General Relativity (2nd part)

Transcription

1 General Relativity (2nd part)

2 Electromagnetism Remember Maxwell equations Conservation

3 Electromagnetism Can collect E and B in a tensor given by And the charge density Can be constructed from and current in 4-vector

4 Electromagnetism Maxwell equations are Equations valid in Minkowski, but it is straight forward to generalise them to curve space time, just

5 Electromagnetism Energy density Poynting vector Maxwell stress tensor They satisfy

6 Perfect Fluids A popular choice of matter in GR is a perfect fluid, defined to be a fluid with NO viscocity or heat flow and isotropic in its rest frame! It is completely specified by it energy density and isotropic pressure where is the fluid's 4-velocity. If Bianchi idendity ( momentum conservation ) implies local energy-

7 Einstein Equations Geometría del Espacio-tiempo Materia/Energía

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

9 Einstein Equations Cosmological Constant Its divergence is zero. Must be small otherwise we would have seen it in lab experiments!

10 Einstein Equations Lovelock 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

11 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

12 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!

13 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.

14 Variational Principles Action? Need to understand integration on a manifold (will not look at exterior calculus, but just the basics)

15 Tensorial densities Important objects in physics which are not tensors 1) volume factor Under transforms as

16 Tensorial densities Important objects in physics which are not tensors 2) determinants Transforms as 1) and 2) are call tensor densities because transform as some powers of the Jacobian

17 Tensorial densities Tensor density of weight w where is the Jacobian! The covariant derivative of a tensor density is arbitrary and not a tensor density in general.

18 Tensorial densities Important objects in physics which are not tensors Notice Is invariant under a change of coordinate, since f(x) is a scalar

19 Tensorial densities 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

20 Hilbert-Einstein action Einstein equations can be obtained from Einstein equations

21 Hilbert-Einstein action Can include a Cosmological Constant

22 Hilbert-Einstein action Matter Lagrangians (examples) 1) Scalar field

23 Hilbert-Einstein action Matter Lagrangians (examples) 2) Electromagnetism

24 Alternative approaches À la Palatini Consider the connection and the metric as independent The metric connection

25 3+1 Formalism Also known as Hamiltonian or ADM Consider the space-time splitting N lapse function Ni shift function gij spatial metric Schematically

26 3+1 Formalism 2) Hamiltonian (ADM or 3+1 formalism) Extrinsic curvature Then one gets the Hamiltonian Hamiltonian Momentum Canonical momenta

27 3+1 Formalism 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

28 Weak field Linearised gravity Gravitational waves

29 Linearised gravity Informal way Use and do an expansion in But does not make any sense formally, since we are comparing tensors (two metrics) on different manifolds!

30 Linearised gravity Formal way Take a 5d spacetime, with a foliation of 4d hypersurfaces that are connected by a congruence (map).

31 Linearised gravity Formal way Gauges (choice of map in the 5d setting choice of coordinates from 4d!) If Then Only gauge invariant quantities are physically relevant!

32 Linearised gravity Inverse metric is And the Christoffel symbols

33 Linearised gravity Hence Is invariant under the change of gauge

34 Linearised gravity It is useful to define the trace-reversed metric perturbation where h is the trace of The inverse is

35 Linearised gravity Einstein's field equations become

36 Linearised gravity Donder or harmonic gauge Under a coord. transformation one gets So one can always choose

37 Linearised gravity The Einstein equations reduce to (Together with the gauge condition )

38 Gravitational radiation In vacuum, the linearised equations are Which can be solved using a single Fourier mode

39 Gravitational radiation Then, these two equations reduce to the conditions

40 Gravitational radiation Then, these two equations reduce to the conditions In particular:

41 Gravitational radiation There is still some gauge freedom left, which can be use to set The two polarization modes of a gravitational wave!

42 Exact Solutions Maximally symmetric Black Holes

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

44 Detour: 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

45 Detour: Penrose diagrams Consider Minkowski spacetime Change variables to

46 Detour: Penrose diagrams Bring infinite to fine distance with

47 Detour: Penrose diagrams Back to the Cartesian coords. Einstein Static Universe (ESU)

48 Detour: Penrose diagrams Einstein Static Universe (ESU) Solution of Einstein equations with a perfect fluid with Or a cosmological constant with a particular value! Minkowski ESU

49 Detour: Penrose diagrams Half of 3-sphere

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

51 Detour: Penrose diagrams Future timelike infinity Future null infinity Spatial infinity Past null infinity Past timelike infinity

52 Detour 2: Horizons Particle horizon Event horizon

53 Minkowski Penrose diagram

54 de Sitter (ds) Easier to picture in 5d

55 de Sitter (ds) Geodesics Planes through the origin 1) Hyperbola -> Timelike 2) Straight Line -> Null 3) Ellipse -> Spacelike

56 de Sitter (ds) By choosing the following useful frames: Static coordinates one can write the metric in

57 de Sitter (ds) Or using cosmological coordinates (steady-state Universe) Only covers half of hyperboloid Exponential expansion! Bondi, Gold, Hoyle

58 de Sitter (ds) Penrose diagram Both timelike and null future infinity

59 de Sitter (ds) Penrose diagram

60 de Sitter (ds) Penrose diagram for expanding Universe (SSU)

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

62 Recipes: Anti de Sitter (AdS) Closed Timelike Curves (travel back in time?) Choose the covering space (unwrap the circle to R) and the closed timelike curves disappear (relevant to physics!)

63 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

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

65 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

66 Schwarzschild solution In detail Choose In vacuum, we have

67 Schwarzschild solution Therefore The metric is By changing time, can always make

68 Schwarzschild solution Now take the combination Can always choose the constant to be zero. Now

69 Schwarzschild solution Since it is unique, it describes from black holes to planets! Ro If r>>1 BH or planet Vacuum Schwarzschild sol.

70 Schwarzschild solution Schwarzschild solution is then Mass inside Ro

71 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)

72 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,

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

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

75 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

76 Schwarzschild solution However, something special still happens at

77 Schwarzschild solution Penrose diagram

78 Schwarzschild solution Penrose diagram (completion of spacetime)

79 Kerr BH Add rotation

80 Kerr BH Momentum bound Extremal case (BPS, stable, in nature?)

81 Reissner Nordström BH Electric charge Extremal case when Charged+Rotating Kerr-Newman solution

82 Kerr, KN or RN

83 Kerr, KN or RN

84 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).

85 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

86 BH thermodynamics Implication Temperature Surface gravity Entropy Area of horizon

87 Cosmology Observations Hot Big Bang Theory Early Universe ideas

88 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

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

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

91 Cosmology Small perturbations of 1 in Simple UNIVERSE Gaussian Scale invariant (tilt)

92 Cosmology Power spectrum (two point correlation function)

93 Cosmology Power spectrum (two point correlation function)

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

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

96 Emitted light does not say it all...

97 Dark Matter Up to 80% of matter in the Universe today is dark!

98 Dark Matter Gravitational lensing

99 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?

100 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

101 Dark Matter Or, could it be a modification of gravity?

102 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

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

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

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

106 Cosmic Pie 5% 27%?? Materia Visible Materia Obs. Energía Obs. 68%

107 Theory Cosmological Principle Assume an isotropic and homogeneous metric FRWL Scale factor Open Flat Closed

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

109 Theory Critical Mass Today's critical mass is Where according to WMAP

110 Theory Define in terms of critical mass today (and including a cosmological constant) Friedmann eqn reduce today to

111 Theory The general solution for a flat spatial slicing (k=0) and a perfect fluid with equation of state where the equations reduce to is

112 Theory 1) Dust (p=0) Age of the Universe 2) Radiation (p=1/3)

113 Theory Solutions with curvature (k)

114 Theory Cosmological constant domination

115 Hot Big Bang Scale factor evolution Dark energy Radiation Matter t

116 Hot Big Bang Scale factor evolution Dark energy Radiation Matter Model fits ALL observations to great accuracy t

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

118 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

119 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

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

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

122 Inflation 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

123 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 of when the approximation breaks down (inflation ends). After that one needs a mechanism of reheating.

124 Inflation Quantum fluctuations Scalar perturbations Changing variables

125 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

126 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.

127 Inflation As a bonus, it explains CMB anisotropies But introduces new (or keeps some) 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)?

128 Inflation However, Planck's results put some pressure

129 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

130 Cyclic Model

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

132 End of the course