General Relativity in a Nutshell

General Relativity in a Nutshell (And Beyond) Federico Faldino Dipartimento di Matematica Università degli Studi di Genova 27/04/2016

1 Gravity and General Relativity 2 Quantum Mechanics, Quantum Field Theory and All That... 3 An insight into QFT on Curved Backgrounds

Newtonian Gravity Newton's Law of Gravitation F = G m G M 11 N m2 r 3 r G = 6.67 10 kg 2 Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 1 / 27

Newtonian Gravity Newton's Law of Gravitation F = G m G M 11 N m2 r 3 r G = 6.67 10 kg 2 This formulation provides problems (e.g. precession of Mercury's orbit perihelion, wrong deviation of light rays, instantaneous propagation) Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 1 / 27

Galilean Relativity Principle (Galilean Relativity) The laws of Mechanics are invariant under a change of inertial frame (IF). Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 2 / 27

Galilean Relativity Principle (Galilean Relativity) The laws of Mechanics are invariant under a change of inertial frame (IF). Galileo's Transformations x = x vt y = y z Velocity Transormation Law: u = u v = z t = t Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 2 / 27

Galilean Relativity Principle (Galilean Relativity) The laws of Mechanics are invariant under a change of inertial frame (IF). Galileo's Transformations x = x vt y = y z Velocity Transormation Law: u = u v = z t = t Maxwell Equations are not invariant under Galilean Relativity Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 2 / 27

Special Relativity Principle (Einstein's Relativity) The laws of Physics are invariant under a change of inertial frame. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 3 / 27

Special Relativity Principle (Einstein's Relativity) The laws of Physics are invariant under a change of inertial frame. Lorentz Transformations For two IF R e R in x-standard conguration, assuming space and time isotropy and homogeneity: x = y = y z = z t = x vt 1 v2 2 c t v c x 1 v2 c 2 Velocity Transformation Law: u = u v 1 uv c 2 Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 3 / 27

Minkowski Spacetime M Denition We call Minkowski Spacetime M the vector space R 4 endowed with Orientation; Metric η with signature (+ ) (or ( + ++)); Time Orientation. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 4 / 27

Causal Structure on M Timelike η(u, u) > 0; Spacelike η(u, u) < 0; Lightlike η(u, u) = 0; Causal η(u, u) 0. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 5 / 27

Equivalence Principle Let us turn on gravity and remember Newton's Second Law F = m I a F = G Mm G r 3 r Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 6 / 27

Equivalence Principle Let us turn on gravity and remember Newton's Second Law F = m I a F = G Mm G r 3 r In principle m I m G since they correspond to two dierent physical properties! Weak Equivalence Principle Existence of Local Inertial Frames Principle (Equivalence Principle) In small enough regions of space-time, the laws of physics reduce to those of special relativity; it is impossible to detect the existence of a gravitational eld by means of local experiments. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 6 / 27

The Einstein' way to General Relativity We want to obtain an analogue for the Poisson Equation for the classic gravitation potential: ϕ G = 4πGρ. We look for: 2 nd order tensorial equations, linear in the derivatives of greater order; We need to reach the Newtonian theory in a suitable limit; The equations must grant the condition i T i = 0 (freely gravitating k mass-energy). Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 7 / 27

The Einstein' way to General Relativity We want to obtain an analogue for the Poisson Equation for the classic gravitation potential: ϕ G = 4πGρ. We look for: 2 nd order tensorial equations, linear in the derivatives of greater order; We need to reach the Newtonian theory in a suitable limit; The equations must grant the condition i T i = 0 (freely gravitating k mass-energy). Einstein Equations: R µν 1 2 g µνr + (Λg µν ) = 4πG c 4 T µν Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 7 / 27

A Pictorial Viewpoint Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 8 / 27

Some well-known solutions to the Einstein Equations Schwarzschild: ( ds 2 = 1 2M r ) ( dt 2 + 1 2M r ) 1 dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 Friedmann - Lemaitre - Robertson - Walker (FLRW): Kerr de Sitter ] ds 2 = a 2 (t) [ dt2 a 2 (t) + dx 2 + dy 2 + dz 2 Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 9 / 27

Quantum Gravity? R µν 1 2 g µνr = 8πT µν Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 10 / 27

Quantum Gravity? R µν 1 2 g µνr = 8πT µν Possible Quantum Gravity Theory: String Theory Loop Quantum Gravity Other Approaches (Twistors, Non-Commutative Geometry, etc.) Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 10 / 27

Quantum Gravity? R µν 1 2 g µνr = 8πT µν Possible Quantum Gravity Theory: String Theory Loop Quantum Gravity Other Approaches (Twistors, Non-Commutative Geometry, etc.) Up to now we are far from the solution... Semi-classical theory of Gravity and QFT over Curved Backgrounds Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 10 / 27

1 Gravity and General Relativity 2 Quantum Mechanics, Quantum Field Theory and All That... 3 An insight into QFT on Curved Backgrounds

What a Quantum Theory is? Oxford Dictionary: a Quantum is a discrete amount of any physical quantity, such as energy, momentum or electric charge. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 11 / 27

What a Quantum Theory is? Oxford Dictionary: a Quantum is a discrete amount of any physical quantity, such as energy, momentum or electric charge. States: Vectors ψ in a suitable Hilbert spaces H; Observables: Self-adjoint operators on H; Time Evolution: Two ways for implementing it: Schrödinger Picture Evolution equation for vector states; Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 11 / 27

What a Quantum Theory is? Oxford Dictionary: a Quantum is a discrete amount of any physical quantity, such as energy, momentum or electric charge. States: Vectors ψ in a suitable Hilbert spaces H; Observables: Self-adjoint operators on H; Time Evolution: Two ways for implementing it: Schrödinger Picture Evolution equation for vector states; Heisenberg Picture Evolution equation for the observables. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 11 / 27

What a Quantum Theory is? Oxford Dictionary: a Quantum is a discrete amount of any physical quantity, such as energy, momentum or electric charge. States: Vectors ψ in a suitable Hilbert spaces H; Observables: Self-adjoint operators on H; Time Evolution: Two ways for implementing it: Schrödinger Picture Evolution equation for vector states; Heisenberg Picture Evolution equation for the observables. Everything can be made rigorous using the Algebraic Formulation! Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 11 / 27

Canonical Quantisation In Schrödinger picture the vector states are represented by ψ(x; t) L 2 (R 3 ), which are the components of ψ(t) = dx ψ(x; t) x. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 12 / 27

Canonical Quantisation In Schrödinger picture the vector states are represented by ψ(x; t) L 2 (R 3 ), which are the components of ψ(t) = dx ψ(x; t) x. We associate a quantum observable to every classical one via the promotion to operator prescription. In Schrödinger picture: x ˆx p ˆp. = i x E Ê. = i t. ˆx and ˆp satisfy the commutation relation [ˆx, ˆp] = i. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 12 / 27

Canonical Quantisation In Schrödinger picture the vector states are represented by ψ(x; t) L 2 (R 3 ), which are the components of ψ(t) = dx ψ(x; t) x. We associate a quantum observable to every classical one via the promotion to operator prescription. In Schrödinger picture: x ˆx p ˆp. = i x E Ê. = i t. ˆx and ˆp satisfy the commutation relation [ˆx, ˆp] = i. From the classical energy-dispersion relation we obtain the Schrödinger Equation: E = p2 2m + V (x) i t ψ = 1 2m d 2 ψ + V (ˆx). dx 2 Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 12 / 27

Quantisation of the Harmonic Oscillator The Schrödinger equation for a 1-dimensional harmonic oscillator reads i t ψ(x; t) = 1 d 2 2m dx ψ(x; t) + 1 2 2 ω2ˆx 2 Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 13 / 27

Quantisation of the Harmonic Oscillator The Schrödinger equation for a 1-dimensional harmonic oscillator reads i t ψ(x; t) = 1 d 2 2m dx ψ(x; t) + 1 2 2 ω2ˆx 2 Its solution is ψ n (x; t) = e 1 2 ωx 2 H n (x ( ω)e ie nt, E n = n + 1 ) ω. 2 Important features of Quantum Mechanics: Discrete spectrum of energy eigenstates; E 0 = 1 ω: there is a ground state with non-zero energy ( Heisenberg 2 Principle) Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 13 / 27

Second Quantisation Let us introduce the Lowering and Raising Operators: â = 1 2ω (ωˆx + iˆp) â = 1 2ω (ωˆx iˆp) [ â, â ] = 1. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 14 / 27

Second Quantisation Let us introduce the Lowering and Raising Operators: â = 1 2ω (ωˆx + iˆp) â = 1 2ω (ωˆx iˆp) [ â, â ] = 1. They allow us to reformulate the Hamiltonian as ( H = ˆN + 1 ). ω ˆN = â â = Number Operator. 2 Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 14 / 27

Second Quantisation Let us introduce the Lowering and Raising Operators: â = 1 2ω (ωˆx + iˆp) â = 1 2ω (ωˆx iˆp) [ â, â ] = 1. They allow us to reformulate the Hamiltonian as ( H = ˆN + 1 ). ω ˆN = â â = Number Operator. 2 Calling n the eigenstates of ˆN we obtain that a generic state vector is given by: ψ(t) = c n e ie nt n. n Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 14 / 27

Special Relativity and Quantum Mechanics: the Klein-Gordon Equation Switching to the relativistic energy dispersion relation E 2 = k 2 + m 2 (c = 1) we get the the Klein-Gordon Equation: ( + m 2 ) ψ = 0. One can derive it computing the Euler-Lagrange Equations of the Klein-Gordon Lagrangian: S KG = d 4 xl KG, L KG = 1 2 ηµν µ φ ν φ 1 2 m2 φ 2 Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 15 / 27

Special Relativity and Quantum Mechanics: the Klein-Gordon Equation Switching to the relativistic energy dispersion relation E 2 = k 2 + m 2 (c = 1) we get the the Klein-Gordon Equation: ( + m 2 ) ψ = 0. One can derive it computing the Euler-Lagrange Equations of the Klein-Gordon Lagrangian: S KG = d 4 xl KG, L KG = 1 2 ηµν µ φ ν φ 1 2 m2 φ 2 Problems: Possible negative-energy solutions (Fourier transform); Violation of causality; ψ 2 can not be interpreted as a probability amplitude. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 15 / 27

A World Made of Fields A solution to these problems can be found considering a system with innitely many degrees of freedom, i.e. a eld (Dirac sea). Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 16 / 27

A World Made of Fields A solution to these problems can be found considering a system with innitely many degrees of freedom, i.e. a eld (Dirac sea).the KG Hamiltonian is (π = φ) H = 1 2 π2 + 1 2 ( φ)2 + 1 2 m2 φ 2 Harmonic Oscillator! (x; p) (φ(x µ ); π(x µ )) φ(x µ ) has no more to be read as a wave function, but xed-time initial value of the KG equation. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 16 / 27

Space of Solutions The KG Equation admits plane-wave solutions: φ(x µ ) = φ 0 e ikµxµ = φ 0 e iωt i k x ω 2 = k 2 + m 2. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 17 / 27

Space of Solutions The KG Equation admits plane-wave solutions: φ(x µ ) = φ 0 e ikµxµ = φ 0 e iωt i k x ω 2 = k 2 + m 2. We look for a complete o.n. set of solutions, hence we need a scalar product on the solutions' space: (φ 1, φ 2 ) = i (φ 1 t φ 2 φ 2 t φ 1 ) d 3 x Σ t so that we get the set (k µ : ω 2 = k 2 + m 2 ): µ f k (x µ ) = eikµx 2π 2ω, (f k1, f k2 ) = δ(3) (k1 k2). Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 17 / 27

Positive- and Negative-Frequencies Solutions The solution are labelled by the continuous parameter k and are determined up to the sign of ω. Since energy (E = hω) is a positive-denite quantity we would like to consider only solutions with positive frequency. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 18 / 27

Positive- and Negative-Frequencies Solutions The solution are labelled by the continuous parameter k and are determined up to the sign of ω. Since energy (E = hω) is a positive-denite quantity we would like to consider only solutions with positive frequency. This is done by introducing, for all ω > 0, positive-frequency solutions and negative-frequency solutions t f k = iωf k t f k = iωf k. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 18 / 27

Canonical Quantisation In total analogy with the harmonic oscillator in QM we promote φ and π to operator, imposing the equal-time commutation relations (Heisenberg Picture) [ φ(x), φ(x ) ] t = iδ(3) (x x ). Expanding as a function of the modes [ ] φ(t; x) = d 3 x â k f k (t; x) + â f (t; x) k k which leads to the Canonical Commutation Relations [â k, â k ] = δ (3) (k k ). Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 19 / 27

Particles and Anti-Particles Positive-frequencies modes are the coecients of the Annihilation operator â k ; Negative-frequencies modes are the coecients of the Creation operator â k. n k = 1 nk! (â k )n k 0 We are creating n particles with momentum k. 0 is the Vacuum State. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 20 / 27

Particles and Anti-Particles Positive-frequencies modes are the coecients of the Annihilation operator â k ; Negative-frequencies modes are the coecients of the Creation operator â k. n k = 1 nk! (â k )n k 0 We are creating n particles with momentum k. 0 is the Vacuum State. Likewise in QM, we can introduce the Number Operator ˆNk = â kâk, whose eigenvectors constitute the Fock Basis. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 20 / 27

What's Missing? Renormalization Subtraction of (innite) the point-zero energy Interactions and Scattering Theory Other kind of Fields (QED, Gauge Theories)... Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 21 / 27

1 Gravity and General Relativity 2 Quantum Mechanics, Quantum Field Theory and All That... 3 An insight into QFT on Curved Backgrounds

Fundamental Ideas The interactions do not inuence the xed background; Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 22 / 27

Fundamental Ideas The interactions do not inuence the xed background; Investigation of the back-reaction; Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 22 / 27

Fundamental Ideas The interactions do not inuence the xed background; Investigation of the back-reaction; Black Holes Physics; Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 22 / 27

Fundamental Ideas The interactions do not inuence the xed background; Investigation of the back-reaction; Black Holes Physics; States (Vacua, Thermal...) and Renormalization; Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 22 / 27

Fundamental Ideas The interactions do not inuence the xed background; Investigation of the back-reaction; Black Holes Physics; States (Vacua, Thermal...) and Renormalization; Making the theory mathematically rigorous. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 22 / 27

Choice of the background L = g ( 12 µ φ µ φ 12 ) m2 φ 2 ξrφ 2 g φ m 2 φ ξrφ = 0. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 23 / 27

Choice of the background L = g ( 12 µ φ µ φ 12 ) m2 φ 2 ξrφ 2 g φ m 2 φ ξrφ = 0. For consistency of the Causchy problem and for causality issues we need to x ourselves on a Globally Hyperbolic Space-Time: Theorem (Bernal - Sanchez) Let (M, g) be a 4-dimensional, time-oriented space-time. Then the following statements are equivalent: 1 (M, g) is globally hyperbolic; 2 (M, g) is hysometric to R Σ with ds 2 = +βdt 2 h ij dx i dx j. Here (t, x i ) 3 i=1 is a suitable coordinate system s.t. β C (M; (0, )), h is a Riemannian metric on Σ depending smoothly on t and each locus {t = const} Σ is a smooth spacelike Cauchy hypersurface embedded in M. Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 23 / 27

What about particles? We dene the conjugate momentum π = g 0 φ and dene the scalar product on a spacelike 3-surface Σ and take over the quantisation imposing impose the CCR: (φ 1, φ 2 ) =. i (φ 1 µ φ 2 φ 2 µ φ 1 ) n µ γd 3 x Σ [ φ(t, x), φ(t, x ) ] = i g δ (3) (x x ) Problem In a general space-time it is impossible to dene positive-frequency solutions because there is no unique notion of time. Hence the concept of particle is ill-dened! Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 24 / 27

Hadamard States We are dealing with semiclassical Einstein equations: R µν 1 2 g µνr = 8π < T µν >, < T µν >. = 0 T µν 0 =? Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 25 / 27

Hadamard States We are dealing with semiclassical Einstein equations: R µν 1 2 g µνr = 8π < T µν >, < T µν >. = 0 T µν 0 =? The lack of the modes expansion implies that there is no denite notion of vacuum state. This lead to the introduction of Hadamard States: H(x, x ) = U(x, x ) (2π) 2 σ ɛ + V (x, x ) log(σ ɛ ) + W (x, x ), σ ɛ. = σ + 2iɛ(t t ) + ɛ 2 < T µν > well-dened Expectation values with nite uctuations Compatible with the unique Minkowski vacuum state Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 25 / 27