Transcript High Performance Computing 811

General Relativity –
PHYS4473
Dr Rob Thacker
Dept of Physics (301-C)
[email protected]
Today’s lecture
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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
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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
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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
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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
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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
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“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
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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
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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
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Working with colleagues to help mutually
understand something is acceptable
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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
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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%
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I plan to set a total of 5 assignments,
approximately one every two weeks
Class Survey
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In a course with a small student intake there is
some freedom for organizing material
Why study GR? - Applications of GR
in modern astrophysics
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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
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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
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White dwarfs and neutron stars
support themselves against
contraction via nonthermal
pressure sources (electron and
neutron degeneracy respectively)
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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
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The description of curved spacetimes
obviously requires GR
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This necessarily implies we are
considering scales far larger than a galaxy
or cluster of galaxies
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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 massive compact
object
Strong lensing by a diffuse mass
distribution in a cluster of galaxies
Planck Scale & Quantum Gravity
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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
G
35

1
.
6

10
m
3
c
lP 
mp 
tP 
c
 2.2 10 8 kg
G
lP
G
 44


5
.
4

10
s
5
c
c
Still of course the great
“unsolved problem” of
modern physics
Gravitational waves
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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)
When is GR important?
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A naïve argument can be constructed as follows:
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Consider a Newtonian approximation with a test particle in a
closed orbit (speed v, radius R) around a mass M
2
GM v
GM
2
 v 
2
R
R
R
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If we divide v2 by c2 then we have a dimensionless ratio
2
v
GM

2
2
c
Rc
Comparison of
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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
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The diagonal line is
2GM=Rc2
2
GM/Rc
values
Successes & failure of Newtonian
picture
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Updated Aristotelian picture that,
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Newton’s First Law provided a step towards relativity
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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
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
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The Galilean transformation (x,y,z,t)→(x’,y’,z’,t’)
y
y’
Boosted by speed v
along x axis
relative to frame S
x
x’
z
z’
Observer 1, frame S
Observer 2, frame S’
x'  x  vt
y'  y
z'  z
t'  t
 
d 2 x d 2 x'
Thus 2  2 and F  F '
dt
dt '
Special Relativity
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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
Spacelike
separation
Past light cone
x
Coordinate transformations in
special relativity
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The Lorentz transformation* (x,y,z,t)→(x’,y’,z’,t’)
y
y’
Boosted by speed v
along x axis
relative to frame S
x
x  vt
x' 
1  (v / c ) 2
y'  y
x’
z'  z
t' 
z
z’
Observer 1, frame S
Observer 2, frame S’
t  vx / c 2
1  (v / c ) 2
*Strictly speaking the Lorentz boost
Space-time diagram under Lorentz
transformations
ct
ct’
Note that ct’,x’
is still an orthogonal
coordinate system
x’
q
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
Electrostatics
Gravity
  GMm 

qQ 
Fg 
eMm Fe 
e
2
2 qQ
r
40 r
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Fg  m g ( xm ) Fe  q e ( xq )
GM
g  
r
Q
e  
 2  g  4G m
 2 e  e /  0
40 r
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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
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The Lorentz force describes how moving charges feel a velocity
dependent force from magnetic fields
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  
Fe  q( E  v  B)
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The velocity dependent term is absent in Newtonian gravity
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Clearly Newtonian gravity is not relativistic as in all frames the
acceleration depends upon mass only
Could we add a Bg term?
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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
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We can establish an inertial frame using neutral
charges
Then particle initially at rest can be used to measure
E
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Fe  qE
Once in motion can then measure B
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Fe  q( E  v  B)
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Does the same line or argument apply in gravity?
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No! No neutral charges! Everything feels gravity
General Relativity as a stepping
stone from SR
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In the presence of gravity freely falling frames are locally inertial – this is the
Principle of Equivalence
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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
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This is often described in terms of Einstein standing in an elevator
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
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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!
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We can’t solve for a unique vacuum structure!
Next lecture
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Special relativity reviewed