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. Topics of Discussion
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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

3. INTRODUCTION

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

5. THE INCOMPATIBILITY OF STR AND
NEWTONIAN GRAVITY

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. EINSTEIN’S ‘HAPPIEST THOUGHT’:
The Principle of Equivalence

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. 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. 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.

11. LOCALIZING GRAVITY

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’.

13. THE ‘PATH OF LEAST RESISTANCE’?

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. 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’.

16. A BRIEF LOOK AT VARIATIONAL
TECHNIQUES

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.

18. Q: BUT WHAT ABOUT TIDAL EFFECTS?
A: SPACETIME IS CURVED!

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. 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. REPRESENTING SPACETIME:
Riemannian Manifolds

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. 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. THE ALPHABET OF GTR:
Vectors, 1-Forms and Tensors

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. The Alphabet of GTR:
Vectors, 1-Forms and Tensors
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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. 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. 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. THE TENSOR FACTORY:
Making new ones from old

30. The Tensor Factory:
Making New Ones from Old
• Addition:
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(A + B)(u, λ) = A(u, λ) + B(u, λ)
(A + B)ij = (Aij + Bij)
• Multiplication with scalar
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(mA)(u, λ) = m(A(u, λ))
(mA)ij = m(Aij)
• Contraction
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Contr A(u, λ) = Σ A(u, ej ,σj, λ)
Einstein Summation convention
Aik = Aijjk
• Tensor Product
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(A B)(u, λ, β,v) = A(u, λ)B(β,v)
(AB)ijkl = AijBkl
• Inner product
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Product followed by contraction.

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. The Tensor Factory:
Making New Ones from Old
• Exterior Derivative:
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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
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Operation is not geometric.
Aij,k= kAij = ( / xk)Aij
• Gradient, or Covariant Derivative:
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Also called ‘torsion-free covariant derivative’.
We will do this later.
• Divergence
• Gradient followed by contraction

33. ENCODING THE GEOMETRY OF SPACETIME:
The Metric Tensor

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. ‘WHO’S WHO’ OF DIFFERENTIAL GEOMETRY
Geodesic, Parallel Transport, Gradient, Connection Coefficient

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. ‘Who’s who’ of Differential Geometry
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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. GEOMETRY TELLS MATTER HOW TO MOVE:
Riemann and Bianchi

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. 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. MATTER TELLS GEOMETRY HOW TO CURVE:
Stress-Momentum-Energy, Einstein, EFEs

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. 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. 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. TESTING THE THEORY:
Eddington’s 1919 Expedition to Principe

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. Testing the Theory:
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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. Testing the Theory:
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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. WHAT NEXT:

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. 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. 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. ‘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. Thank you!