MATH 492: Junior/Senior Honors Seminar in Mathematical General
Relativity, Spring 2014
Lecture: Tue 18:40-20:00 Hill-525
Instructor: Shabnam Beheshti
Office: #214, Hill Centre, Busch Campus
e-mail: beheshti[at]math[dot]rutgers[dot]edu
Graduate TA: Moulik
Balasubramanian
Office: #622, Hill Centre, Busch Campus
e-mail:
moulik[at]math[dot]rutgers[dot]edu
"Office" Hours: We will be holding workshop-style office hours in
the Busch Campus Center on Fridays, 14:00-15:30. You can stop
by to discuss weekly readings, work on suggested exercises, or
practice your presentations with us (over coffee)!
492-RUMA GUEST LECTURES IN MATHEMATICAL PHYSICS,
13:30 -- 17:00, TUE 06 MAY in HILL 705
- 13:30 -- 14:30 Lunch and Informal Q & A (Courtesy of
Chairman Simon Thomas and the Mathematics Department--Thank You!)
- 14:30 -- 15:15 Shadi Tahvildar-Zadeh, Department of
Mathematics
The Dirac Equation in Zero-Gravity Spacetimes
In this talk we will explore whether the following is science or
sorcery:
...The magician held up a large hoop and snapped his fingers. The tiger
got up on the stool, jumped through the hoop... and disappeared! The
magician explained to the audience that the hoop is actually a portal that
leads into a parallel universe, where another copy of him is right now
standing, holding up the same hoop, out of which the same tiger has just
emerged, to the wild applause of a parallel audience in a parallel circus
in that parallel universe. To prove his point, he held up the hoop again,
and in a flash, the tiger re-emerged on the other side of the hoop, made a
soft landing on the stage, and licked his lips...
- 15:15 -- 15:30 Q & A, Tea/Coffee Break
- 15:30 -- 16:15 Christoph Keller, NHETC/Department of
Physics
Quantum Gravity in 3 Dimensions
Quantum Gravity, that is finding a quantum version of general relativity
, is an extremely difficult problem. I will discuss some aspects of it in
three dimensions, where the situation is slightly simpler. If time
permits, I will point out a connection to the Monster group and its
moonshine.
- 16:15 -- 16:30 Q & A, Coffee/Tea Break
- 16:30 -- 17:00 Open Discussion (and more Coffee!)
Course Information
Required Course Notes: Curvature of Space and Time, with an Introduction to Geometric
Analysis, by I. Stavrov Allen
Recommended Course Textbook: Introducting Einstein's
Relativity, by Ray d'Inverno, Oxford University Press 1992
(ISBN: 9780198596868; softcover, 400 pages, $89.95)
A Remark on the Recommended Text: I will be following lecture
notes given at an advanced undergraduate summer school on geometric analysis,
highlighting the relevant segments of d'Inverno's book for further
study. A copy of this text is on reserve at
the Mathematics Library (ground floor of the Hill Centre). I will be
compiling a reference list for each lecture (see below).
Prerequisites for 640:492: Differential & Integral Calculus,
Multivariable Calculus, Ordinary Differential Equations and Linear
Algebra. Lectures and coursework will involve computation as
well as proofs, so courses such as 640-300 (Math Reasoning), 311/312
(Real Analysis I/II) are also extremely helpful. Any background
in Differential Geometry and/or Partial Differential Equations will also
be supremely useful. I have included the supplementary background reading
list above for your use throughout the term, all of which can be found in our
library (or a similar book is available). The "Recommended Readings" will
assume you know most of the basic knowledge that this list
encompasses.
Background Reading (for above Prerequisites):
1. Vector Calculus, by J. Marsden and A. Tromba. This, or any
other calculus textbook covering Green's, Gauss'/Divergence and Stokes'
Theorem is highly suggested for a review of chain rule, directional
derivatives, gradient, line and surface integrals. Rutgers uses
Calculus: Early Transcendentals, by Rogawski (2nd edition). I urge
you to have some vector calculus book handy to quickly do the easier
calculations suggested in class.
2. Elementary Differential Equations, by W. Boyce and R. Di Prima.
Often used as a course textbook for ODEs.
3. Partial Differential Equations, by W. Strauss. A classic
textbook in PDE, on which the current Rutgers 423 course is largely based.
4. Introduction to Real Analysis, by R. Bartle and D.
Sherbert. Useful for basic calculus proofs on the real line.
5. Differential Geometry of Curves and Surfaces, by M. P. Do Carmo.
Beautiful introduction to the fundamental objects of study in differential
geometry. His other books are also excellent. Warning: Uses different
sign conventions than us.
6. ABCs of Gravity, by X.Y. Zee.
Our (rather ambitious) Course Syllabus:
- Part I. Riemannian Geometry
- Framework of Riemannian Geometry
- Geodesics
- Differential Calculus
- Vector Calculus
- Differential Calculus with Tensors
- Introduction to Curvature
- Part II. Spacetime Curvature
- Jacobi's Equations
- Comparison Theorems
- Framework of General Relativity
- Geometry of Schwarzschild Spacetime
- Part III. Selected Topics
- Penrose Diagrams
- A Geometric Analyst's toolbox
- Evolution Problems in GR
- Cosmology
This course will be run in lecture-seminar style. I will describe this
setup more clearly on the first day of classes. You will be expected to
prepare and present lecture material (typically in groups), complete
exercises of varying difficulty, and complete assigned readings in
advance of each weekly meeting.
Since this course only meets once a week, attendance AND participation are
mandatory. You will be responsible for all the
material covered in lecture, including announcements, changes to
course material as well as exercises and readings.
Grading Scheme: You will submit three homework assignments (Feb, Mar, Apr)and give two presentations.
There will be no midterm or final examinations.
It is also important for you to know the Rutgers University Academic Integrity Policy.
Lecture Schedule
- Lecture 0, 21 Jan
Snow Day!
- Lecture 1, 28 Jan (SB)
Course outline, introductions,
first reading assignment
- Lecture 2, 04 Feb (SB)
Introduction: Coordinatizing
R^n, Line elements, the real/complex projective plane
Notes:
- How do you know that at each point in R^n the basis {del_j} forms a
n-dimensional space?
- Label the axes and other notable features in each diagram appearing
in this section (and future sections, as you read).
- Write a small paragraph on what d and del are. What does drdt mean,
for instance?
- In the reading, the change of coordinates x = r cos theta, y = r sin
theta can be used to transform the line element ds^2 = dx^2 + dy^2
to ds^2 = dr^2 + r^2 dtheta^2. What happnes if we use the inverse
transformation, namely y/x = tan theta, x^2+y^2 = r^2?
- Why is the Mercator projection the most popular? When was it
created? Be sure you understand the diagram just below the world map. See
page 426, exercises #45-48 in Rogawski for some interesting exercises on
this topic!
- On page 8, you are asked to check the transformation of line elements
under a spherical coordinate transformation. Can you relate the
quantities you encounter to the Jacobian from Calc III/Math 251?
- Implement stereographic projection on Mathematica or Maple to
visualize coordinate grids on the sphere, as seen on page 9.
- On the projective plane, page 9: Why is the following statement
important? "Note that l_u
= l_v if and only if u and v are collinear nonzero vectors." Why do we
need three maps for RP^2 (could we use fewer?). What is the dimension of
the manifold RP^2? Can you come up with the line element for RP^2
appearing in the middle of page 10?
References:
- Calculus, Early Transcendentals,
by Rogawski for review of Math 251/Calc III material
- Calculus on Manifolds by Spivak provides a readable, intuitive
introduction to the subject
- Differential Forms and
Applications by doCarmo for precise definitions of d and del
- Functions of One Complex Variable by Conway for treatment of
stereographic projection in C (first chapter)
- Your Suggestions for Exercise #2c on projective spaces over
quaternions and octonions
Lecture 2 Exercises: #5, 6, 8, 9
- Lecture 3, 11 Feb (Fay, Smith, Wong)
Framework
of Riemannian Geometry: Manifolds,
Riemannian metrics, lengths and volumes
Notes:
- Look up the Frenet formulas for curvature and torsion of curves in
R^3 to compare and contrast with the statements in the introduction of
this section.
- For V = sum V^i del_i, where do each of these quantities "live"
(i.e., are they functions, vectors, operators)?
- Consult Boothby (see References below) pages 52, 59-64 for details on
Disclaimer No. 2
- In classical differential geometry, the inner product on the tangent
space inherited from Euclidean 3-space is called the First Fundamental
Form of a Surface (see doCarmo)
- Work out the circular cylinder example with a piece of paper!
- Be sure you understand the linear algebra argument given in the first
paragraph of volume measurements. Can you give a second, geometric proof
for vectors in R^3?
- On page 18, what does "Scaling to the infinitesimal level" mean?
- At the end of this section, compare the definition of a compact
manifold with the definition of compactness from your Analysis course(s)
References:
- Differential Geometry of Curves and Surfaces by doCarmo for a
thorough treatment of metrics and curvature on curves/surfaces.
- Introduction to Differentiable Manifolds and Riemannian
Geometry by Boothby is a standard first-year graduate textbook you can
consult for any/all precise definitions on manifolds
Lecture 3 Exercises: #1, 3, 6, 7, 10
- Lecture 4, 18 Feb (Ma, Tamanas, Wasserman)
Geodesics: Length and
Energy functionals, geodesic equations, hyperbolic space
Notes:
- On the discussion of the length functional, there is the statement
"Note that for a given geometric family of curves there are infinitely
many representations of such a form" Can you give several
(nontrivially different) representations of a given curve?
- Write a brief paragraph explaining how the length of
gamma-dot(epsilon, t) formula compares with exercise 2b from the previous
lecture set. Why is this quantity set to equal L(gamma)/(b-a) and not
normalized to 1?
- Be sure you understand how epsilon --> L(gamma_epsilon) and gamma -->
L(gamma) differ from each other (and that we can do calculus with one of
these)
- Once we have found a critical point of the energy functional (or
length functional, for that matter), how do we actually know we have found
a minimizer and not a maximizer? Can you try to think of some ways to
devise a "second derivative" test?
- Under what conditions are we allowed to differentiate under the
integral sign at the bottom of page 22?
- Calculate the Christoffel symbols (and then the geodesic equations)
for R^2 in polar coordinates. Compare your equations with those of
Cartesian coordinates.
- Note that Theorem 1 on page 24 should remind you of well-posedness
from PDE 423 (if you have taken it). Irrespective, be sure that you
understand the quantities in the geodesic equations in the context of
this theorem. For a proof, see Boyce and diPrima, for instance
(References, below). Put a picture to Theorem 2 for surfaces in R^3
- Read and outline the proof of the Hopf-Rinow Theorem (see References)
- In the sphere example of page 25, why does it suffice to to study the
great circles in the subspace x^3 = ...= x^n = 0? What does script(I) do
to points on the sphere?
- For the hyperbolic plane example H^2, try to visualize the spacetime
by reproducing the curves in the diagram. Check that the mappings given
are, in fact, isometries. Again, why does it "suffice to analyze the
geodesic problem in H^2" (bottom of page 25) in order to understand it in
H^n?
- Solve the ODEs in the middle of page 26 for the H^2 geodesic
equations. Can you exhibit an isomorphism between the upper half plane
model H^2 and the Poincare disc model (on pages 27-28)?
- Look up Euclid's Fifth Postulate (original) and compare with
Playfair's Axiom.
References:
- Ordinary Differential Equations, by Boyce and DiPrima, for
basic techniques and theorems of ODEs
- Riemannian Geometry and Geometric Analysis by Jost for
a nice discussion of the Hopf-Rinow Theorem
- Elements by Euclid
- Theorie des groupes Fuchsiens by Poincare in Acta Math. 1
(1882)
Lecture 4 Exercises: #1, 4, 6, 9
- Lecture 5, 25 Feb (Harper, Lipman, Parikh)
Differential
Calculus: generalizing
directional derivatives, connections/parallel transport, covariant
differentiation
Notes:
- Be sure you understand the following statements by the end of your
reading/study of this Lecture: " It turns out that when differentating
vector fields one cannot work with coordinate functions alone: coordinate
systems change with respect to themselves and thus contricute to the
overall rate of change of a vector field"
- Where are the additions and multiplications in the properties
possessed by directional derivatives of functions and connections (the 3
bullet points in each definition) happening? Be careful to note when we
are addiing vectors, functions, and what the output of each derivative is
giving
- "prima facie" means "at first glance"
- What is happening on page 32?!?!!
- Convince yourself of the paragraph on page 33 just before the section
on Connections
- Why do we want/why is it "intuitive" to have the condition that the
directional derivative of gamma-dot in the direction of gamma-dot equals
zero for geodesics? This about this statement carefully
- Look up the Fundamental Theorem of Riemannian Geometry (see
References)
- Check all of the details of the parallel transport equation in the
final example of the section
References:
- Boothby has a nice presentation of the Fundamental Theorem of
Riemannian Geometry (and connections in general)
- A Latin Dictionary of your choice ^_^
Lecture 5 Exercises: #2, 3, 4, 8, 9
- February Assignment: Submit any 10 of the
18 selected problems from Lectures 1 - 5 (plus your rewrite of page 32) by
Friday 14 March
- Lecture 6, 04 Mar (Deneroff, Chi, Lee)
Vector
Calculus:
vector/covector fields, divergence theorem and IBP, weak/strong maximum principle for the
Laplacian
Notes:
- Consider the last sentence in the first paragraph of this Lecture: "Although one might be
tempted to think of nabla f as the gradient vector field of f this would be incorrect: nable f is
not a vector field. Instead it is what is known as a covector field" In this context, why do
we call nable f a vector field in Calculus III/251? Reconcile this with out notions of nabla x F and
nabla . F (cross and dot products with a vector/vector field F).
- Using local definitions of nabla_V f and nabla_V W, check that restricting nabla f to a point
produces a linear map from R^n to R^1 and restricting nabla_V X to a point produces a linear map
from R^n to R^n.
- Look up the definition of the dual of a vector space.
- Note that the mapping "b-flat" sends the vector field V to the mapping flat(V) defined for
each W as flat(V)(W) = . Check that this mapping is onto, so that for each covector field w, there
exists a vector field V such that =w(W).
- On the bottome of page 40, check the following statement: "It can easily be checked that V =
sum_ij g^ij w(del_j) del_i satisfies the relationship g(V,W)=w(W)."
- Can we say something about the composition of flats and sharps? For a vector field V, does
sharp(flat(V)) = V? Analogously, for a covector field w, does flat(sharp(w))=w?
- Give examples of what grad f is in two metrics, euclidean 2-space and the hyperbolic upper half
plane.
- What does it mean to be a geometric invariant of an operator?
- Write down the various definition of Tr(L) for n=2 and L=rotation by a fixed angle alpha and
verify that they all agree. How is the integral definition of Tr(L) compatible with the previous
ones? Where does this formula come from?
- Check the two chains of equalities in the middle/bottom of page 41 for div(X).
- Look up the alternative proof of Divergence Theorem given in Spivak (see references below).
- In Theorem 8, notice the symmetry of the expression in alpha and beta. Can you relate this
quantity to something you may have seen in Quantum Mechanics?
- Use mean value properties to prove the Weak and Strong Maximum Principles (see references).
References:
- Calculus on Manifolds by Spivak has a nice discussion and proof of the Divergence
Theorem (and Stokes' Theorem)
- For brief proofs of the Weak/Strong Maximum Principles, see page 27 of Partial Differential
Equations by Evans.
Lecture 6 Exercises: #3, 4, 6, 8,
9
- Lecture 7, 11 Mar (Oana, Sung, Velan)
Tensors:
definition, differentiation of tensors, the curvature tensor
Notes:
- Define a (k,l)-tensor (see references).
- To understand covariant differentiation of tensors, reformulate the notation in linear algebra
terms.
- On the bottom of page 45, why is it "clear from the definition that nabla w(V;W) is tensorial in
the W-entry" but less clear that it is tensorial in the V-entry?
- Concerning the definition of the covariant derivative of a (k,i)-tensor field, see also the
first part of the discussion on "differentiation of tensors" on Wikipedia.
- Work out the two examples on page 46 in full detail, using the fact that nabla_del_i del_j =
sum_k Gamma_ij ^k del_k, and the definiton of Gamma_ij ^k in terms of derivatives of g.
- Calculate the Christoffel symbols in the Exampe on page 47 explicitly to verify the calculation.
- Double check that R(V,W)X is tensorial and antisymmetric in V and W and tensorial in X (bottom
of page 47).
- Using the dual formulation, reformulate the (3,1)-tensor version of the Riem as a (4,0)-tensor.
- Check the details of the R(del_i, del_j) del_k = sum_l {...} del_l calculation on page 48.
References:
- For a concise and well-written discussion on tensors in manifold theory, see section V.2,
page 192 in Boothby.
- Lecture Notes on General Relativity by S. Carroll give an working-physicists' definiton
for differentiation of tensors is a useful reference when performing calculations.
Lecture 7 Exercises: #1, 3, 4, 6, 7,
10
(Lecture 7, exercise #8, page 61 has a typo: "gamma_0" should be
everywhere replaced with "gamma_0-dot" in the integral--thank you,
Yianni!)
- Lecture 8, 25 Mar (Osterman, Sewell, Lei)
Jacobi's Equations: examples, sectional curvature, worked problems on the hyperbolic
plane
Notes:
- Drawing pictures/diagrams for the quantities in this lecture will really help your understanding
of the section!
- Note that the vector field Y defined as del_s gamma is, in fact, evaluated at s=0, giving a
vector field Y= Y(t), no longer dependent on s. In particular, Y = sum_i (del x^i / del s)_s=0
(del_i)_gamma_0.
- Clarify the examples for R^n S^n and H^n: show, for instance, that gamma_s is also a geodesic
(line, great circle, etc.), draw diagrams for gamma_0 and gamma_s, graph the vector field Y.
- In the H^n example, where does the IVP on page 54 come from? What are the geodesic equations
for this model (what are the nonzero Christoffel symbols)? Solve the IVP using standard techniques
of integration.
- Recall that a vector field Y being parallel along gamma_0 means that nabla_(dot(gamma_0)) Y =0
(see page 35).
- Rewrite Theorem 15 (Jacobi Equation) more clearly and use the fact that [V , W] = nabla_V W
- nabla_W V and nabla_V nabla_W X + nabla_[V,W] X + R(V,W)X = nabla_V nabla_W X (on page 35 and 47)
to prove the chain of equalities in the proof. Why is [del_t, del_s] =0? Why is each del_t a
geodesic?
- Check that for a two-dimensional manifold, there is only ONE independent, non-vanishing
curvature component.
- How does exercise 7 from the previous Lecture prove that sectional curvature determines the
curvature tensor?
- For Theorem 17, recall the definition of a complete manifold from pages 24-25. To prove the
result, draw a diagram and recall that the energe E(gamma_s) can be viewed as a function of s (hence
the second derivative test is invoked), but that when moving the derivative inside the integral, d/ds
needs to act on an inner product of vector fields (and hence must be viewed as covariant
differentiation). Check the integral formula for d^2/ds^2 E(gamma_s) appearing in the proof of this
result.
- In the definition of Ricci, recall R(V,W,X,Y) = from page 48. and Exericses 3, 4
from Lecture 5 for the alternate formulations on the top of page 59.
- Compare the result of Myers and Cheng directly with Theorem 17. Why is this indeed a
generalization?
- Remark: Rigidity Theorems are of great importance in General Relativity: Positive Mass
Theroems and some recent well-posedness results have been successfully cast as rigidity therems.
References:
- See the final remarks on pages 401-402 of Boothby for other directions in which such a
discussion can go.
- Proofs of the Remark on page 58 appear in Milnor's Morse Theory (Ann. of Math. Studies,
vol. 51, 1963) and Bishop & Crittenden's Geometry of Manifolds (AMS Chelsea Pub. 2001).
Lecture 8 Exercises: #1, 2, 3, 7a, 8
(Lecture 8, Exercise #10, page 69 has a typo: E should equal
3F/2, and not "F = 3E/2" as written--thank you,
Richard!)
- March Assignment: Submit any 5 of the
16 selected problems from Lectures 6 - 8 (plus your "feedback
questionnaire") by Mon 14 Apr
- Lecture 9, 01 Apr (Blumberg, Deneroff, Ma, Wong)
Manifolds of positive
sectional curvature, Ricci Curvature, Comparison Theorems: Motivation, asymptotic formulas,
Gauss-Bonnet Theorem
Notes:
- On page 62,we define scalar curvature, "Scal" (also often denoted by just "R" or "R(g)" where g
is the metric). Compare the integral formulation of "Scal" with exercise 4 in Lecture 5. Can you
calculate "Scal" in some simple examples? Further down the page, Stavrov says "it is easy to see that
scalar curvature of the standard n-dimensional sphere of radius rho is n(n-1)rho^-2, while the scalar
curvature of the hyperbolic space H^n is -n(n-1)." Verify these statements by performing the
calculation explicity.
- Once we have learned what "Scal" is, we are in a position to define a very important class of
manifolds called Einstein manifold. See exercises 6, 7 and 8 in this lecture.
- Concerning the footnotes appearing on page 63: footnote 4 is in a misleading place, as P is simply a
point. In that paragraph, is Y a Jacobi field? Also, you should check the details of the claim in
footnote 5 by applying Taylor's Theorem to the coefficients of the constructed vector field.
- Note that upon using the Taylor approximation of the vector field Y= rV + r^2/2V_2 + r^3/3! v_3 +
... in the Jacobi Equation Y_tt+R(Y,U)U=0, we use the fact that knowing W_2 at a point determines W_2
(see top of page 40).
- Notice that we can view sectional curvatures in terms of a "defect" or "higher order
correction" to traditional quantities such as circumference/surface area of small Euclidean
circles/discs.
- Assume the sectional curvature of a given 2-dimensional manifold is positive. Why can we rewrite
it's metric ds^2 = dr^2 + (...)^2 dtheta^2 in geodesic
polar coordinates? That is, verify the three equalities appearing at the top of page 65.
- Although we did not cover the Asymptotic formulas for volumes of small balls in arbitrary
dimension (pages 65--67), read through the estimates carefully and try to relate them to the
previously defined quantities.
References:
- Gauss' Theorem and the Gauss-Bonnet Theorem are nicely presented in Riemannian Manifolds:
An Introduction to Curvature, by J. Lee. There is also a very nice presentation online, given
in H. Gluck's Differential Geometry 501 course at UPenn by (graduate student) Shiying Dong here.
- Interesting an far-reaching consequences of a Generalized Gauss-Bonnet Theorem may also be
studied.
- For further reading on the Kulkarni-Nomizu product and the Weyl Tensor (exercise 9), see
Einstein Manifolds, by A. Besse and Riemanian Geometry, by Gallot, Hulin and Lafontaine.
The latter is a more encylopedic reference, which can be quite useful for standard quantities defined
in Riemannian Geometry.
Lecture 9 Exercises: #2, 3b, 5, 6, 7,
8, 10
- Lecture 10, 08 Apr (Fay, Lipman, Tamanas, Parikh)
Framework
of GR: special relativity, Minkowski spacetime, review of
Newtonian gravity, Poisson's equation
Notes:
- Why do we refer to the trajectory of light as a "cone" in spacetime?
Can you confirm that two inertial observers must observe the same value of
-(delta t)^2 + (delta x)^2 + (delta y)^2 + (delta z)^2?
- Verify that m(u_1,u_2) = -t_1t_2 + x_1x_2 + y_1y_2 + z_1z_2 defines
a non-degenerate inner product on R^4. Use this inner product to deduce
time dilation
from the geometry of R^4 equipped with this metric (see also Exercise 2).
- Check that each non-degenerate inner product admits a
pseudo-orthonormal basis and that the signature of the inner product/basis
does not depend on the choice of basis.
- On page 71, the author says "for reasons we do not have time to go
into, massive particle move at speeds slower than the speed of light" and
she describes their motion using a condition on their velocity vector
fields in terms of proper time. Try to understand these statements better
by looking up what "proper time" is (in Schutz's book).
- Check the equalities appearing at the top of page 72 for the
gravitational field F(y). Be "at least a little bit concerned about" the
signularities in the last two expressions. Given our integrands, then
why, in the next equation, can we move the partial derivative inside the
integral?
- Carefully check the details in the proof of Gauss' Law, Theorem 23.
This will involve (the very useful) reviewing your defitions of div, grad
and inner products from the earlier lectures. Why can be place an upper
bound on the size of the gradient? Are we justified in taking that last
limit on page 73?
References:
- A First Course in General Relativity, by Bernard Schutz. The
notion of simultenaety, accelerated observers and the interval of special
relativity are nicely described in here.
- For further discussion on Special Relativity, both the
theoretical and experimental aspects, see Modern Physics, by F.
Blatt and Invitation to Contemporary Physics, by Ho-Kim, Kumar and
Lam.
Lecture 10 Exercises: #1, 2, 3, 5, 9, 10
- Lecture 10-11, 15 Apr (Chi, Velan, Wasserman, Sewell)
Framework of General Relativity: Discussion on GR, from Poisson to
Einstein, non-vacuum Einstein Equations
Geometry of Schwarzschild spacetime: solving EVE under spherical
symmetry
Notes:
- Poisson's Equation on page 73 gives rise to the equation for the
acceleration of a particle in free fall. From this equation, one can
deduce Kepler's Laws. Review the "calculus III derivation" of
Kepler's Laws appearing in--you guessed it--your calculus book,
Chapter 13 of Rogawski!
- Think carefully about how the notions of relativity and gravity are
unified by way of GR using four-dimensional manifolds whose tangent spaces
are equipped with Minkowski geometry (and obey special relativity).
- It is possible to develop the geometry of pseudo-Riemannian manifolds
in parallel to what we have done in class so far, including geodesics,
Christoffel symbols, connectures and the curvature tensor. How do these
quantities differ, given that nonzero tangent vectors can have zero or
negative lengths and that the volume element is the one given at the top
of page 75?
- Why can we view Ricci(gamma-dot, gamma-dot) as "matter along"
gamma-dot?
- How many equations comprise the Einstein-vacuum system, prior to any
symmetry reductions or simplifications (i.e., given only the existing
properties of Ric_g)?
- It is important to note that to transition from the vacuum Einstein
equations to the the non-vacuum case, Ricci = T, an auxiliary condition is
placed on the energy-momentum tensor, namely that tracing nablaT gives
zero. Why must we require that T be divergence-free?
- Under what physical considerations can we reduce the metric g in the
vacuum equations to the form appearing on page 79, that is, when is it
physically reasonable to assume spherical symmetry?
- Roll up your sleeves and calculate Ric(g) for the metric static,
spherically symmetric metric g given in
local coordinates by ds^2 = -E(r) dt^2 + F(r) dr^2 + r^2 dS_2^2. You can
check your answer by writing a Mathematica or MAPLE protocol (or use any
one of the existing programs out there for both systems)... but try doing
this by hand first!
- Can you reason why the integration constant C should be written as 2M
by re-inserting the appropriate constants we have normalized to 1 at the
beginning of the problem and performing some dimensional analysis?
References:
- Calculus: Early Transcendentals by J. Rogawski. In the
second edition, Chapter 13, Section 13.6 has a very readable discussion of
Kepler's Laws.
- Semi-Riemannian Geometry with Applications to Relativity, by
Barrett O'Neill is a classic text and addresses nicely many of
the parallels (pun intended!) between the Riemannian and semi-Riemannian
analysis/geometries.
- The Large-Scale Structure of Spacetime by Hawking and Ellis
contains an appendix in which the Schwarzschild solution is derived very
nicely. Do take a look at this reference if you plan to continue studying
GR, as it is the "physics bible" for many classical results!
Lecture 11 Exercises: #1, 2, 3, 4, 5
- Lecture 11, 22 Apr (Lee, Osterman, Sung, Harper)
Geometry of Schwarzschild spacetime: the Schwarzschild solution, Newtonian
celestial mechanics, orbits
in Schwarzschild (equations), orbits in Schwarzschild (conservation
laws)
Notes:
- At the bottom of page 80, invariance of the classical point-particle
motion equations under reflection around the equatorial plane reduces the
system to two equations, expressed in polar coordinates. Check this
carefully!
- Can you justify why equation (7) for curly E is called an energy,
and (why J in the earlier paragraph is angular momentum)? Solve the
second-order differential equation for u(theta) explicitly to obtain u= A
cos (theta - theta_0) + (M/J)^2 and conclude that the resulting formula
for r is precisely that of a conic section.
- Redraw all of the graphs for r, u and V_eff and their corresponding
orbits diagrams. This is key in your understanding of this
lecture!
- In all of the subsequent discussion past page 82, try to remember
that we would like to reconcile our solutions with Special Relativity,
Minkowski spacetime and Newtonian Theory by taking the appropriate
perspective (or limits). For example, when r --> infty, the Schwarzschild
metric approaches Minkowski, as expected (is that true? do you expect
it?).
- Write down the geodesic equations for the Schwarzschild metric,
explicating the terms referred to as "(mess)" in the middle of page 82.
Compare these equations to the Newtonian system. Does curly-E-tilde
compare with curly-E as an energy? If not, what about the new conserved
quantity J-tilde compared with J?
- Can you solve the nonlinear differential equation in u explicitly
using power series techniques (or other methods)? Verify, at least
numerically that solutions follow the kinds of orbits our analysis
suggests. Do you observe critical behaviour for (J/M)^2 = 12?
References:
- See any calculus textbook for a discussion on conic sections, e.g.,
see Rogawski, Chapter 11, section 11.5.
- A full discussion of ODEs from a sophisticated manifolds/dynamical
systems approach appears in V. I. Arnold's text, Ordinary Differential Equations.
- Boyce and DiPrima have some very basic introduction to power series
methods for solving ODEs and discuss nonlinear ODEs briefly near the end
of their textbook, Elementary Differential Equations and Boundary Value
Problems (millionth edition at this point).
Your exercise for this week is to
browse through the final readings from Chapter 3: A PDE Toolbox
- Lecture 11+, 29 Apr (Oana, Blumberg,
Balasubramanian)
Geometry of Schwarzschild spacetime: orbits in Schwarzschild (cases),
Perihelion advance
Introduction to Penrose Diagrams
Notes:
- Try to reconstruct the trajectory diagrams in the critical J^2 =
12M^2 and noncritical settings, by combining analysis of the graphs of
V_eff with numerical solutions to the nonlinear ODE in u.
- Can you perform a similar analysis to what we have discussed in class
for the Schwarzschild-anti de Sitter solution ds^2 given in exercise #5,
page 86 to the non-vacuum Einstein equations? This will involve even more complicated geodesic
equations, but notice that if you set Lambda to zero, you should recover
exactly the equations we have derived in class.
- Look up Einstein's original 1915 paper, "Explanation of the
Perihelion Motion of Mercury from the General Theory of Relativity" and
see if it makes sense after all of our hard work!
References:
- For further discussion on the derivation and physics of the
Schwarzschild solution as well as Moulik's discussion on Penrose
Diagrams, see S. Carroll's book,
Lecture Notes on General Relativity. In fact, (his page) has some
other excellent discussions worth reading as well as useful links for
further studies.
Selected Topics Lecture Exercises: none!
- April Assignment: Submit any 5 of the
18 selected problems from Lectures 9 onward by Tue 06 May
Below, I have compiled a variety of other GR-related resources around the web with some brief commentary.
Please note that this is by no means an exhaustive list, but some suggestions for places you can go next to
learn more!
- International Society on General Relativity and Gravitation (self
explanatory).
- The Marcel Grossman Meeting on General Relativity is one of the
biggest meetings of pure and applied scientists discussing various aspects of GR (every 3 years in
various locations).
- hyperspace@aei is a site hosted by the Max Planck
Institute-Albert Einstein Institute (hence "aei") in order to foster interaction among scientists working on
General Relativity and gravitation-related problems. Their page includes conference listings, people working in
the field, and job/postdoc/graduate study announcements.
- The Mathematical Sciences Research Institute (MSRI) in Berkeley, CA hosted a MSRI Summer School on Mathematical General
Relativity in 2012 for graduate students to attend. Not only are there streaming, beautifully delivered
lectures, but all of the supplementary pdf files (including exercise sets) are also available for your use.
- The Summer School mentioned above was part of a parent program, the Theme Semester in Mathematical General Relativity. Of particular
note are the extensive bibliography available and
the very accessible Introductory Program. Not to be missed in
the Introductory Program are the first two lectures by Daniel Pollack on Spacetime geometry, and Lan-Hsuan
Huang's Thursday lecture on Density Theorems for the Einstein Constraint Equations. For those of you interested
in understanding the interface between PDE problems in GR, numerical relativity, and the modeling of binary
black hole mergers, I would also suggest viewing the very nice talks of Manuela Campanelli and Dierdre
Shoemaker in the Connections for Women Program. Yours truly
was also a plenary speaker in that subprogram, so you may see me asking questions after their talks (if they
didn't cut off the video)!
- J. Shapiro from the Physics Department here in Rutgers gave a GR talk
for the SPS (Society of Physics Students) and posted a pdf file of the
whole
thing. Many of the facts he states are ones which we have either
discussed carefully or proved in our course, so I think you will enjoy
these notes (thanks, Aditya)!
- J. Valiente-Kroon has taught multiple courses in General Relativity at Queen Mary, University of London,
many of which have very nice lecture notes and
summaries here.
- Brainchild of famous string theorist Brian Greene is a fascinating
"science on the web" project which has now come to fruition. He very eloquently
summarizaes some of the most difficult concepts in theoretical physics in a manner which is accessible to a
layperson. It is a bit of a BG love-fest (as far as I can tell, he is the star of every video) but
nonetheless, definitely worth a visit sometime this summer on your study breaks. Speaking of which...
- Searching for more things to learn and feeling ambitious? I suggest taking a look at this and making a study plan.
You too can work towards a better-than-ivy-league
education!