Introduction to the General Theory of Relativity


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Introduction to the General Theory of Relativity

  1. 1. The General Theory of Relativity Arpan Saha 1st year Engineering Physics DD IIT Bombay Monday, November 9, 2009 Room 202, Physics Dept. IIT Bombay
  2. 2. Topics of Discussion • • • • • • • • • • • • • • • • • • Introduction The Incompatibility of STR and Newtonian Gravity Einstein’s ‘Happiest Thought’: The Equivalence Principle Localizing Gravity What exactly do we mean by the ‘Path of Least Resistance’? Digression: a Brief Look at Variational Techniques Q: But what about tidal effects? A: Spacetime is curved! Representing Spacetime: Manifolds The Alphabet of GTR: Vectors, 1-forms and Tensors The Tensor Factory: Making New Ones from Old Encoding the Geometry of Spacetime: the Metric ‘Who’s who’ of Differential Geometry: Geodesics, Parallel Transport, Covariant Derivative and Connection Coefficients ‘Geometry tells matter how to move’: Riemann and Bianchi ‘Matter tells geometry how to curve’: Stress-Energy-Momentum, Einstein, EFEs Testing the Theory: Eddington’s 1919 Expedition to Principe What next? Question Session
  4. 4. Introduction Universal Law of Gravitation • Any two bodies in the Universe attract each other with a force proportional to the masses and inverse of the square of the separation, along the line joining them. Sir Isaac Newton • F = Gm1m2/r2
  6. 6. The Incompatibility of STR and Newtonian Gravity Newtonian Gravity Changes in gravitational fields instantaneously transmitted across arbitrary stretches of space. Einstein’s STR But according to STR, a physical law has to respect Lorentz invariance. Ergo instantaneous transmission is impossible.
  7. 7. EINSTEIN’S ‘HAPPIEST THOUGHT’: The Principle of Equivalence
  8. 8. Einstein’s ‘Happiest Thought’: The Principle of Equivalence • It is impossible, by means of any local experiment, to distinguish between the frame of a falling body and an inertial one.
  9. 9. Einstein’s ‘Happiest Thought: The Principle of Equivalence • Likewise it also impossible to distinguish between between an accelerated frame of reference and a frame at rest in a gravitational field.
  10. 10. Einstein’s Happiest Thought • GMm/r2 = mg • The Principle of Equivalence is not a consequence of the equality of gravitational and inertial mass. • The equality of gravitational and inertial mass is a consequence of the Principle of Equivalence.
  12. 12. Localizing Gravity • Physics is simple only when viewed locally. • The idea of gravity as action at a distance must be discarded. • Instead, gravity must be treated as a local phenomenon. • The principle of equivalence enables us to do this. • Free fall is the natural state of all bodies. • Falling bodies always follow ‘path of least resistance’.
  14. 14. The ‘Path of Least Resistance’? • The proper time measured in an inertial frame colocal with two events A and B, is always greater than that in a non-inertial frame colocal with A and B.
  15. 15. The ‘Path of Least Resistance’? • Therefore, a freely falling body traces out a world curve that extremizes proper time. • Geodesics are the ‘Path of Least Resistance’.
  17. 17. A Brief Look at Variational Techniques • The Calculus of Variations, developed by Euler, the Bernoulli brothers, Lagrange and others, deals with extremization of functionals rather then functions. • Same principle: First order variations for extremal functions vanish.
  19. 19. Q: But what about tidal effects? A: Spacetime is curved! Shoemaker-Levy breaking up due to tidal forces in Jupiter’s vicinity. • Bodies in gravitational fields experience tidal forces. • PoE is not violated – tidal phenomena are not local. • But we’re interested in an explanation other than one involving forces exerted on the body.
  20. 20. Q: But what about tidal effects? A: Spacetime is curved! Elementary, my dear Newton. Just as two ants on the surface of a sphere, starting some distance apart at the equator along ‘parallel’ routes, eventually meet at the poles, the inertial frames experience relative acceleration in the presence of a gravitating mass as the mass warps spacetime. How would this be explained within Einstein’s framework?
  21. 21. REPRESENTING SPACETIME: Riemannian Manifolds
  22. 22. Representing Spacetime: Riemannian Manifolds • How do we represent spacetime mathematically? • Answer was first offered by Bernhardt Riemann in his Habilitationvorlesung ‘On the Hypotheses Underlying the Foundations of Geometry’.
  23. 23. Representing Spacetime: Riemannian Manifolds • Start with a set of points , each point representing an event. • Specify a collection of ‘open sets’ or ‘neighborhoods’ to go along. We have a ‘topological space’. • Establish ‘homeomorphism’ between the set and Rn. We’ll call these maps ‘co-ordinate charts’. • Relax the need for a global co-ordinate chart. • Set may be covered by an ‘atlas’ of charts on open sets, provided covering is complete, there are regions of overlap and the transformation map in those regions is infy differentiable. • We will then have a Riemannian manifold.
  24. 24. THE ALPHABET OF GTR: Vectors, 1-Forms and Tensors
  25. 25. The Alphabet of GTR: Vectors, 1-Forms and Tensors • GTR is entirely formulated in terms of ‘tensors’. • The calculus of tensors and differential forms (special kinds of tensors) was developed in 1890 by Tullio Levi-Civita and Gregorio Ricci-Curbastro, about two decades before GTR. • Hence, Einstein was not very familiar with the math himself and had to take the help of classmate Marcel Grossmann and LeviCivita. • ‘There are two things I like about Italy. One, spaghetti the other Levi-Civita.’ – Albert Einstein
  26. 26. The Alphabet of GTR: Vectors, 1-Forms and Tensors • • • • • Vectors as directed line segments. Vectors as limiting case of line segments. Vectors as equivalence class of curves. Vectors as directional derivative operators. Vectors in terms of components i.e. coordinate basis vectors. • Tangent Vectors/ Vector Fields
  27. 27. The Alphabet of GTR: Vectors, 1-Forms and Tensors • 1-forms – the intuitive idea. • 1-forms as gradients/ ‘exterior derivative’ of scalar functions df. • Isomorphism, anyone? • 1-forms as linear maps from tangent space to R. • Vector space structure for 1-forms. • 1-forms in terms of components i.e. co-ordinate basis 1-forms. • Cotangent vectors/ 1-forms
  28. 28. The Alphabet of GTR: Vectors, 1-Forms and Tensors • Tensors – the machine with slots picture. • Rank/ type/ valence of a Tensor • A (p, q)-rank tensor as a (p + q)-linear map from q copies of tangent space and p copies of cotangent space. • Empty slots? • Vectors and 1-forms – (1,0)-rank and (0,1)rank tensors respectively. • Tensors/ Tensor fields
  29. 29. THE TENSOR FACTORY: Making new ones from old
  30. 30. The Tensor Factory: Making New Ones from Old • Addition: • • (A + B)(u, λ) = A(u, λ) + B(u, λ) (A + B)ij = (Aij + Bij) • Multiplication with scalar • • (mA)(u, λ) = m(A(u, λ)) (mA)ij = m(Aij) • Contraction • • • Contr A(u, λ) = Σ A(u, ej ,σj, λ) Einstein Summation convention Aik = Aijjk • Tensor Product • • (A B)(u, λ, β,v) = A(u, λ)B(β,v) (AB)ijkl = AijBkl • Inner product • Product followed by contraction.
  31. 31. The Tensor Factory: Making New Ones from Old • Wedge Product: • A B = (A B) – (B A) • p-vectors and p-forms • Transposition: • This is merely exchange of slots • N(u, v) = S(v, u) • Symmetrization and Antisymmetrization: • This involves constructing symmetric or antisymmetric tensors by appropriate linear combination of original tensor with its transposes. • Duals: • This involves contracting with Levi-Civita tensor accompanied by normalization.
  32. 32. The Tensor Factory: Making New Ones from Old • Exterior Derivative: • • • • For scalars this is the gradient. For general differential forms, defined inductively as: d(α β) = (dα) β + (-1)pα (dβ) Where α and β are p-form and q-form respectively. • Component Derivatives • • Operation is not geometric. Aij,k= kAij = ( / xk)Aij • Gradient, or Covariant Derivative: • • Also called ‘torsion-free covariant derivative’. We will do this later. • Divergence • Gradient followed by contraction
  34. 34. Encoding the Geometry of Spacetime: the Metric Tensor • Einstein’s GTR provides for an additional condition on the Riemannian manifold of spacetime – the metric tensor g(u,v). • It is a symmetric (0,2)-rank tensor that when fed with two vectors returns the ‘dot-product’. • It establishes an isomorphism between vectors and 1-forms allowing us to ‘raise or lower indices’. • But more importantly, it provides a notion of norm i.e. distance.
  35. 35. ‘WHO’S WHO’ OF DIFFERENTIAL GEOMETRY Geodesic, Parallel Transport, Gradient, Connection Coefficient
  36. 36. ‘Who’s who’ of Differential Geometry • In GTR, there are certain intrinsic properties of spacetime we are interested in. • We must work without assuming spacetime is embedded in higher dimensions. • The absolute differential geometry developed by Riemann, Elie Cartan, LeviCivita, Gregorio Ricci-Curbastro, Elwin Bruno Christoffel and others is perfectly suited for our purposes. • We will hence acquaint ourselves with some of the principal characters.
  37. 37. ‘Who’s who’ of Differential Geometry • • • • • • Geodesic – path of extremum metric Parallel transport via Schild’s Ladder Parallel transport via geodesic approximation Covariant derivative aka gradient via PT Equation for PT in terms of gradient Connection Coefficients aka Christoffel symbols – gradients of basis vectors/ 1-forms • Gradient in terms for CC • Geodesic in terms of PT – path PTing its tangent vector along itself.
  38. 38. GEOMETRY TELLS MATTER HOW TO MOVE: Riemann and Bianchi
  39. 39. Geometry tells matter how to move: Riemann and Bianchi • The Riemann tensor is a fourth-rank tensor which contains complete information about the curvature of spacetime. • Riemann as change in vector paralleltransported around closed loop. • Riemann as a commutator . • Riemann as geodesic deviation.
  40. 40. Geometry tells matter how to move: Riemann and Bianchi • The symmetries and antisymmetries of the Riemann tensor. • Bianchi’s first identity. • In 4D, Riemann has 20 independent components. • Bianchi’s second identity. • The boundary of a boundary is zero.
  41. 41. MATTER TELLS GEOMETRY HOW TO CURVE: Stress-Momentum-Energy, Einstein, EFEs
  42. 42. Matter tells geometry how to curve: Stress-Momentum-Energy, Einstein, EFEs • SME tensor is second rank tensor describing the distribution of mass and energy in spacetime. • Interpretation using the cornflakes box example. • What do the components represent? • SME is symmetric and divergence-free. • First guess at EFE
  43. 43. Matter tells geometry how to curve: Stress-Momentum-Energy, Einstein, EFEs • What is second-rank symmetric, divergencefree and derived completely from the Riemann and metric tensor? • Einstein – a contracted double-dual of Riemann, fits in as the unique candidate. • The vanishing of its divergence is a restatement of Bianchi’s second identity. • Einstein, as moment of rotation.
  44. 44. Matter tells geometry how to curve: Stress-Momentum-Energy, Einstein, EFEs • What about the constant of curvature? • We can find it through correspondence to Newtonian theory. • The constant in geometrized units is 8π.
  45. 45. TESTING THE THEORY: Eddington’s 1919 Expedition to Principe
  46. 46. Testing the Theory: Eddington’s 1919 Expedition to Principe • Experiment always has the final say. • GTR could be confirmed by measuring the deflection of light passing close to a massive body. • For the deflection to be sizeable, the massive body would have to be the sun. • However, observing deflection is difficult as the brightness of the sun blots out the stars ‘close by’. • Solution? Observe shifts in apparent positions of stars during solar eclipse.
  47. 47. Testing the Theory: ` Eddington’s 1919 Expedition to Principe • In May 1919, British astronomer Arthur Eddington sailed to Principe, of the coast of Africa, where a total solar eclipse would be observed. • He took a series of photographs of the sun, as the eclipse progressed. • The plates clearly showed a shift in the apparent position of the background stars by an amount as predicted by Einstein. • GTR had been proved.
  48. 48. Testing the Theory: ` Eddington’s 1919 Expedition to Principe • Mercurial perihelion precession was more of a postdiction . • So, Eddington’s expedition, as well as a series of similar expeditions carried out elsewhere were the first confirmations of GTR. • The front pages of newspapers proclaimed Einstein’s victory. • Einstein had progressed from being an unknown Swiss patent clerk to influential physicist to international celebrity. • But the journey was far from complete.
  49. 49. WHAT NEXT:
  50. 50. What Next? • Though GTR has been extensively verified by local experiments, we are not sure whether we can make the leap to the Universe as a whole. • Einstein himself attempted this. • The results he obtained were surprising – the Universe, the EFEs implied, could not be static.
  51. 51. What Next? • Hence, he modified them by including a cosmological constant, so as to allow for a static solution. • Later, when the Universe was shown to be expanding, he regretted it as his biggest mistake. • And like many other ideas in the history of science, the cosmological constant was handed the pink slip. • But it is finally making its comeback, in an entirely new avatar.
  52. 52. What Next? • The Universe is expanding at an accelerating rate and we are not sure why. • According to current trends, this might be attributed to a scalar field called quintessence arising due to ‘dark energy’. • It is clear GTR requires some major revision before it can be applied to the Universe as a whole, and before we come to know how everything came to be.
  53. 53. ‘But the years of anxious searching in the dark, with their intense longing, their alternations of confidence and exhaustion, and the final emergence into the light – only those who have experienced it can understand that.’
  54. 54. Thank you!