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Course OutlineReasons to Study and Use GR

Some Interesting Issues in SR and GR

Phys 4390: General Relativity

Dr. David McNutt ,1(call me Dave)

1Department of PhysicsSaint Mary’s University

January 9, 2015

1 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity



My Background

I am a mathematician with an interestin differential geometry, dynamicalsystems and their applications inphysics and engineering.

My PhD focused on the classificationof space-times in General Relativity(GR), in particular my focus was oncertain "toy-models" of gravitationalradiation: the pp-wave and Kundtwave solutions.

At the moment my work in GR hasbeen limited, however I do haveconsiderable experience with thetheory.

Figure: A purely electromagnetic plane-wavespacetime causing a null cone from event Q to befocused at a second vertex R




Outline

1 Course OutlineCourse GoalsCourse OutlineCourse TextsAcademic Policies

2 Reasons to Study and Use GRPrecision Gravity In the Solar SystemRelativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!)Cosmological ModelingGravitational...Quantum Gravity

3 Some Interesting Issues in SR and GRWhen is Gr ImportantSome ContextSpecial RelativityFrom SR to GR




Course GoalsCourse OutlineCourse TextsAcademic Policies

Course Goals

At completion of this course, students will be able to:

Employ tensor analysis to approach problems in general relativity.

Understand and explain the underlying physical principles of general relativity(GR)

Have a quantitative understanding of the application of GR in modern astrophyics.





Course Outline

Introduction.

Review of Special Relativity (SR), and use of tensor notation.

Tensor algebra and Calculus: metrics, curvature, covariant differentiation.

Fundamental concepts in GR: Principle of Equivalence, Mach’s Principle, Principleof Covariance, Principle of Minimal Coupling.

Energy momentum tensor and Einstein’s equations.

Scwarzchild solutions and black holes.

Applications of GR in astrophysics: compact objects, gravitational waves, lensingcosmology.

This outline, especially the last point, is subject to change depending on student’sinterest and time constraints





Course Texts

I will primarily use the textbook "Introducing Einstein’s Relativity" by Ray D’Inverno,as this is a good introduction to GR with some advanced topics covered.

That said, I may also take material from:

Robert M. Wald’s "General Relativity",

John Stewart’s "Advanced General Relativity",

James Hartle’s "Gravity: An Introduction to Einstein’s General Relativity".





Approach to Teaching

After this class I will not be using slides. Instead I will work on the board and followclass notes.

As I am left handed and calligraphy was never a strong point, I will provide typedclass notes.

These notes may follow after the class is done, but it is my sincere hope toprepare them prior to class.

These notes will be posted on my personal website, and blackboard.





Academic Integrity

I encourage students to discuss work with each other in order to mutuallyunderstand the topics taught.

Similarly, I see no harm with people working together on assignment questions.

I object to solutions being copied word-for-word without any attempt to rehash toreflect the student’s understanding.





Marking Scheme

The current marking scheme is:

Assignments − 60%

Final Exam − 40%

There will be a total of 5 assignments, with one every two weeks.




Precision Gravity In the Solar SystemRelativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!)Cosmological ModelingGravitational...Quantum Gravity

A Big List

1 Precision gravity in the solar system2 Relativistic stars3 Black holes4 Cosmological Models5 Gravitational lensing6 Gravitational waves7 Quantum gravity





Precision Gravity

To measure the gravity field of theearth, NASA and the GermanAerospace Center launchedGRACE:Gravity Recovery and ClimateExperiment.

This has become an important tool forstudying the ocean, geology andclimate of the earth.

Combining the data from GRACE withthat of LAGEOS:LAser GEOdynamicsSatellites, it is hoped that therelativistic effect of frame-draggingcould be measured.

These satellites are intended toprovide an orbiting laser rangingbenchmark for geodynamical studiesof the Earth.

Figure: Gravity Anomaly map from GRACE;taken from Wikipedia

Figure: LAGEOS-1 satellite; taken from Wikipedia


http://www.csr.utexas.edu/grace/

http://www.csr.utexas.edu/grace/

http://science.nasa.gov/missions/lageos-1-2/

http://science.nasa.gov/missions/lageos-1-2/




Relativistic Stars

White dwarfs and neutron starsemploy non-thermal pressure sources,i.e., electron and neutron degeneracyrespectively, to resist contraction.

At low masses white dwarves may beanalyzed without relativity, however athigher masses these models areinaccurate.

Neutron stars are relativistic.

Recent computational models on theignition of supernovae are consideringgeneral relativistic effects.

Figure: Chandrasekhar Limit Graph; taken fromwikiepdia





"Global" Cosmology

In order to describe curvedspacetimes we must use GR

This requires working on scales muchlarger than galactic clusters; often dustor perfect fluid models are consideredfor the spacetimes.In the weak field approximation it canbe shown that the Newtoniandescription is sufficient, with greataccuracy.

The Friedmann equations dictatecosmic expansion and allow for one tostudy a number of possible scenariosfor curvature of the universe.

The cosmological constant, whichoriginally frustrated Einstein’s vision ofa static universe has been shown tobe a necessary part of cosmology.

Figure: Two dimensional examples of surfaceswith negative, vanishing, and positive curvaturerespectively; taken from Cosmology tutorial


http://www.astro.ucla.edu/~wright/cosmo_03.htm




Gravitational Lensing

While originally discussed byChwolson (1924) and Klin (1936),Einstein published a canonical articleon the subject in 1936.

Three classes of gravitational lensing:strong lensing, weak lensing, andmicrolensing.

Figure: Einstein’s Cross: An example of stronggravitational lensing, where four images of thesame distant quasar appear around a foregroundgalaxy.

Figure: Artist conception of gravitational lensingfrom a galaxy cluster; taken from NASA





Gravitational Waves

GR predicts that ripples in spacetimecan propagate at the speed of light,these ripples are called gravitationalwaves.

Mergers of compact objects shouldproduce immense amounts ofgravitational radiation

All mass produces gravitational waves,in this sense if one could detectgravitational waves, the universewould be very "bright".

Unfortunately these are incrediblydifficult to detect due to weak couplingof matter Fgrav

Felec∼ 10−36.

Figure: Artist conception of LISA (LaserInterferometer Space Antenna) spacecraft; takenfrom NASA





Planck Scale and Quantum Gravity

Using the fundamental constants of nature, it is possible to derive units associatedwith an era when quantum gravity is relevant on the "Planck" Scale.

Taking ~, G and c, we can produce Planck length, mass and time.

lp =√

~Gc3 = 1.6× 10−33m

mp =√

~cG = 2.2× 10−8kg

tp =lpc =

√~Gc5 = 5.4× 10−44s




When is Gr ImportantSome ContextSpecial RelativityFrom SR to GR

Hand-waving argument

As a simple argument consider the following:

In the Newtonian approximation of a test particle in a closed orbit with speed v,radius R around a mass M

GMR2

=v2

R→ v2 =

GMR

Dividing v2 by c2 yields a dimensionless ratio

v2

c2=

GMRc2





Comparison of Gm/Rc2 values

Blackholes ∼ 1

Neutron stars ∼ 10−1

Sun ∼ 10−6

Earth ∼ 10−9

Figure: Taken from Fig 1.1 of Hartle provides acomparison of masses and distances.18 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity




Successes and Failures of Newtonian Picture

Replaced the Aristotelian picture that:Objects move when acted upon by a force, and tend to a stationary state when force isremoved.Could not explain the force of gravity, which is a constant force but objects wereaccelerated.

Newton’s First Law gave us the first clue about relativity.If the force is such that F = 0 then v = C where C is a constant vectorThis allows for the concept of inertial frames of reference. Any frame for which v = C isdefined to be an inertial frame of reference

Newton’s Laws cannot impose the constancy of the speed of light, which led to anerroneous believe in absolute simultaneity instead of a relative one.





Newtonian Transformations between Inertial Frames of Reference

Making the transformation:

xn = x − vt , yn = y , zn = z, tn = t

it is easily shown that the second derivative of x w.r.t. t satisfies d2xdt2 = d2xn

dt2n

and F = Fn





The Light Cone in Special Relativity

Speed of light is the same in all inertial frames

Normal matter is restricted to speeds less than c

New concept of simultaneity, namely Relative Simultaneity





Coordinate Transformations in Special Relativity

Consider the Lorentz transformation,

xn = x−vt√1−(v/c)2

, yn = y , zn = z, tn = t−vx/c2√

1−(v/c)2

As before, the frame will be boosted by speed v along x axis relative to the originalframe at O





Space-time Diagram Under Lorentz Boost





Correspondence of electric and (Newtonian) Gravitational Force

Newtonian Gravity Electrostatics

Forces BetweenSources Fg = −GMm

r2 eMm Fe = qQ4πε0r2 eqQ

Force DerivedFrom Potential Fg = −m∇Φg(xm) Fe = −q∇Φe(xq)

Potential Outsidea Spherical Source Φg = −GM

r Φe = − Q4πε0r

Field Equation ∇2Φg = 4πGρm ∇2Φe = ρe/ε0

Note: if g(x = −∇Φg(x) then ∇g(x) = −4πρm(x), which is similar to∇E = −ρe(x)/ε0.





Moving Charges: Maxwell’s Equations and Lorentz Force

The Lorentz force describes how a collection of moving charges are affected by avelocity dependent force from magnetic fields.

Fe = q(E + v × B)

The velocity dependent term is missing in Newtonian Gravity. For any framechosen, acceleration depends on mass alone in Newtonian gravity, this implies itis not relativistic.

It is possible to add Bg term, however this becomes very complicated.





Measuring E and B Fields

Using neutral charges one can construct an inertial frame.

Any particle at rest can be used to measure E , since

Fe = qE

Once in motion one can measure B through the expression

Fe = q(E + v × B)

Sadly this approach cannot be used to measure gravity, since there is no form ofmatter that is "neutral" to gravity.





Steppin’ Stone: SR to GR

In the presence of gravity, freely falling frames are locally inertial, we call this thePrinciple of Equivalence.

These particles will follow geodesics, which can be seen as the path of leastresistance.

Returning to the point of no "neutral" charges. Any particle is a source ofgravitational field, so as they move through spacetime they also bend it.

We can approach GR by formulating SR in this new frame and taking our physicallaws and applying the Principle of Covariance (Physical Laws are preserved underchanges of coordinates.)

Some people believe we need additional principles.


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