general relativity - universidad de...

General RelativityIV Escuela Mexicana

de Cuerdas y Supersimetría

Gustavo NizJune 2015

General Relativity

Based on:

●Sean Carrol's notes (short ones)● John Stewart course/book●Personal notes/talks

GR as a Cooking Book

● Ingredients– Physics & Maths

● Cookware– Equations

● Recipes– Solutions

If time allows it … ”BEYOND GR”

Ingredients

● Why GR?

Newtonian gravity

Vs

Einstein's gravity

Particle moves in straight line until hit by a force

Gravity is not a force but a result of space-time geometry

Ingredients

● Why GR?

Newtonian gravity

Vs

Einstein's gravity(GR)

Particle moves in straight line until hit by a force

Gravity is not a force but a result of space-time geometry

Keywords

Ingredients

Need to understand

a) Space and time together

b) Curved space

Ingredients

Need to understand

a) Space and time together

b) Curved space

TensorNotation

Ingredients: Special Relativity

● Speed of light is constant in any frame (c=1)● Space and time are entangled

Coordinates of space-time


Special Relativity (SR) lives in

Minkowski space-time

4-vector

Event P


Minkowski metric

Distance (dot product)

Infinitesimal distance (line element)

Summation convention

Same information as in the metric


Lorentzian signature ( - , + , + , + )

[ sometimes (+,-,-,-) ]

C.f. Euclidean 4d space

Euclidean signature ( + , + , + , + )


Lorentz group

– Transformations that leave ds unchanged

e.g. x-directionboost

Exercise


For a particle moving only

in time (xi=const.) the elapsed time is

Define proper time as


Spacetime diagram


Spacetime diagram

Null vector

Defines light-cone


Spacetime diagram

Null vector

Timelike vector

Spacelike vector


Trajectory specified by

is timelike/spacelike/null

if tangent vector

is timelike/spacelike/null


● 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


Massive particles

4-velocity

4-momentum

m rest frame mass


Massive particles

4-velocity 4-momentum

Energy is , which in the rest frame, is


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

Ingredients: Curved Space

Minkoswki Manifold

Cartesian coords. Cannot use Cartesian

coords. globally


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

Manifold


In simple words


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

Ingredients: Tensors

Under a coordinate transformation a

tensor changes in the following way

(summation convention)


Under a coordinate transformation a

tensor changes in the following way

(n,m) - Tensor

(summation convention)


Observations

(0,0) – tensor scalar (function)

(0,1) – tensor vector

(1,0) – tensor co-vector

Defin

ition

s


Observations

Trace over indices

Symmetric

Antisymmetric

Mat

hs


Observations

An equation that holds in one

coordinate system holds in all

coordinate systems

Equivalence Principle!

Physi

cs


In GR the most important tensor is the metric

a generalisation of Minkowski's metric

which encodes the geometrical information of

spacetime


At a given point, there is always a coordinate system such that

and also first derivatives of vanish!

Second derivatives cannot be made to vanish,

a manifestation of curvature

i.e. the spacetime

looks locally flat


Dot product

Suggests the concept of raising/lowering indices, namely

where is the inverse of

so


Important objects in physics which are not tensors

1) determinants

Under transforms as



2) volume factor

Transforms as

1) and 2) are call tensor densities because transform as some powers of the Jacobian



Notice

Is invariant under a change of coordinate, since f(x) is a scalar



Notice

Is invariant under a change of coordinate, since f(x) is a scalar

Therefore the right way of writing integrals is using



3) Partial derivatives

On scalar are OK

...but on vectors,


Define the Covariant Derivative

Connection transforms to cancel this


transforms a tensor and

defines parallel transport


transforms a tensor and

defines parallel transport

Similarly for lower indices


Christoffel Symbols

There is a particular choice of connection that

In components, it is

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)


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 tensormeasures the

difference!


Properties of

1)

2)

(Bianchi identity)

3)

Antisymmetric

Symmetric


Other tensors derived from

1) Ricci tensor

It satisfies

2) Ricci scalar


Other tensors derived from

3) Einstein tensor

It obeys

Cookware: Geodesic Equation

A geodesic is a curve that extremises

The path is a geodesic if it obeys

where is the affine parameter

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!

Cookware: Einstein Equations

Einstein (1915)


Non-linear equations of the metric components!

EnergyMomentum

tensor


Dimension-full parameters

Newton's Constant

If c=h=1 then


Geometría del Materia/Energía

Espacio-tiempo


El espacio-tiempo es como una “manta” invisible

deformado por la materia o energía


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


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!


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.


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


Newtonian limit

Recall Poission equation for the Newtonian potential

2nd order diff.operator over

a field


Newtonian limit



a field

In a relativistic theorywe expect the energy

momentum tensor


Newtonian limit



a field

In a relativistic theorywe expect the energy

momentum tensor

Einstein equations


Newtonian limit

Now formally consider (perturbation around Minkowski)

Geodesic Equation + slow motion

Newton's 2nd law


Newtonian limit

Now formally consider (perturbation around Minkowski)

Geodesic Equation + slow motion

00-component of Einstein eqns Poisson's eq.

Newton's 2nd law

Cookware: Hilbert-Einstein action

Einstein equations can be obtained from

Einstein equations

Cookware: Hilbert-Einstein action

Can include a Cosmological Constant

Cookware: Alternative approaches

1) á la Palatini

Consider the connection and the metric as independent

The metricconnection


2) Hamiltonian (ADM or 3+1 formalism)

Consider the space-time splitting

N lapse function

Ni shift function

gij spatial metric

Schematically



Extrinsic curvature

Then one gets the Hamiltonian Canonical momenta

Hamiltonian

Momentum



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

Recipes: content

● Maximally symmetric solutions

– Penrose Diagrams● Black Hole solutions

– Schwarzschild– Others– Thermodynamics

● Cosmology. In other course(s)!!

Recipes: Maximally symmetric solns

Consider vacuum with a cosmological constant

There are 3 maximally symmetric solutions

– Minkowski

– De Sitter (dS)

– Anti-de Sitter (AdS)

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


Consider Minkowski spacetime

Change variables to


Bring infinite to fine distance with


Back to the Cartesian coords.

Conformal metric


Half of3-sphere


Take half of triangle

and attacha full

3-sphere ateach point


Future timelike infinity

Future null infinity

Spatial infinity

Past null infinity

Past timelike infinity

Recipes: de Sitter (dS)

Easier to picture in 5d


By choosing one can write the metric in the following useful frames:

Static coordinates

Cosmological coordinates

Exponential expansion


Penrose diagram

Both timelike and null future infinity


Penrose diagram

Recipes (detour): Horizons

Particle horizon

Event horizon

Recipes: Anti de Sitter (AdS)

Maximal symmetry

5d picture

Two times


Closed Timelike Curves (travel back in time?)

On the Universal covering space we avoid them!


By choosing one can write the metric as

Static coordinates

Penrose diagram: cannot be drawn well...

Relevant for AdS/CFT

Recipes: Solutions so far...

Solution Symmetry

Minkowski 0 0 Maximal

de Sitter >0 0 Maximal

Anti de Sitter <0 0 Maximal

Schwarzschild 0 Spherical

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. do not depend on time)

● Characterised by one integration constant alone (M)

It is called the Schwarzschild solution


Schwarzschild solution is

Since it is unique, it describes from

black holes to planets!

BHor

planetVacuum Schwarzschild sol.

Mass inside Ro

Ro


Test particle in the plane

For the Schwarzschild metric the potential is

Particle of unitmass and energyE in 1d potential

Kinetic potential energy conservation

Const . Newtonian angular potential momentum

Only GRcorrection(Mercury)


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,


Two singularities

1) True singularity diverge

2) Coordinate singularity bad frame

If

Nothing happens at


However, something special still happens at

Event horizon

Sees (1) moving slowerand slower, taking infinitetime to reach



Event horizon

Sees (1) moving slowerand slower, taking infinitetime to reach

(1) takes a finite timeto cross then r and t swap

roles and (1) goes straight into r=0


Penrose diagram


Penrose diagram (completion of spacetime)

Recipes: Kerr BH

Add rotation

Recipes: Kerr BH

Momentum bound

Extremal case

(BPS, stable, in nature?)

Recipes: Reissner–Nordström BH

Electric charge

Extremal case when

Charged+Rotating Kerr-Newman solution

Recipes: Kerr, KN or RN

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).

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

Recipes: BH thermodynamics

Implication

Temperature Surface gravity

Entropy Area of horizon

Beyond GR ...

● Desser's construction. Bottom up!

● Lovelock's theorem

● Problems with GR

● UV modifications

● IR modifications

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)

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/0411023, 0910.2975

Bottom up approach

Linear Gravity(Written in )

à la Palatini

Generates its own

Energy-MomentumTensor

(Belifante)

FULLGR

EOMWritten in

Are the same!!!

Bottom up approach

Start from (Fierz Pauli) action

with independent Minkowski tensors

Want to recover

Totalderivative

Bottom up approach

Start from (Fierz Pauli) action

with independent Minkowski tensors

Want to recover

Totalderivative

Full GR only needs an extra cubic term!!!

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 newaction

Bottom up approach

Finally, one can check the EOM from

are

LL L

Theorems

Lovelock's Theorem (1971-72)

If we want second order differential field equations for the metric with a 4d space-time, the only form is

To recover Newtonian physics

Theorems

Lovelock's Theorem

Implications:

1) Terms in the Lagrangian such as

which give second order field equations in 5d, are topological in 4d.

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

Why do we need to modify GR

● Non-renormalisable (perturbatively)● Essential singularities

– Black holes

– Big bang singularity

● Early Universe physics– Hot big bang problems (horizon, Universe size,

monopoles/defects, flatness)

in the UV ?

Why do we want to modify it

Cosmology's dark sector– Dark matter (particle physics?)

– Dark energy ● Small observed value● Why today (coincidence problem)● Vacuum energy (“old cosmological constant problem”)

in the IR ?

Why do we want to modify it

Cosmology's dark sector– Dark matter (particle physics?)

– Dark energy ● Small observed value● Why today (coincidence problem)● Vacuum energy (“old cosmological constant problem”)

in the IR ?

Non-renormalisable (perturbatively) in the UV

IRideas

Dark Sector in Cosmology

Observational evidence of dark energy

Simple solution: Λ (Vacuum energy?)

Complicated solution:New gravitational degrees of freedom!

Theoryconsistency ObservationsBeauty

New physics?

Dark Energy

Cosmological Constant (CC) & vacuum energy

Vacuum energy gravitates as a CC

QFT

Lorentz invariance

Energy-Momentum tensor


To get one needs

an important cancellation

The effective CC is

QFT

Dark Energy


Estimate perturbatively

By adding the normal modes of a free scalar with mass m we can see the problem

QFT

Dark Energy


Estimate perturbatively

By adding the normal modes of a free scalar with mass m, we can see the problem

QFT

Dark Energy


For some many species of matter

Fermionicnumber

E.g. electrons

QFT

Dark Energy


One of the most important problems in theoretical physics

Dark Energy

Possible ways outCancellation very specific among contributions (eg. SUSY)

Relaxation of its value as Universe expands is impossible

(“no-go” theorem by Weinberg, 1989)

Dark Energy

Possible ways outCancellation very specific among contributions (eg. SUSY)

Relaxation of its value as Universe expands is impossible

(“no-go” theorem by Weinberg, 1989)

Dark Energy

Modified

?

Baker et al, 2014

Where canbe modified?

Eot-Wash

Baker et al, 2014

Baker et al, 2014

Baker et al, 2014, Koyama's talk 2015

Screening mechanisms

EOM for scalar d.o.f.

Kineticterm

Potential(mass)

MatterInteraction

Only consider a coupling to non-relativistic matter


Chameleon


Symmetron


Vainshtein effect (Galileon)


Horndeski Lagrangian

Horndeski (70's)

Most general scalar-tensor theory with second order EOMS

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)

UVideas

UV: Non-renormalisable

Where is the problem?

Consider

Re-scaling

Lorentz invariance


Potential

Diverges in the UV if n>4


Dimensional analysis

Propagator


In General Relativity the linear theory is

Propagator (schematically)

Differential operator of 2nd order

Dimensionfulcoupling


● We need an infinite number of counter-terms

● Due to Lorentz symmetry the have to be powers of curvature tensors


So what's wrong with ?

Propagator (schematically)

...


But

Propagator

Masslessgraviton

Massiveghost!!


But

Propagator

Masslessgraviton

Massiveghost!!

Usual problem withhigher derivatives

(Ostrogradski)


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


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.

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