General Relativity. IV Escuela Mexicana de Cuerdas y Supersimetría. Gustavo Niz June 2015

Transcription

1 General Relativity IV Escuela Mexicana de Cuerdas y Supersimetría Gustavo Niz June 2015

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

3 GR as a Cooking Book Ingredients Physics & Maths Cookware Equations Recipes Solutions If time allows it BEYOND GR

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 Exercise

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

14 Ingredients: Special Relativity Spacetime diagram

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

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

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

18 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

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

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

21 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

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

23 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

24 Ingredients: Curved Space In simple words

25 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

26 Ingredients: Tensors Under a coordinate transformation tensor changes in the following way a (summation convention)

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

28 Ingredients: Tensors Observations (0,0) tensor scalar (function) Definitions (0,1) tensor vector (1,0) tensor co-vector

29 Ingredients: Tensors Observations Trace over indices Maths Symmetric Antisymmetric

30 Ingredients: Tensors Observations Physics 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 is the inverse of so

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

50 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!

51 Cookware: Einstein Equations Einstein (1915)

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

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

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

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

56 Cookware: Einstein Equations Cauchy Problem. e.g. Maxwell eqns. No t derivatives 1st order t der. Constraints 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

57 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!

58 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.

59 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

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

61 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

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 Einstein equations

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

64 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.

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

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

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

68 Cookware: Alternative approaches 2) Hamiltonian (ADM or 3+1 formalism) Consider the space-time splitting N lapse function N i shift function g ij spatial metric Schematically

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

70 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

71 Recipes: content Maximally symmetric solutions Penrose Diagrams Black Hole solutions Schwarzschild Others Thermodynamics Cosmology. In other course(s)!!

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

73 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

74 Recipes: Penrose diagrams Consider Minkowski spacetime Change variables to

75 Recipes: Penrose diagrams Bring infinite to fine distance with

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

77 Recipes: Penrose diagrams Half of 3-sphere

78 Recipes: Penrose diagrams Take half of triangle and attach a full 3-sphere at each point

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

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

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

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

83 Penrose diagram Recipes: de Sitter (ds)

84 Recipes (detour): Horizons Particle horizon Event horizon

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

86 Recipes: Anti de Sitter (AdS) Closed Timelike Curves (travel back in time?) On the Universal covering space we avoid them!

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

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

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

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

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

92 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,

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

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

95 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

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

97 Recipes: Schwarzschild solution Penrose diagram

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

99 Add rotation Recipes: Kerr BH

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

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

102 Recipes: Kerr, KN or RN

103 Recipes: Kerr, KN or RN

104 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).

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

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

107 Beyond GR... Desser's construction. Bottom up! Lovelock's theorem Problems with GR UV modifications IR modifications

108 Bottom up approach Is GR unique? (Kraichnan, Feynman, Weinberg, Boulware, Deser, etc.) Electromagnetism The vector field (spin 1) couples to conserved current, but it does not contribute to it (photons do not carry charge!) Gravity The graviton field (spin 2) couples to conserved current (energy-momentum tensor), and it does contribute to it (c.f. Yang-Mills)

109 Bottom up approach Basic idea To recover Einstein's theory from linear theory of gravitons coupled to the gravitational energy momentum tensor Assumptions: Symmetric stress-energy tensor Second order derivatives Total derivatives (boundary terms) irrelevant Key one: local Lorentz symmetry Infinite series in metric field, but finite in the right variables Deser, gr-qc/ ,

110 Bottom up approach Linear Gravity (Written in ) à la Palatini Generates its own Energy-Momentum Tensor (Belifante) FULL GR EOM Written in Are the same!!!

111 Bottom up approach Start from (Fierz Pauli) action with independent Minkowski tensors Want to recover Total derivative

112 Bottom up approach Start from (Fierz Pauli) action with independent Minkowski tensors Want to recover Total derivative Full GR only needs an extra cubic term!!!

113 Bottom up approach Add coupling With Belifante-Rosenfield Procedure (1940) Covariantise linear theory w.r.t. fictitious metric and derive associated energy momentum tensor. Set at the end. Defines new action

114 Bottom up approach Finally, one can check the EOM from are L L

115 Theorems Lovelock's Theorem ( ) If we want second order differential field equations for the metric with a 4d space-time, the only form is To recover Newtonian physics

116 Theorems Lovelock's Theorem Implications: 1) Terms in the Lagrangian such as which give second order field equations in 5d, are topological in 4d.

117 Theorems Lovelock's Theorem 2) To modify Einstein's theory we need Consider more fields (beyond the spin-2) Accept higher order derivatives in field equations Higher dimensions Give up rank (0,2) field eqns, symmetry of the indices in eqns, or divergence free equations Give up locality

118 Why do we need to modify GR in the UV? Non-renormalisable (perturbatively) Essential singularities Black holes Big bang singularity Early Universe physics Hot big bang problems (horizon, Universe size, monopoles/defects, flatness)

119 Why do we want to modify it Cosmology's dark sector in the IR? Dark matter (particle physics?) Dark energy Small observed value Why today (coincidence problem) Vacuum energy ( old cosmological constant problem )

120 Why do we want to modify it Cosmology's dark sector Dark matter (particle physics?) Dark energy Small observed value in the IR? Why today (coincidence problem) Vacuum energy ( old cosmological constant problem ) Non-renormalisable (perturbatively) in the UV

121 IR ideas

122 Dark Sector in Cosmology Observational evidence of dark energy Simple solution: Λ (Vacuum energy?) Complicated solution: New gravitational degrees of freedom! Beauty Theory consistency Observations New physics?

123 Dark Energy Cosmological Constant (CC) & vacuum energy Lorentz invariance QFT Energy-Momentum tensor Vacuum energy gravitates as a CC

124 Dark Energy Cosmological Constant (CC) & vacuum energy The effective CC is QFT To get one needs an important cancellation

125 Dark Energy Cosmological Constant (CC) & vacuum energy Estimate perturbatively By adding the normal modes of a free scalar with mass m we can see the problem QFT

126 Dark Energy Cosmological Constant (CC) & vacuum energy Estimate perturbatively By adding the normal modes of a free scalar with mass m, we can see the problem QFT

127 Dark Energy Cosmological Constant (CC) & vacuum energy For some many species of matter QFT Fermionic number E.g. electrons

128 Dark Energy Cosmological Constant (CC) & vacuum energy One of the most important problems in theoretical physics

129 Dark Energy Possible ways out Cancellation very specific among contributions (eg. SUSY) Relaxation of its value as Universe expands is impossible ( no-go theorem by Weinberg, 1989)

130 Dark Energy Possible ways out Cancellation very specific among contributions (eg. SUSY) Relaxation of its value as Universe expands is impossible ( no-go theorem by Weinberg, 1989) Modified?

131 Where can be modified? Baker et al, 2014

132 Eot-Wash Baker et al, 2014

133 Baker et al, 2014

134 Baker et al, 2014, Koyama's talk 2015

135 Screening mechanisms EOM for scalar d.o.f. Kinetic term Potential (mass) Matter Interaction

136 Screening mechanisms Only consider a coupling to non-relativistic matter

137 Chameleon Screening mechanisms

138 Symmetron Screening mechanisms

139 Screening mechanisms Vainshtein effect (Galileon)

140 Horndeski Lagrangian Horndeski (70's) Most general scalar-tensor theory with second order EOMS

141 Fab Four Charmousis et al, 2011 Sector of Horneski's Lagrangian which 1) Minkowski is a vacuum solution independently of the bare cosmological constant (c.f. De Sitter in GR) 2) Solution remains flat after phase transition (change in the bare C.C.) 3) Composed by four different terms (Paul, Ringo, John, George)

142 UV ideas

143 UV: Non-renormalisable Where is the problem? Consider Re-scaling Lorentz invariance

144 UV: Non-renormalisable Potential Diverges in the UV if n>4

145 UV: Non-renormalisable Dimensional analysis Propagator

146 UV: Non-renormalisable In General Relativity the linear theory is Differential operator of 2nd order Propagator (schematically) Dimensionful coupling

147 UV: Non-renormalisable We need an infinite number of counter-terms Due to Lorentz symmetry the have to be powers of curvature tensors

148 UV: Non-renormalisable So what's wrong with? Propagator (schematically)...

149 UV: Non-renormalisable But Propagator Massive ghost!! Massless graviton

150 UV: Non-renormalisable But Propagator Massive ghost!! Massless graviton Usual problem with higher derivatives (Ostrogradski)

151 UV: Non-renormalisable Possible ways out 1)SUSY SUGRA (helps but not a solution) 2)Lorentz Horava-Liftshitz gravity 3)Non-perturbative methods Asymptotic safety 4)Path integral Wave function of the Universe Spin-foams

152 UV: Non-renormalisable Possible ways out 4)Canonical quantization (background independent) Wheeler-deWitt Loop quantum gravity 5)New physics/mathematics Algebraic structures (Kac-Moody) BCJ amplitudes? String Theory Etc.