general relativity in a nutshell - (and beyond) · general relativity in a nutshell (and beyond)...

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 = GmGM

r3r G = 6.67 · 10−11 N ·m

This formulation provides problems (e.g. precession of Mercury's orbitperihelion, wrong deviation of light rays, instantaneous propagation)

Federico Faldino (PhD Seminars) GR in a Nutshell 27/04/2016 1 / 27

Newtonian Gravity

Newton's Law of Gravitation

F = GmGM

r3r G = 6.67 · 10−11 N ·m

This formulation provides problems (e.g. precession of Mercury's orbitperihelion, wrong deviation of light rays, instantaneous propagation)

Galilean Relativity

Principle (Galilean Relativity)

The laws of Mechanics are invariant under a change of inertial frame (IF).

Galileo's Transformationsx ′ = x − vt

y ′ = y

z ′ = z

t ′ = t

Velocity Transormation Law: u′ = u − v

Maxwell Equations are not invariant under Galilean Relativity

Galilean Relativity

y ′ = y

z ′ = z

t ′ = t

Galilean Relativity

y ′ = y

z ′ = z

t ′ = t

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 con�guration, assuming space and timeisotropy and homogeneity:

x ′ = x−vt√1− v

y ′ = y

z ′ = z

t ′ =t− v

1− v2

Velocity Transformation Law: u′ =u − v

1− uvc2

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 con�guration, assuming space and timeisotropy and homogeneity:

x ′ = x−vt√1− v

y ′ = y

z ′ = z

t ′ =t− v

1− v2

Velocity Transformation Law: u′ =u − v

1− uvc2

Minkowski Spacetime M

De�nition

We call Minkowski Spacetime M the vector space R4 endowed with

Orientation;

Metric η with signature (+−−−) (or (−+ ++));

Time Orientation.

Causal Structure on M

Timelike η(u, u) > 0;

Spacelike η(u, u) < 0;

Lightlike η(u, u) = 0;

Causal η(u, u) ≥ 0.

Causal Structure on M

Timelike η(u, u) > 0;

Spacelike η(u, u) < 0;

Lightlike η(u, u) = 0;

Causal η(u, u) ≥ 0.

Equivalence Principle

Let us turn on gravity and remember Newton's Second Law

F = mIa F = GMmG

In principle mI 6= mG since they correspond to two di�erent physicalproperties!

Weak Equivalence Principle ⇒ Existence of Local Inertial Frames

Principle (Equivalence Principle)

In small enough regions of space-time, the laws of physics reduce to thoseof special relativity; it is impossible to detect the existence of agravitational �eld by means of local experiments.

F = mIa F = GMmG

The Einstein' way to General Relativity

We want to obtain an analogue for the Poisson Equation for the �classic�gravitation potential:

∆ϕG = 4πGρ.

We look for:

2nd 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 ∇iTik = 0 (freely gravitating

mass-energy).

Einstein Equations: Rµν −1

2gµνR + (Λgµν) = −4πG

c4Tµν

The Einstein' way to General Relativity

We want to obtain an analogue for the Poisson Equation for the �classic�gravitation potential:

∆ϕG = 4πGρ.

We look for:

2nd 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 ∇iTik = 0 (freely gravitating

mass-energy).

Einstein Equations: Rµν −1

2gµνR + (Λgµν) = −4πG

c4Tµν

A Pictorial Viewpoint

Some well-known solutions to the Einstein Equations

Schwarzschild:

ds2 = −(1− 2M

)dt2 +

(1− 2M

)−1dr2 + r2dθ2 + r2 sin2 θdφ2

Friedmann - Lemaitre - Robertson - Walker (FLRW):

ds2 = a2(t)

[− dt2

a2(t)+ dx2 + dy2 + dz2

de Sitter

Quantum Gravity?

Rµν −1

2gµνR = −8πTµν

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

Quantum Gravity?

Rµν −1

String Theory

Quantum Gravity?

Rµν −1

String Theory

What a �Quantum Theory� is?

Oxford Dictionary: a �Quantum� is �a discrete amount of any physicalquantity, 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!

States: Vectors |ψ〉 in a suitable Hilbert spaces H;

Observables: Self-adjoint operators on H;Time Evolution: Two ways for implementing it:

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.

Canonical Quantisation

In Schrödinger picture the vector states are represented byψ(x ; t) ∈ L2(R3), 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 → x p → p.

= −i∂x E → E.

= i∂t .

x and p satisfy the commutation relation

[x , p] = i .

From the classical energy-dispersion relation we obtain the SchrödingerEquation:

2m+ V (x)⇒ −i∂t |ψ〉 = − 1

dx2|ψ〉+ V (x).

|ψ(t)〉 =

∫dx ψ(x ; t) |x〉 .

x → x p → p.

= −i∂x E → E.

= i∂t .

[x , p] = i .

2m+ V (x)⇒ −i∂t |ψ〉 = − 1

dx2|ψ〉+ V (x).

|ψ(t)〉 =

∫dx ψ(x ; t) |x〉 .

x → x p → p.

= −i∂x E → E.

= i∂t .

[x , p] = i .

2m+ V (x)⇒ −i∂t |ψ〉 = − 1

dx2|ψ〉+ V (x).

|ψ(t)〉 =

∫dx ψ(x ; t) |x〉 .

x → x p → p.

= −i∂x E → E.

= i∂t .

[x , p] = i .

2m+ V (x)⇒ −i∂t |ψ〉 = − 1

dx2|ψ〉+ V (x).

Quantisation of the Harmonic Oscillator

The Schrödinger equation for a 1-dimensional harmonic oscillator reads

−i∂tψ(x ; t) = − 1

dx2ψ(x ; t) +

2ω2x2

Its solution is

ψn(x ; t) = e−12ωx2Hn(x

√ω)e−iEnt , En =

Important features of Quantum Mechanics:

Discrete spectrum of energy eigenstates;

E0 = 12ω: there is a ground state with non-zero energy ( Heisenberg

Principle)

−i∂tψ(x ; t) = − 1

dx2ψ(x ; t) +

2ω2x2

Its solution is

Principle)

−i∂tψ(x ; t) = − 1

dx2ψ(x ; t) +

2ω2x2

Its solution is

Principle)

Second Quantisation

Let us introduce the Lowering and Raising Operators:

a =1√2ω

(ωx + i p) a† =1√2ω

(ωx − i p)[a, a†

They allow us to reformulate the Hamiltonian as

.= a†a = Number Operator.

Calling |n〉 the eigenstates of N we obtain that a generic state vector isgiven by:

|ψ(t)〉 =∑n

cne−iEnt |n〉 .

Second Quantisation

a =1√2ω

(ωx + i p) a† =1√2ω

(ωx − i p)[a, a†

|ψ(t)〉 =∑n

cne−iEnt |n〉 .

Second Quantisation

a =1√2ω

(ωx + i p) a† =1√2ω

(ωx − i p)[a, a†

|ψ(t)〉 =∑n

cne−iEnt |n〉 .

Special Relativity and Quantum Mechanics: the

Klein-Gordon Equation

Switching to the relativistic energy dispersion relation E 2 = k2 + m2

(c = 1) we get the the Klein-Gordon Equation:

(�+ m2) |ψ〉 = 0.

One can derive it computing the Euler-Lagrange Equations of theKlein-Gordon Lagrangian:

∫d4xLKG , LKG = −1

2ηµν∂µφ∂νφ−

2m2φ2

Problems:

Possible negative-energy solutions (Fourier transform);

Violation of causality;

||ψ〉|2 can not be interpreted as a probability amplitude.

Special Relativity and Quantum Mechanics: the

Klein-Gordon Equation

Switching to the relativistic energy dispersion relation E 2 = k2 + m2

(c = 1) we get the the Klein-Gordon Equation:

(�+ m2) |ψ〉 = 0.

One can derive it computing the Euler-Lagrange Equations of theKlein-Gordon Lagrangian:

∫d4xLKG , LKG = −1

2ηµν∂µφ∂νφ−

2m2φ2

Problems:

Possible negative-energy solutions (Fourier transform);

Violation of causality;

||ψ〉|2 can not be interpreted as a probability amplitude.

A World Made of Fields

A solution to these problems can be found considering a system within�nitely many degrees of freedom, i.e. a �eld (Dirac sea).

The KGHamiltonian is (π = φ)

2π2 +

2(∇φ)2 +

2m2φ2 Harmonic Oscillator!

(x ; p) 7→ (φ(xµ);π(xµ))

φ(xµ) has no more to be read as a wave function, but �xed-time initialvalue of the KG equation.

A World Made of Fields

A solution to these problems can be found considering a system within�nitely many degrees of freedom, i.e. a �eld (Dirac sea).The KGHamiltonian is (π = φ)

2π2 +

2(∇φ)2 +

2m2φ2 Harmonic Oscillator!

(x ; p) 7→ (φ(xµ);π(xµ))

φ(xµ) has no more to be read as a wave function, but �xed-time initialvalue of the KG equation.

Space of Solutions

The KG Equation admits plane-wave solutions:

φ(xµ) = φ0eikµx

µ= φ0e

iωt−ik·x ω2 = k2 + m2.

We look for a complete o.n. set of solutions, hence we need a scalarproduct on the solutions' space:

(φ1, φ2) = −i∫

(φ1∂tφ∗2 − φ∗2∂tφ1) d3x

so that we get the set (kµ : ω2 = k2 + m2):

fk(xµ) =e ikµx

2π√2ω, (fk1 , fk2) = δ(3)(k1 − k2).

Space of Solutions

The KG Equation admits plane-wave solutions:

φ(xµ) = φ0eikµx

µ= φ0e

iωt−ik·x ω2 = k2 + m2.

We look for a complete o.n. set of solutions, hence we need a scalarproduct on the solutions' space:

(φ1, φ2) = −i∫

(φ1∂tφ∗2 − φ∗2∂tφ1) d3x

so that we get the set (kµ : ω2 = k2 + m2):

fk(xµ) =e ikµx

2π√2ω, (fk1 , fk2) = δ(3)(k1 − k2).

Positive- and Negative-Frequencies Solutions

The solution are labelled by the continuous parameter k and are determinedup to the sign of ω. Since energy (E = hω) is a positive-de�nite quantitywe would like to consider only solutions with positive frequency.

This is done by introducing, for all ω > 0, positive-frequency solutions

∂t fk = −iωfk

and negative-frequency solutions

∂t f∗k

= iωf ∗k.

Positive- and Negative-Frequencies Solutions

The solution are labelled by the continuous parameter k and are determinedup to the sign of ω. Since energy (E = hω) is a positive-de�nite quantitywe would like to consider only solutions with positive frequency.This is done by introducing, for all ω > 0, positive-frequency solutions

∂t fk = −iωfk

and negative-frequency solutions

∂t f∗k

= iωf ∗k.

In total analogy with the harmonic oscillator in QM we promote φ and π tooperator, imposing the equal-time commutation relations (HeisenbergPicture) [

φ(x), φ(x′)]t

= iδ(3)(x− x′).

Expanding as a function of the modes

φ(t; x) =

∫d3x

[akfk(t; x) + a†

kf ∗k

(t; x)]

which leads to the Canonical Commutation Relations[ak, a

†k′

]= δ(3)(k− k

In total analogy with the harmonic oscillator in QM we promote φ and π tooperator, imposing the equal-time commutation relations (HeisenbergPicture) [

φ(x), φ(x′)]t

= iδ(3)(x− x′).

Expanding as a function of the modes

φ(t; x) =

∫d3x

[akfk(t; x) + a†

kf ∗k

(t; x)]

which leads to the Canonical Commutation Relations[ak, a

†k′

]= δ(3)(k− k

Particles and Anti-Particles

Positive-frequencies modes are the coe�cients of the Annihilationoperator ak;

Negative-frequencies modes are the coe�cients of the Creationoperator a†

|nk〉 =1√nk!

(a†k)nk |0〉

We are creating n particles with momentum k. |0〉 is the Vacuum State.

Likewise in QM, we can introduce the Number Operator Nk = a†kak, whose

eigenvectors constitute the Fock Basis.

|nk〉 =1√nk!

(a†k)nk |0〉

|nk〉 =1√nk!

(a†k)nk |0〉

What's Missing?

Renormalization Subtraction of (in�nite) the point-zero energy

Interactions and Scattering Theory

Other kind of Fields (QED, Gauge Theories)

Fundamental Ideas

The interactions do not in�uence the �xed background;

Investigation of the back-reaction;

Black Holes Physics;

States (Vacua, Thermal...) and Renormalization;

Making the theory mathematically rigorous.

Fundamental Ideas

Choice of the background

L =√−g(−1

2∇µφ∇µφ−

2m2φ2 − ξRφ2

)⇒ �gφ−m2φ− 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 ds2 = +βdt2 − hijdxidx j . Here

(t, xi )3i=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

L =√−g(−1

2∇µφ∇µφ−

2m2φ2 − ξRφ2

)⇒ �gφ−m2φ− ξRφ = 0.

L =√−g(−1

2∇µφ∇µφ−

2m2φ2 − ξRφ2

)⇒ �gφ−m2φ− ξRφ = 0.

What about particles?

We de�ne the conjugate momentum π =√−g∇0φ and de�ne the scalar

product on a spacelike 3-surface Σ and take over the quantisation imposingimpose the CCR:

(φ1, φ2).

= −i∫

Σ(φ1∇µφ∗2 − φ∗2∇µφ1) nµ

√γd3x[

φ(t, x), φ(t, x′)]

=i√−g

δ(3)(x− x′)

Problem

In a general space-time it is impossible to de�ne positive-frequencysolutions because there is no unique notion of time. Hence the concept ofparticle is ill-de�ned!

What about particles?

We de�ne the conjugate momentum π =√−g∇0φ and de�ne the scalar

product on a spacelike 3-surface Σ and take over the quantisation imposingimpose the CCR:

(φ1, φ2).

= −i∫

Σ(φ1∇µφ∗2 − φ∗2∇µφ1) nµ

√γd3x[

φ(t, x), φ(t, x′)]

=i√−g

δ(3)(x− x′)

Problem

In a general space-time it is impossible to de�ne positive-frequencysolutions because there is no unique notion of time. Hence the concept ofparticle is ill-de�ned!

Hadamard States

We are dealing with semiclassical Einstein equations:

Rµν −1

2gµνR = −8π < Tµν >, < Tµν >

.= 〈0|Tµν |0〉 =?

The lack of the modes expansion implies that there is no de�nite notion ofvacuum state.This lead to the introduction of Hadamard States:

H(x , x ′) =U(x , x ′)

(2π)2σε+V (x , x ′) log(σε) +W (x , x ′), σε

.= σ+2iε(t− t ′) + ε2

< Tµν > well-de�ned

Expectation values with �nite �uctuations

Compatible with the unique Minkowski vacuum state

Hadamard States

Rµν −1

2gµνR = −8π < Tµν >, < Tµν >

.= 〈0|Tµν |0〉 =?

The lack of the modes expansion implies that there is no de�nite notion ofvacuum state.

This lead to the introduction of Hadamard States:

H(x , x ′) =U(x , x ′)

.= σ+2iε(t− t ′) + ε2

Hadamard States

Rµν −1

2gµνR = −8π < Tµν >, < Tµν >

.= 〈0|Tµν |0〉 =?

The lack of the modes expansion implies that there is no de�nite notion ofvacuum state.This lead to the introduction of Hadamard States:

H(x , x ′) =U(x , x ′)

.= σ+2iε(t− t ′) + ε2

general relativity in a nutshell - (and beyond) · general relativity in a nutshell (and beyond)...

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