Transcript Introduction to General Relativity

Introduction to
General Relativity
Lectures by Pietro Fré
Virgo Site May 26th 2003
The issue of reference frames
Since oldest
and observers
antiquity the
Who is at motion? The Sun or the Earth?
A famous question with a lot of history behind it
humans have
looked at the sky
and at the motion
of the Sun, the
Moon and the
Planets. Obviously
they always did it
from their
reference frame,
namely from the
EARTH, which is
not at rest, neither
in rectilinear
motion with
constant velocity!
The Copernican Revolution....
According to Copernican
and Keplerian theory , the
orbits of Planets are Ellipses
with the Sun in a focal
point. Such elliptical orbits
are explained by NEWTON’s
THEORY 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
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:
y
x
This angle is the
perihelion advance,
predicted by G.R.
3mm
For the planet Mercury it is
  43" of arc per century
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
F  ma
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
The idea of General Covariance
• 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....
Equivalence Principle: a first
approach
Newton’s Law
Inertial and
gravitational masses
are equal
Constant gravitational
field
Accelerated frame
Gravity has been
Locally suppressed
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 is crucial in this
context!!
G.R. model of the physical world
Physics
• The when and the where of any
physical physical phenomenon
constitute an event.
• The set of all events is a
continuous space, named spacetime
• 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
Geometry
• 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
Hence the mathematical model
of space time is a pair:
M , g 
Differentiable
Manifold
Metric
We 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:
X2
X12  X 22  X 32  1
The hyperboloid:  X 2  X 2  X 2  1
0
2
3
The sphere:
X0
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:
X1
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
The transition function
on
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
Under change of local
coordinates

A tangent vector is a 1st order
differential operator
Parallel Transport
A 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
In a flat manifold,
curved manifolds
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
a
a  b g  
b
g
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
This surface is the locus of
points satisfying the equation
 X  X  X 1
2
0
2
1
2
2
We can solve the equation
parametrically by setting:
Then we obtain the induced metric
The metric: a rule to calculate the
lenght of curves!!
A curve on the surface is
described by giving the
coordinates as functions of a
B
single parameter t
X 0 (t )  Sinh a(t )
a  a (t )
A
Answer:
X 1 (t )  Cosh a(t ) Cos (t )
   (t )
tB
l AB  
tA
How long is this
curve?
X 2 (t )  Cosh a(t ) Sin  (t )


dX 0
dt
   
2

dX1
dt
2

This integral is a rule ! Any such rule is a Gravitational Field!!!!
dX 2
dt
 dt
2
Underlying our rule for lengths is
the induced metric:
ds 
2
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:
ds 
2
using the parametric solution for X0 , X1 , X2
  a  
0    2
What do particles do in a
gravitational field?
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 sheet
summarizes 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 spacetime is bended under the weight of
matter moving inside it!
What are the straight lines
They are the geodesics, curves that do not change length under small
deformations. These are the curves along which we have parallel
transported our vectors
On a sphere
geodesics are
maximal 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
• ds2 < 0 space-like geodesics: cannot be
Three different
types of
geodesics
Relativity
= Lorentz signature
-,+
space
time
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
l   

da 2
dt
 Cosh a
Is the rule to calculate lengths
2
 dt
d 2
dt
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
Space-like
tg  p
Sinh a
p  Cosh a
2
2
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.
The shape of geodesics is a
consequence of our rule to calculate
the length of curves, namely of the
metric
This latter is a constant
of motion, a first
integral
Time-like
tg 2  1
Cosh a  E
2
2
tg   E
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!
a


T anh  T an   a 
2
2

Light like
X2
X0
Light like geodesics are conserved
under conformal transformations
X1
These curves
lie on the
hyperboloid ,
are straight
lines and are
characterized
by a first
integral of the
motion which
is the angle
shift a
Let us now review the general
case
Christoffel
symbols
=
Levi Civita
connection
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 A connection is a map
 : TM  TM  TM
From the product of the
tangent bundle with itself to
the tangent bundle
with defining
properties:
1
 X Y  Z    X Y   X Z
2
3
 fX Y  f X Y
4
 X Y  Z   X Z  Y Z
 X  fY   X  f Y  f X Y
In a basis...
This defines the covariant derivative of a (controvariant) vector field
Torsion and Curvature
T  X ,Y    X  Y   X ,Y 
Torsion
Tensor
R X ,Y , Z   X Y Z Y  X Z  X ,Y Z
Curvature
Tensor
aa
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 p  M of space-time we can find a
reference frame where the Levi Civita connection vanishes
at that point. Such a frame is provided by the harmonic or
locally inertial coordinates and it is such that the
gravitational field is locally removed. Yet the gradient of the
gravitational field cannot be removed if it exists.
In other words Curvature can never be removed, since it is
tensorial
Harmonic Coordinates and the
exponential map

 v  Tp M
exp: Tp M  Vp  M

exptv   g v (t )
g v (t )
Follow the geodesics
that admits the vector
v as tangent and
passes through p up
to the value t=1 of the
affine parameter. The
point you reach is the
image of v in the
manifold
 v t
a
a
Are the harmonic
coordinates
A view of the locally inertial
frame
d 

0
2
dt
2
The geodesic equation, by definition,
reduces in this frame to:
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
Local
inertial
frame at q
Local
inertial
frame at p
p
We can
construct the
family of
locally inertial
frames
attached to
each point of
the manifold
q
M
 |x   ( x)
a
a
a


Ea ( x)  
x

E  E ( x)dx
a
a
The vielbein encodes the metric
Indeed we can write:
Mathematically the vielbein is part of a connection on a
Poincarè bundle, namely it is like part of a Yang—Mills
gauge field for a gauge theory with the Poincaré group as
gauge group
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
conservation of
the stress energy
tensor
It suffices that the field equations be
of the form:
Gab  4 G Tab
D Tab  0
a
•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
Action
Principle
S
grav
 
 
1

R g det  g d 4 x
16G
1
=
Rab  E c  E d  abcd
64G

plus the action of matter
S tot  S grav  S matter
where
S matter
 L
matter
Lagrangian density of matter being a 4-form
TORSION EQUATION
it varying
the action
with respect to the spin connection:
inWe
theobtain
absence
of matter
we get
matter
 abcd DE c abE d 10 
dL
c
d
  Sc  
 32G ab
 abcd DE  E +
d
ab
DE  T  0    LeviCivita connection

(
d
) 0
=
S
Action
Principle
grav
 
 
1

R g det  g d 4 x
16G
1
=
Rab  E c  E d  abcd
64G
=

plus the action of matter
S tot  S grav  S matter
where
S matter
 L
matter
Lagrangian density of matter being a 4-form
EINSTEIN EQUATION
We obtain it varying the action with respect to the vielbein
Expanding on the vielbein basis we obtain
G
2 8RGT
 E  E  abcd    E L
E S 
ab
ab
c
ab
Where Gab is the Einstein tensor
d
matter
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
WEFIND
HAVE
TOSOLUTION
SOLVE:
WE
THE
And from this, in few straightforward steps we obtain the EINSTEIN
TENSOR
The solution
Boundary conditions for
asymptotic flatness
ar    0
r
br    0
r
G 0
a
b
a   b  0  a  b because of boundary condition


b
1
b e2b
2b 
b  2b  2  0 ; 2  2  2  0  e r  1
r
r r
r
2
This yields the final form of
the Schwarzschild solution
m

e2br   1 2 
r

m

e2ar   1 2 
r

1
The Schwarzschild metric and
its orbits
THE METRIC
WHICH
MEANSIS:
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 EFFECTS
The effects: Periastron Advance
Numerical solution of
orbit
equation in
G.R.
Keplerian
orbit
Bending of Light rays
More to come in
next lectures....
Thank you for your
attention