general relativity - universidad de...

General Relativity

III Escuela Mexicanade Cuerdas y Supersimetría

Gustavo NizJune 2013

General Relativity

Based on:

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

GR as a Cooking Book

● Ingredients– Physics & Maths

● Cookware– Equations

● Recipes– Solutions

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


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

(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

Ingredients: exercises

1) Work out the Christoffel symbols for the metric

2) Using the previous result, write down the covariant derivative of the following vector

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

– The Hot Big Bang– Dark Matter/Energy– Early Universe (inflation, cyclic)

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 are

to 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?)

Careful analysis shows there are not


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

Recipes

Cosmology

Observations

Hot Big Bang Theory

Early Universe ideas

Recipes: 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

Recipes: Cosmology

Now we know:

0) Dark

1) Bigger

2) Expanding

Velocity = H (distance)

Hubble Diagram

Recipes: Cosmology

Implies a denser/hotter past

Cosmic Microwave

Background (CMB)

T=2.7 K

Recipes: Cosmology

Small perturbations of 1 in 100000

Simple UNIVERSE

GaussianScale invariant (tilt)

Recipes: Cosmology

Power spectrum (two point correlation function)

Recipes: Large Scale Structure

On sufficiently large scales (>200Mpc) the Universe is isotropic and homogeneous

Filaments

Voids

SLOAN

Recipes: Large Scale Structure

But, what else can we say

about its energy/matter content?

Recipes: Further Observations

Emitted light does not say it all...

Recipes: Dark Matter

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

Gravitational

lensing


¡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?



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

Or, could it be a modification of gravity?


There are other objects (called supernovae IA) which

also belong to the distance ladder. The supernovae are

big star explosions.

SN 1987A

Remanentede SN 1572 SN 1987A

Recipes: Dark Energy


Hubble diagram:

Supernovae tell us

the Universe presents

accelerated expansion

We understand even less what it is!

Only acts gravitationally, but in a repulsive way

Best candidate: Cosmological Constant


But...Quantum vacuum gravitates

as a cosmological constant

(zero point energy)

Theory = 100000000000000000000000000000000000000000000000000

0000000000000000000000000000000000000000000000000000000000000

0000000000000000000 Observation!!


Recipes: Cosmic Pie

5%

27%

68%

Materia VisibleMateria Obs.Energía Obs.

??

Recipes: Theory

Cosmological Principle

Assume an isotropic and homogeneous metricFRWL

Open

Flat

Closed

Scalefactor

Recipes: Theory

Friedmann equations

Einstein equations with a perfect fluid reduce to

Open k = -1

Flat k = 0

Closed k = +1

Recipes: Theory

Critical Mass

Define in terms of critical mass

Friedmann eqn reduce to

Recipes: Theory

Scale factor evolution

t

MatterRadiation

Dark energy

Recipes: Theory

Scale factor evolution

t

MatterRadiation

Dark energy

Model fits ALL observations to great accuracy

Recipes: Hot Big Bang

But not this...


Other observations support this model (age

of stars, light element

abundances, etc)


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

Expansion13.7 Gyrs


Problems


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

Expansion13.7 Gyrs

5) The initial (big bang) singularity

Singularity theorems

Initial data (assuming some energy conditions) can lead, unavoidable, to geodesically incomplete space-times.

Global statement.

What about the analytical structure of fields near the singularity?

Penrose, Hawking (60-70's)

Recipes: The singularity in GR

Belinski, Khalatnikov and Lifshitz (BKL)

Assumed ultralocality :

spatial gradients are not as important as time derivatives!

Big Bang

System reduces to 1d, but may have strong dependence on the initial conditions!

Chaos

(e.g. Mixmaster)


All depends on matter content: *scalar fields tend to remove chaos *gauge fields (p-forms) may restore it

Cosmological billiards

Damour et al '03Hamiltonian:

Near t=0,


Away from walls:

Kasner metric



*General Relativity (d<11)

KacMoody algebras(formally integrable)

*Low energy limits of 5 string theories and Mtheory


Note that there are potentials for the scalar fields which can overtake this oscillating behaviour, providing a deterministic evolution of the Universe.

However, for generic cases, we found a unpredictable behaviour of the metric near the spacetime singularity. We can even forget a description of the singularity itself.

Recipes: Beyond GR

Need to go beyond GR to solve all these problems of the early Universe

1) Inflation

2) Cyclic Universe

3) Others

But bare in mind that the Universe is SIMPLELatest results show only a Higgs field (nothing else) at LHCand Gaussian, almost scale invariant perturbations in CMB

Recipes: Inflation

Phase of exponential acceleration

Simplest realisation: canonical scalar field slowly rolling down a potential

Recipes: Inflation

Friedmann equations read

If slow-roll is assumed (potential energy dominates over kinetic), then

which imply the following solution


Solves the Problems


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

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

Recipes: Inflation

“Quantum” fluctuations

Scalar perturbations

Changing variables

Recipes: Inflation

In slow-roll

It is a harmonic oscillator! Easy to solve providing consistent initial conditionsThe curvature perturbations (which eventually get imprinted in the CMB) are

We want to calculate the two-point correlation function

Recipes: 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.

Recipes: Cyclic model

As a bonus, it explains CMB anisotropies

But introduces new (or keeps) 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)?

Recipes: Inflation

Moreover, Planck's results put some pressure

Recipes: Cyclic Model

Alternative model where a PreBig bang phase creates the CMB pertns.

Two flavours: singular & nonsingular

Contracting phase

t=0

Expanding phase

Big Crunch/Big Bang

Recipes: Cyclic Model

Recipes: Cyclic model

Creates perturbations in a very similar way to inflation

But... again some pressureby Planck

Recipes: alternatives

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