2 +, (3.214) where A k is understood in terms of matrix multiplication. Show that J 1 (α) = exp(α L 1 ), J 2 = exp(β L 2 ), J 3 = exp(γ L 3 ). (3.215) Hint: calculate first few powers of matrix L 1 and observe the pattern. The exercise shows how an arbitrary element of SO(3) can be generated by the multiplication of exponentials of the generators. It is interesting to calculate the commutators of generators, where commutator of matrices is defined simply by [A, B] = A B B A; (3.216) obviously, [A, B] = [B, A]. Direct calculation reveals [L 1, L 2 ] = L3, [L 2, L 3 ] = L 1, [L 3, L 1 ] = L 2, (3.217) which can be compactly written in terms of the Levi-Civita symbol as [L i, L j ] = k ɛ ijk L k. (3.218) Not by accident, this is exactly the algebra of the operators of angular momentum in quantum mechanics. 48

49 In general, generators of a Lie group are not elements of the group itself. Maybe it is not obvious here, because both J i s and L i s are matrices, but these matrices have different properties. Defining property of orthogonal matrices is (3.205), which in the index form reads R ij R kj = δ ik. j (3.219) Suppose that matrix R is parametrized by some angle θ, so that R ij = R ij (θ) and corresponding generator is a matrix with elements L ij = Ṙij(0). Rotation with zero angle is an identity, so R ij (0) = δ ij. Now, differentiate equation (3.219) with respect to θ and set θ = 0: j [ drij dθ ] R dr kj kj + R ij = dθ θ=0 j (L ij δ kj + δ ij L kj ) = L ik + L ki = 0. (3.220) Hence, the generators of SO(3) are antisymmetric. Any antisymmetric matrix L generates a rotation R = exp L, (3.221) i.e. a matrix which is orthogonal. All generators of the group G are said to form a Lie algebra g of the group G. Notice that Lie algebra is indeed an algebra, in the sense that it is a vector space whose elements can be multiplied (they are still matrices), but these elements do not form a group anymore. In particular, matrices (3.213) are not invertible. Lie algebra of group SO(3) is the space so(3) of 3 3 antisymmetric matrices called generators of group SO(3). If f i so(3), i = 1, 2, 3, (3.222) is arbitrary basis of the Lie algebra, the commutator [f i, f j ] can be again expanded in this basis as [f i, f j ] = k c ijk f k, (3.223) where functions c ijk are called structure coefficients of the Lie algebra. In this terminology, ɛ ijk are the structure coefficients of Lie algebra of rotational group SO(3), cf. (3.218). Metric and curvature. Consider a unit sphere with the metric g = dθ dθ + sin 2 θ dφ dφ g ij dx i dx j, (3.224) so that the matrix g ij = diag(1, sin 2 θ). The nonzero Christoffel symbols are Γ θ φφ = sin θ cos θ, Γ φ θφ = cot θ. (3.225) The scalar curvature is R = 2, (3.226) hence we say that the unit sphere is a manifold of constant curvature. 49

50 Killing vectors on the sphere. Let us find the Killing vectors of the metric (3.224), that is, let us solve the Killing equations (3.204). Killing vectors are, by definition, symmetries of the metric. Since the sphere is invariant under rotations about any axis, we expect to find three independent Killing vectors corresponding to three independent axes. Let K = K i dx i = A(θ, φ) dθ + B(θ, φ) dφ (3.227) be a Killing 1-form, i.e. a form satisfying (3.204) and its components A, B are the unknown functions. Using the Christoffel symbols (3.225), we find that the Killing equations read θ A = 0, φ A + θ B 2 B cot θ = 0, A cos θ sin θ + φ B = 0. (3.228) First equation tells us that A does not depend on θ. We also know that one solution must be A = 0, B = 1, because the metric (3.224) does not depend on the angle φ, i.e. the first (but we denote it by subscript 3) Killing vector is K 3 = φ, K 3 = (K 3 ) i dx i = sin 2 θ dφ, (3.229) which corresponds to A = 0, B = sin 2 θ. So, we still look for two other independent Killing vectors. Second equation in (3.228) can be rewritten as (A = A(φ), A = da/dφ) A + sin 2 θ θ (sin 2 θ B) = 0, (3.230) which easily integrates to sin 2 θ B = A dθ sin 2 θ = A cot θ + f(φ), (3.231) where f is an arbitrary function which we set to zero: B = A sin θ cos θ. (3.232) Then the third equation in (3.228) reduces to A + A = 0, (3.233) which has two independent solutions. For A = cos θ we get the Killing vector K1 = cos φ dθ sin φ sin θ cos θ dφ, K 1 = cos φ θ + cot θ sin φ φ. (3.234) and for the choice A = sin θ we obtain K2 = sin φ dθ cos φ sin θ cos θ dφ, K 2 = sin φ θ cot θ cos φ φ, (3.235) Exercise 11 With the Killing vectors at hand, verify that they satisfy [K i, K j ] = k ɛ ijk K k. (3.236) This is an amazing result because it shows that the Killing vectors of the sphere satisfy the same commutators like the generators of the group SO(3). Algebra of the Killing vectors of the sphere S 2 is isomorphic to the Lie algebra of the group SO(3). In general, if the spacetime exhibits some symmetry under the Lie group of transformations, then the metric tensor admits Killing vectors whose algebra is the same like the algebra of the generators of the group. In this sense, Killing vectors are generators of the symmetry and the nature of the group is encoded in the algebra of the Killing vectors. 50

51 Chapter 4 General relativity Einstein s equations: R µν 1 2 g µν R = 8 π T µν. (4.1) 51

52 52

53 Chapter 5 Schwarzschild spacetime In this chapter we describe in detail the so-called Schwarzschild solution discovered by Karl Schwarzschild almost immediately after Einstein published his general theory of relativity [Schwarzschild(1916)]. Since Einstein s equations written in terms of the metric tensor and its derivatives are very complicated, Schwarzschild was looking for a vacuum solution describing the idealized situation with high degree of symmetry, which is still however of some physical interest. Schwarzschild assumed that the solution is static and spherically symmetric. Later it turned out [Birkhoff and Langer(1923)] that this solution is unique and that any solution of vacuum Einstein s equations is in fact static; this is the celebrated Birkhoff theorem. Simple as it is, Schwarzschild solution exhibits several remarkable features and quite rich structure and the attempts to understand it fully inspired a lot of advances in general relativity and black hole physics. The Schwarzschild metric in the original coordinates (in which Schwarzschild discovered it) is singular for two values of the radial coordinate, namely r = 0 and r = 2 M, where M will be interpreted as the mass of an object described by the Schwarzschild metric. At this stage, by singularity we mean the fact that the components of the metric go to infinity. The presence of such singularities usually signals some breakdown of the physical theory, as in the experiment we never measure infinite values. A natural question therefore arose, what is the significance of these singularities? Is it a breakdown of a theory? Is it just a mathematical artifact, or does it have any physical implications? A striking consequence of the aforementioned Birkhoff theorem is that the Schwarzschild solution describes the gravitational field around any spherically symmetric object. Imagine, for example, a spherically symmetric star. The interior of the star certainly is not described by the Schwarzschild solution, because inside the star there is no vacuum. We can model the interior by a perfect fluid which has energy density and pressure. The interior can even by dynamical in principle, if the dynamics preserves spherical symmetry. Outside the star, however, there is a vacuum and therefore, by the Birkhoff theorem, the exterior of the star will be described by the Schwarzschild solution, independently of what happens inside the star. We can easily estimate the typical value of the Schwarzschild radius r S where the singularity r = 2 M appears. In full SI units, it can be expressed as r S = 2 M G c 2 = 2.95 ( M M ) km. (5.1) In other words, for the object of the Solar mass, the Schwarzschild radius is about three kilometers. Clearly, this is a tiny distance compared to the radius of the Sun, which means that the singularity of the Schwarzschild solution lies deeply in the interior of the object, where the Schwarzschild solution is not valid anymore. One can therefore adopt the view that the singularity at r = r S is indeed just a mathematical artifact: the singularity is safely hidden in the region when the Schwarzschild solution does not apply. In [Oppenheimer and Snyder(1939)], the authors considered a dynamical situation of a collapse of spherically symmetric dust, i.e. a perfect fluid without pressure. They have shown that the description of the collapse will differ for different observers. A static observer outside the collapsing object will see how the dust shrinks due to the gravitational attraction, but eventually the dust particles will slow down, approach- 53

54 ing the Schwarzschild radius only asymptotically. The light emitted by the dust will arrive to an outside observer with gradually increasing redshift, and the dust particles will never cross this magical surface of the Schwarzschild radius. On the other hand, an observer who is comoving with the collapsing dust will cross this surface in a finite proper time, but signals he emits after that will never escape. This result clearly indicates that under certain circumstances also the part of Schwarzschild radius containing the singularity r = 2 M must be taken seriously. It could be still possible, however, that a more realistic model of collapse will produce a different result. For example, a real star, rather than pressure-less dust, would try to resist the gravitational collapse by its own pressure. Today we know, however, that for a sufficiently massive star the collapse is inevitable, independently of the internal structure of the star. Since here we are interested mainly in the black hole aspects of the Schwarzschild solution, we will not discuss the details of the collapse and its dynamics here. The upshot of the above considerations is that the gravitational collapse is a very realistic process which is inevitable for sufficiently massive objects. If such collapse occurs, one cannot argue anymore that the singularity r = 2 M is unphysical, because it is located in the interior of the object: once the object collapses below the critical radius r S, the Schwarzschild solution is valid even in the region containing this singular surface, and therefore we have to understand the properties of such collapsed object and the nature of both aforementioned singularities. As it turns out, and as we will analyze in detail, the nature of the two singularities is quite different. The one discussed so far, r = 2 M, turns out to be the so-called coordinate singularity. It does not describe singular properties of the spacetime, it just signals the breakdown of the coordinates in which Schwarzschild discovered his solution. Roughly speaking, this is the reason why we had to introduce the notion of manifold as a topological space equipped with the atlas, i.e. a collection of several coordinate charts. Schwarzschild coordinate do not cover entire spacetime, just its portion. Nevertheless, by a different choice of coordinates we can cover a different portion of the spacetime which contains also the dangerous place with the singularity and its neighborhood. In such coordinates, there is no singularity in the metric tensor. Hence, the spacetime is not singular at r = 2 M, only the original coordinates are ill-defined there. For this reason we talk about the coordinate singularity, which is just apparent singularity, an artifact of the bad choice of the coordinates. The fact that r = 2 M is merely a coordinate singularity is already indicated in the Oppenheimer-Snyder analysis. There we saw that an observer comoving with the collapsing dust passes through the surface r = 2 M without noticing anything special. Yet, it is a special surface, for it is impossible to send any signal outside once you cross that surface! External observer can never receive any signal emitted from the region r < 2 M. The surface r = 2 M is called the event horizon and we will justify this name in what follows. On the other hand, the point r = 0, which is another singular point of the Schwarzschild metric, turns out to be a real, physical singularity of the spacetime. We also say that it is a curvature singularity, because the Riemann tensor describing the curvature of the spacetime blows up there. Again, a question arises whether this singularity is a mathematical artifact, or a consequence of the spherical symmetry which is never realized exactly in nature. The answer to this question is provided by the celebrated Penrose and Hawking singularity theorems and will be discussed in a separate chapter. Nevertheless, the basic result of these theorems is that formation of the singularity is inevitable under appropriate circumstances and that the singularity of a spacetime is a real, physical prediction of general relativity. Still, we expect that in Nature real physical quantities cannot acquire infinite values. It is generally believed today that the presence of the singularities signals the breakdown of general theory of relativity. We expect that under such extreme conditions also the rules of quantum mechanics play a significant role and some yet unknown mechanism prevents the formation of the singularity. In this chapter we will derive the Schwarzschild metric and proof the Birkhoff theorem guaranteeing its uniqueness in spherically symmetric case. Then we will study the geodesics of the Schwarzschild metric and compare them with the motion of test particles in the Newtonian gravitational field of a spherically symmetric body. Then we will discuss the nature of coordinate singularity and introduce the notion of the event horizon. Then we will study the causal structure of the Schwarzschild metric and introduce the so-called Kruskal extension and corresponding Penrose diagram. These considerations will justify the term black hole. 54

55 For standard treatments on the subject, see [Carrol(2010), Hawking and Ellis(2010), Wald(2010)]. The reader not interested in mathematical details of the derivation of the Schwarzschild metric can skip the first section and consult just shaded box on the page Derivation of the Schwarzschild metric Vacuum Einstein equations. Now we wish to derive the vacuum solution of Einstein s equations which is spherically symmetric. In vacuum, the energy-momentum tensor vanishes and, hence, Einstein s equations read R µν 1 2 g µν R = 0. (5.2) We can contract the equation with g µν, arriving at R = 0, (5.3) since g µν g µν = n = 4 is the dimension of the manifold, and R = g µν R µν. Inserting this back to (5.2), we find that vacuum Einstein s equations acquire the form 1 R µν = 0. (5.4) Hence, we need to solve equation (5.4) with respect to the metric which enters (5.4) through the second derivatives; moreover, first derivatives enter in a nonlinear fashion. This makes the solution of (5.4) unfeasible in a general situation. We have to restrict metric by additional assumptions in order to find an exact solution. Spherical symmetry. We are looking for a spherically symmetric solution of Einstein s equations. In order to define what we mean by spherical symmetry, recall that also the flat Minkowski spacetime is spherically symmetric. That means that the Minkowski metric is invariant under the action of the group SO(3) which is a subgroup of the Poincaré group. In the flat spacetime we can say that the group SO(3) consists of rotation about the point r = 0, t = constant. In the Schwarzschild spacetime, however, this point is a point with curvature singularity (as we will demonstrate) and hence is not well-defined. In general curved spacetime we cannot define the rotational symmetry in terms of transformations which preserve distance of a point from some fixed axis or point. Instead, we have to extract the important properties of rotational symmetry, i.e. of group SO(3). Recall that in section we discovered that the group of symmetry, SO(3), can be reconstructed from its Lie algebra, which is however isomorphic to the algebra of the Killing vectors of the sphere. These Killing vectors do not commute, but their algebra is closed 2 which means that these Killing vectors are surface forming, i.e. they are tangent to a surface. Hence, the orbits of these Killing vectors form a surface with the metric (3.224), which is therefore invariant under the flow induced by these Killing vectors. 1 Hence, vacuum Einstein s equations imply vanishing of the Ricci tensor which, in turn, means that the only nonvanishing part of the Riemann tensor is given by the Weyl tensor. 2 This means that any commutator can be expressed in the form (3.223), i.e. as the linear combination of the generators. If the algebra was not closed, commutator of some generators would be an element which is not included in the set of generators or their linear combinations. 55

56 Spacetime M is said to be spherically symmetric, if there exists a group of isometries generated by the Killing vectors K i, i = 1, 2, 3 satisfying the algebra [K i, K j ] = k ɛ ijk K k, (5.5) where ɛ ijk is the (3-dimensional) Levi-Civita symbol. The orbits of such Killing vectors form a 2-dimensional closed surfaces and there exist coordinates θ (0, π) and φ [0, 2π) such that the metric acquires the form (3.224), (2) g dθ dθ + sin 2 θ dφ dφ dσ 2. (5.6) By the left superscript (2) we indicate that the metric is not the metric of full spacetime, only on one 2-surface of spherical symmetry. The coordinate θ breaks down at the poles θ = 0 and θ = π, so we excluded them from the range of the coordinates. The breakdown can be seen, for example, from the fact that metric becomes singular at the poles (singular in the sense that the metric is not invertible, since (2) g = dθ dθ at the poles). The symbol dσ 2 = dθ dθ + sin 2 θ dφ dφ (5.7) here is used as an abbreviation and merely for convenience. We interpret dσ 2 as the metric on the unit sphere. This notation is widely (and sometimes wildly) used in physics, but do not be confused: this symbol does not mean the differential of function Σ 2. The Killing vectors of the 2-sphere S 2 have been explicitly found in the section , recall equations (3.234), (3.235) and (3.229). The orbits of these Killing vectors form 2-dimensional surfaces which are diffeomorphic to S 2. Each point P M of the spacetime M lies on some orbit which we denote by S(P ) = S 2. We say that the full spacetime is foliated by the orbits of spherical Killing vectors. Since the full spacetime is 4-dimensional, while the spheres are 2-dimensional, it means that the spacetime (as a topological space) can be written as a Cartesian product M = N S 2, (5.8) where N is a 2-dimensional set. Intuitively, one dimension of N represents time and the other one represents the distance from the center of the sphere. Mathematically this means that on N we can introduce the pair of coordinates T (, ) and R (0, ), which are constant on the spheres of symmetry (i.e. on the orbits S(P ) for any P ). Our task is to find the metric on M = N S 2, provided that the metric on S 2 is given by (5.6). The considerations leading to (5.6) determine the metric up to a multiplication factor, so that the metric in fact reads (2) g = f(t, R) ( dθ dθ + sin 2 θ dφ dφ ) f(t, R) dσ 2. (5.9) However, (2) g is a metric on S 2 and does not tell us how the full metric looks like. Consider, for example, surface of constant T, which is a 3-dimensional space foliated by the spheres of symmetry, and each sphere is labeled by the value of coordinate R. Up to now, we can choose coordinates θ and φ arbitrarily on each sphere. As an illustrative example, consider the following exercise. Exercise 12 Consider 3-dimensional Euclidean space with the Cartesian coordinates x, y, z and the usual spherical coordinates x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ, (5.10) 56

57 in which the Euclidean metric reads g = dr dr + r 2 ( dθ dθ + sin 2 θ dφ dφ ). (5.11) In particular, the induced metric on a sphere r = constant is (2) g = r 2 ( dθ dθ + sin 2 θ dφ dφ ). (5.12) Perform the coordinate transformation θ = θ + r, (5.13) and show that the new metric is (2) g = (1 + r 2 )dr dr r 2 ( dθ 2 dr + dr dθ ) + r 2 ( dθ dθ + sin 2 (θ r)dφ dφ ). (5.14) What is the geometrical meaning of the transformation (5.13)? Draw the figure! Find the induced metric on a surface r = r 0 in the new coordinates. On this surface (and only on this surface!), introduce new coordinate θ = θ r 0 and show that the induced metric is (2) g = r 2 ( d θ d θ + sin 2 θ dφ dφ ). (5.15) What happened in the previous exercise? We started from the standard spherical coordinates, in which the 3-dimensional space is foliated by the spheres r = constant and the induced metric on each sphere has the form (5.12). In these coordinates, θ = 0 is the north pole which lies on the z-axis for any radius r: Schwarzschild metric is a vacuum solution of Einstein s equations which is static, spherically symmetric and asymptotically flat. In the Schwarzschild coordinates, this metric reads ( ds 2 = 1 2 M ) dt 2 + dr2 r 1 2 M + r 2 ( dθ 2 + sin 2 θ dφ 2). (5.16) r Here, for convenience, we use the usual physicist s notation dt 2 = dt dt and similarly for the other coordinates. 57

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59 Chapter 6 Thermodynamics of black holes and singularities 6.1 Causal structure Curve γ is called causal if the tangent vector γ is everywhere timelike or null.. If two events P and Q of spacetime M are connected by a causal curve, one can send signal from P to Q which means there can be a causal relation between the two events. Two causally separated events (i.e. connected by a causal curve) cannot be simultaneous for any observer. All events Q which can be reached by a timelike curve starting from a given event P, is called chronological future of the event P and is denoted by I + (P ). Similarly, all events which can be reached by null or timelike curves is the causal future. If spacetime is time orientable, if there is a consistent splitting of the light cone to the future part and the past part. Causal vectors pointing to the future part of the light cone are said to be future-directed; similarly for past-oriented vectors. Roughly speaking, causal future J + (P ) is the future part of the light cone (including the interior) from point P, see figure 6.1. Chronological future of event P M is the set I + (P ) = {Q M there exists a future-directed timelike curve from P to Q}, (6.1) and similarly for the chronological past I (P ). Causal future J + (P ) is, analogously, the set of all events Q which can be reached by a future-oriented causal curve; similarly for the causal past. Causal future of a subset S M is the union J + (S) = J + (P ), (6.2) P S and similarly for the other notions. A subset S M is called achronal if I + (S) S =, (6.3) where is an empty set. Achronal set means that no two events P, Q S can be connect by a timelike curve. Such events are spacelike separated or null separated. 59

60 Figure 6.1: Future and past parts of the light cone. 60