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NOTES ON GENERAL RELATIVITY (GR) AND GRAVITY
ERNEST YEUNG
Abstract. 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 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 write these notes to build upon and supplement the video lectures and tutorials already created by them. This includes my corrections, comments, relations to other aspects of theoretical physics, and code implementing calculations in GR. It should be noted that for symbolic computation, I heavily use the SageManifolds v.0.7 package for Sage Math. My goal in 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 con- tribution at Tilt/Open or Patreon, which is a subscription service). I want to keep these notes openly accessible because I want 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.com The stated goal of the WE Heraeus International Winter School on Gravity and Light is to take the student from an introduction to the research frontier (cf. http://www.gravity-and-light.org/lectures). I want to get myself and other students 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 and passionate about physics and want to contribute) equipped with all the tools available to do research, do calculations, to design experiments or collect data. Again, we’re not here to reinvent the wheel. I’m not trying to make a General Relativity appreciation class, but this is a serious attempt towards training to do research.
Part 1. WE Heraeus International Winter School on Gravity and Light 2 Introduction (from EY) 2
- Lecture 1: Topology 2 Topology Tutorial Sheet 3
- 4
- 4
- 4
- 4
- 4
- Lecture 7: Connections 4
- Lecture 8: Parallel Transport & Curvature (International Winter School on Gravity and Light 2015) 8 Tutorial 8 Parallel transport & Curvature 9
- Lecture 9: Newtonian spacetime is curved! 11
- Lecture 10: Metric Manifolds 15
- Symmetry 15
- Integration 19
Date: 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. With your support, we can keep education and research material available online, openly accessible, and free for anyone, anytime. If you like what I’m trying to do for physics education research, please go to my Tilt/Open or Patreon crowdfunding campaign, read the mission statement, share the page, and contribute financially if you can. ernestyalumni.tilt.com https://www.patreon.com/ernestyalumni.
- Lecture 13: Relativistic spacetime 21
- Lecture 14: Matter 25
- Lecture 15: Einstein gravity 28 Tutorial 13 Schwarzschild Spacetime 31
- 32
- 32
- 32
- 32
- 32
- 32
- Lecture 22: Black Holes 32 References 35
Contents
Part 1. WE Heraeus International Winter School on Gravity and Light
Introduction (from EY)
The International Winter School on Gravity and Light held central lectures given by Dr. Frederic P. Schuller. These lectures on General Relativity and Gravity are unequivocally and undeniably, the best and most lucid and well-constructed lecture series on General Relativity and Gravity. The mathematical foundation from topology and differential geometry from which General Relativity arises from is solid, well-selected in rigor. The lectures themselves are well-thought out and clearly explained.
Even more so, the International Winter School provided accompanying Tutorial Sessions for each of the lectures. I had given up hopes in seeing this component of the learning process ever be put online so that anyone and everyone in the world could learn through the Tutorial process as well. I was afraid that nobody would understand how the Tutorial or “Office Hours” session was important for students to digest and comprehend and work out-doing exercises-the material presented in the lectures. This International Winter School gets it and shows how online education has to be done, to do it in an excellent manner, moving forward.
For anyone who is serious about learning General Relativity and Gravity, I would simply point to these video lectures and tutorials.
What I want to do is to build upon the material presented in this International Winter School. Why it’s important to me, and to the students and practicing researchers out there, is that the material presented takes the student from an introduction to the research frontier. That is the stated goal of the International Winter School. I want to dig into and help contribute to the cutting edge in research and this entire program with lectures and tutorials appears to be the most direct and sensible route directly to being able to do research in General Relativity and Gravity. -EY 20150323
- Lecture 1: Topology
1.1. 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 =⇒ U
V ∈ O
(iii) Uα ∈ O, α ∈ A =⇒
α∈A Uα
∈ O
What I won’t do here is retype up the solutions presented in the Tutorial (cf. https://youtu.be/_XkhZQ-hNLs): the presenter did a very good job. If someone wants to type up the solutions and copy and paste it onto this LaTeX file, in the 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 numerical computation because, first, we’re in the 21st century, second, to set the stage for further applications in research. I use Python and Sage Math alot, mostly because they are open-source software (OSS) and fun to use. Also note that the structure 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 each of 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.sage Download topology.sage
Loading topology.sage, after changing into (with the usual Linux terminal commands, cd, ls) by
sage : load ( ‘ ‘ topology. sage ’ ’)
Exercise 2: Topologies on a simple set.
Question Does O 1 :=... constitute a topology... ?.
Solution: Yes, since we check by typing in the following commands in Sage Math:
emptyset in O_ Axiom2check ( O_1 ) # True Axiom3check ( O_1 ) # True
Question What about O 2... ?.
Solution: No since the 3rd. axiom fails, as can be checked by typing in the following commands in Sage Math:
emptyset in O_ Axiom2check ( O_2 ) # True Axiom3check ( O_2 ) # False
- 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 )
∇X : C∞(M ) C∞(M )
∇X : T M^
p (^) ⊗ T ∗M q (^) i.e. ( p q
tensor field
T M p^ ⊗ T ∗M q^ i.e. ( p q
tensor field
7.1. Directional derivatives of tensor fields. manifold with connection is quadruple (M, O, A, ∇)
topology O
atlas A
Consider chart (U, x) ∈ A
Definition 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.
As
T ⊗ 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∞-linear
7.2. New structure on (M, O, A) required to fix ∇. There are (dimM )^3 many Γijk
Γijk : U → R
p 7 →
dxi(∇ (^) ∂x∂
∂xj^
(p)
Now ∇ (^) ∂x∂m (dxi) =?
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 by definition, Γ = 0.
∇X Y = Xj^
∂xj^ (Y i)
∂xi^
∂xi^ = Xj^ ∂Y i ∂xj
∂xi^
and similarly for any (p, q)-tensor field, i.e.
∇X T = Xj^
∂T (^) ji 11 ...j...ipq ∂xj
- ∇X f = Xj^ ∂f ∂xj^ = X · grad(f )
- ∀ (U, x) ∈ A, locally (after working out the first few cases, and doing induction, one can look up the expression for the local form; I found it in Nakahara’s Geometry, Topology and Physics, Eq. 7.26, and it needs to be modified for the convention of order of bottom indices for Γ: ∇ν tλ μ^11 ...λ...μpq = ∂ν tλ μ^11 ...λ...μpq + Γλ κν^1 tκλ μ 1 ...μ^2 ...λq p+ · · · + Γλ κνp tλμ^11 ...λ...μpq− 1 κ− Γκμ 1 ν tλ κμ^1 ...λ 2 ...μpq − · · · − Γκμq ν tλ μ^11 ...λ...μpq− 1 κ Clearly, ∇X is uniquely fixed ∀ p ∈ M by choosing each of the (dimM )^3 many connection coefficient functions Γ.
(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 coefficient function Γ that is symmetric in the bottom indices.
- ∇X f = X(f ) = Xi^ ∂f ∂xi
- (∇X Y )a^ = Xi^ ∂Y a ∂xi^ + Γ
ajkY j (^) Xk
- (∇X ω)a = Xi^ ∂ωa ∂xj^ −^ Γ
iakωiXk
∂xm^ (T^
abc) + ΓaimT (^) bci − ΓibmT (^) ica − ΓjcmT (^) bja
- (∇[m A) (^) n] = (∇mA)n − (∇nA)m = ∂An ∂xm^ − ΓinmAi −
∂Am ∂xn^ − ΓimnAi
= ∂Am ∂xm^ − ∂Am ∂xn
(∇mω)nr = ∂ωnr ∂xm^ − Γinmωir − Γirmωni
Exercise 3. : Connection coefficients
Question.
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) = ∂yi ∂xq
∂^2 xq ∂yj^ ∂yk^ +^
∂yi ∂xq
∂xs ∂yj
∂xp ∂yk^ Γ
q sp(x)
- Lecture 8: Parallel Transport & Curvature (International Winter School on Gravity and Light
8.1. Parallelity of vector fields.
Definition 4. (1) parallely transported along smooth curve γ : R → M
if
(8.1) ∇vγ X = 0
(2) A slightly weaker condition is “parallel”
(∇vγ,γ(λ) X)γ(λ) = μ(λ)Xγ(λ)
8.2. Autoparallely transported curves.
Definition 5. curve γ : R → M is called autoparallely transported if
(8.2) ∇vγ vγ = 0^!
8.3. Autoparallel equation. ∇vγ vγ = 0
in summary:
(8.3) ¨γ (mx)(λ) + (Γm (x))ab(γ(λ)) ˙γa (x)(λ) ˙γb (x)(λ) = 0
8.4. 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 entry
T (ω, f X, Y ) = ω(∇f X Y − ∇Y (f X) − [f X, Y ])
Now
[X, Y ]i^ = Xj^
∂xj^ Y^
i (^) − Y j ∂X i ∂xj For coordinate vectors, [∂i, ∂j ] = 0 ∀ i, j = 0, 1... d.
Thus
Rijab =
∂xa^ Γijb −
∂xb^ Γija + ΓiαaΓαjb − ΓiαbΓαja
Question :Ric(X, Y ) := RiemmambXaY 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 Introduction Second Ed. 2003, Chapter 9 Covariant Differentiation and Curvature, Section 9.3b. The Covariant Differential of a Vector Field. P.S. EY : 20150320 I would like a copy of the Third Edition but I don’t have the funds right now to purchase the third edition: go to my tilt crowdfunding campaign, http://ernestyalumni.tilt.com, and help with your financial support if you can or send me a message on my various channels and ernestyalumni gmail email address if you could help me 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,
∇ (^) ∂x∂k Y =
∂Y i ∂xk^ + Γ
ijkY j
∂xi
Make the ansatz (approche, impostazione) that the connection ∇ acts on Y , the vector field, first:
∇Y (X) =
Xk^ ∂Y i ∂xk^
∂xi^ = Xk^
∇ (^) ∂x∂k Y
)i (^) ∂ ∂xi^ = (∇X Y )i^
∂xi^
= ∇X Y
Now from Lecture 7, Definition for Γ,
dxi
∇ (^) ∂x∂k
∂xj
= Γijk
Make this ansatz (approche, impostazine)
∇
∂xj^
Γijkdxk
∂xi^
∈ Ω^1 (M, T M ) = T ∗M ⊗ T M
where Ω^1 (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 follow- ing:
Γijkdxk^ = Γij ∈ Ω^1 (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^ ) ⊗
∂xi
So the connection is a (smooth) map from T M to the set of all vector-valued 1-forms on M , Ω^1 (M, T M ), and then, after “eating” a vector Y , yields the “covariant derivative”:
∇ : T M → Ω^1 (M, T M ) = T ∗M ⊗ T M ∇ : Y 7 → ∇Y ∇Y : T M → T M ∇Y (X) 7 → ∇Y (X) = ∇X (Y )
Now (^) [ ∂ ∂xi^
∂xj
]
f =
∂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) = Ω^1 (M, E)
∇ea = ωcabf b^ ⊗ ec f b^ ∈ T ∗M (this is the dual basis for T M and, note, this is for the manifold, M ∇fb ea = ωcabec ∈ E
ωac = ωcabf b^ ∈ Ω^1 (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. Correct me if I’m wrong: ernestyalumni at gmail and various social media.
Let σ ∈ Γ(E), σ = σaea
∇σ = (dσc^ + ωabcσaf b) ⊗ ec with
dσc^ = ∂σc ∂xb^ f b
=⇒ ∇X σ =
Xb^ ∂σc ∂xb^
ec = Xb
∂σc ∂xb^
ec
- Lecture 9: Newtonian spacetime is curved!
Axiom 1 (Newton I:). A body on which no force acts moves uniformly along a straight line
Axiom 2 (Newton II:). Deviation of a body’s motion from such uniform straight motion is effected by a force, reduced by a 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 be considered a force if Newton I is supposed to remain applicable.
Yes, choosing Γ^0 ab = 0
Γαβγ = 0 = Gammaα 0 β = Γαβ 0
only: Γα 00 =! −f α
Question: Is this a coordinate-choice artifact?
No, since Rα 0 β 0 = − (^) ∂x∂β f α^ (only non-vanishing components) (tidal force tensor, − the Hessian of the force compo- nent)
Ricci tensor =⇒ R 00 = Rm 0 m 0 = −∂αf α^ = 4πGρ
Poisson: −∂αf α^ = 4πG · ρ
writing: T 00 = 12 s
=⇒ R 00 = 8πGT 00
Einstein in 1912 (^) ((((((hhhhhhRab = 8πGT(hab
Conclusion: Laplace’s idea works in spacetime
Remark Γα 00 = −f α Rαβγδ = 0 α, β, γ, δ = 1, 2 , 3 R 00 = 4πGρ
Q: What about transformation behavior of LHS of
x¨a^ + Γabc X˙b^ X˙c ︸ ︷︷ ︸ (∇vX vX )a ︸ ︷︷ ︸ :=aa^ “acceleration vector”
9.3. The foundations of the geometric formulation of Newton’s axiom. new start
Definition 9. A Newtonian spacetime is a quintuple
(M, O, A, ∇, t)
where (M, O, A) 4-dim. smooth manifold
t : M → R smooth function (i) “There is an absolute space” (dt)p 6 = 0 ∀ p ∈ M
(ii) “absolute time flows uniformly” ∇dt (^) ︸︷︷︸= space of (0, 2)-tensor fields
0 everywhere
∇dt is a (0, 2)-tensor field (iii) add to axioms of Newtonian spacetime ∇ = 0 torsion free
Definition 10. absolute space at time τ
Sτ := {p ∈ M |t(p) = τ } −^ dt−−^6 =0→ M = ∐^ S τ
Definition 11. A vector X ∈ TpM is called
(a) future-directed if dt(X) > 0
(b) spatial if dt(X) = 0
(c) past-directed if dt(X) < 0
picture
Newton I: The worldline of a particle under the influence of no force (gravity isn’t one, anyway) is a future-directed autoparallel i.e.
∇vX vX = 0 dt(vX ) > 0
Newton II:
∇vX vX =
F
m ⇐⇒ m · a = F
where F is a spatial vector field:
dt(F ) = 0
Convention: restrict attention to atlases Astratefied whose charts (U, x) have the property
x^0 : U → R x^1 : U → R .. .
x^3
x^0 = t|U =⇒
0 “absolute time flows uniformly” = ∇dt 0 = ∇ (^) ∂x∂a dx^0 = −Γ^0 ba a = 0, 1 , 2 , 3
Let’s evaluate in a chart (U, x) of a stratified atlas Asheet: Newton II:
∇vX vX = F m in a chart. (X^0 )′′^ + (^) ((((((
Γ^0 cd(Xa)′(Xb)′stratified atlas^ = 0
(Xα)′′^ + Γαγδ Xγ ′ Xδ ′
- Γα 00 X^0 ′ X^0 ′
- 2Γαγ 0 Xγ ′ X^0 ′ = F α m α^ = 1,^2 ,^3
=⇒ (X^0 )′′(λ) = 0 =⇒ X^0 (λ) = aλ + b constants a, b with X^0 (λ) = (x^0 ◦ X)(λ) stratified = (t ◦ X)(λ)
convention parametrize worldline by absolute time
d dλ = a d dt a^2 X¨α^ + a^2 Γαγδ X˙γ^ X˙δ^ + a^2 Γα 00 X˙^0 X˙^0 + 2Γαγ 0 X˙γ^ X˙^0 = F α m =⇒ X¨α^ + Γαγδ X˙γ^ X˙δ^ + Γα 00 X˙^0 X˙^0 + 2Γαγ 0 X˙γ^ X˙^0 ︸ ︷︷ ︸ aα
a^2
F α m
Then for fixed λ ∈ R hXλ : M → M smooth
picture hXλ (S) 6 = S( if X 6 = 0)
11.4. Lie subalgebras of the Lie algebra (Γ(T M ), [·, ·]) of vector fields.
(a) Γ(T M ) = { set of all vector fields } C∞(M )-module = R-vector space =⇒ [X, Y ] ∈ Γ(T M ) [X, Y ]f := X(Y f ) − Y (Xf ) (i) [X, Y ] = −[Y, X] (ii) [λX + Z, Y ] = λ[X, Y ] + [Z, Y ] (iii) [X, [Y, Z]] + [Z, [X, Y ]] + [Y, [Z, X]] = 0 (Γ(T M ), [·, ·]) Lie algebra (b) Let X 1... Xs for s (many) vector fields on M , such that
Tutorial 11 Symmetry. Exercise 1. : True or false?
(a) •
- φ∗^ : T ∗N → T ∗ M i.e. φ∗ν(X) = ν(φ∗X) for smooth φ : M → N , so the pullback of a covector ν ∈ T ∗N maps to a covector in T ∗ M.
(b) (c)
Exercise 2. : Pull-back and push-forward
Question. Let’s check this locally
φ∗(df )(X) = (df )(φ∗X) = (df )(Xi^ ∂y
j ∂xi
∂yj^ ) = Xi^ ∂y
j ∂xi
∂f ∂yj^ where φ∗X = Xi^ ∂y
j ∂xi
∂yj
d(φ∗f )(X) = d(f (φ))(X) = ∂f ∂yj
∂yj ∂xi^ dx
i(X) = Xi ∂y j ∂xi
∂f ∂yj
So φ∗(df ) = d(φ∗f ) ∀ p ∈ M, ∀ X ∈ X(M )
The big idea is that this is a showing of the naturality of the pullback φ∗^ with d, i.e. that this commutes:
Ω^1 (M ) Ω^1 (N )
C∞(M ) C∞(N )
φ∗
d
φ∗
d
Question.
(φ∗)ab := (dya)(φ∗(
∂xb^
Let g ∈ C∞(N )
φ∗
∂xb
g = ∂xb g φ(p) =
∂xb^ gφx−^1 x(p) =
∂xb^ (gyy−^1 φx−^1 )(x) =
∂xb^ (gy−^1 (yφx−^1 (x(p)))) = ∂g ∂y
b∣∣∣ ∣∣ y
∂ya ∂xb
x
= ∂y
a ∂xb
∂g ∂ya
Then
φ∗
∂xb
= ∂y
a ∂xb
∂ya
and so
(φ∗)ab = ∂ya ∂xb
Question.
Exercise 3. :Lie derivative-the pedestrian way
Question. While it is true that ∀ p ∈ S^2 , for x(p) = (θ, ϕ), and (yix−^1 )(θ, ϕ) = (y^1 , y^2 , y^3 ) ∈ R^3 and that, at this point p, (y^1 )^2 /a^2 + (y^2 )^2 /b^2 + (y^3 )^2 /c^3 = 1, this doesn’t imply (EY: 20150321 I think) that, globally, it’s an ellipsoid (yet). In the familiar charts given, spherical chart (U, x) ∈ A and (R^3 , y = idR 3 ) ∈ B it looks like an ellipsoid, but change to another choice of charts, and it could look something very different.
Question.
Equip (R^3 , Ost, B) with the Euclidean metric g, and pullback g.
Note that the pullback of the inclusion from R^3 onto S^2 for the Euclidean metric is the following:
i∗g
∂θi^
∂θj
= g
i∗^ ∂ ∂θi^ , i∗^ ∂ ∂θj
= g
∂xa ∂θi
∂xa^ , ∂x
b ∂θj
∂xb
= gab^ ∂x
a ∂θi
∂xb ∂θj
With gab = δab, the usual Euclidean metric, this becomes the following:
gellipsoid ij = ∂xa ∂θi
∂xa ∂θj
At this point, one should get smart (we are in the 21st century) and use some sort of CAS (Computer Algebra System). I like Sage Math (version 6.4 as of 20150322). I also like the Sage Manifolds package for Sage Math.
I like Sage Math for the following reasons:
- Open source, so its open and freely available to anyone, which fits into my principle of making online education open and freely available to anyone, anytime
- Sage Math structures everything in terms of Category Theory and Categories and Morphisms naturally correspond to Classes and Class methods or functions in Object-Oriented Programming in Python and theyve written it that way
and I like Sage Manifolds for roughly the same reasons, as manifolds are fit into a category theory framework thats written into the Python code. e.g.
sage: S2 = Manifold(2, ’S^2’, r’\mathbb{S}^2’, start_index=1) ; print S sage: print S 2-dimensional manifold ’S^2’ sage: type(S2) <class ’sage.geometry.manifolds.manifold.Manifold_with_category’>
sage: X_3.lie_der(X_2).display() sin(phi) d/dthe + cos(phi)*cos(the)/sin(the) d/dphi
Indeed, one can check on a scalar field feps ∈ C∞(S^2 ):
sage: f_eps = S2.scalar_field({eps: function(’f’, the, phi ) }, name=’f’ ) sage: (X_1( X_2(f_eps)) - X_2(X_1(f_eps) ) ).display() U_{ep} --> R (the, phi) |--> -D1(the, phi) sage: X_2.lie_der(X_1) == -X_ True sage: X_3.lie_der(X_1) == X_ True sage: X_3.lie_der(X_2) == -X_ True
=⇒ [Xi, Xj ] = −ijkXk
So spanR{X 1 , X 2 , X 3 } equipped with [ , ] constitute a Lie subalgebra on S^2 (It’s closed under [ , ]
- Integration
12.1.
12.3. Volume forms.
Definition 16. On a smooth manifold (M, O, A) a (0, dimM )-tensor field Ω is called a volume form if
(a) Ω vanishes nowhere (i.e. Ω 6 = 0 ∀ p ∈ M )
(b) totally antisymmetric Ω(... , (^) ︸︷︷︸X ith
,... , ︸︷︷︸Y
jth
... ) = −Ω(... , ︸︷︷︸Y
ith
,... , ︸︷︷︸X
jth
In a chart:
Ωi 1 ...id = Ω[i 1 ...id]
Example (M, O, A, g) metric manifold
construct volume form Ω from g
In any chart: (U, x)
Ωi 1 ...id :=
det(gij (x))i 1 ...id
where Levi-Civita symbol i 1 ...id is defined as 123 ...d = +
1 ...d = [i 1 ...id]
Proof. (well-defined) Check: What happens under a change of charts
Ω(y)i 1 ...id =
det(g(y)ij )i 1 ...id =
det(gmn(x) ∂x
m ∂yi
∂xn ∂yj^ ) ∂y
m 1 ∂xi^1
... ∂y
md ∂xid [m 1 ...md] =
|detgij (x)|
∣∣det
∂x ∂y
∣∣ det
∂y ∂x
i 1 ...id =
detgij (x)i 1 ...id sgn
det
∂x ∂y
EY : 20150323
Consider the following:
Ω(y)(Y(1)... Y(d)) = Ω(y)i 1 ...id Y (^) (1)i^1... Y (^) (idd) =
=
det(gij (y))i 1 ...id Y (^) (1)i^1... Y (^) (idd) =
det(gmn(x)) ∂x
m ∂yi
∂xn ∂yj^ i 1 ...id^ ∂y
i 1 ∂xm^1
... ∂y
id ∂xmd Xm^1... Xmd^ =
det(gmn(x)) ∂x
m ∂yi
∂xn ∂yj^ det
∂y ∂x
m 1 ...md Xm^1... Xmd^ =
det(gmn(x))
∣∣det
∂x ∂y
∣∣ det
∂y ∂x
m 1 ...md Xm^1... Xmd^ =
=
det(gmn(x))m 1 ...md sgn
det
∂x ∂y
Xm^1... Xmd^ = sgn(det
∂x ∂y
)Ωm 1 ...md (x)Xm^1... Xmd
If det
∂y ∂x
Ω(y)(Y(1)... Y(d)) = Ω(x)(X(1)... X(d))
This works also if Levi-Civita symbol i 1 ...id doesn’t change at all under a change of charts. (around 42:43 https://youtu. be/2XpnbvPy-Zg)
Alright, let’s require, restrict the smooth atlas A to a subatlas (A↑^ still an atlas) A↑^ ⊆ A
s.t. ∀ (U, x), (V, y) have chart transition maps y ◦ x−^1
x ◦ y−^1
s.t. det
∂y ∂x
such A↑^ called an oriented atlas
(M, O, A, g) =⇒ (M, O, A↑, g)
Note: associated bundles.
Note also: det
∂yb ∂xa
= det(∂a(ybx−^1 )) ∂y b ∂xa^ is an endomorphism on vector space^ V^.^ ϕ^ :^ V^ →^ V detϕ independent of choice of basis
g is a (0, 2) tensor field, not endomorphism (not independent of choice of basis)
|det(gij (y))|
