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 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! Who is at motion? The Sun or the Earth? A famous question with a lot of history behind it
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
This angle is the perihelion advance, predicted by G.R. 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
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
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....
Newton’s Law Inertial and gravitational masses are equal Constant gravitational field Accelerated frame Gravity has been Locally suppressed Equivalence Principle: a first approach
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!!
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 ametricgonM Straight linesaregeodesics Field equationsare generally covariant under diffeomorphisms G.R. model of the physical world Physics Geometry
Hence the mathematical model of space time is a pair: Differentiable Manifold Metric We need to review these two fundamental concepts
The sphere: The hyperboloid: 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: 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
Stereographic projection The transition function on Gluing together a Manifold: the example of the sphere
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
Under change of local coordinates A tangent vector is a 1st order differential operator Tangent spaces and vector fields
A vector field is parallel transported along a curve, when it mantains a constant angle with the tangent vector to the curve Parallel Transport
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.
a b g 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, namelygeodesics for this manifold
The hyperboloid: a space with negative curvature and lorentzian signature This surface is the locus of points satisfying the equation We can solve the equation parametrically by setting: Then we obtain the induced metric
How long is this curve? B A Answer: 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 single parametert This integral is a rule ! Any such rule is a Gravitational Field!!!!
Underlying our rule for lengths is the induced metric: Where a and qare the coordinates of our space. This is a Lorentzian metric and it is just induced by the flat Lorentzian metric in three dimensions: using the parametric solution forX0 , X1 , X2
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 space-time 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.
Relativity = Lorentz signature - , + space time Let us see what are the straight lines (=geodesics) on the Hyperboloid • ds2 < 0space-like geodesics: cannot be followed by any particle (it would travel faster than light) • ds2 > 0time-like geodesics. It is a possible worldline for a massive particle! • ds2 = 0light-like geodesics. It is a possible world-line for a massless particle like a photon Three different types of geodesics 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
Still continuing Let us now study the shapes and properties of these curves
The shape of geodesics is a consequence of our rule to calculate the length of curves, namely of the metric Space-like 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
Time-like 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!
Light like geodesics are conserved under conformal transformations Light like 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
Christoffel symbols = Levi Civita connection Let us now review the general case
the Christoffel symbols are: wherefrom do they emerge and what is their meaning? • They are the coefficients of an affine connection, namely the proper mathematical concept underlying the concept of parallel transport. ANSWER: Let us review the concept of connection
1 2 3 4 Connection and covariant derivative A connection is a map From the product of the tangent bundle with itself to the tangent bundle with defining properties:
In a basis... This defines the covariant derivative of a (controvariant) vector field
a a Torsion and Curvature 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!!!!!
At any event 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 Now we can state the....... Appropriate formulation of the Equivalence Principle:
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 Are the harmonic coordinates Harmonic Coordinates and the exponential map
A view of the locally inertial frame 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.
Local inertial frame atq Local inertial frame at p The vielbein or Repère Mobile We can construct the family of locally inertial frames attached to each point of the manifold q p M
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 This 1- form substitutes the affine connection Poincaré connection