General Relativity

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General Relativity PHYS4473

• Dr Rob Thacker
• Dept of Physics (301-C)
• thacker_at_ap.stmarys.ca

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Todays lecture

• My background
• Course outline
• Reasons to study GR, and when is it important
• Brief overview of some interesting issues in SR
  and GR
• I will pull a few terms out of the hat this
  morning, dont worry, well come back and meet
  them later

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My background

• Im a computational cosmologist, I work on
  computer modelling of galaxy formation
• I started my PhD working on quantum gravity, but
  then diverted into working on inflation, and
  finally I ended working on computer simulations
• I am not at this time a GR researcher, but I do
  have quite a bit of experience with it

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Course Goals

• When completed, students enrolled in the course
  should be able to
• Use tensor analysis to attempt straightforward
  problems in general relativity
• Understand and explain the underlying physical
  principles of general relativity
• Have a quantitative understanding of the
  application of general relativity in modern
  astrophysics

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Course Outline

• Introduction (today)
• Review of special relativity, and use of tensor
  notation (including scalars, vectors)
• Tensor algebra calculus metrics, curvature,
  covariant differentation
• Fundamental concepts in GR Principle of
  Equivalence, Machs Principle, Principle of
  Covariance, Principle of Minimal Coupling
• Energy momentum tensor and Einsteins (Field)
  Equations
• Schwarzschild solution black holes
• Applications of GR in astrophysics (depending on
  scheduling, compact objects, gravitational waves,
  lensing, cosmology)

I reserve the right to make changes to order and
or content if necessary
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Course text

• Introducing Einsteins Relativity by Ray
  DInverno
• Medium to advanced text there is a lot of
  material in here for a more advanced course, so
  if you carry on in GR you should find the text
  very useful
• Good stepping stone to the GR bible The
  large-scale structure of space-time by Hawking
  and Ellis
• This is a very difficult text though, definitely
  grad material
• Gravity An Introduction to Einsteins General
  Relativity by James Hartle is also excellent and
  has perhaps more physical intuition

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Teaching methodology

• I find it difficult to use powerpoint for
  advanced courses
• I prefer to work on the board, which helps pace
  the course
• Because the course is a new preparation it is
  going to be virtually impossible for me to
  provide notes ahead of time sorry!
• I will look into scanning the notes to post them
  on the web

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Academic Integrity

• Working with colleagues to help mutually
  understand something is acceptable
• Discuss approaches, ideas
• However, wrote copying of solutions will not be
  tolerated!

Personal note GR can be tough, but it is a lot
of fun and richly rewarding to work through some
of the harder problems!
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Marking scheme

• I prefer not to give a mid term (but if enough
  people want one I will do so)
• My current marking scheme is as follows
• Assignments 30
• Final 70
• I plan to set a total of 5 assignments,
  approximately one every two weeks

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Class Survey

• In a course with a small student intake there is
  some freedom for organizing material

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Why study GR? - Applications of GR in modern
astrophysics

• Precision gravity in the solar system
• Relativistic stars (white dwarfs, neutron stars,
  supernovae)
• Black holes (!)
• (Global) Cosmology (but not formation of
  galaxies)
• Gravitational lensing
• Gravitational waves
• Quantum gravity (including string theory)

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Precision Gravity

• Climate change and General Relativity in the same
  experiment?
• Yep Gravity Recovery And Climate Experiment
  (GRACE http//www.csr.utexas.edu/grace/)
• Designed to measure changes in shape of the Earth
  geodesy
• Data has been used to test the theory of frame
  dragging in GR where rotating bodes actually
  distort spacetime around them (drag it)

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Relativistic stars

• White dwarfs and neutron stars support themselves
  against contraction via nonthermal pressure
  sources (electron and neutron degeneracy
  respectively)
• Note that a white dwarf can be analyzed from a
  non-relativistic perspective at low masses, but
  becomes increasing inaccurate at high masses
• Neutron stars are fairly strongly relativistics
• New computational work on the ignition of
  supernovae is including general relativistic
  effects

White dwarf mass-radius Non-relativistic
(green) Relativistic (red)
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Global Cosmology

• The description of curved spacetimes obviously
  requires GR
• This necessarily implies we are considering
  scales far larger than a galaxy or cluster of
  galaxies
• In a weak field approximation we can get away
  with a Newtonian description that is surprisingly
  accurate!
• The Friedmann equations govern cosmic expansion
  and allow us to study a number of different
  possible Universe curvatures
• Einsteins biggest blunder, the Cosmological
  Constant, was shown in the late 1990s to be a
  necessary part of cosmology

Adding global is a tautology, but Cosmology is
now taken to include galaxy formation, which
doesnt have much dependence on GR
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Gravitational lensing (1936)
Strong lensing, by massive compact object
Strong lensing by a diffuse mass distribution in
a cluster of galaxies
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Planck Scale Quantum Gravity

• Combining the fundamental constants of nature, we
  can derive units associated with an era when
  quantum gravity is important the Planck Scale
• h,G,c can be combined to give the Planck length,
  mass and time

Still of course the great unsolved problem
of modern physics
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Gravitational waves

• GR predicts that ripples in spacetime propagate
  at the speed of light gravitational waves
• Mergers of compact objects (e.g. black holes)
  produce immense amounts of gravitational
  radiation
• Note that the universe is not dim in terms of
  gravitational radiation all mass produces it
• Exceptionally difficult to detect because of the
  weak coupling to matter Fgrav/Felec10-36

Laser Interferometer Gravitational Wave
Observatory LIGO (Livingston, Louisiana)
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When is GR important?

• A naïve argument can be constructed as follows
• Consider a Newtonian approximation with a test
  particle in a closed orbit (speed v, radius R)
  around a mass M
• If we divide v2 by c2 then we have a
  dimensionless ratio

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Comparison of GM/Rc2 values

• Black holes 1
• Neutron stars 10-1
• Sun 10-6
• Earth 10-9
• Fig 1.1 of Hartle gives an interesting comparison
  of masses and distances
• The diagonal line is 2GMRc2

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Successes failure of Newtonian picture

• Updated Aristotelian picture that,
• Objects move when acted on by force, but tend to
  a stationary state when force is removed
  (friction!)
• Contradicted by force of gravity constant force
  but objects accelerate
• Newtons First Law provided a step towards
  relativity
• if force is such that F0 then vC where C is a
  constant vector
• This adds the concept of inertial frames of
  reference, whereby any frame for which vC is
  defined to be an inertial frame of reference
• However, Newtons Laws do not impose the
  constancy of the speed of light and thus
  encourage the belief in absolute simultaneity,
  rather than relative

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(Newtonian) transformation between inertial
frames of reference

• The Galilean transformation (x,y,z,t)?(x,y,z,t
  )

Boosted by speed v along x axis relative to
frame S
Observer 1, frame S
Observer 2, frame S
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Special Relativity

• Speed of light is the same in all inertial frames
• Speeds are also restricted to be less than c
• Necessarily introduces relative simultaneity

Future light cone
ct
Objects on tconstant are simultaneous in frame S
Timelike separation
x
Spacelike separation
Past light cone
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Coordinate transformations in special relativity

• The Lorentz transformation (x,y,z,t)?(x,y,z,t
  )

Boosted by speed v along x axis relative to
frame S
Observer 1, frame S
Observer 2, frame S
Strictly speaking the Lorentz boost
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Space-time diagram under Lorentz transformations
ct
ct
Note that ct,x is still an orthogonal
coordinate system
x
S has a new line of simultaneity
q
x
Hyperbolic angle is a measure of the relative
velocity between frames
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Correspondence of electric and (Newtonian)
gravitational force
Newtonian Gravity Electrostatics
Forces between sources
Force derived from potential
Potential outside a spherical source
Field equation
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Moving charges Maxwells equations Lorentz
force

• The Lorentz force describes how moving charges
  feel a velocity dependent force from magnetic
  fields
• The velocity dependent term is absent in
  Newtonian gravity
• Clearly Newtonian gravity is not relativistic as
  in all frames the acceleration depends upon mass
  only
• Could we add a Bg term?
• Well kind of, but rather lengthy and complicated,
  much better to look at full GR theory
• There has been renewed interest in this
  gravitomagnetic formalism of late

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Measuring E B fields

• We can establish an inertial frame using neutral
  charges
• Then particle initially at rest can be used to
  measure E
• Once in motion can then measure B
• Does the same line or argument apply in gravity?
• No! No neutral charges! Everything feels gravity

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General Relativity as a stepping stone from SR

• In the presence of gravity freely falling frames
  are locally inertial this is the Principle of
  Equivalence
• This is often described in terms of Einstein
  standing in an elevator
• Such particles will follow the path of least
  resistance (minimize action), which are termed
  geodesics
• Notice that since particles are sources of
  gravitational field as they move through
  spacetime they also bend it
• From this point if we can formulate SR in our new
  frame then we can almost create GR by taking all
  our physical laws and applying the Principle of
  General Covariance
• Physical Laws are preserved under changes of
  coordinates, implies all equations should be
  written in a tensorial form
• This will introduce all the background curvature
  into our equations
• (Note that there is discussion over whether you
  need a couple of additional principles)

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Quantum Gravity Joke

• In Newtonian gravity we can solve the two-body
  problem analytically, but we cant solve the
  three-body problem
• In GR we can solve the one-body problem
  analytically, but we cant solve the two-body
  problem
• In quantum gravity/string theory it isnt even
  clear that we can solve the zero-body problem!
• We cant solve for a unique vacuum structure!

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Next lecture

• Special relativity reviewed