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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
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
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
2 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
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
3 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and 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.
4 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
Course GoalsCourse OutlineCourse TextsAcademic Policies
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
5 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
Course GoalsCourse OutlineCourse TextsAcademic Policies
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".
6 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
Course GoalsCourse OutlineCourse TextsAcademic Policies
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.
7 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
Course GoalsCourse OutlineCourse TextsAcademic Policies
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.
8 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
Course GoalsCourse OutlineCourse TextsAcademic Policies
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.
9 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
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
10 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
Precision Gravity In the Solar SystemRelativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!)Cosmological ModelingGravitational...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
11 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
Precision Gravity In the Solar SystemRelativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!)Cosmological ModelingGravitational...Quantum Gravity
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
12 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
Precision Gravity In the Solar SystemRelativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!)Cosmological ModelingGravitational...Quantum Gravity
"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
13 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
Precision Gravity In the Solar SystemRelativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!)Cosmological ModelingGravitational...Quantum Gravity
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
14 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
Precision Gravity In the Solar SystemRelativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!)Cosmological ModelingGravitational...Quantum Gravity
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
15 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
Precision Gravity In the Solar SystemRelativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!)Cosmological ModelingGravitational...Quantum Gravity
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
16 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
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
17 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
When is Gr ImportantSome ContextSpecial RelativityFrom SR to GR
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
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
When is Gr ImportantSome ContextSpecial RelativityFrom SR to GR
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.
19 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
When is Gr ImportantSome ContextSpecial RelativityFrom SR to GR
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
20 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
When is Gr ImportantSome ContextSpecial RelativityFrom SR to GR
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
21 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
When is Gr ImportantSome ContextSpecial RelativityFrom SR to GR
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
22 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
When is Gr ImportantSome ContextSpecial RelativityFrom SR to GR
Space-time Diagram Under Lorentz Boost
23 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
When is Gr ImportantSome ContextSpecial RelativityFrom SR to GR
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.
24 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
When is Gr ImportantSome ContextSpecial RelativityFrom SR to GR
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.
25 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
When is Gr ImportantSome ContextSpecial RelativityFrom SR to GR
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.
26 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity
Course OutlineReasons to Study and Use GR
Some Interesting Issues in SR and GR
When is Gr ImportantSome ContextSpecial RelativityFrom SR to GR
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.
27 / 27 Dr. David McNutt , (call me Dave) Phys 4390: General Relativity