Introduction to General Relativity

Lectures by Pietro Fré

Virgo Site May 26th 2003

The issue of reference frames and observers Since oldest

antiquity the humans have looked at the sky and at the motion of the Sun, the the Sun, the Moon and the Moon and the PlanetsPlanets. Obviously they always did it from their reference frame, namely from the EARTHEARTH, which is not at rest, neither in rectilinear motion with constant velocity!

Who is at motion? The Sun or the Earth?

A famous question with a lot of history behind it

The Copernican Revolution....

x

y

z

x

According to Copernican According to Copernican and Keplerian theory , the and Keplerian theory , the orbits of Planets are Ellipses orbits of Planets are Ellipses with the Sun in a focal with the Sun in a focal point. Such elliptical orbits point. Such elliptical orbits are explained by are explained by NEWTON’s NEWTON’s THEORY of GRAVITYTHEORY of GRAVITY

But Newton’s Theory works if we choose the Reference frame of the SUN. If we used the reference frame of the EARTH, as the ancient always did, then Newton’s law could not be applied in its simple form

Seen from the EARTH

x

y

z

x

The orbit of a Planet is much more complicated

Actually things are worse than that..

• The true orbits of planets, even if seen from the SUN are not ellipses. They are rather curves of this type:

For the planet Mercury it is

This angle is the perihelion advance, predicted by G.R.

m3mx

y

yper centurarcof "43

Were Ptolemy and the ancients so much wrong?• Who is right: Ptolemy or Copernicus?

• We all learned that Copernicus was right

• But is that so obvious?

• The right reference frame is defined as that where Newton’s law applies, namely where

aF m

Classical Physics is founded.......

• on circular reasoning

• We have fundamental laws of Nature that apply only in special reference frames, the inertial ones

• How are the inertial frames defined?

• As those where the fundamental laws of Nature apply

• It would be better if Natural Laws were formulated the same in whatever reference frame

• Whether we rotate with respect to distant galaxies or they rotate should not matter for the form of the Laws of Nature

• To agree with this idea we have to cast Laws of Nature into the language of geometry....

The idea of General Covariance

Constant gravitational field

Gravity has been Locally suppressed

Equivalence Principle: a first approach

Inertial and gravitational masses

are equal

Newton’s Law

Accelerated frame

This is the Elevator Gedanken Experiment of Einstein

There is no way to decide whether we are in an accelerated frame or immersed in a locally constant gravitational field

The word local local is crucial in this context!!

G.R. model of the physical world

• The when and the where of any physical physical phenomenon constitute an event.

• The set of all events is a continuous space, named space-time

• Gravitational phenomena are manifestations of the geometry of space—time

• Point-like particles move in space—time following special world-lines that are “straight”

• The laws of physics are the same for all observers

• An event is a point in a topological space

• Space-time is a differentiable manifold M

• The gravitational field is a metric g on M

• Straight lines are geodesics

• Field equations are generally covariant under diffeomorphisms

PhysicsPhysics GeometryGeometry

Hence the mathematical model of space time is a pair:

gM ,

Differentiable Differentiable Manifold Manifold

MetricMetric

We need to review these two fundamental conceptsWe need to review these two fundamental concepts

Manifolds are:Topological spaces whose points can be labeled by coordinates.Sometimes they can be globally defined by some property.

For instance as algebraic loci:

The sphere:The sphere: 123

22

21 XXX

The hyperboloid:The hyperboloid: 123

22

20 XXX

X1

X2

X0

X1

X2

In general, however, they can be built, only by patching together an Atlas of open charts

The concept of an Open Chart is the Mathematical formulation of a local Reference Frame. Let us review it:

Open Charts:

The same point (= event) is contained in more than one open chart. Its description in one chart is related to its description in another chart by a transition function

Gluing together a Manifold: the example of the sphere

on

The transition function

Stereographic projection

We can now address the proper Mathematical definitions• First one defines a Differentiable

structure through an Atlas of open Charts

• Next one defines a Manifold as a topological space endowed with a Differentiable structure

Differentiable structure

Differentiable structure continued....

Manifolds

Tangent spaces and vector fields

A tangent vector is a 1st order differential operator

Under change of local coordinates

Parallel TransportA vector field is parallel transported along a curve, when it mantains a constant angle with the tangent vector to the curve

The difference between flat and curved manifolds In a flat manifold,

while transported, the vector is not rotated. In a curved manifold it is rotated:

To see the real effect of curvature we must consider.....Parallel transport along LOOPS

After transport along a loop, the vector does not come back to the original position but it is rotated of some angle.

On a sphere The sum of the internal angles of a triangle is larger than 1800

This means that the curvature

is positive

How are the sides of the this traingle drawn?

They are arcs of maximal circles, namely geodesics

for this manifold

The hyperboloid: a space with negative curvature and lorentzian signature

X1

X2

X0

X1

X2

122

21

20 XXX

This surface is the locus of points satisfying the equation

Then we obtain the induced metric

We can solve the equation parametrically by setting:

The metric: a rule to calculate the lenght of curves!!

A

B

)(

)(

t

taa

)(Sin)(Cosh)(

)(Cos)(Cosh)(

)(Sinh)(

2

1

0

ttatX

ttatX

tatX

A curve on the surface is described by giving the coordinates as functions of a

single parameter t

This integral is a rule ! Any such rule is a This integral is a rule ! Any such rule is a Gravitational Field!!!!Gravitational Field!!!!

Answer: dtt

tAB

B

Adt

dXdt

dXdt

dXl 2

22

12

0

How long is this curve?

Underlying our rule for lengths is the induced metric:

2ds

Where a and are the coordinates of our space. This is a Lorentzian metric and it is just induced by the flat Lorentzian metric in three dimensions:

20 a

2ds

using the parametric solution for X0 , X1 , X2

What do particles do in a gravitational field?Answer:Answer: They just go straight as in empty space!!!!

It is the concept of straight line that is modified by the presence of gravity!!!!The metaphor of Eddington’s sheetsummarizes General Relativity.In curved space straight lines are different from straight lines in flat space!! The red line followed by the ball falling in the throat is a straight line (geodesics). On the other hand space-time is bended under the weight of matter moving inside it!

What are the straight linesThey are the geodesics, curves that do not change length under small deformations. These are the curves along which we have parallel These are the curves along which we have parallel transported our vectorstransported our vectors

On a sphere On a sphere geodesics are geodesics are maximal circlesmaximal circles

In the parallel transport the angle with the tangent vector remains fixed. On geodesics the tangent vector is transported parallel to itself.

Let us see what are the straight lines (=geodesics) on the Hyperboloid

Three different types of geodesics

Relativity

= Lorentz signature - , +

time

space

dtal dtd

dtda 222 Cosh

• ds2 < 0 space-like geodesics: cannot be followed by any particle (it would travel faster than light)

• ds2 > 0 time-like geodesics. It is a possible worldline for a massive particle!

• ds2 = 0 light-like geodesics. It is a possible world-line for a massless particle like a photon

Is the rule to calculate lengths

Deriving the geodesics from a variational principle

The Euler Lagrange equations are

The conserved quantity p is, in the time-like or null-like cases, the energy of the particle travelling on the geodesic

Continuing...

This procedure to obtain the differential equation of orbits extends from our toy model in two dimensions to more realistic cases in four dimensions: it is quite general

Still continuing

Let us now study the shapes and properties of these curves

X1

X2

X0

X2

Space-likeap

aptg

22 Cosh

Sinh

These curves lie on the hyperboloid and are space-like. They stretch from megative to positive infinity. They turn a little bit around the throat but they never make a complete loop around it . They are characterized by their inclination p.

This latter is a constant of motion, a first integral

The shape of geodesics is a consequence of our rule to calculate the length of curves, namely of the metric

X1

X2

X0

X1

X2

X1

X2

X0

X1

X2

X1

X2

X0

X1

X2

Time-like 22

2 1 Cosh

Etg

tgEa

These curves lie on the hyperboloid and they can wind around the throat. They never extend up to infinity. They are also labeld by a first integral of the motion, E, that we can identify with the energy

Here we see a possible danger for causality:

Closed time-like curves!

X1X2

X0

X2

Light like

2 Tan

2Tanh

a

These curves lie on the hyperboloid , are straight lines and are characterized by a first integral of the motion which is the angle shift Light like geodesics are conserved

under conformal transformations

X1

X2

X0

X1

X2

Let us now review the general case

Christoffel Christoffel symbolssymbols

==

Levi Civita Levi Civita connectionconnection

the Christoffel symbols are:

wherefrom do they emerge and what is their meaning?

ANSWER: They are the coefficients of an affine connection, namely the proper mathematical concept underlying the concept of parallel transport.

Let us review the concept of connection

Connection and covariant derivative

TMTMTM :A connection is a map

From the product of the tangent bundle with itself to the tangent bundle

X X XY Z Y Z1 X Y X YZ Z Z2

fX XY f Y3 X XfY X f Y f Y4

with defining properties:

In a basis...

This defines the covariant derivative of a (controvariant) vector field

aa

Torsion and Curvature

T X Y X YX Y, ,

R X Y Z Z Z ZX Y Y X X Y, , ,

Torsion Tensor

Curvature Tensor

The Riemann curvature tensor

If we have a metric........An affine connection, namely a rule for the parallel transport can be arbitrarily given, but if we have a metric, then this induces a canonical special connection:

THE LEVI CIVITA CONNECTION

This connection is the one which emerges from the variational principle of geodesics!!!!!

Now we can state the....... Appropriate formulation of the Equivalence Principle:

At any event of space-time we can find a At any event of space-time we can find a reference frame where the Levi Civita connection vanishes reference frame where the Levi Civita connection vanishes at that point. Such a frame is provided by the harmonic or at that point. Such a frame is provided by the harmonic or locally inertial coordinates and it is such that the locally inertial coordinates and it is such that the gravitational field is locally removed. Yet the gradient of the gravitational field is locally removed. Yet the gradient of the gravitational field cannot be removed if it exists.gravitational field cannot be removed if it exists.

In other words Curvature can never be removed, since it is tensorial

Mp

Harmonic Coordinates and the exponential map

MVMT pp :expFollow the geodesics Follow the geodesics that admits the vector that admits the vector vv as tangent and as tangent and passes through passes through pp up up to the value to the value t=1t=1 of the of the affine parameter. The affine parameter. The point you reach is the point you reach is the image of v in the image of v in the manifoldmanifold

MTv p

)(tv

)(exp tvt v

tvaa Are the harmonic Are the harmonic coordinatescoordinates

A view of the locally inertial frame

The geodesic equation, by definition,

reduces in this frame to:0

2

2

dt

d

The structure of Einstein Equations• We need first to set down the items entering the

equations• We use the Vielbein formalism which is simpler,

allows G.R. to include fermions and is closer in spirit to the Equivalence Principle

• I will stress the relevance of Bianchi identities in order to single out the field equations that are physically correct.

The vielbein or Repère Mobile

p q

Local inertial frame at p

Local inertial frame at q

M

)(| xax

a x

xEa

a

)( dxxEE aa )(

We can construct the family of locally inertial frames attached to each point of the manifold

Mathematically the vielbein is part of a connection on a Mathematically the vielbein is part of a connection on a Poincarè bundle, namely it is like part of a Poincarè bundle, namely it is like part of a Yang—Mills Yang—Mills gauge fieldgauge field for a for a gauge theorygauge theory with the with the Poincaré groupPoincaré group as as gauge groupgauge group

The vielbein encodes the metricIndeed we can write:

Poincaré connection

This 1- form substitutes the affine connection

Using the standard formulae for the curvature 2-form:

The Bianchi Identities

The Bianchis play a fundamental role in building the physically correct field equations. It is relying on them that we can construct a tensor containing the 2nd derivatives of the metric, with the same number of components as the metric and fulfilling a conservation equation

Bianchi’s and the Einstein tensor

Allows for the Allows for the conservation of conservation of the the stress energy stress energy tensortensor

It suffices that the field equations be of the form:

abab TGG 4 0abaTD

•Source of gravity in Newton’s theory is the mass

•In Relativity mass and energy are interchangeable. Hence Energy must be the source of gravity.

•Energy is not a scalar, it is the 0th component of 4-momentum. Hence 4—momentum must be the source of gravity

•The current of 4—momentum is the stress energy tensor. It has just so many components as the metric!!

•Einstein tensor is the unique tensor, quadratic in derivatives of the metric that couples to stress-energy tensor consistently

S R g g d x

R E E

gravG

Gab c d

abcd

1

164

1

64

det =

=

Action Principle

TORSION EQUATION

We obtain it varying the action with respect to the spin connection:

S DE E d

dab

G abcdc d

ab 1

320( ) + L matter

 

where

Lagrangian density of matter being a 4-form

plus the action of matter

S S Stot grav matter Smatter matterL

abcdc d

c d

DE E

DE T

0

0

LeviCivita connectionab

in the absence of matter we get

S R g g d x

R E E

gravG

Gab c d

abcd

1

164

1

64

det =

=

Action Principle

plus the action of matter

S S Stot grav matter where Smatter matterL

Lagrangian density of matter being a 4-form

 

EINSTEIN EQUATION

We obtain it varying the action with respect to the vielbein

matterEabcd

dcabE LEERS 2

abab GTG 8Expanding on the vielbein basis we obtain

Where Gab is the Einstein tensor

We have shown that.......

• The vanishing of the torsion and the choice of the Levi Civita connection is the yield of variational field equation

• The Einstein equation for the metric is also a yield of the same variational equation

• In the presence of matter both equations are modified by source terms.

• In particular Torsion is modified by the presence of spinor matter, if any, namely matter that couples to the spin connection!!!

A fundamental example: the Schwarzschild solution

Using standard polar coordinates plus the time

coordinate t

Is the most general static and spherical symmetric metric

Finding the solution

WE HAVE TO SOLVE:WE FIND THE SOLUTION

And from this, in few straightforward steps we obtain the EINSTEIN TENSOR

The solution

a b a b

b b br r

br

e

re r

bb

0

22

2

222 2 0 1 2 0 1

because of boundary condition

;

a rr

0

b rr

0

Boundary conditions for asymptotic flatness

0abG

em

r

em

r

b r

a r

21

2

1 2

1 2

This yields the final form of

the Schwarzschild solution

The Schwarzschild metric and its orbits

THE METRIC IS:THE METRIC IS:WHICH MEANS THE LAGRANGIANWHICH MEANS THE LAGRANGIAN

Energy & Angular Momentum

Newtonian Potential.Is present for time-like but not for null-like

Centrifugal barrier

G.R. ATTRACTIVE TERM: RESPONSIBLE FOR NEW EFFECTSG.R. ATTRACTIVE TERM: RESPONSIBLE FOR NEW EFFECTS

Keplerian orbit

The effects: Periastron Advance

Numerical solution of orbit equation in G.R.

Bending of Light rays

More to come in next lectures....Thank you for your attention