NOTES ON GENERAL RELATIVITY (GR) AND GRAVITY
NOTES ON GENERAL RELATIVITY (GR) AND GRAVITYERNEST YEUNGAbstract. These are notes on General Relativity (GR) and Gravity.As of March 23, 2015, I find that the Central Lectures given by Dr. Frederic P. Schuller for the WE Heraeus InternationalWinter School to be, unequivocally, the best, most lucid, and well-constructed lecture series on General Relativity andGravity. Instead of reinventing the wheel, I write these notes to build upon and supplement the video lectures and tutorialsalready created by them. This includes my corrections, comments, relations to other aspects of theoretical physics, and codeimplementing calculations in GR.It should be noted that for symbolic computation, I heavily use the SageManifolds v.0.7 package for Sage Math. My goalin this area is this: we see a concept or idea from GR and we go from the equation on the blackboard or textbook and into(Python/Sage Math) code that immediately computes a calculation.I keep these notes available online, openly accessible, and free for anyone, anytime (with your (financial) help and contribution at Tilt/Open or Patreon, which is a subscription service). I want to keep these notes openly accessible because Iwant to encourage anyone to freely edit, copy, and make their own notes in the spirit of open-source software.I continuously update these notes and post them here ernestyalumni.wordpress.comThe stated goal of the WE Heraeus International Winter School on Gravity and Light is to take the student from anintroduction to the research frontier (cf. http://www.gravity-and-light.org/lectures). I want to get myself and otherstudents or ambitious non-academic (maybe he or she is a working professional who had studied physics before in college,went to work in another field, maybe even, gasp, investment banking or mobile app developer, but still is curious andpassionate about physics and want to contribute) equipped with all the tools available to do research, do calculations, todesign experiments or collect data. Again, we’re not here to reinvent the wheel. I’m not trying to make a General Relativityappreciation class, but this is a serious attempt towards training to do research.Part 1. WE Heraeus International Winter School on Gravity and LightIntroduction (from EY)1. Lecture 1: TopologyTopology Tutorial Sheet2.3.4.5.6.7. Lecture 7: Connections8. Lecture 8: Parallel Transport & Curvature (International Winter School on Gravity and Light 2015)Tutorial 8 Parallel transport & Curvature9. Lecture 9: Newtonian spacetime is curved!10. Lecture 10: Metric Manifolds11. Symmetry12. Integration22234444448911151519Date: 23 mars 2015.1991 Mathematics Subject Classification. General Relativity.Key words and phrases. General Relativity, Gravity, Differential Geometry, Manifolds, Integration, MIT OCW, education, mathematics,physics.I write notes, review papers, and code and make calculations for physics, math, and engineering to help with education and research. Withyour support, we can keep education and research material available online, openly accessible, and free for anyone, anytime. If you like what I’mtrying to do for physics education research, please go to my Tilt/Open or Patreon crowdfunding campaign, read the mission statement, sharethe page, and contribute financially if you can. ernestyalumni.tilt.com https://www.patreon.com/ernestyalumni.1
13. Lecture 13: Relativistic spacetime14. Lecture 14: Matter15. Lecture 15: Einstein gravityTutorial 13 Schwarzschild Spacetime16.17.18.19.20.21.22. Lecture 22: Black t 1. WE Heraeus International Winter School on Gravity and LightIntroduction (from EY)The International Winter School on Gravity and Light held central lectures given by Dr. Frederic P. Schuller. Theselectures on General Relativity and Gravity are unequivocally and undeniably, the best and most lucid and well-constructedlecture series on General Relativity and Gravity. The mathematical foundation from topology and differential geometryfrom which General Relativity arises from is solid, well-selected in rigor. The lectures themselves are well-thought out andclearly explained.Even more so, the International Winter School provided accompanying Tutorial Sessions for each of the lectures. I hadgiven up hopes in seeing this component of the learning process ever be put online so that anyone and everyone in the worldcould learn through the Tutorial process as well. I was afraid that nobody would understand how the Tutorial or “OfficeHours” session was important for students to digest and comprehend and work out-doing exercises-the material presentedin the lectures. This International Winter School gets it and shows how online education has to be done, to do it in anexcellent manner, moving forward.For anyone who is serious about learning General Relativity and Gravity, I would simply point to these video lectures andtutorials.What I want to do is to build upon the material presented in this International Winter School. Why it’s important tome, and to the students and practicing researchers out there, is that the material presented takes the student from anintroduction to the research frontier. That is the stated goal of the International Winter School. I want to dig into andhelp contribute to the cutting edge in research and this entire program with lectures and tutorials appears to be the mostdirect and sensible route directly to being able to do research in General Relativity and Gravity. -EY 201503231. Lecture 1: Topology1.1. Lecture 1: Topological Spaces.Definition 1. Let M be a set.A topology O is a subset O P(M ), P(M ) power set of M : set of all subsets of M . satisfying(i) O, M O(ii) U O, V O UTV O S(iii) Uα O, α A α A Uα O2
O} utterly uselessDefinition 2. Ostandard P(Rd )EY : 20150524I’ll fill in the proof that Ostandard is a topology.Proof. Ostandardsince p , r R : Br (p) (i.e. satisfied “vacuously”)Suppose U, V Ostandard .Let p UTV . Then r1 , r2 R s.t. Br1 (p) UBr2 (p) VLet r min {r1 , r2 }.TTClearly Br (p) U and Br (p) V . Then Br (p) U V . So U V Ostandard .Suppose, Uα Ostandard , α A.SLet p α A Uα . Then p Uα for at least 1 α A.SS rα R s.t. Brα (p) Uα α A Uα . So α A Uα Ostandard 1.2. 2. Continuous maps.1.3. 3. Composition of continuous maps.1.4. 4. Inheriting a topology. EY : 20150524I’ll fill in the proof that given f continuous (cont.), then the restriction of f onto a subspace S is cont. If you want areference, check out Klaus Jänich [2, pp. 13, Ch. 1 Fundamental Concepts, Sec. Continuous Maps]If cont. f : M N , S M , then f S cont.Proof. Let open V N , i.e. V ON i.e. V in the topology ON of N . 1f S (V ) {m M f S (m) V }Now f 1 (V ) {m M f (m) V }.T 1So f 1 (V ) S f S (V )Now f cont. So f 1 (V ) ON .Tand recall OS : {U S U OM }.T 1 1so f 1 (V ) S f S (V ) OS i.e. f S (V ) open. f S cont. Topology Tutorial Sheetfilename : main.pdfThe WE-Heraeus International Winter School on Gravity and Light: TopologyEY : 201505243
What I won’t do here is retype up the solutions presented in the Tutorial (cf. https://youtu.be/ XkhZQ-hNLs): thepresenter did a very good job. If someone wants to type up the solutions and copy and paste it onto this LaTeX file, inthe spirit of open-source collaboration, I would encourage this effort.Instead, what I want to encourage is the use of as much CAS (Computer Algebra System) and symbolic and numericalcomputation because, first, we’re in the 21st century, second, to set the stage for further applications in research. I usePython and Sage Math alot, mostly because they are open-source software (OSS) and fun to use. Also note that thestructure of Sage Math modules matches closely to Category Theory.In checking whether a set is a topology, I found it strange that there wasn’t already a function in Sage Math to check eachof the axioms. So I wrote my own; see my code snippet, which you can copy, paste, edit freely in the spirit of OSS here,titled topology.sage:gist github ernestyalumni topology.sageDownload topology.sageLoading topology.sage, after changing into (with the usual Linux terminal commands, cd, ls) bysage : load ( ‘ ‘ topology . sage ’ ’)Exercise 2: Topologies on a simple set.Question Does O1 : . . . constitute a topology . . . ?.Solution: Yes, since we check by typing in the following commands in Sage Math:emptyset in O 1Axiom2check ( O 1 ) # TrueAxiom3check ( O 1 ) # TrueQuestion What about O2 . . . ?.Solution: No since the 3rd. axiom fails, as can be checked by typing in the following commands in Sage Math:emptyset in O 2Axiom2check ( O 2 ) # TrueAxiom3check ( O 2 ) # False2.3.4.5.6.7. Lecture 7: Connections X f Xf (df )(X) but (not quite)X : C (M ) C (M )df : Γ(T M ) C (M ) X : C (M ) C (M )4
X : C (M )C (M ). X :T M p T M q i.e. ptensor fieldq.T M p T M q i.e. ptensor fieldq7.1. Directional derivatives of tensor fields. manifold with connection is quadruple (M, O, A, )topology Oatlas AConsider chart (U, x) ADefinition 3. pair (X, (p, q) tensor field) (X, (p, q) T F ),connection on smooth manifold (M, O, A) : (X, (p, q) T F ) (p, q) T F s.t.(i) X f Xf(ii) X (T S) X T X S(iii) X (T (ω, Y )) ( X T )(ω, T ) T ( X ω, Y ) T (ω, X Y )“Leibnitz” rule.AsT S(ω(1) . . . ω(p r) , Y(1) . . . Y(q s) ) T (ω(1) . . . ω(p) , Y(1) . . . Y(q) ) · S(ω(p 1) . . . ω(p r) , Y(q 1) . . . Y(q s) )so X (T S) ( X T ) S T X S(iv) f X Z T f X T Z T C -linear7.2. New structure on (M, O, A) required to fix . There are (dimM )3 many ΓijkΓijk : U R p 7 dxi ( ) (p) x xjNow xm(dxi ) ?5
) m (dx(δ i ) 0j x x xm j {z} iδ ij{z }(iii) dxi ( m j ) 0 dxj xm x x {z x }i Γqjm xq xm xmdxi Γijm xjdxi Γijm dxjHence( X Y )i X(Y i ) ΓijY jXmm {z}last entry goes in direction of X( X ω)i X(ωi ) Γjim ωj X mabove expression for ( X Y )i , in theNote that for the immediatelysecond term on the right hand side, Γijm has the lastentry at the bottom, m going in the direction of X, so that it matches up with X m . This is a good mnemonic to memorizethe index positions of Γ.summary so far:( X Y )i X(Y i ) Γijm Y j X m( X ω)i X(ωi ) Γjim ωj X msimilarly, by further application of LeibnitzT a (1, 2)-TF (tensor field)( X T )ijk X(T ijk ) Γism T sjk X m Γsjm T isk X m Γskm T ijs X mWhat is a Euclidean space:(M Rn , Ost , A) smooth manifold.Assume (Rn , idRn ) A and(Γi(x) )jk dx ( E )i xk xj ! 07.3. Change of Γ’s under change of chart. (U, x), (V, y) A and U V 6 Γijk (y) : dy i y k y j y i q xs dx xp pjs y k x y x xqNote f X is C -linear for f Xcovector dy i is C -linear in its argument Γijk (y) y i q dx xq (7.1) xp y k xp xs y j xs xs y j xp xs y i xp xs q y i xp xs qδ Γ (x) xq y k xp y j s xq y k y j spΓijk (y) y i 2 xq y i xs xp q Γ (x)qjk x y y xq y j y k spEq. (7.1) is the change of connection coefficient function under the change of chart (U V, x) (U V, y)6
7.4. Normal Coordinates.Tutorial 7 Connections. Exercise 1. : True or false?(a) f X Y f X Y by definition so f X f X i.e. X is C (M )-linear in X f C (M ) is a (0, 0)-tensor field. X f Xf X(f ) by definition. If the manifold is flat, I’m assuming that means that the manifold is globally a Euclidean space, and bydefinition, Γ 0. Y i (Y i ) i Γijk Y k X k i X j j 0j x x x x xiand similarly for any (p, q)-tensor field, i.e. X Y X ji .i X T X X f X jj Tj11.jqp xj f X · grad(f ) xj (U, x) A, locally (after working out the first few cases, and doing induction, one can look up the expressionfor the local form; I found it in Nakahara’s Geometry, Topology and Physics, Eq. 7.26, and it needs tobe modified for the convention of order of bottom indices for Γ:λ1 .λpλp λ1 .λp 1 κλ1 κλ2 .λpppp · · · Γκµq ν tµλ11 .λ Γκµ1 ν tλκµ1 .λ ν tλµ11 .λ.µq 1 κ.µq ν tµ1 .µq Γ κν tµ1 .µq · · · Γκν tµ1 .µq2 .µqClearly, X is uniquely fixed p M by choosing each of the (dimM )3 many connection coefficient functionsΓ.(b) : X(M ) X(M ) : (p, q)-tensor field 7 (p, q)-tensor field By definition, satisfies the Leibniz rule. Exercise 2. : Practical rules for how acts Torsion-free covariant derivative boils down to a connection coefficientfunction Γ that is symmetric in the bottom indices. X f X(f ) X i f xi( X Y )a X i Y a Γajk Y j X k xi( X ω)a X i ωa Γiak ωi X k xj ( m T )abc iaa(T a ) Γaim Tbc Γibm Tic Γjcm Tbj xm bc ( [m A) n] ( m A)n ( n A)m An Γinm Ai xm7 Am Γimn Ai xn Am Am xm xn
( m ω)nr ωnr Γinm ωir Γirm ωni xmExercise 3. : Connection coefficientsQuestion .The connection coefficient functions Γ in chart (U V, y) is given, in terms of chart (U V, x) as follows:Recall Eq. (7.1)Γijk (y) y i 2 xq y i xs xp q Γ (x) xq y j y k xq y j y k sp8. Lecture 8: Parallel Transport & Curvature (International Winter School on Gravity and Light2015)8.1. Parallelity of vector fields.Definition 4.(1) parallely transported along smooth curve γ : R Mif vγ X 0(8.1)(2) A slightly weaker conditionis “parallel”( vγ,γ(λ) X)γ(λ) µ(λ)Xγ(λ)8.2. Autoparallely transported curves.Definition 5. curve γ : R M is calledautoparallely transported if! v γ vγ 0(8.2)8.3. Autoparallel equation. v γ vγ 0in summary:(8.3)mabγ̈(x)(λ) (Γm(x) )ab (γ(λ))γ̇(x) (λ)γ̇(x) (λ) 08.4. Torsion.Definition 6. torsion of a connection is the (1, 2)-tensor field(8.4)T (ω, X, Y ) : ω( X Y Y X [X, Y ])(Inside a cloud)[X, Y ] vector field defined by[X, Y ]f : X(Y f ) Y (Xf )Proof. check T is C -linear in each entryT (ω, f X, Y ) ω( f X Y Y (f X) [f X, Y ]) 8
Definition 7. A (M, O, A, ) is called torsion-free if T 0In a chartT iab : T dx , a , b x xi dxi (. . . ) Γiab Γiba 2Γi[ab]From now on, in these lectures, we only use torsion-free connections.8.5. Curvature.Definition 8. Riemann curvature of a connection is the (1, 3)-tensor fieldRiem(ω, Z, X, Y ) : ω( X Y Z Y X Z [X,Y ] Z)(8.5)Proof. do it: C -linear in each slot. Tutorials Riemijab . . .Tutorial 8 Parallel transport & CurvatureExercise 1.Exercise 2. : Where connection coefficients appearIt was suggested in the tutorial sheets and hinted in the lecture that the following should be committed to memory.Question : Recall the autoparallel equation for a curve γ.(a) v γ vγ 0(b) vγ vγ γ̇ xµνvγ γ̇ ν vγ γ̇ν ρ vγµ ρµν γ̇ρµ Γµν vγ γ̇ Γµν γ̇ 0 xν xρ xν xρ γ̈ ρ Γρµν γ̇ µ γ̇ νas, for example, for F (x(t)),dF (x(t)) Fd ẋ Fdt xdtso thatγ̇ ν vγµd µd2 µ v γγ xνdλdλ2Question : Determine the coefficients of the Riemann tensor with respect to a chart (U, x).Recall this manifestly covariant definitionRiem(ω, Z, X, Y ) ω( X Y Z Y X Z [X,Y ] Z)We wantRijab .now X Y Z X ((Y µ ρ µ νβZ Γρµν Z µ Y ν ) ρ ) (X α α (Y µ µ Z ρ Γρµν Z µ Y ν ) Γραβ (Y µ µ Z α Γαµν Z Y )X )µ x x x x x xρFor X a , Y b , Z j , then the partial derivatives of the coefficients of the input vectors become zero. a b j (Γi ) Γiαa Γαjb xa jb9
Now[X, Y ]i X ji ij XY Y xj xjFor coordinate vectors, [ i , j ] 0 i, j 0, 1 . . . d.ThusRijab i iiαΓ Γ Γiαa Γαjb Γαb Γja xa jb xb jaQuestion :Ric(X, Y ) : Riemmamb X a Y b define (0, 2)-tensor?.Yes, transforms as such:EY developments. I roughly follow the spirit in Theodore Frankel’s The Geometry of Physics: An IntroductionSecond Ed. 2003, Chapter 9 Covariant Differentiation and Curvature, Section 9.3b. The Covariant Differential of a VectorField. P.S. EY : 20150320 I would like a copy of the Third Edition but I don’t have the funds right now to purchasethe third edition: go to my tilt crowdfunding campaign, http://ernestyalumni.tilt.com, and help with your financialsupport if you can or send me a message on my various channels and ernestyalumni gmail email address if you could helpme get a hold of a digital or hard copy as a pro bono gift from the publisher or author.The spirit of the development is the following:“How can we express connections and curvatures in terms of forms?” -Theodore Frankel.From Lecture 7, connection on vector field Y , in the “direction” X, Y iij Γjk Y k Y x xk xiMake the ansatz (approche, impostazione) that the connection acts on Y , the vector field, first: i Y i k Y (X) X k k Γijk Y j X k X ( X Y )i i X Y Y xk x xi xi xNow from Lecture 7, Definition for Γ, dx i xk xj ΓijkMake this ansatz (approche, impostazine) Γijk dxk Ω1 (M, T M ) T M T Mj x xiwhere Ω1 (M, T M ) T M T M is the set of all T M or vector-valued 1-forms on M , with the 1-form being the following:Γijk dxk Γij Ω1 (M )i 1 . . . dim(M )j 1 . . . dim(M )So Γij is a dimM dimM matrix of 1-forms (EY !!!).Thus Y (d(Y i ) Γij Y j ) 10 xi
So the connection is a (smooth) map from T M to the set of all vector-valued 1-forms on M , Ω1 (M, T M ), and then, after“eating” a vector Y , yields the “covariant derivative”: : T M Ω1 (M, T M ) T M T M : Y 7 Y Y : T M T M Y (X) 7 Y (X) X (Y )Now ,f 0 xi xj xi xj xj xi(this is okay as on p (U, x); x-coordinates on same chart (U, x))EY : 20150320 My question is when is this nontrivial or nonvanishing (i.e. not equal to 0).[ea , eb ] ?for a frame (ec ) and would this be the difference between a tangent bundle T M vs. a (general) vector bundle?Wikipedia helps here. cf. wikipedia, “Connection (vector bundle)” : Γ(E) Γ(T M E) Ω1 (M, E)c b ea ωabf ecf b T M (this is the dual basis for T M and, note, this is for the manifold, Mc fb ea ωabec Ec bωac ωabf Ω1 (M )is the connection 1-form, with a, c 1 . . . dimV . EY : 20150320 This V is a vector space living on each of the fibers of E.I know that Γ(T M E) looks like it should take values in E, but it’s meaning that it takes vector values of V . Correctme if I’m wrong: ernestyalumni at gmail and various social media.Let σ Γ(E), σ σ a eac a b σ (dσ c ωabσ f ) ec with σ c bf xb c c σc a bbc ab σ X σ X ωab σ X ec X ωab σ ec xb xbdσ c 9. Lecture 9: Newtonian spacetime is curved!Axiom 1 (Newton I:). A body on which no force acts moves uniformly along a straight lineAxiom 2 (Newton II:). Deviation of a body’s motion from such uniform straight motion is effected by a force, reduced bya factor of the body’s reciprocal mass.Remark:(1) 1st axiom - in order to be relevant - must be read as a measurement prescription for the geometry of space . . .(2) Since gravity universally acts on every particle, in a universe with at least two particles, gravity must not beconsidered a force if Newton I is supposed to remain applicable.11
9.1. Laplace’s questions. Laplace 1749†1827Q: “Can gravity be encoded in a curvature of space, such that its effects show if particles under the influence of (no other)force we postulated to more along straight lines in this curved space?”Answer: No!Proof. gravity is a force point of viewmẍα (t) F α (x(t))mẍα (t) mf α (x(t)) {z }Fα α f α 4πGρ (Poisson)ρ mass density of matter(EY : 20150330) You know this, F Gm1 m2 /r2ẍα (t) f α (x(t)) 0Laplace asks: Is this (ẍ(t)) of the formẍα (t) Γαβγ (x(t))ẋβ (t)ẋγ (t) 0Conclusion: One cannot find Γ s such that Newton’s equation takes the form of an autoparallel. 9.2. The full wisdom of Newton I. use also the information from Newton’s first law that particles (no force) moveuniformlyintroduce the appropriate setting to talk about the difference easilyinsight: in spacetimeuniform & straight motionis simply straight motionSo let’s try in spacetime:let x : R R3be a particle’s trajectory in space worldline (history) of the particleX : R R4t 7 (t, x1 (t), x2 (t)
As of March 23, 2015, I nd that the Central Lectures given by Dr. Frederic P. Schuller for the WE Heraeus International Winter School to be, unequivocally, the best, most lucid, and well-constructed lecture series on General Relativity and Gravity. Instead of reinventing the wheel, I wri
1 RELATIVITY I 1 1.1 Special Relativity 2 1.2 The Principle of Relativity 3 The Speed of Light 6 1.3 The Michelson–Morley Experiment 7 Details of the Michelson–Morley Experiment 8 1.4 Postulates of Special Relativity 10 1.5 Consequences of Special Relativity 13 Simultaneity and the Relativity
Sean Carroll, “Spacetime and Geometry” A straightforward and clear introduction to the subject. Bob Wald, “General Relativity” The go-to relativity book for relativists. Steven Weinberg, “Gravitation and Cosmology” The go-to relativity book for particle physicists. Misner, Thorne and Wheeler, “Gravitation”
Theory of Relativity. Einstein's General Theory of Relativity by Asghar Qadir. Einstein'sGeneralTheoryofRelativity ByAsgharQadir . Relativity: An Introduction to the Special Theory (World Scientific 1989) or equivalent, but do not have a sound background of Geometry. It can be used
Introduction Special Relativity General Relativity Curriculum Books The Geometry of Special Relativity Tevi
The theory of relativity is split into two parts: special and general. Albert Einstein came up with the spe-cial theory of relativity in 1905. It deals with objects mov-ing relative to one another, and with the way an observer's experience of space and time depends on how she is mov-ing. The central ideas of special relativity can be formu-
General Relativity and Newtonian Gravitation Theory "530-47773_Ch00_2P.tex" — 1/23/2012 — 17:18 — page ii 1 0 1 . by Robert Geroch (1984) Useful Optics, by Walter T. Welford . This manuscript began life as a set of lecture notes for a two-quarter (twenty-week) course on the foundations of general relativity that I taught at .
Topics in the foundations of general relativity and Newtonian gravitation theory / David Malament. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-226-50245-8 (hardcover : alkaline paper) ISBN-10: 0-226-50245-7 (hardcover : alkaline paper) 1. Relativity (Physics) 2. Gravitatio
Differential Topology in Relativity (Philadelphi3' Society for Industrial and Applied Mathe matics. 1972). More accessible than either is Robert .Geroch, "Space-Time Structure from a Global Viewpoint," in B. K. Sachs, ed., General Relativity and Cosmology (New York Academic Press, 1971). 3. A future end point need not be a point on the curve.
