General Relativity. III Escuela Mexicana de Cuerdas y Supersimetría. Gustavo Niz June 2013

Transcription

1 General Relativity III Escuela Mexicana de Cuerdas y Supersimetría Gustavo Niz June 2013

2 General Relativity Based on: Sean Carrol's notes (short ones) John Stewart course/book Personal notes

3 GR as a Cooking Book Ingredients Cookware Physics & Maths Equations Recipes Solutions

4 Ingredients Why GR? Newtonian gravity Particle moves in straight line until hit by a force Vs Einstein's gravity Gravity is not a force but a result of space-time geometry

5 Ingredients Why GR? Newtonian gravity Particle moves in straight line until hit by a force Vs Einstein's gravity (GR) Gravity is not a force but a result of space-time geometry Keywords

6 Ingredients Need to understand a) Space and time together b) Curved space

7 Ingredients Need to understand a) Space and time together b) Curved space Tensor Notation

8 Ingredients: Special Relativity Speed of light is constant in any frame (c=1) Space and time are entangled Coordinates of space-time

9 Ingredients: Special Relativity Special Relativity (SR) lives in Minkowski space-time 4-vector Event P

10 Ingredients: Special Relativity Minkowski metric Distance (dot product) Summation convention Infinitesimal distance (line element) Same information as in the metric

11 Ingredients: Special Relativity Lorentzian signature (-,+,+,+) [ sometimes (+,-,-,-) ] C.f. Euclidean 4d space Euclidean signature (+,+,+,+)

12 Ingredients: Special Relativity Lorentz group Transformations that leave ds unchanged e.g. x-direction boost

13 Ingredients: Special Relativity Lorentz group Transformations that leave ds unchanged e.g. x-direction boost

14 Ingredients: Special Relativity For a particle moving only in time (xi=const.) the elapsed time is Define proper time as

15 Ingredients: Special Relativity Spacetime diagram

16 Ingredients: Special Relativity Spacetime diagram Null vector Defines light-cone

17 Ingredients: Special Relativity Spacetime diagram Null vector Timelike vector Spacelike vector

18 Ingredients: Special Relativity Trajectory specified by is timelike/spacelike/null if tangent vector is timelike/spacelike/null

19 Ingredients: Special Relativity Null trajectories represent light or massless particles Spacelike trajectories are causally disconnected points (or particles with v>1) Timelike trajectories describe massive particles (observers) More convenient to use proper time

20 Ingredients: Special Relativity Massive particles 4-velocity 4-momentum m rest frame mass

21 Ingredients: Special Relativity Massive particles 4-velocity 4-momentum Energy is, which in the rest frame, is

22 Ingredients: Special Relativity Massive particles One can boost the frame with a Lorentz transf., say in the x-dir. Which for small v reduces to Rest mass + kinetic energy Newtonian momentum

23 Ingredients: Curved Space Minkoswki Cartesian coords. Manifold Cannot use Cartesian coords. globally

24 Ingredients: Curved Space Manifold Def. A topological space M is an n-dimensional manifold if there is a collection (atlas, of maps (charts, ), such that each map is continuous, bijective and invertible )

25 Ingredients: Curved Space In simple words

26 Ingredients: Curved Space Equivalence Principle. A choice of chart (coordinates) is arbitrary and the physics should not depend on this! A convenient way to describe chart-independent equations is to use tensors, which can be thought of generalisations of vectors, with possibly more indices

27 Ingredients: Tensors Under a coordinate transformation tensor way a changes in the following (n,m) - Tensor (summation convention)

28 Ingredients: Tensors (0,0) tensor scalar (function) (0,1) tensor vector (1,0) tensor co-vector De fin iti o ns Observations

29 Ingredients: Tensors Observations M at hs Trace over indices Symmetric Antisymmetric

30 Ingredients: Tensors Observations Ph ys i cs An equation that holds in one coordinate system holds in all coordinate systems Equivalence Principle!

31 Ingredients: Tensors In GR the most important tensor is the metric a generalisation of Minkowski's metric which encodes the geometrical information of spacetime

32 Ingredients: Tensors At a given point, there is always a coordinate system such that i.e. the spacetime looks locally flat and also first derivatives of vanish! Second derivatives cannot be made to vanish, a manifestation of curvature

33 Ingredients: Tensors Dot product Suggests the concept of raising/lowering indices, namely where so is the inverse of

34 Ingredients: Tensors Important objects in physics which are not tensors 1) determinants Under transforms as

35 Ingredients: Tensors Important objects in physics which are not tensors 2) volume factor Transforms as 1) and 2) are call tensor densities because transform as some powers of the Jacobian

36 Ingredients: Tensors Important objects in physics which are not tensors Notice Is invariant under a change of coordinate, since f(x) is a scalar

37 Ingredients: Tensors Important objects in physics which are not tensors Notice Is invariant under a change of coordinate, since f(x) is a scalar Therefore the right way of writing integrals is using

38 Ingredients: Tensors Important objects in physics which are not tensors 3) Partial derivatives On scalar are OK...but on vectors,

39 Ingredients: Tensors Important objects in physics which are not tensors 3) Partial derivatives On scalar are OK...but on vectors,

40 Ingredients: Tensors Define the Covariant Derivative Connection transforms to cancel this

41 Ingredients: Tensors transforms a tensor and defines parallel transport

42 Ingredients: Tensors transforms a tensor and defines parallel transport Similarly for lower indices

43 Ingredients: Tensors Christoffel Symbols There is a particular choice of connection that In components, it is

44 Ingredients: Curvature Information about the curvature is contained in the metric tensor, and it is the Riemann tensor which explicitly accounts for it (note: it has second derivatives of the metric)

45 Ingredients: Curvature Geometrical interpretation In Euclidean space if a vector is parallel transported around a closed loop, it returns unchanged. In curved space, this is not necessarily true. The Riemann tensor measures the difference!

46 Ingredients: Curvature Properties of Antisymmetric 1) Symmetric 2) (Bianchi identity) 3)

47 Ingredients: Curvature Other tensors derived from 1) Ricci tensor It satisfies 2) Ricci scalar

48 Ingredients: Curvature Other tensors derived from 3) Einstein tensor It obeys

49 Ingredients: exercises 1) Work out the Christoffel symbols for the metric 2) Using the previous result, write down the covariant derivative of the following vector

50 Cookware: Geodesic Equation A geodesic is a curve that extremises The path where is a geodesic if it obeys is the affine parameter

51 Cookware: Geodesic Equation In GR test particles always move along geodesics. Massless particles follow null geodesics (null trajectory) Massive bodies follow timelike geodesics (timelike trajectory) Objects move along geodesics until a force knocks them off!

52 Cookware: Einstein Equations Einstein (1915)

53 Cookware: Einstein Equations Energy Momentum tensor Non-linear equations of the metric components!

54 Cookware: Einstein Equations Dimension-full parameters Newton's Constant If c=h=1 then

55 Cookware: Einstein Equations Geometría del Espacio-tiempo Materia/Energía

56 Cookware: Einstein Equations El espacio-tiempo es como una manta invisible deformado por la materia o energía

57 Cookware: Einstein Equations Cauchy Problem. e.g. Maxwell eqns. No t derivatives Constraints 1st order t der. Evolution eqns. Give suitable initial data so that the system has only one solution. We need E(0,x), B(0,x) which satisfy the constraints, then E(t,x), B(t,x) are obtained from the evolution equations

58 Cookware: Einstein Equations Cauchy Problem. Similar for RG 1) No t derivatives Constraints 2) 1st order t der. Constraints 3) 2nd order t der. Evolution eqns. Find for, which satisfy the constraints initially, and use the evolution equations to solve for The equations are linear in second derivatives!

59 Cookware: Einstein Equations Relevant components of the metric has 10 independent components (remember it is symmetric), but there are 6 evolution equations to determine them (?). Actually, we have an arbitrary choice of coordinates. Therefore, there are only 10-4=6 variables to determine, which can be found using the evolution equations.

60 Cookware: Einstein Equations Energy momentum tensor A popular choice is a perfect fluid Fluid with no viscosity, or heat flow, and isotropic in its rest frame Completely specified by energy density and pressure where is the unitary 4-velocity Bianchi identity implies conservation

61 Cookware: Einstein Equations Newtonian limit Recall Poission equation for the Newtonian potential 2nd order diff. operator over a field

62 Cookware: Einstein Equations Newtonian limit Recall Poission equation for the Newtonian potential 2nd order diff. operator over a field In a relativistic theory we expect the energy momentum tensor

63 Cookware: Einstein Equations Newtonian limit Recall Poission equation for the Newtonian potential 2nd order diff. operator over a field In a relativistic theory we expect the energy momentum tensor Einstein equations

64 Cookware: Einstein Equations Newtonian limit Now formally consider (perturbation around Minkowski) Geodesic Equation + slow motion Newton's 2nd law

65 Cookware: Einstein Equations Newtonian limit Now formally consider (perturbation around Minkowski) Geodesic Equation + slow motion Newton's 2nd law 00-component of Einstein eqns Poisson's eq.

66 Cookware: Hilbert-Einstein action Einstein equations can be obtained from Einstein equations

67 Cookware: Hilbert-Einstein action Can include a Cosmological Constant

68 Cookware: Alternative approaches 1) á la Palatini Consider the connection and the metric as independent The metric connection

69 Cookware: Alternative approaches 2) Hamiltonian (ADM or 3+1 formalism) Consider the space-time splitting N lapse function Ni shift function gij spatial metric Schematically

70 Cookware: Alternative approaches 2) Hamiltonian (ADM or 3+1 formalism) Extrinsic curvature Then one gets the Hamiltonian Hamiltonian Momentum Canonical momenta

71 Cookware: Alternative approaches 2) Hamiltonian (ADM or 3+1 formalism) Observations Only is dynamical are Lagrange multipliers which lead to (in vacuum) Dynamical D.O.F. 6 (1+3) = 2 two polarization modes of the graviton

72 Recipes: content Maximally symmetric solutions Penrose Diagrams Black Hole solutions Schwarzschild Others Thermodynamics Cosmology The Hot Big Bang Dark Matter/Energy Early Universe (inflation, cyclic)

73 Recipes: Maximally symmetric solns Consider vacuum with a cosmological constant There are 3 maximally symmetric solutions Minkowski De Sitter (ds) Anti-de Sitter (AdS)

74 Recipes: Penrose diagrams Causal and topological structure Infinite is brought to a finite distance Light rays propagate at 45 Conformal factors do not change the lightcone structure, so can be dropped Useful to study black holes

75 Recipes: Penrose diagrams Consider Minkowski spacetime Change variables to

76 Recipes: Penrose diagrams Bring infinite to fine distance with

77 Recipes: Penrose diagrams Back to the Cartesian coords. Conformal metric

78 Recipes: Penrose diagrams Half of 3-sphere

79 Recipes: Penrose diagrams Take half of triangle are to attach a full 3-sphere at each point

80 Recipes: Penrose diagrams Future timelike infinity Future null infinity Spatial infinity Past null infinity Past timelike infinity

81 Recipes: de Sitter (ds) Easier to picture in 5d

82 Recipes: de Sitter (ds) By choosing following useful frames: Static coordinates Cosmological coordinates Exponential expansion one can write the metric in the

83 Recipes: de Sitter (ds) Penrose diagram Both timelike and null future infinity

84 Recipes: de Sitter (ds) Penrose diagram

85 Recipes (detour): Horizons Particle horizon Event horizon

86 Recipes: Anti de Sitter (AdS) Maximal symmetry 5d picture Two times

87 Recipes: Anti de Sitter (AdS) Closed Timelike Curves (travel back in time?) Careful analysis shows there are not

88 Recipes: Anti de Sitter (AdS) By choosing one can write the metric as Static coordinates Penrose diagram: cannot be drawn well... Relevant for AdS/CFT

89 Recipes: Solutions so far... Solution Symmetry Minkowski 0 0 Maximal de Sitter >0 0 Maximal Anti de Sitter <0 0 Maximal Schwarzschild 0 Spherical

90 Recipes: Schwarzschild solution Birkhoff's theorem If one assumes spherical symmetry in GR in vacuum (using the gauge freedom can fix ), then The solution is unique Static (i.e. Characterised by one integration constant alone (M) do not depend on time) It is called the Schwarzschild solution

91 Recipes: Schwarzschild solution Mass inside Ro Schwarzschild solution is Since it is unique, it describes from black holes to planets! Ro BH or planet Vacuum Schwarzschild sol.

92 Recipes: Schwarzschild solution Test particle in the plane Particle of unit mass and energy E in 1d potential Kinetic potential energy conservation For the Schwarzschild metric the potential is Const. Newtonian angular potential momentum Only GR correction (Mercury)

93 Recipes: Schwarzschild solution Re-write V(r) as Particle cannot scape the gravitational potential unless is traveling faster than light For a star one hits the surface before A black hole is such that matter has collapse inside its Schwarzschild radius,

94 Recipes: Schwarzschild solution Two singularities 1) 2) True singularity Coordinate singularity If Nothing happens at diverge bad frame

95 Recipes: Schwarzschild solution However, something special still happens at Event horizon Sees (1) moving slower and slower, taking infinite time to reach

96 Recipes: Schwarzschild solution However, something special still happens at (1) takes a finite time to cross then r and t swap roles and (1) goes straight into r=0 Event horizon Sees (1) moving slower and slower, taking infinite time to reach

97 Recipes: Schwarzschild solution However, something special still happens at

98 Recipes: Schwarzschild solution Penrose diagram

99 Recipes: Schwarzschild solution Penrose diagram (completion of spacetime)

100 Recipes: Kerr BH Add rotation

101 Recipes: Kerr BH Momentum bound Extremal case (BPS, stable, in nature?)

102 Recipes: Reissner Nordström BH Electric charge Extremal case when Charged+Rotating Kerr-Newman solution

103 Recipes: Kerr, KN or RN

104 Recipes: Kerr, KN or RN

105 Recipes: Kerr, KN or RN Black holes have no hair If matter collapses into a black hole, then all information would be lost, since the later is described by its Mass, Charge and Angular Momentum only. Moreover, due to quantum processes, it emits isotropic (Hawking) radiation in a rate proportional to its mass (M). Therefore, it looses energy and evaporates in a time of the order The associated radiation temperature is related to its surface gravity (acceleration of observer on its horizon).

106 Recipes: BH thermodynamics 0th law Thermodynamics Black Holes Constant T Constant surface in thermal eq. gravity (at horizon) 1st law 2nd law 3rd law No system with S=0 in finite series of steps No BH with

107 Recipes: BH thermodynamics Implication Temperature Surface gravity Entropy Area of horizon

108 Recipes Cosmology Observations Hot Big Bang Theory Early Universe ideas

109 Recipes: Cosmology Until the XX century, the Universe was though to be finite and as big as the Milky Way (problem: how to measure distances) Distance ladder Cepheids Real luminocity depends on period Apparent luminocity provides distance

110 Recipes: Cosmology Now we know: 0) Dark 1) Bigger 2) Expanding Velocity = H (distance) Hubble Diagram

111 Recipes: Cosmology Implies a denser/hotter past Cosmic Microwave Background (CMB) T=2.7 K

112 Recipes: Cosmology Small perturbations of 1 in Simple UNIVERSE Gaussian Scale invariant (tilt)

113 Recipes: Cosmology Power spectrum (two point correlation function)

114 Recipes: Cosmology Power spectrum (two point correlation function)

115 Recipes: Large Scale Structure On sufficiently large scales (>200Mpc) the Universe is isotropic and homogeneous Filaments Voids SLOAN

116 Recipes: Large Scale Structure

117 Recipes: Further Observations But, what else can we say about its energy/matter content?

118 Emitted light does not say it all...

119 Recipes: Dark Matter Up to 80% of matter in the Universe today is dark!

120 Recipes: Dark Matter Gravitational lensing

121 Recipes: Dark Matter We do not understand what it is! Only acts gravitationally and does not emit light Best candidate: a weakly interacting particle that we have not seen yet Or have we?

122 Recipes: Dark Matter Some experiments (Dama, CoGeNT, CDMS) have signals which are hard to explain with know physcis, but others (Xenon) have not seen anything in the same regions

123 Recipes: Dark Matter Or, could it be a modification of gravity?

124 Recipes: Dark Energy There are other objects (called supernovae IA) which also belong to the distance ladder. The supernovae are big star explosions. Remanente de SN 1572 SN 1987A

125 Recipes: Dark Energy Hubble diagram: Supernovae tell us the Universe presents accelerated expansion

126 Recipes: Dark Energy We understand even less what it is! Only acts gravitationally, but in a repulsive way Best candidate: Cosmological Constant

127 Recipes: Dark Energy But... Quantum vacuum gravitates as a cosmological constant (zero point energy) Theory = Observation!!

128 Recipes: Cosmic Pie 5% 27%?? Materia Visible Materia Obs. Energía Obs. 68%

129 Recipes: Theory Cosmological Principle Assume an isotropic and homogeneous metric FRWL Scale factor Open Flat Closed

130 Recipes: Theory Friedmann equations Einstein equations with a perfect fluid reduce to Open k = -1 Flat k = 0 Closed k = +1

131 Recipes: Theory Critical Mass Define in terms of critical mass Friedmann eqn reduce to

132 Recipes: Theory Scale factor evolution Dark energy Radiation Matter t

133 Recipes: Theory Scale factor evolution Dark energy Radiation Matter Model fits ALL observations to great accuracy t

134 Recipes: Hot Big Bang But not this...

135 Recipes: Hot Big Bang Other observations support this model (age of stars, light element abundances, etc)

136 Recipes: Hot Big Bang Problems 1) Magnetic monopoles With many phase transitions why there are any topological relics? 2) Horizon problem Why disconnected region of space have same CMB temperature 3) Size 4) Flatness Expansion 13.7 Gyrs

137 Recipes: Hot Big Bang Problems 1) Magnetic monopoles With many phase transitions why there are any 5) The initial (big bang) singularity topological relics? 2) Horizon problem Why disconnected region of space have same CMB temperature 3) Size 4) Flatness Expansion 13.7 Gyrs

138 Recipes: The singularity in GR Singularity theorems Initial data (assuming some energy conditions) can lead, unavoidable, to geodesically incomplete space-times. Penrose, Hawking (60-70's) Global statement. What about the analytical structure of fields near the singularity?

139 Recipes: The singularity in GR Belinski, Khalatnikov and Lifshitz (BKL) Assumed ultralocality : spatial gradients are not as important as time derivatives! System reduces to 1d, but may have strong dependence on the initial conditions! Chaos Big Bang (e.g. Mixmaster)

140 Recipes: The singularity in GR All depends on matter content: *scalar fields tend to remove chaos *gauge fields (p-forms) may restore it Cosmological billiards Hamiltonian: Damour et al '03 Near t=0,

141 Recipes: The singularity in GR Away from walls: Kasner metric

142 Recipes: The singularity in GR *General Relativity (d<11) *Low energy limits of 5 string theories and M theory Kac Moody algebras (formally integrable)

143 Recipes: The singularity in GR Note that there are potentials for the scalar fields which can overtake this oscillating behaviour, providing a deterministic evolution of the Universe. However, for generic cases, we found a unpredictable behaviour of the metric near the spacetime singularity. We can even forget a description of the singularity itself.

144 Recipes: Beyond GR Need to go beyond GR to solve all these problems of the early Universe 1) Inflation 2) Cyclic Universe 3) Others But bare in mind that the Universe is SIMPLE Latest results show only a Higgs field (nothing else) at LHC and Gaussian, almost scale invariant perturbations in CMB

145 Recipes: Inflation Phase of exponential acceleration Simplest realisation: canonical scalar field slowly rolling down a potential

146 Recipes: Inflation Friedmann equations read If slow-roll is assumed (potential energy dominates over kinetic), then which imply the following solution

147 Recipes: Hot Big Bang Solves the Problems 1) Magnetic monopoles The exponential expansion dilutes them 2) Horizon problem A small patch was enlarged beyond the Hubble horizon (ie got causally disconnected because of the expansion) 3) Size 4) Flatness Exponential expansion decreases curvature, if any 5) Singularity problem remains

148 Recipes: Inflation In order to solve problems inflation should last N =50-60 e-foldings Slow-roll paramters Should be roughly <0.01 to achieve inflation. Measure when the approximation breaks down (inflation ends). After that one needs a mechanism of reheating.

149 Recipes: Inflation Quantum fluctuations Scalar perturbations Changing variables

150 Recipes: Inflation In slow-roll It is a harmonic oscillator! Easy to solve providing consistent initial conditions The curvature perturbations (which eventually get imprinted in the CMB) are We want to calculate the two-point correlation function

151 Recipes: Inflation Where k is the wavenumber of the perturbation. After some algebra, one obtains which is constant to leading order in slow-roll. Therefore, the power spectrum is scale-invariant, and corrections to that are given by the slow-roll parameters. These results can be contrasted with the CMB, fitting very well the data.

152 Recipes: Cyclic model As a bonus, it explains CMB anisotropies But introduces new (or keeps) problems: It does not explain the amplitude of the CMB anisotropies What is the inflaton (inflation's scalar field)? Why did we start at the top of the potential? Initial conditions (singularity)?

153 Recipes: Inflation Moreover, Planck's results put some pressure

154 Recipes: Cyclic Model Alternative model where a Pre Big bang phase creates the CMB pertns. Contracting phase Big Crunch/ Big Bang Expanding phase t=0 Two flavours: singular & non singular

155 Recipes: Cyclic Model

156 Recipes: Cyclic model Creates perturbations in a very similar way to inflation But... again some pressure by Planck

157 Recipes: alternatives