Space Time and Gravity: Revision Notes

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1 Space Time and Gavity: Revision Notes This set of notes is a summay of the main fomulae and key concepts intoduced in the couse Space Time and Gavity. They have been thought as a guide fo evising the couse, but they ae not a complete summay of the couse mateial. Students should lean and undestand the physical meaning and the deivation of the concepts and fomulae included in this file as they had been pesented duing the lectues. Einstein s sum convention applies thoughout the pape. Basic concepts in Geneal Relativity Befoe Geneal Relativity NEWTONIAN GRAVITY was poviding a coheent desciption of gavity by descibing: the acceleation of a body in a gavitational potential φ(x): a = φ(x) which tells how gavitation influences matte s behaviou; the gavitational potential geneated by a mass distibution ρ(x): 2 φ(x) = 4πGρ(x) which tells how a matte distibution detemines gavitation SPECIAL RELATIVITY: but it is valid only fo inetial fames. How does special elativity genealise when we deal with acceleated fames? Is thee a connection between gavity and acceleated fames? Encoded by the Weak Equivalence Pinciple: The inetial mass and the gavitational mass of a body do coincide. Expeiments show that the acceleation felt by a body moving feely in a gavitational field is univesal and independent on the paticle s popeties. Thus gavity is an intinsic popety of spacetime athe than being the effect of a popagating field. 1

2 Gavity has only a elative existence: a peson in a small box which is feely falling does not expeience any gavitational field. Futhemoe we can fake theeffectsofagavitationalfieldbyacceleatingaboxinempty space. EINSTEIN S EQUIVALENCE PRINCIPLE: In small enough egions of spacetime the laws of physics educe to those of special elativity; it is impossible to detect the existence of a gavitational field by means of local expeiments. It exists a pefeed set of tajectoies, named inetial o feely falling, which ae those followed by acceleated paticles (i.e., paticles subjected onlyto theactionofgavity),andweshould attibutethe effects ofgavity to the cuvatue of spacetime, namely to the cuvatue of the tajectoies which feely falling paticles follow. The notion of inetial fame is eplaced by that of Feely Falling Fame o Locally Inetial Fame: a fame which is subjected only to the local effects of gavity, namely which is moving feely in the poximity of an object defoming spacetime and soucing gavity, and that is small enough to make the inhomogeneities of the gavitational field undetectable. We can ephase the Einstein s Equivalence Pinciple as: At any point in spacetime, in any abitay gavitational field, it is possible to choose a Locally Inetial Fame such that in an abitay neighbouhood of that fame the laws of Physics assume the fom as in an unacceleated coodinate system in the absence of gavity. We can get a measue of spacetime cuvatue aleady in the Newtonian theoy, by obseving the non-local effect of gavity Obseve the elative motion of two test paticles in the poximity of the Eath. The gavitational potential is φ( x). One paticle sits in x and the othe in y, and they ae ξ apat: y = x+ ξ. the vaiation of the elative velocity of the two paticles is given by the fomula: ξ i K ij ξ j,,i,j = 1,2,3 wheeasumoveepeatedindexesisundestoodandwheek ij = 2 φ( x) x i x j. Notice that: i K ii T [ K ] = 2 φ = 4πρ SPECIAL RELATIVITY Postulates of Special Relativity: The fundamental laws of physics have the same fom in any inetial efeence fame. 2

3 The speed of light in the vacuum is a univesal constant. Loentz Tansfomations Two inetial efeence fames K, descibed by the set of coodinates (t,x,y,z), and K, descibed by the set of coodinates (t,x,y,z ), moving with elative speed v along the x-axes ae elated by the following set of Loentz tansfomations (c = 1): t = t vx 1 v 2 t = coshζt sinhζx x = x vt 1 v x = sinhζt coshζx 2 y = y y = y z = z z = z whee we have intoduced the apidity ζ defined by v = tanhζ. The Loentz tansfomation leave invaiant the spacetime inteval: ( s) 2 = ( t) 2 +( x) 2 +( y) 2 +( z) 2. ( s) 2 isthedistancebetweentwopoints(events)inminkowskyspacetime. If we conside two points in Minkowsky spacetime P in (t P,x P,y P,z P ) and Q in (t P +dt,x P +dx,y P +dy,z P +dz), the (infinitesimal) invaiant distance between P and Q is: ds 2 = dt 2 +dx 2 +dy 2 +dz 2. The (finite o infinitesimal) distance between two events in spacetime can be geate, less, o equal to zeo. Accoding to this, the two events ae: if ( s) 2 < 0 timelike sepaated this fo example is the case of events happening at the same place but at diffeent time; can be connected by a paticle tavelling at speed lowe than the speed of light; if ( s) 2 = 0 lightlike (o null) sepaated can be connected by light ays; if ( s) 2 > 0 spacelike sepaated this fo example is the case of events happening at the same place but at diffeent time; cannot be connected by a paticle moving at speed lowe o equal to the speed of light. The light cone of a point P is the locus of points which ae null sepaated fom P. Each point P in spacetime has its own light cone. The futue light cone is geneated by the light ays moving outwad fom P. 3

4 The past light cone is geneate by the light ays conveging in P. The points that ae timelike sepaated fom P lay inside the light cone. The points that ae spacelike sepaated fom P lay outside the light cone. Paticles with non zeo est mass move along timelike paths which ae always inside the light cones at each point along thei tajectoies. The pope time τ is defined by: dτ 2 = ds 2 (dτ 2 = ds2 ifc 1) c2 and can be defined as the time that would be measued by a clock caied by the obseve himself. Let s intoduce the fou vecto x µ = (t,x,y,z), whee µ is an index taking values fom 0 to 3 (namely x 0 = t, x 1 = x, etc.). Then we can define the scala poduct (Einstein sum convention applies): x x = x µ η µν x ν = t 2 +x 2 +y 2 +z 2 whee η µν = is the Minkowsky metic The infinitesimal Minkowsky spacetime sepaation is ewitten as ds 2 = η µν dx µ dx ν CURVED GEOMETRIES Can be descibed though thei embedding in highe dimensional spaces. Example: a two-dimensional spheical suface can be seen as the set of points in R 3 whose coodinates satisfy the equation x 2 +y 2 +z 2 = 2. Can othewise be descibed though a set of intinsic popeties, which do not depend on anything outside the suface itself. These ae: The set of geodesics of a suface, namely the paths of minimal distance between two points in the cuved space unde consideation. They ae the equivalent of staight lines of Euclidean geomety, because they ae the staightest possible path one can daw on the suface. Example: On the 2-dimensional sphee, the geodesics ae half geat cicles. 4

5 The lenght of segments (potions of geodesics) on a given spacetime, and thei elations povide a way of deiving its intinsic cuvatue. Example: We know fom Euclidean geomety that the ation between the cicumfeence of a cicle dawn on a flat suface and its adius is C = 2π. On the othe hand, if we daw a cicle of adius on the suface of a 2-dimensional sphee (see figue), we get: C a C = 2πsin a a Expanding the elation above fo small values of we get: C 2π(1 1 6 K2 ) whee K = 1 a 2 is the (intinsic) Gaussian cuvatue of a 2 dimensional suface in the neighbouhood of a point P. A geneal local definition fo the intinsic cuvatue of a 2-dimensional suface is then: K = 3 π lim 2π C 0 3 Besides the cicumfeence C, othe geometical popeties of a spacetime which depend on its intinsic cuvatue ae: the aea A of a cicle on a 2-dimensional suface: A π 2 (1 K ) The second vaiation of the spead between two geodesics intesecting at a point P: η = Kη These popeties depend on the sign of K: if K > 0, C and A ae smalle than the flat space case, and two geodesic spead with deceasing speed; if K < 0, C and A ae bigge than the flat space case, and two geodesic spead with inceasing speed; if K = 0, we etieve the flat space esults. 5

6 Metic The metic povides a desciption of a given spacetime geomety which is not estained to the neighbouhood of a point. In an abitay 4-dimensional cuved geomety the line element will be a genealisation of the Minkowsky metic. In geneal, we define: (n-dimensional) Riemannian Space: is an n-dimensional space paametised by a set of n coodinates x µ (µ = 0,...,n 1), whose intinsic geomety is defined by a metic g µν (x) such that: Impotant emaks: ds 2 = g µν (x ρ )dx µ dx ν and detg 0 The invese metic g µν coesponds to the invese matix: g µρ g ρν = δ ρ µ. Example: A 2-dimensional Riemannian space is descibed by the line element ds 2 = x 2 dx 2 + xydy 2. The metic is g µν = ( x xy). The invese metic is g µν = ( x (xy) 1 ). The set of coodinates x µ is not necessay uniquely defined. One can have diffeent coodinate systems elated by a non-degeneate change of coodinates, and the fom of the line element will not in geneal be peseved when switching fom one set of coodinates to anothe. Example: Let s conside the Euclidean plane R 2 : in Catesian coodinate (x, y): in pola coodinates (, θ) ds 2 = dx 2 +dy 2 g ij = ds 2 = d dθ 2 g ij = ( ) ( ) It may happen that a single coodinate set is not sufficient to descibe the full space. We can encounte a coodinate singulaity. This is a point P, of coodinates x P such that one o moe elements of the metic diveges fo x = x P, but such that the cuvatue scalas behave well in that point. Example: The metic ds 2 = 1 ρ (dρ 2 +ρ 2 dθ 2 ) has a singulaity fo ρ = 0. 4 Is this a tue cuvatue singulaity? Does the geomety behaves badly fo ρ = 0? If we pefom the change of coodinates ρ = 1 the metic becomes ds 2 = d dθ 2. This is just the metic on the Euclidean plane in pola coodinates. The Euclidean plane is flat eveywhee, the point ρ = 0 coesponds to all the set of points at infinity which ae taken to a single point by the change of coodinates ρ. This simply means that the set of coodinates (ρ,θ) ae not suitable to descibe this point. Anothe example of coodinate singulaity is the Schwazschild adius (see late). 6

7 If a geomety can be descibed though its embedding into a highe dimensional space, its metic can be deived estaining the line element of this space on the lowe dimensional one. Example: S 2 can be embedded in R 3 R 3 line element: ds 2 = dx 2 +dy 2 +dz 2 ; each point on S 2 can be paametised by two angula coodinates (θ,φ) such that x = Rsinθcosφ, y = Rsinθsinφ, and z = Rcosθ. Implementing this in the line element above yields to the metic on S 2 : ds 2 S2 = R 2 (dθ 2 +sin 2 θdφ 2 ). Spacetime is a 4-dimensional Riemannian space. The equivalence pinciple suggests that locally the popeties of spacetime should be indistinguishable fom those of the flat spacetime of specialelativity. Thismeansthatgivenameticg µν (x)ateachpoint P of spacetime it is possible to intoduce a new set of coodinates x µ (x) such that g µν(x α P ) = η µν whee η µν is the flat Minkowsky metic and x α P of the point P. ae the coodinates Futhemoe one can find a change of coodinates such that locally in a point P: g µν(x α P ) = η µν and g µν x ρ P = 0 This is the mathematical desciption of a feely falling fame o locally inetial fame. Ifspacetimeiscuved,itisnot possibletofindachangeofcoodinates such that the metic becomes flat eveywhee. This can happen only if the whole spacetime is flat, and thee is no cuvatue (thee is no gavity gavity is spacetime cuvatue). The definition of timelike, spacelike and lightlike sepaations hold in cuved space as well. Massive paticles move along timelike tajectoies, and the set of lightlike cuves connected with a point P fom the light cone in P. Given a metic ds 2 = g tt dt 2 +g xx dx 2 +g yy dy 2 +g zz dz 2 the infinitesimal pope lenght along the diections t,x,y, and z ae espectively g tt dt, g xx dx, g yy dy, and g zz dz. the aea element along the diections x and y is da = g xx g yy dxdy; simila fomulae hold fo infinitesimal aea elements along diffeent diections; the infinitesimal 3-volume element is dv = g xx g yy g zz dxdydz; theinfinitesimal4-volumeelementisdw = g tt g xx g yy g zz dtdxdydz; 7

8 Tenso calculus Pinciple of Geneal Covaiance The laws of Physics can be stated in a fom which is independent on the specific choice of spacetime coodinates. The laws of Physics must be valid in any system of coodinates. Thus, they must be expessible as tenso equation. In this way, they ae covaiant: they etain the same fom upon a coodinate tansfomation. Coodinate tansfomations Let s focus on a d-dimensional spacetime geomety and let s conside an invetible coodinate tansfomation: x µ x µ, µ = 0,...,d 1. Unde such a coodinate tansfomation: a scala field is invaiant: φ(x) = φ (x ) a contavaiant vecto tansfoms as: whee Λ µ ν = x µ x ν ; a covaiant vecto tansfoms as: whee Λ ν µ = xν x µ. A µ (x) A µ (x ) = Λ µ ν Aν (x) B µ (x) B µ(x ) = Λ ν µ B ν (x) The contaction of a contavaiant and a covaiant vecto field is a scala unde a coodinate tansfomation, hence: Λ ν µ = (Λµ ν ) 1. Tensos A tenso T µ1µ2...µp ν 1ν 2...ν q is a multi-index mathematical object which tansfoms linealy and homogeneously unde a spacetime coodinate tansfomation: evey contavaiant (up) index tansfoms with Λ µ ν evey covaiant (down) index tansfoms with Λ ν µ Remaks. The total numbe of indexes denotes the ank of the tenso. Hence: 8

9 T µ1µ2...µp is a contavaiant tenso of ank p; T µ1µ 2...µ q is a covaiant tenso of ank q; T µ1µ2...µp ν 1ν 2...ν q is a mixed tenso of ank p+q; The null tenso (evey component equal to zeo) is zeo in any efeence fame. The sum of two tenso of one kind is still a tenso of the same kind: A µ (x)+b µ (x) = C µ (x) The poduct of two tensos is a tenso with ank equal to the sum of the anks of the two factos: A µ1µ2...µp B ν1ν 2...ν q = C µ1µ2...µp ν 1ν 2...ν q The metic is a covaiant tenso of ank 2. Unde a coodinate tansfomation it tansfoms as: Contaction of indexes: g µν (x) g µν(x ) = Λ ρ µ Λ σ ν g ρσ (x) contacting a covaiant and a contavaiant tenso we obtain a scala (e.g A µ B µ = A 0 B 0 +A 1 B 1 +A 2 B 2 +A 3 B 3 in 4 dimensions); we can also contact covaiant and contavaiant indexes in the same tenso: T µ µ T = T 0 0 +T 1 1 +T 2 2 +T 3 3; this genealises to the contaction of any pai of tenso indexes: T ρµ µ T ρ µν is a contavaiant tenso of ank 1, T µνσ T σ is a covaiant tenso of ank 1, etc. we can use the metic g µν and the invese metic g µν to aise and lowe the indexes: T µ = g µν T ν T µ = g µν T ν T µ µ = Tµν g µν = T µν g µν The Covaiant Deivative The odinay deivative of a scala is a tenso, but the deivative of a covaiant (o contavaiant) vecto is not. In fact, it does not tansfoms covaiantly as a tenso unde a coodinate tansfomation x µ x µ : µ φ(x) x x Λµ ν νφ(x) µ V ν (x) x x Λµ ρ Λ ν σ ρv σ (x)+ 2 x ρ x µ x νv ρ(x) 9

10 We ovecome this poblem by intoducing the covaiant deivative: D µ V ν (x) = µ V ν (x) Γ α µνv α (x) whee the set of coefficient Γ α µν fom the connection. The connection is not a tenso: unde a coodinate tansfomation, it tansfoms as: Γ α βγ x x Γ α βγ = Λα µ Λ ν β Λ ρ γ Γµ νρ +Λ ρ α 2 x ρ x β x γ The covaiant deivative of a covaiant vecto D µ V ν (x) tansfoms as a covaiant ank-2 tenso. We define similaly the covaiant deivative of a contavaiant vecto: D µ B ν (x) = µ B ν (x)+γ ν µρb ρ (x) Unde a coodinate tansfomation D µ B ν (x) tansfoms as a mixed tenso of ank 2: D µ B ν (x) x x D µb ν (x) = Λµ ρ Λ ν σd ρ B σ (x) These definitions ae easily genealised to tensos of abitay ank. Remaks. The covaiant deivative is distibutive: D µ (TS) = (D µ T)S+T(D µ S) (T and S ae tensos of abitay ank). The covaiant deivative of scala is equivalent to the odinay deivative: D µ φ(x) = µ φ(x) Demanding that the covaiant deivative commute with the opeation of aising and loweing indexes, D µ (A ρ g ρν ) (D µ A ρ )g ρν, we deive the futhe condition: D µ g ρν = 0 The covaiant deivative of the metic is identically zeo This last condition can be expanded and solved fo the connection coefficient, yielding to the definition of the Chistoffel Symbols: Γ α βγ = 1 2 gαµ ( β g µγ + γ g µβ µ g βγ ) We define the diectional covaiant deivative along a path x µ (s) as: D Ds = dxµ ds D µ. The diectional covaiant deivative does not change the ank of a tenso, it just moves it along the path! AtensoT µ1...µp ν 1...ν q ispaallelytanspotedalongthepathx µ (s)ifitsdiectionalcovaiantdeivativealongthatpathiszeo: dxρ ds D ρt µ1...µp ν 1...ν q = 0 10

11 Geodesics Equations Geodesics ae the equivalent in cuved space of what staight lines ae in flat space. They can then be defined as the cuves whose tangent vecto does not vay when moved along: the Geodesics of a cuved space ae the set of cuves along which the tangent vecto is paallely tanspoted. These ae the set of cuves satisfying the equations: d 2 x µ dx ν dx ρ ds 2 +Γµ νρ ds ds = 0 In agivengeomety, let sconsideacuvepaametisedbyx µ (s). Thevecto tangentto the cuveis u µ = dxµ (s) ds, and it is constantalongthe cuveif duµ ds = 0. But : 0 = duµ ds = dxρ ds ρu µ COVARIANTISE dxρ ds D ρu µ = 0 We obtain the geodesics equations by expanding the expession fo the covaiant deivative in the last fomula. Remaks. Geodesics ae the tajectoies followed by unacceleated paticles, subjected only to the action of gavity. In flat space and in Catesian ectilinea coodinates Γ µ νρ = 0, and the geodesics equations educe to d2 x µ ds = 0, i.e. they descibe the motion of 2 a paticle on a staight line. Fo timelike tajectoies the paamete s can be any affine paamete, namely any paamete elated to the pope time τ by a linea tansfomation: s = aτ +b, a and b constants. It is possible to deive the geodesic equations as the equations of the tajectoies extemising the pope time between the stating point A and end point B of the paticle tajectoy: τ AB = B A g µν dx µ ds dx ν ds ds Along the geodesics the following nomalisation condition holds: g µν dx µ ds dx ν ds = constant If the affine paamete s coincides with the pope time τ, then the constant is equal to 1. 11

12 Locally Inetial Fames Let s conside a Riemannian spacetime whose geomety is descibed by the metic g µν (x), in the set of coodinates x µ. Then, at any point P it is possible to intoduce a new set of coodinates, namely a new efeence fame, which we denote x µ, such that: g µν (x P ) = η µν and g µν x ρ P = 0 whee η µν is the Minkowsky metic and x P ae the coodinates of P in the new set. This is the fomal definition of a Locally Inetial Fame. Remaks: The condition g µν x ρ P = 0 is automatically satisfied if, in the new set of coodinates, Γ µ νρ P = 0. These elations hold locally in a small neighbouhood of a point P: one cannot find a unique coodinate tansfomation unde which they hold eveywhee, unless the geomety unde consideation is flat. The second deivatives of the metic ae non-zeo and they ente in the definition of the spacetime intinsic cuvatue in the neighbouhood of the point P. The spacetime distance between the point P and its neighbouing points can be eithe computed in the old set of coodinates o in the LIF coodinates, andtheycanbeeithenegative,positive, onull. Asusuallightays move along null diections, while massive paticles along timelike woldlines. Locally, light-cone stuctue is the same as in special elativity, it is the distibution of light cones in spacetime which vaies. The Riemann Cuvatue Tenso Paallel tanspot between two points along two diffeent paths yields to two diffeent esults. This is elated to the fact that we cannot exchange the action of two subsequent covaiant deivatives. In a cuved geomety the commutato oftwocovaiantdeivativesactingonafou-vectoa µ isnonzeo,andquantifies the spacetime cuvatue: D σ D ν A µ D σ D ν A µ = A ρ R ρ µνσ The tenso R ρ µνσ is the Riemann Cuvatue Tenso: R ρ µνσ = σ Γ ρ µν ν Γ ρ µσ +Γ α µσγ ρ αν Γ α µνγ ρ ασ. Remaks. 12

13 The components of the Riemann tenso depend on the second deivatives of the metic: in a cuved geomety they cannot be all put to zeo by changing to Riemann Nomal Coodinates; If, in a given spacetime, it exists a coodinate choice fo which R ρ µνσ = 0 fo evey value of the indexes, then that spacetime is flat. Symmeties of the Riemann Tenso: whee R µνρσ = g µα R α νρσ Ricci Tenso R µνρσ = R ρσµν = R νµρσ = R µνσρ Obtained by contacting the fist and thid indexes of the Riemann Tenso: Ricci Scala R µν = g αβ R αµβν Obtained by contacting the Ricci tenso with the metic: Einstein Tenso It is the combination: R = g µν R µν G µν = R µν 1 2 Rg µν Bianchi Identities D λ R ρσµν +D ρ R σλµν +D σ R λρµν = 0 The Bianchi Identities can be veified by ewiting D λ R ρσµν in Riemann Nomal Coodinates (such that Γ = 0 but Γ 0), and then summing ove the cyclical pemutations of λ, ρ and σ. If we contact the Bianchi identities witten above with g ρσ g µν we get: D µ G µλ = D µ( R µλ 1 ) 2 Rg µλ = 0 The Einstein tenso is conseved. Equation of Geodesic Deviation It descibes the elative acceleation between two geodesics x µ (s) and x µ (s) = x µ (s)+ξ µ (s): D 2 ξ µ Ds 2 = dx σ Rµ νρσ ξρdxν ds ds The paamete s is a geneic affine paamete along the geodesics. 13

14 Minimal Coupling Between Matte and Gavity Gavity is univesal: it can be descibed as a featue of the backgound whee paticles move athe than as a popagating field: fee paticles ae those subjected only to the effects of gavity; fee paticles move along geodesics defined by the geodesic equation. The minimal coupling between matte and gavity let us combine these findings with the laws of physics that we gathe fom special elativity, To minimally couple matte and gavity we: 1. take the laws of physics valid in an inetial fame in Minkowsky spacetime; 2. wite them in a covaiant fom; 3. asset that the esults which they descibe emain valid in a cuved geomety chaacteised by the metic g µν, then we get thei cuved space countepat by eplacing: η µν g µν µ D µ Examples. GEODESIC EQUATIONS The minimal coupling is exactly what we did when we deived the equations fo the geodesics. We: wote in a covaiant fom the equation demanding that the tangent vecto is constant along the geodesics: dx ρ ds ρu µ ; eplaced the odinay deivative ρ with the covaiant deivative D ρ. MAXWELL EQUATIONS In covaiant fom in Minkowsky spacetime they ead µ F µν = j ν. Minimally coupled with gavity, they become: D µ F µν = j ν whee all the contactions ae pefomed with the metic g µν. CONSERVATION LAWS. In a covaiant fom and in flat spacetime the consevations of enegy and momentum is encoded in the consevation of the stess enegy tenso T µν. Then: µ T µν = 0 min.coupling D µ T µν = 0 14

15 Newtonian limit of GR We must ecove the esults of Newtonian gavity unde the following conditions: 1. The paticles move slowly with espect to the speed of light. 2. Gavity is weak, and can be teated as a petubation of flat spacetime. 3. Fields ae static: no dependence on the coodinate time t. Newtonian limit of the geodesics equations Let s focus on the tajectoies of timelike paticles, which we paametise with thei pope time τ. At the level of the geodesic equation, the thee conditions above ae implemented by equiing: 1. dx i dτ dt dτ i = 1,2,3 2. g µν = η µν +h µν with h µν << 1 3. t g µν = 0 µ,ν = 0,1,2,3. Implementing this condition in the geodesic equation we get: d 2 x µ dx ν dx ρ dτ 2 +Γµ νρ dτ dτ = 0 ifµ = 0 = dt dτ = const. ifµ = i 0 = d 2 x i dt 2 = 1 2 i h 00 Thefistelationtellsusthatthecoodinatetimecanbeusedaspaamete along the geodesics, while the second elation shows that, upon identifying h 00 2φ( x), whee φ( x) is the Newtonian potential, we get exactly the Newton equation fo gavity: d 2 x dt 2 = φ( (x)) The identification between h 00 and φ holds if we can wite g 00 = (1+ 2φ(x)), namely if this component of the metic is a solution ofthe Einstein equations (the equations that detemine the explicit fom of the metic). Einstein Equations The Einstein equations ae the elativistic genealisation of the Poisson equation fo the Newton potential 2 φ = 4πGρ(x), whee ρ(x) is the mass density. They have the fom: TENSOR DESCRIBING THE SPACETIME GEOMETRY = TENSOR DESCRIBING THE SPACETIME ENERGY CON- TENT 15

16 The Stess-Enegy tenso This is the tenso descibing the matte and enegy content of spacetime. Physical definition: T µν = Flux of fou-momentum p µ acoss a suface of constant x µ Sepaating the 0 coodinate fom i = 1,2,3, we identify: ( ) T 00 T 0j T i0 T ij whee T 00 : mass-enegy density; T i0 : flux of enegy; T 0j : momentum density; T ij : momentum flux: stess Fo a pefect fluid whee T µν fluid = (ρ+p)uµ U ν +pg µν ρ is the mass-enegy density of the fluid; p is the pessue (same in evey diection); U µ is the aveage macoscopic 4-velocity. The stess-enegy tenso is conseved: D µ T µν = 0. If we integate this elation ove a volume V we ecove the equations fo the consevation of enegy (integating the elation fo ν = 0) and fo the consevation of the thee components of the linea momentum (integating the thee elations fo ν = i 0). Complete fom of the Einstein Equations R µν 1 2 Rg µν +Λg µν = 8πGT µν whee: g µν is the metic descibing the geomety of a given spacetime. R µν = R αµβν g αβ is the Ricci tenso. R = R µν g µν is the Ricci scala. 16

17 T µν is the stess-enegy tenso, which descibes the matte and enegy content of a given spacetime. Λ is the cosmological constant, which descibes the vacuum enegy density, namely the enegy density of empty space. Einstein equations in the vacuum In the vacuum T µν = 0, then the Einstein equations educe to: Remaks. R µν = 0 These equations can be deived compaing the equation fo the geodesic deviation with the equation fo the tidal foces in the case when the mass density of the gavitational souce is zeo. In the static and weak field limit they educe to the Poisson equation in empty space: R µν δ µν 2 φ(x) then R 00 = 0 2 φ(x) = 0. The Schwazschild Metic It is the unique static and spheically symmetic solution of the Einstein equations in the vacuum R µν = 0. In spheical coodinates (t,,θ,φ): ( ds 2 = 1 2GM ) ( dt GM ) 1d (dθ 2 +sin 2 θdφ 2 ) whee M is the mass soucing the gavitational field and G is the univesal gavitational constant. The coefficients of the metic becomes singula fo = 0 and = 2GM. The quantity S 2GM is called Schwazschild adius and it is a coodinate singulaity of the Schwazschild metic (if c 1 S 2GM c ). When 2 = 2GM the coefficient g of the metic becomes singula, howeve this is due just to the paticula coodinate set we ae using, and the geomety behaves well. To pove it one must conside scalas built out of the cuvatue Riemann tenso, and check if they ae finite o if they divege in the singula points of the metic. Fo example one can conside R αβγδ R αβγδ = 48G2 M 2 We see that R αβγδ R αβγδ 0, so = 0 is a tue cuvatue singulaity, while R αβγδ R αβγδ R=2GM = 3 4G 4 M 4 is finite at the Schwazschild adius. 17 6

18 The Schwazschild adius has a physical meaning only fo ulta-compact objects, such that this distance is still in the vacuum, and the neighbouing geomety is still descibed by the Schwazschild metic (e.g., the Schwazschild adius of the Sun 2GM c is about 2.9km, well inside the 2 suface of the sta, whee the geomety is no longe descibed by the Schwazschild solution). The Schwazschild adius is the distance of the event hoizon of a black hole. The weak field limit of the Schwazschild metic holds fa fom the mass soucingthegeomety,namelyfo 2GM. Atsuchdistancesthemetic can be seen as a petubation of the Minkowsky metic: g µν = η µν +h µν. But we can ewite g 00 = 1 + 2GM = η 00 2φ(), whee φ() is the Newtonian potential. Similaly, if we take the weak field limit of g we get: ( g = 1 2GM and the metic becomes: ) 1 2GM 1+ 2GM = η 11 2φ() ds 2 2GM (1+2φ())dt 2 +(1 2φ())d (dθ 2 +sin 2 θdφ 2 ) To wite this as a petubation of Minkowsky, we can choose h µν = 2φδ µν. The metic tends to Minkowsky fo. This phenomenon is called asymptotic flatness, and it is a consequence of the metic, not an ansatz. The metic tends to Minkowsky fo M 0 (no mass soucing the geomety). Tests of Geneal Relativity Thee phenomena pedicted by Geneal Relativity which can be consideed as expeimental tests of the theoy ae: The pecession of planet s peihelia The bending of light ays in poximity of a gavitational souce The gavitational edshift To quantify the effects ofthe fist twophenomenawe need toknowthe obits of test paticles in the Schwazschild geomety. Spheical symmety constaints the obits to lay on planes of constant θ. Let s fix θ = π 2. 18

19 Obits of massive paticles Timelike obits can be computed by exploiting the following conditions: Consevation of enegy (because of no time dependence): ( 1 2GM ) dt dτ = constant = k Consevation of angula momentum(because of spheical symmety): 2dφ dτ = constant = l nomalisation condition along timelike geodesics g µν dx µ ( 1 2GM ) d 2 t dτ 2 ( 1 2GM dτ ) 1d 2 dτ 2 2d2 φ dτ 2 = 1 dx ν dτ = 1: Combining these thee conditions we find that paticles move along the obits of the effective potential V eff () = MG + l2 2 2 l2 MG 3 We have bigge discepancy with espect to the Newtonian effective potential V N () = MG + l2 2 fo smalle values of : seach fo measuable 2 effects in the obits of innemost planets of the Sola System. Pecession of planets peihelia Intoducing u = 1, the vaiation of u with is encoded by the equation: d 2 u MG +u = dφ2 l 2 +3MGu 2 while the Newtonian effective potential gives: d 2 u dφ 2 +u = MG l 2. The exta tem is small fo sola system planets, and can be teated as a petubation, leading to the appoximate solution u GR = GM l 2 [ 1+ecos [ φ ( 1 3G2 M 2 l 2 )] ] u GR is a peiodic function of peiod Fo Mecuy 2π 1 3G2 M 2 l 2 2π 1 3G2 M 2 l 2 > 2π. 2π 42.9 pe centuy. 19

20 Photons obits and deflection of light ays Lightlike tajectoies can be woked out exactly in the same way as fo timelike, but now consideing that fo lightlike tajectoies ds 2 = 0, then the nomalisation condition along the geodesic eads: ( 1 2GM ) d 2 t dτ 2 ( The equation fo u(φ) now eads: 1 2GM d 2 u +u = 3MGu2 dφ2 ) 1d 2 dτ 2 2d2 φ dτ 2 = 0 If M = 0 the equation this is the equation fo a staight line When M 0 it is solved by: whee R is an integation constant. 1 = sinφ R + MG 2R (3+cos2φ) in Catesian coodinates the distance fom the gavitational souce along the y axes vaies as: y = R GM R y 2 +2x 2 x2 +y 2, and the total deflection fom x to x + is equal to Θ 2 y 4GM x x + R (if c 1 4GM Rc 2 ) The deflection of stalight by the Sun is about Gavitational edshift Light climbing a gavitational field gets edshifted. Let s conside an atom in a gavitational field sitting at a distance e, which emits adiation with fequency ν e. The emitted pulse tavels adially towads an obseve sitting at a distance 0. If the metic is static (i.e., it does not depend on time), the obseved adiation will have fequency ν o such that: ν 0 ν e = g 00 ( e ) g 00 ( o ) whee g 00 is the coefficient of dt 2 in the metic. 20

21 In Schwazschild geomety: Remaks. ν 0 ν e = 1 2GM e 1 2GM o To deive the fomula fo the gavitational edshift one must explicitly use the fact that the metic does not depend on the coodinate time t, and then in coodinate time the peiod of the emitted and of the obseved wave is the same. If we ae fa fom the souce ( 2GM), and νo ν e 1 GM e + GM o = 1+φ( e ) φ( o ), whee φ() is the Newton potential. Since φ( e ) > φ( 0 ), ν o < ν e, i.e., the adiation gets edshifted. Effective speed of light The cuvatue induced aound a massive body inceases the tavel time of light ays elative to what it would be in flat space. Fo timelike geodesic ds 2 = 0, and if we focus on adial tajectoies dφ = dθ = 0, then: ( 1 2GM Time delay ) ( dt 2 1 2GM ) 1d 2 = 0 d dt = ( 1 2GM ) < 1 Time uns slowe in a gavitational field is compaed to flat spacetime. Fo a stationay obseve (d = dθ = dφ = 0), the pope time is in Schwazschild: dτ = 1 2GM dt in flat spacetime: dτ = dt whee t is the coodinate time measued by an obseve at infinity. Black Holes The Schwazschild solution descibes static black holes with no electic chage. The Schwazschild adius R s = 2GM is a coodinate singulaity of the geomety: spacetime cuvatue is finite fo = R S, and this featue is an atifact of the coodinate system we ae using, which ae not suitable to descibe this geomety fo 2GM. What does it happen to the metic when < 2GM? if > 2GM: g tt < 0 and g > 0 21

22 if < 2GM: g tt > 0 and g < 0 Thus consideing a woldline along the t-axes (descibing two events happening in the same position at diffeent time): ( ds 2 = 1 2GM 1 2GM ) dt 2 ր < 0fo > 2GM TIMELIKE ց > 0fo < 2GM SPACELIKE thus passing fom > 2GM to < 2GM the time and space chaacte of the coodinates do evese. ( ) 2GM g tt = 0 suface of infinite edshift. Behaviou of light cones along adial paths Along adial lines dθ = dφ = 0, and along the path of light ays ds 2 = 0, then the slope of the light cone is: dt ( d = ± 1 2GM ) 1 ( and + efe to outgoing and ingoing light ays) ±1 fo descibes light cones in odinay Minkowsky spacetime dt d Focusing only on the outgoing cuves (sign +), dt d + fo 2GM the light cones become moe and moe naow while getting close to the event hoizon. Passing fom > 2GM to < 2GM flips the inne and the oute pat of the light cones: the ole of and t is exchanged, and fo outgoing paticles deceases as the time flows: the paticles ae bound to move towads = 0. Path of a adially infalling paticle We want to descibe what happens to an obseve falling feely and adially into a spheical black hole fom a fixed distance 0 > 2GM. We want to descibe his motion: fom his point of view; fom the point of view of anothe obseve, sitting still at the same distance 0. Let s suppose also that the fee-falling obseve caies a lamp which emits a sequence of light pulses at constant pope time sepaation τ. We see that: 22

23 The pope time that the obseve takes to each a given distance < 0 is: τ = 1 3( 3 3) GM 2 Fo the infalling obseve, nothing special happens at = 2GM. Neglecting the action of tidal foces, he will just coss the hoizon and keep on falling until when, afte a finite amount of pope time, he will each the cental singulaity at = 0. The otheobsevesitting still at 0 measuesthe time with the coodinate time t. He sees the feely-falling obseve eaching the distance such that 2GM < < 0 afte a time: 2 t = 3 2GM This means that: [ ( ) ( 0 + 2GM 2GM) 0 +6GM ( ) ] 0 +2GM ln ( 0 + ( + ) 2GM) 2GM if both and 0 aemuch biggethan the Schwazschildadius 2GM, than t τ, and the light pulses will each the still obseve at sepaation t τ; t fo 2GM. So, the still obseve will measue an infinite Schwazschild coodinate time fo the fee-falling obseve to each the hoizon, and only the light pulses emitted befoe the falling obseve cosses the hoizon will each him. The still obseve will eceive the light pulses at intevals t < t < t... which gow lage and lage, and the pulses of light will be moe and moe edshifted. Eddington-Finkelstein coodinates Rewitten in this set ofcoodinatesthe metic is nolongesingulaat = 2GM. They ae defined as the set of coodinates ( t,,θ,φ) such that the new time coodinate t is: t = t+2gm ln 2GM The metic becomes: ( ds 2 = 1 2GM ) d t 2 + ( 1+ 2GM ) d 2 + 4GM dd t+ 2 (dθ 2 +sin 2 θdφ 2 ) and it is egula fo = 2GM. Studying the behaviou of adial light cones we find that: d t d = 1 d t d = +2GM 2GM (1a) (1b) 23

24 (1a) is always oiented at 45deg with espect to the vetical axis (1b) tends to +1 (flat space) fo and it tends to infinity fo 2GM, and when 0 < < 2GM it deceases until the limit value 1. the futue light cone is always diected towads 0 Remaks: The suface = 2GM is a coodinate singulaity which can be emoved via a change of coodinates. = 2GM defines a membane called event hoizon of the black hole. It is possible fo futue diected tajectoies to coss it fom > 2GM to < 2GM, but the evese is not possible. Moving towads smalle, the light cones begin to tip ove. At = 2GM lightlike diections ae stationay. Fo < 2GM all futue diected lightlike and timelike tajectoies ae diected towads = 0. Cosmology Cosmology is the study of the univese as a whole. It is based upon the: Cosmological Pinciple: The univese is homogeneous and isotopic. Homogeneous: the metic is the same thoughout space. Isotopic: fom any point space look the same in any diection. Remaks. The cosmological pinciple is suppoted by obsevation Obsevation shows that distant galaxies ae ecessing fom us: the univese is not static, hence it is homogeneous and isotopic only in space, and not in time. The metic can be put in the fom: whee ds 2 = dt 2 +a 2 (t)γ ij (x)dx i dx j t is the cosmological time, x i,i = 1,2,3 ae the thee spatial coodinates, othogonal to the time diection fo an obseve sitting at constant x i (comoving). a(t) is the scale facto: it says how much big the space pat of spacetime is at any given coodinate time t. γ ij (x) is the maximally symmetic metic on the thee-dimensional space. 24

25 Implementing the homogeneity and isotopy conditions one finds: [ ds 2 = dt 2 +a 2 d 2 ] (t) 1 K 2 +2 (dθ 2 +sin 2 θdφ 2 ) This metic is called Robetson-Walke metic. The paamete K can assume the following values: K = 1 the spatialpathasconstantpositivecuvatue, it is closed; K = 0 the spatial pat has zeo cuvatue, it is flat; K = 1 the spatial pat has constant negative cuvatue, it is open. The Einstein equations become a set of diffeential equations fo the scale facto a(t). Matte and enegy content of the univese The univese is modelled as a pefect fluid at est in comoving coodinates: T µν = (ρ+p)u µ U ν +pg µν with U µ = (1,0,0,0) U µ = ( 1,0,0,0). It is convenient to conside T µ ν = g µα T αν ρ T µ ν = 0 p p p The tace of the stess-enegy tenso is: T µ µ = ρ+3p The stess-enegy tenso is conseved: D µ T µ ν = 0. The equation fo the consevation of enegy is obtained selecting ν = 0. One gets: ρ+3ȧ a (ρ+p) = 0 whee the dot indicates a deivative by the cosmological time t. An equation of state elates the pessue p and the enegy density ρ: p = ωρ whee ω = constant The consevation of enegy leads to: ρ ρ = 3ȧ (ω +1) a ρ a 3(ω+1) Examples: 25

26 Dust: collisionless and non-elativistic matte no pessue: p = 0. Then ω = 0 and: ρ M a 3 The numbe density of paticles deceases as the univese expands. Radiation: eithe pope EM adiation o paticles moving with velocity close to the speed of light. One has p R = 1 3 ρ R and: ρ M a 4 The enegy density of adiation falls faste than the enegy density of dust. Vacuum: epesented by a vacuum stess-enegy tenso T (vac) µν = Λ 8πG g µν. Is like a pefect fluid with ω = 1, fom which we get: ρ (vac) = constant the vacuum enegy density is independent on size and shape of the univese. If ρ M pevails ove the othes: univese is matte dominated. If ρ R pevails, univese is adiation dominated. If ρ (vac) pevails: univese is vacuum dominated. Einstein equations Neglecting the cosmological constant tem, the Einstein equations become a pai of coupled equations detemining the evolution of the scale facto: ä a = 4πG 3 ( ρ+3p ) (ȧ a ) 2 = 8πG 3 ρ K a 2 These equations ae called Fiedmann equations. The Robetson-Walke metic whose scale facto is a solution of these equations is called Fiedmann-Robetson- Walke metic. Cosmological paametes Hubble paamete: defines the expansion ate of the univese: H(t) = ȧ(t) a(t) Hubble constanth 0 isthehubblepaameteatpesenttime: 70km/sec/Mpc < H 0 < 100 km/sec/mpc. Hubble lenght: d H = H 0 c h 1 Mpc; 26

27 Hubble time: t H = H 1 0 = h 1 y. Deceleation paamete: measues the ate of change of the expansion velocity: q = aä ȧ 2 Density paamete: Ω = 8πG 3H 2ρ = ρ ρc whee the citical density ρc = 3H2 8πG detemines the sign of K, i.e., univese s geomety: Hubble Law Ω < 1 ( ρ < ρc) K < 0 (open univese); Ω = 1 ( ρ = ρc) K = 0 (flat univese); Ω > 1 ( ρ > ρc) K > 0 (closed univese); Obtained expeimentally, elates the ecessing velocity of distant galaxies to thei distance: v = H 0 d Can be obtained diectly fom the Fiedmann-Robetson-Walke metic by computing the adial distance between distant galaxies. Scale Facto evolution Obtained by solving the Fiedmann-Robetson-Walke equations once they ae given the spatial cuvatue K and the enegy densities ρ R, ρ M, and ρ (vac). If ρ > 0 AND p 0, and Λ = 0 (enegy o matte dominated univese), we get: ä < 0 The univese is deceleating, and we know fom obsevations that the univese is expanding. It was expanding faste in the past, and we can tace this expansion back to the singula state at t = 0: big bang. In paticula if K 0 (flat and open cases): ȧ 2 = 8πG 3 ρa2 + K The ight hand side is geate than zeo. ȧ has neve passed though the zeo, and it must have been positive all the time: open and flat univeses expand foeve. But ȧ 2 a(t) K 27

28 thus K = 1: evolution appoaches the limiting value ȧ = 1; K = 0: univese keeps expanding but moe and moe slowly. K = +1 (closed univese). We get: ȧ 2 = 8πG 3 ρa2 1 Fo ȧ 2 not to become negative, the univese cannot expand foeve, but a(t) must have an uppe bound avac. As a appoaches avac, ä is always negative: the univese keeps on contacting. Closed univeses ae closed in time too. The pecise fom of a(t) changes accodingly if the univese is matte o adiation dominated: Radiation dominated: a(t) t. Matte dominated: a(t) t 2 3. Empty univeses (only vacuum enegy): have eithe p < 0 o ρ < 0, and pevious consideations don t hold anymoe. We can distinguish: Λ < 0: it exists only an open (K = 1) solution with a(t) e Ht. This is called anti de Sitte univese. Λ > 0: they exist solutions fo K = 1, K = 0, and K = +1, all called de Sitte univeses. 28