corso di relatività generale i parte
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Privileged observers and affine manifolds
Both Newtonian Physics and Special Relativityhave privileged observers
Affine Manifold
Curved Manifolds and Atlases
The intuitive idea of an atlas of open charts, suitably reformulated in mathematical terms,provides the very definition of a differentiable manifold
The transition function
There are just two open charts and the transition function is the following one
Tangent vectors at a point p 2 M
Intuitively the tangent in p at a curve that starts from p is the curve’s initial direction
Fibre bundles
Fibre Bundles
Definition: A Fibre--bundle E M F G, , , , is a geometrical
structure that consists of the following list of elements: 1. A differentiable manifold E named the total space
2. A differentialble manifold M named the base space
3. A differentiable manifold F named the standard fibre
4. A Lie group G named the structure group which acts as a transformation group on the standard fibre:
g G ; g : F F i e f F g f F. . ,
5. A surjection map :E M named the projection . If n=dim M and m=dim F, then we have dim E = n+m and p E , Fp= 1 p is an
m-dimensional manifold diffeomorphic to the standard fibre F . The manifold Fp is named the fibre at point p
6. A covering of the base space UA = M realized by a collection
U M of open subsets equipped with a diffeomorphism:
:U F U 1
such that p U f F , : p f p,
The map is named a local trivialization of the bundle
7 If we write p f fp, ( ), the map , :p pF F is the
diffeomorphism required by point 5) of the present definition. For all points p U U in the intersection of two local trivialization
domains, the composite map t p F Fp p , , :1 is an
element of the structure group: t p G named the transition
function. Furthermore, the transition function realizes a smooth map: t : U U G ; p f p t p f, ,
Il concetto di Spazio Fibrato
M denota lo spazio di base P denota lo spazio totale denota la proiezione Udenota un aperto dello spazio base -1(U) é il fascio di fibre sopra U. Esso é omeomorfo
al prodotto di U con la fibra standard F.
I fibrati
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 LOOPSAfter transport along a loop, the vector does not come back to the original position but it is rotated of some angle.
La 1-forma di connessione La definizione di connessione su di un fibrato vettoriale E M può essere riassunta nel modo seguente: Una connessione é una mappa:
: E M E M T M, , che ad ogni sezione s del fibrato vettoriale associa una 1-forma a valori sezioni del fibrato s in maniera tale che X TM M , ,
X s s X
In questa formulazione, le proprietà soddisfatte dalla connessione sono: a) a s a s a s a s1 1 2 2 1 1 2 2
b) fs df s f s f C M s E M , ,
Riferimenti e potenziali vettoriConsideriamo ora una trivializzazione locale: : F U U 1
DEFINIZIONE: Un riferimento su U é un insieme di sezioni s sk 1 , ,
tale che p U M i k vettori s p s pk 1 , , formano una base
per lo spazio vettoriale 1 p , cioè per la fibra al di sopra del punto p .
Dato un riferimento sopra U la 1--forma di connessione in quel riferimento può
essere data ponendo:
s A sij
i j ; A A dx Tj
i II j
i
dove le matrici k k TI ji
sono una base di generatori per l’algebra di Lie
del grupppo strutturale G del fibrato vettoriale, nella rappresentazione D sopportata dalla fibbra standard F . In altre parole la connessione è una 1--forma a valori elementi dell’algebra di Lie del gruppo strutturale. IN GERGO FISICO è un potenziale vettore per il gruppo di gauge.
Funzioni di transizione tra trivializzazioni locali diverse in uno spazio fibrato
Le funzioni di transizione
Trasformazioni di gauge = cambio di trivializzazione locale
Se consideriamo due trivializzazioni locali U ed U sull’intersezione
U U abbiamo due definizioni della 1--forma di connessione A ed A
che sono legate dalla formula
tAttdtA 11
In ogni trivializzazione locale alla 1--forma di connessione possiamo associare una 2--forma di curvatura :
F dA A A dA A A 12
,
la relazione tra F ed F nell’intersezione di due trivializzazioni locali é:
F t F t 1
Curvatura e Torsione di una connessione affine
Una connsessione affine é una connessione sul fibrato tangente ad una varietà differenziabile. Per comodità di notazione l’insieme delle sezioni del fibrato tangente viene denotato X M e forma un algebra di Lie infinito dimensionale rispetto al
commutatore. Possiamo quindi dire che la connesione affine é una mappa: : X M X M X M
che soddisfa le proprietà di una connessione date precedentemente:
X X XY Z Y Z ; X Y X YZ Z Z
fX XY f Y ; X XfY X f Y f Y
Data una connessione affine si definiscono la 2--forma di torsione T e la due forma di curvatura R che sono a valori nello spazio delle sezioni del fibrato tangente cioè in
X M . Abbiamo:
Torsione: T X M X M X M: :
T X Y X YX Y, ,
Curvatura: R X M X M X M X M: :
R X Y Z Z Z ZX Y Y X X Y, , ,
Essenzialmente, la curvatura esprime il commutatore di due derivate covarianti. Essa é leagata al fatto che in uno spazio curvo il trasporto parallelo lungo curve diverse da risultati diversi. Alla fine dei giochi la curvatura esprime il fatto che in uno spazio curvo la geometria non é più quella euclidea. La somma degli angoli dei triangoli non é più 180 gradi!
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 triangle 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
)()(
ttaa
)(Sin)(Cosh)()(Cos)(Cosh)(
)(Sinh)(
2
1
0
ttatXttatX
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!!!!
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
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
X1
X2
X0
X2
Space-likeap
aptg22 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 CoshEtg
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:
Where from 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:
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!!!!!
S R g g d x
R E E
gravG
Gab c d
abcd
116
4
164
det =
=
plus the action of matter
S S Stot grav matter where Smatter matterL the
Lagrangian density of matter being a 4-form.
We obtain it varying the action with respect to the spin connection:
S DE E dd
abG abcd
c dab 1
320( ) + L matter
in the absence of matter we get
abcdc d
c d
DE EDE T
00
LeviCivita connectionab
We obtain it varying the action with respect to the Vielbein
EINSTEIN EQUATIONS IN DIFFERENTIAL FORM LANGUAGE
Action Principle
TORSION EQUATION
EINSTEIN EQUATION
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