General Relativity – General Relativity – PHYS4473PHYS4473

Dr Rob ThackerDr Rob Thacker

Dept of Physics (301-C)Dept of Physics (301-C)

thacker@ap.stmarys.cathacker@ap.stmarys.ca

Today’s lectureToday’s lecture

My backgroundMy background Course outlineCourse outline Reasons to study GR, and when is it Reasons to study GR, and when is it

importantimportant Brief overview of some interesting Brief overview of some interesting

issues in SR and GRissues in SR and GR I will pull a few terms “out of the hat” I will pull a few terms “out of the hat”

this morning, don’t worry, we’ll come this morning, don’t worry, we’ll come back and meet them laterback and meet them later

My backgroundMy background I’m a computational I’m a computational

cosmologist, I work on computer cosmologist, I work on computer modelling of galaxy formationmodelling of galaxy formation

I started my PhD working on I started my PhD working on quantum gravity, but then quantum gravity, but then diverted into working on diverted into working on “inflation”, and finally I ended “inflation”, and finally I ended working on computer working on computer simulations simulations

I am not at this time a GR I am not at this time a GR researcher, but I do have quite a researcher, but I do have quite a bit of experience with itbit of experience with it

Course GoalsCourse Goals

When completed, students enrolled in When completed, students enrolled in the course should be able to:the course should be able to: Use tensor analysis to attempt Use tensor analysis to attempt

straightforward problems in general straightforward problems in general relativityrelativity

Understand and explain the underlying Understand and explain the underlying physical principles of general relativityphysical principles of general relativity

Have a quantitative understanding of the Have a quantitative understanding of the application of general relativity in modern application of general relativity in modern astrophysicsastrophysics

Course OutlineCourse Outline Introduction (today)Introduction (today) Review of special relativity, and use of tensor notation Review of special relativity, and use of tensor notation

(including scalars, vectors)(including scalars, vectors) Tensor algebra & calculus: metrics, curvature, Tensor algebra & calculus: metrics, curvature,

covariant differentationcovariant differentation Fundamental concepts in GR: Principle of Fundamental concepts in GR: Principle of

Equivalence, Mach’s Principle, Principle of Equivalence, Mach’s Principle, Principle of Covariance, Principle of Minimal CouplingCovariance, Principle of Minimal Coupling

Energy momentum tensor and Einstein’s (Field) Energy momentum tensor and Einstein’s (Field) Equations Equations

Schwarzschild solution & black holesSchwarzschild solution & black holes Applications of GR in astrophysics (Applications of GR in astrophysics (depending on depending on

schedulingscheduling, compact objects, gravitational waves, , compact objects, gravitational waves, lensing, cosmology)lensing, cosmology)I reserve the right to make changes to order and or content if necessary

Course textCourse text ““Introducing Einstein’s Introducing Einstein’s

Relativity”Relativity” by Ray D’Inverno by Ray D’Inverno Medium to advanced text – there is Medium to advanced text – there is

a lot of material in here for a more a lot of material in here for a more advanced course, so if you carry on advanced course, so if you carry on in GR you should find the text very in GR you should find the text very usefuluseful

Good stepping stone to the GR Good stepping stone to the GR bible “The large-scale structure of bible “The large-scale structure of space-time” by Hawking and Ellisspace-time” by Hawking and Ellis This is a very difficult text though, This is a very difficult text though,

definitely grad materialdefinitely grad material ““Gravity: An Introduction to Gravity: An Introduction to

Einstein’s General Relativity” by Einstein’s General Relativity” by James Hartle is also excellent and James Hartle is also excellent and has perhaps more physical has perhaps more physical intuitionintuition

Teaching methodologyTeaching methodology

I find it difficult to use powerpoint for I find it difficult to use powerpoint for advanced coursesadvanced courses

I prefer to work on the board, which I prefer to work on the board, which helps pace the coursehelps pace the course

Because the course is a new Because the course is a new preparation it is going to be virtually preparation it is going to be virtually impossible for me to provide notes impossible for me to provide notes ahead of time: ahead of time: sorry!sorry!

I will look into scanning the notes to I will look into scanning the notes to post them on the webpost them on the web

Academic IntegrityAcademic Integrity

Working with colleagues to help Working with colleagues to help mutually understand something is mutually understand something is acceptableacceptable Discuss approaches, ideasDiscuss approaches, ideas

However, wrote copying of solutions However, wrote copying of solutions will not be tolerated!will not be tolerated!

Personal note: GR can be tough, but it is a lot offun and richly rewarding to work through some ofthe harder problems!

Marking schemeMarking scheme

I prefer not to give a mid term (but if I prefer not to give a mid term (but if enough people want one I will do so)enough people want one I will do so)

My current marking scheme is as My current marking scheme is as follows:follows: Assignments 30%Assignments 30% Final 70% Final 70%

I plan to set a total of 5 assignments, I plan to set a total of 5 assignments, approximately one every two weeksapproximately one every two weeks

Class SurveyClass Survey

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

Why study GR? - Why study GR? - Applications of GR in Applications of GR in modern astrophysicsmodern astrophysics

Precision gravity in the solar systemPrecision gravity in the solar system Relativistic stars (white dwarfs, neutron Relativistic stars (white dwarfs, neutron

stars, supernovae)stars, supernovae) Black holes (!)Black holes (!) (Global) Cosmology (but not formation of (Global) Cosmology (but not formation of

galaxies)galaxies) Gravitational lensingGravitational lensing Gravitational wavesGravitational waves Quantum gravity (including string theory)Quantum gravity (including string theory)

Precision GravityPrecision Gravity

Climate change and General Relativity Climate change and General Relativity in the same experiment?in the same experiment?

Yep: GYep: Gravity ravity RRecovery ecovery AAnd nd CClimate limate EExperiment (GRACE; xperiment (GRACE; http://www.csr.utexas.edu/grace/)http://www.csr.utexas.edu/grace/) Designed to measure changes in shape of Designed to measure changes in shape of

the Earth “geodesy”the Earth “geodesy” Data has been used to test the theory of Data has been used to test the theory of

“frame dragging” in GR where rotating “frame dragging” in GR where rotating bodes actually distort spacetime around bodes actually distort spacetime around them (“drag it”) them (“drag it”)

Relativistic starsRelativistic stars White dwarfs and neutron White dwarfs and neutron

stars support themselves stars support themselves against contraction via against contraction via nonthermal pressure sources nonthermal pressure sources (electron and neutron (electron and neutron degeneracy respectively)degeneracy respectively) Note that a white dwarf can be Note that a white dwarf can be

analyzed from a non-relativistic analyzed from a non-relativistic perspective at low masses, but perspective at low masses, but becomes increasing inaccurate becomes increasing inaccurate at high massesat high masses

Neutron stars are fairly Neutron stars are fairly strongly relativisticsstrongly relativistics

New computational work on New computational work on the ignition of supernovae is the ignition of supernovae is including general relativistic including general relativistic effectseffects

White dwarf mass-radiusNon-relativistic (green)Relativistic (red)

““Global*” CosmologyGlobal*” Cosmology The description of curved The description of curved

spacetimes obviously requires GRspacetimes obviously requires GR This necessarily implies we are This necessarily implies we are

considering scales far larger than considering scales far larger than a galaxy or cluster of galaxiesa galaxy or cluster of galaxies

In a In a weak fieldweak field approximation we approximation we can get away with a Newtonian can get away with a Newtonian description that is surprisingly description that is surprisingly accurate! accurate!

The The Friedmann equationsFriedmann equations govern govern cosmic expansion and allow us to cosmic expansion and allow us to study a number of different study a number of different possible Universe curvatures possible Universe curvatures

Einstein’s “biggest blunder”, the Einstein’s “biggest blunder”, the Cosmological ConstantCosmological Constant, was , was shown in the late 1990s to be a shown in the late 1990s to be a necessary part of cosmology necessary part of cosmology

*Adding global is a tautology, but Cosmology is now taken to include galaxy formation, *Adding global is a tautology, but Cosmology is now taken to include galaxy formation, which doesn’t have much dependence on GRwhich doesn’t have much dependence on GR

Gravitational lensing Gravitational lensing (1936)(1936)

Strong lensing, by massive compact object

Strong lensing by a diffuse massdistribution in a cluster of galaxies

Planck Scale & Quantum Planck Scale & Quantum GravityGravity

Combining the fundamental constants Combining the fundamental constants of nature, we can derive units of nature, we can derive units associated with an era when quantum associated with an era when quantum gravity is important: the “Planck” Scalegravity is important: the “Planck” Scale

h,G,c can be combined to give the h,G,c can be combined to give the Planck length, mass and timePlanck length, mass and time

s 104.5

kg 102.2

m 106.1

445

8

353

c

G

c

lt

G

cm

c

Gl

PP

p

P

Still of course the great“unsolved problem” ofmodern physics

Gravitational wavesGravitational waves GR predicts that ripples in GR predicts that ripples in

spacetime propagate at the spacetime propagate at the speed of light – gravitational speed of light – gravitational waveswaves

Mergers of compact objects Mergers of compact objects (e.g. black holes) produce (e.g. black holes) produce immense amounts of immense amounts of gravitational radiationgravitational radiation

Note that the universe is not Note that the universe is not “dim” in terms of gravitational “dim” in terms of gravitational radiation – all mass produces itradiation – all mass produces it

Exceptionally difficult to detect Exceptionally difficult to detect because of the weak coupling because of the weak coupling to matter Fto matter Fgravgrav/F/Felecelec~10~10-36-36

Laser Interferometer GravitationalWave Observatory: LIGO(Livingston, Louisiana)

When is GR important?When is GR important?

A naïve argument can be constructed as A naïve argument can be constructed as follows:follows: Consider a Newtonian approximation with a Consider a Newtonian approximation with a

test particle in a closed orbit (speed v, radius test particle in a closed orbit (speed v, radius R) around a mass MR) around a mass M

If we divide vIf we divide v22 by c by c22 then we have a then we have a dimensionless ratiodimensionless ratio

R

GMv

R

v

R

GM 2

2

2

22

2

Rc

GM

c

v

Comparison of GM/RcComparison of GM/Rc22 valuesvalues

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

Successes & failure of Successes & failure of Newtonian pictureNewtonian picture

Updated Aristotelian picture that,Updated Aristotelian picture that, Objects move when acted on by force, but tend to Objects move when acted on by force, but tend to

a stationary state when force is removed (friction!)a stationary state when force is removed (friction!) Contradicted by force of gravity: constant force Contradicted by force of gravity: constant force

but objects acceleratebut objects accelerate Newton’s First Law provided a step towards Newton’s First Law provided a step towards

relativityrelativity if force is such that if force is such that FF==00 then then vv=C where =C where CC is a is a

constant vectorconstant vector This adds the concept of inertial frames of This adds the concept of inertial frames of

reference, whereby any frame for which reference, whereby any frame for which vv==CC is is defined to be an inertial frame of reference defined to be an inertial frame of reference

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

(Newtonian) transformation (Newtonian) transformation between inertial frames of between inertial frames of

referencereference The Galilean transformation The Galilean transformation

(x,y,z,t)(x,y,z,t)→→(x’,y’,z’,t’)(x’,y’,z’,t’)

tt

zz

yy

vtxx

'

'

'

'

x

y

z

x’

y’

z’

Observer 1, frame S Observer 2, frame S’

Boosted by speed valong x axis relative to frame S

' and '

' Thus

2

2

2

2

FFdt

xd

dt

xd

Special RelativitySpecial Relativity

Speed of light is the same in all Speed of light is the same in all inertial framesinertial frames

Speeds are also restricted to be less Speeds are also restricted to be less than cthan c

Necessarily introduces Necessarily introduces relative relative simultaneitysimultaneity

ct

x

Objects on t=constantare simultaneous in frame S

Future light cone

Past light cone

Spacelike separation

Timelikeseparation

Coordinate transformations Coordinate transformations in special relativityin special relativity

The Lorentz transformation* The Lorentz transformation* (x,y,z,t)(x,y,z,t)→→(x’,y’,z’,t’)(x’,y’,z’,t’)

2

2

2

)/(1

/'

'

'

)/(1'

cv

cvxtt

zz

yy

cv

vtxx

x

y

z

x’

y’

z’

Observer 1, frame S Observer 2, frame S’

Boosted by speed valong x axis relative to frame S

*Strictly speaking the Lorentz boost

Space-time diagram under Space-time diagram under Lorentz transformationsLorentz transformations

ct ct’

x

x’

S’ has a new line of simultaneity

Note that ct’,x’is still an orthogonal coordinate system

Hyperbolic angle isa measure of therelative velocitybetween frames

Correspondence of electric Correspondence of electric and (Newtonian) and (Newtonian)

gravitational forcegravitational forceNewtonian Newtonian GravityGravity

ElectrostaticsElectrostatics

Forces Forces between between sourcessourcesForce derived Force derived from potentialfrom potential

Potential Potential outside a outside a spherical spherical sourcesource

Field equationField equation

022

0

20

2

/ 4

4

)( )(

4

eemg

eg

qeemgg

qQeMmg

G

r

Q

r

GM

xqFxmF

er

qQFe

r

GMmF

0/)(-E. akin to ),(4)(. then )()( If xxGxgxxg emg

Moving charges: Maxwell’s Moving charges: Maxwell’s equations + Lorentz forceequations + Lorentz force

The Lorentz force describes how moving charges The Lorentz force describes how moving charges feel a velocity dependent force from magnetic fieldsfeel a velocity dependent force from magnetic fields

The velocity dependent term is absent in Newtonian The velocity dependent term is absent in Newtonian gravitygravity Clearly Newtonian gravity is not relativistic as in all frames Clearly Newtonian gravity is not relativistic as in all frames

the acceleration depends upon mass onlythe acceleration depends upon mass only

Could we add a Could we add a BBgg term? term? Well kind of, but rather lengthy and complicated, much Well kind of, but rather lengthy and complicated, much

better to look at full GR theorybetter to look at full GR theory There has been renewed interest in this There has been renewed interest in this gravitomagneticgravitomagnetic

formalism of lateformalism of late

)( BvEqFe

Measuring E & B fieldsMeasuring E & B fields We can establish an inertial frame using We can establish an inertial frame using

neutral chargesneutral charges Then particle initially at rest can be used to Then particle initially at rest can be used to

measure Emeasure E

Once in motion can then measure BOnce in motion can then measure B

Does the same line or argument apply in Does the same line or argument apply in gravity?gravity? No! No neutral charges! No! No neutral charges! EverythingEverything feels gravity feels gravity

EqFe

)( BvEqFe

General Relativity as a General Relativity as a stepping stone from SRstepping stone from SR

In the presence of gravity freely falling frames are locally In the presence of gravity freely falling frames are locally inertial – this is the inertial – this is the Principle of EquivalencePrinciple of Equivalence This is often described in terms of Einstein standing in an This is often described in terms of Einstein standing in an

elevatorelevator Such particles will follow the path of least resistance Such particles will follow the path of least resistance

(minimize action), which are termed (minimize action), which are termed geodesicsgeodesics Notice that since particles are sources of gravitational field Notice that since particles are sources of gravitational field

as they move through spacetime they also bend itas they move through spacetime they also bend it From this point if we can formulate SR in our new frame From this point if we can formulate SR in our new frame

then we can almost create GR by taking all our physical then we can almost create GR by taking all our physical laws and applying the Principle of General Covariancelaws and applying the Principle of General Covariance Physical Laws are preserved under changes of coordinates, Physical Laws are preserved under changes of coordinates,

implies all equations should be written in a tensorial formimplies all equations should be written in a tensorial form This will introduce all the background curvature into our This will introduce all the background curvature into our

equationsequations (Note that there is discussion over whether you need a (Note that there is discussion over whether you need a

couple of additional principles)couple of additional principles)

Quantum Gravity JokeQuantum Gravity Joke In Newtonian gravity we can solve the In Newtonian gravity we can solve the

two-body problem analytically, but we two-body problem analytically, but we can’t solve the three-body problemcan’t solve the three-body problem

In GR we can solve the one-body In GR we can solve the one-body problem analytically, but we can’t solve problem analytically, but we can’t solve the two-body problemthe two-body problem

In quantum gravity/string theory it isn’t In quantum gravity/string theory it isn’t even clear that we can solve the zero-even clear that we can solve the zero-body problem!body problem! We can’t solve for a unique vacuum We can’t solve for a unique vacuum

structure!structure!

Next lectureNext lecture

Special relativity reviewedSpecial relativity reviewed