Introduction to General Relativity

Transcription

1 Introduction to General Relativity Lectures by Igor Pesando Slides by Pietro Fré Virgo Site May 22nd 2006

2 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. Planets Obviously they always did it from their reference frame, namely from the EARTH, EARTH which is not at rest, neither in rectilinear motion with constant velocity!

3 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 z y x 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

4 Keplerian Orbits The differential equation and its geometrical solution are consequences of Newton s law valid in the sun frame considered to be inertial and of the attractive central potential

5 From an Earthly viewpoint... if we call: The cartesian coordinates of the planet in the Earth frame are: and the motion is as follows

6 Seen from the EARTH The orbit of a Planet is much more complicated

7 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. 3m m For the planet Mercury it is Δ =43 ital of ital arc ital per century } {}

8 Let us see a Movie

9 This is a consequence of a new attractive term in the potential... Follows from GR and leads to a new term in the orbit differential equation

10 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 =m a

11 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

12 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...

13 Equivalence Principle: a first approach Newton s Law Inertial and gravitational masses are equal Constant gravitational field Accelerated frame Gravity has been Locally suppressed

14 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!!

15 G.R. model of the physical world Physics Geometry 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

16 Hence the mathematical model of space time is a pair: M, g Differentiable Manifold Metric We need to review these two fundamental concepts

17 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 The hyperboloid : X 20 X 22 X 23 =1 The sphere: X0 X1 X 21 X 22 X 23 =1 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:

18 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

19 Gluing together a Manifold: the example of the sphere The transition function on Stereographic projection

20 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

21 Differentiable structure

22 Differentiable structure continued...

23 Manifolds

24 Tangent spaces and vector fields Under change of local coordinates A tangent vector is a 1st order differential operator

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

26 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:

27 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.

28 On a sphere The sum of the internal angles of a triangle is larger than 1800 This means that the curvature α is positive α β γ π β γ How are the sides of the this triangle drawn? They are arcs of maximal circles, namely geodesics for this manifold

29 The hyperboloid: a space with negative curvature and lorentzian signature This surface is the locus of points satisfying the equation X X 0 X 1 X 2 =1 We can solve the equation parametrically by setting: X0 X1 Then we obtain the induced metric

30 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 parameter A t a =a t θ=θ t B How long is this curve? X 0 t =Sinh a t X 1 t =Cosh a t Cos θ t X 2 t =Cosh a t Sin θ t This integral is a rule! Any such rule is a Gravitational Field!!!!

31 Underlying our rule for lengths is the induced metric: 2 ds = Where a and θ are the coordinates of our space. This is a Lorentzian metric and it is just induced by the flat Lorentzian metric in three dimensions: 2 ds = using the parametric solution for X0, X1, X2 a 0 θ 2π

32 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!

33 The Methaphor as a Movie

34 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.

35 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 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 2 l = time da dt 2 Cosh a Is the rule to calculate lengths dθ 2 dt dt

36 Deriving the geodesics from a variational principle

37 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

38 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

39 Still continuing Let us now study the shapes and properties of these curves

40 Space-like tg θ = p X2 X1 X2 X0 X1 The shape of geodesics is a consequence of our rule to calculate the length of curves, namely of the metric X0 Sinh a 2 p Cosh 2 a 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

41 2 tg θ 1 Cosh a=e tg 2 θ E 2 Time-like X2 X2 X0 X0 X1 X1 Here we see a possible danger for causality: Closed time-like curves! 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

42 a θ Tanh =Tan α 2 2 [ ] Light like X2 X2 X0 X0 X1 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 α

43 Let us now review the general case Christoffel symbols = Levi Civita connection

44 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

45 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 Ñ X Y Z =Ñ X Y +Ñ X Z 2 3 Ñ fx Y = fñ X Y 4 Ñ X Y Z =Ñ X Z +ÑY Z Ñ fy = X [ f ] Y fñ Y X X

46 In a basis... This defines the covariant derivative of a (controvariant) vector field

47 Torsion and Curvature T X,Y ºÑ X -Ñ Y [ X, Y ] Torsion Tensor R X,Y, Z =Ñ X Ñ Y Z -Ñ Y Ñ X Z -Ñ [ X,Y ] Z Curvature Tensor aa The Riemann curvature tensor

48 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!!!!!

49 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

50 Harmonic Coordinates and the exponential map γ v t v T p M exp: T p M V p M 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 exp [ t v ] =γ v t a a ξ =v t Are the harmonic coordinates

51 A view of the locally inertial frame 2 The geodesic equation, by definition, reduces in this frame to: d ξ =0 2 dt

52 The effect of curvature: let us compare two metrics in 3 dimensions A) Flat Euclidean metric B) An instance of Bianchi 2 metric How are geodesics, namely straight lines, in the two metrics? For the metric A) the answer is easy. Straight lines are straight For the metric B) they are instead circular spirals...!

53 Geodesics for the metric A y x z These are the familiar straight lines in 3D space

54 Geodesics in the metric B x y z D view Projection onto the xy plane Spirals with circular projection onto the xy plane 8

55 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.

56 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 a a ξ x =ξ x a ξ a E μ x = μ x E a a μ =E μ x dx

57 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

58 Using the standard formulae for the curvature 2-form:

59 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

60 Bianchi s and the Einstein tensor Allows for the conservation of the stress energy tensor

61 It suffices that the field equations be of the form: G ab =4π G T ab a D T ab =0 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

62 S Action Principle grav =- 1 R [ g ] det g d x = 4 16 pg 1 = 64 pg R E E e abcd ab c d plus the action of matter S tot = S grav S matter S where matter = L matter Lagrangian density of matter being a 4-form TORSION EQUATION We obtain it varying the action with respect to the spin connection: dl d w S = dw eabcd DE E + ab 32 pg dw ab 1 matter c d =0

63 TORSION EQUATION We obtain it varying the action with respect to the spin connection: dl d w S = dw Ù e abcd DE E + ab 32 pg dw ab 1 matter c d =0 in the absence of matter we get eabcd DE E =0 c d ab DE =T =0 w = LeviCivita connection c d

64 S Action Principle grav = pg R [ g ] det g d x = 4 1 = 64 pg R E E eabcd ab c d plus the action of matter S tot = S grav S matter S where matter = L matter Lagrangian density of matter being a 4-form EINSTEIN EQUATION We obtain it varying the action with respect to the vielbein ab c d δ E S =2 R E δe ε abcd δ E L matter

65 EINSTEIN EQUATION We obtain it varying the action with respect to the vielbein ab c d δ E S =2 R E δe ε abcd δ E L Expanding on the vielbein basis we obtain G ab=8πgt ab Where Gab is the Einstein tensor matter

66 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!!!

67 A fundamental example: the Schwarzschild solution Using standard polar coordinates plus the time coordinate t Is the most general static and spherical symmetric metric

68 Finding the solution WE FIND THE SOLUTION And from this, in few straightforward steps we obtain the EINSTEIN TENSOR

69 The solution Boundary conditions for asymptotic flatness a r r 0 b r r 0 a G b =0 ' a b =0 a =- b because of boundary condition ' ' ' b 1 b e 2b b 2 b 2 =0 ; =0 e r =1 r r r r '' ' 2 This yields the final form of the Schwarzschild solution 2b m 2b r e = 1 2 r m 2a r e = 1 2 r 1

70 Let us consider the retrieval of the Schwarzschild solution by Computer Calculations schw arzsolut.nb

71 The Schwarzschild metric and its orbits THE METRIC IS: WHICH MEANS THE LAGRANGIAN

72 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

73 The effects: Periastron Advance Numerical solution of orbit equation in G.R. Keplerian orbit

74 As we already saw...:

75 Bending of Light rays

76 Gravitational effects in Schwarzschild metric Bending of a laser beam Coming close to the horizon the image of the companion star is doubled by gravitational lensing

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