General Relativity – PHYS4473

Dr Rob Thacker Dept of Physics (301-C) [email protected]

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

My background

   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

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

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, 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

Course text

    “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

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

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!

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

Class Survey

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

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)

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”)

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)

“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 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

Gravitational lensing (1936)

Strong lensing by a diffuse mass distribution in a cluster of galaxies Strong lensing, by massive compact object

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

l P

 

G c

3  1 .

6  10  35 m

m p



t P



l P c



c

 2 .

2  10  8 kg

G

 

G c

5  5 .

4  10  44 s Still of course the great “unsolved problem” of modern physics

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 F grav /F elec ~10 -36 Laser Interferometer Gravitational Wave Observatory: LIGO (Livingston, Louisiana)

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

GM R

2 

v

2

R



v

2 

GM R

 If we divide v 2 by c 2 then we have a dimensionless ratio

v

2

c

2 

GM Rc

2

Comparison of GM/Rc

2

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 2GM=Rc 2

  

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 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

(Newtonian) transformation between inertial frames of reference

 The Galilean transformation (x,y,z,t) → (x’,y’,z’,t’) y x

Boosted by speed v along x axis relative to frame S

y’ x’

x

' 

x



vt y

' 

z

' 

z y t

' 

t

z Observer 1, frame S z’ Observer 2, frame S’ Thus 2

d dt

2

x



d dt

2

x

' and 

F

' 2  

F

'

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 Objects on t=constant are simultaneous in frame S Future light cone ct

Timelike separation Spacelike separation

x Past light cone

Coordinate transformations in special relativity

 The Lorentz transformation* (x,y,z,t) → (x’,y’,z’,t’) y z Observer 1, frame S x

Boosted by speed v along x axis relative to frame S

y’ x’

t x

' 

x



vt y

' 

z

' 

z y

1  (

v

/

c

) 2 ' 

t



vx

/

c

2 1  (

v

/

c

) 2 z’ Observer 2, frame S’ *Strictly speaking the Lorentz boost

Space-time diagram under Lorentz transformations

ct ct’ Note that ct’,x’ is still an orthogonal coordinate system q x’ S’ has a new line of simultaneity x Hyperbolic angle is a measure of the relative velocity between frames

Correspondence of electric and (Newtonian) gravitational force

Forces between sources Force derived from potential Potential outside a spherical source Field equation Newtonian Gravity 

F g

 

GMm r

2 

F g

 

m

 

g

( 

x m

) Electrostatics 

e Mm



F e



F e

  4 

qQ



q

0 

r

2 

e



e qQ

( 

x q

) 

g

 

GM r



e

 2 

g

 4 

G

  

m

 2 

e



Q

4  0

r



e

/  0 If 

g

( 

x

)   

g

( 

x

) then  .



g

( 

x

)   4 

G



m

( 

x

), akin to   .

E  

e

( 

x

) /  0

Moving charges: Maxwell’s equations + Lorentz force

   The Lorentz force describes how moving charges feel a velocity 

F e



q

( 

E



v

  

B

) 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 B g  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

Measuring E & B fields

    We can establish an inertial frame using neutral charges Then particle initially at rest can be used to measure E 

F e



q



E

Once in motion can then measure B 

F e



q

( 

E

 

v

 

B

) Does the same line or argument apply in gravity?

 No! No neutral charges! Everything feels gravity

    

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)

Quantum Gravity Joke

   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!

Next lecture

 Special relativity reviewed