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General Relativity – PHYS4473

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- 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, don’t worry, we’ll come back and meet them later

- I’m 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

- 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

- 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, Mach’s Principle, Principle of Covariance, Principle of Minimal Coupling
- Energy momentum tensor and Einstein’s (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

- “Introducing Einstein’s Relativity” by Ray D’Inverno
- 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 Einstein’s General Relativity” by James Hartle is also excellent and has perhaps more physical intuition

- 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

- 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!

- 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

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

- 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)

- 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”)

- 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)

- 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!

- This necessarily implies we are considering scales far larger than a galaxy or cluster of galaxies
- The Friedmann equations govern cosmic expansion and allow us to study a number of different possible Universe curvatures
- Einstein’s “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 doesn’t have much dependence on GR

Strong lensing,

by massive compact

object

Strong lensing by a diffuse mass

distribution in a cluster of galaxies

- 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

- 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/Felec~10-36

Laser Interferometer Gravitational

Wave Observatory: LIGO

(Livingston, Louisiana)

- 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

- 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 2GM=Rc2

- 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

- Newton’s First Law provided a step towards relativity
- if force is such that F=0 then v=C where C is a constant vector
- This adds the concept of inertial frames of reference, whereby any frame for which v=C is defined to be an inertial frame of reference

- However, Newton’s Laws do not impose the constancy of the speed of light and thus encourage the belief in absolute simultaneity, rather than relative

y

y’

x

x’

z

z’

- 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’

- 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 t=constant

are simultaneous in frame S

Timelike

separation

x

Spacelike

separation

Past light cone

y

y’

x

x’

z

z’

- 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

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

- 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

- 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

- 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)

- In Newtonian gravity we can solve the two-body problem analytically, but we can’t solve the three-body problem
- In GR we can solve the one-body problem analytically, but we can’t solve the two-body problem
- In quantum gravity/string theory it isn’t even clear that we can solve the zero-body problem!
- We can’t solve for a unique vacuum structure!

- Special relativity reviewed