A naive approach to Tensors on Manifolds
Cho, Yong-Hwa
Department of Mathematical Sciences, KAIST
1 / 35 Manifolds?
2 / 35 Manifolds?
2 / 35 Manifolds?
2 / 35 =
Topological Nature
3 / 35 Topological Nature
=
3 / 35 “Flat Space”
Rescaling the plane
Studying Manifolds: How?
4 / 35 “Flat Space”
Rescaling the plane
Studying Manifolds: How?
4 / 35 “Flat Space”
Rescaling the plane
Studying Manifolds: How?
4 / 35 “Flat Space”
Studying Manifolds: How?
Rescaling the plane
4 / 35 Studying Manifolds: How?
“Flat Space”
Rescaling the plane
4 / 35 = 126
“Oriented Line density”
Vectors and Covectors (2D)
· ? = Scalar
5 / 35 ? Scalar = 12 “Oriented Line density”
Vectors and Covectors (2D)
· = = 6
5 / 35 ? Scalar = 6 “Oriented Line density”
Vectors and Covectors (2D)
· = = 12
5 / 35 ? Scalar = 12
Vectors and Covectors (2D)
· = = 6
“Oriented Line density”
5 / 35 Codimension 1 oriented plane density
· = = 3
Dimension 1 oriented plane capacity
Vectors and Covectors (3D)
6 / 35 Codimension 1 oriented plane density
Dimension 1 oriented plane capacity
Vectors and Covectors (3D)
· = = 3
6 / 35 Vectors and Covectors (3D)
Codimension 1 oriented plane density
· = = 3
Dimension 1 oriented plane capacity
6 / 35 Forms
2-form 3-form → & ↑= ↑ & =
2-vector 3-vector Multivectors
Multivectors and Forms
1-form
1-vector
7 / 35 Forms
3-form ↑ & =
3-vector Multivectors
Multivectors and Forms
1-form 2-form → & ↑=
1-vector 2-vector
7 / 35 Forms
Multivectors
Multivectors and Forms
1-form 2-form 3-form → & ↑= ↑ & =
1-vector 2-vector 3-vector
7 / 35 Multivectors and Forms
Forms
1-form 2-form 3-form → & ↑= ↑ & =
1-vector 2-vector 3-vector Multivectors
7 / 35 contravariant parts(vectors) covariant parts(forms)
A tensor T with n covariant and m contravariant parts
b1b2···bm Ta1a2···an
ωa ωab ωabc
v a v ab v abc
Abstract Index Notation
8 / 35 contravariant parts(vectors) covariant parts(forms)
ωa ωab ωabc
v a v ab v abc
Abstract Index Notation
A tensor T with n covariant and m contravariant parts
b1b2···bm Ta1a2···an
8 / 35 covariant parts(forms)
ωa ωab ωabc
v a v ab v abc
Abstract Index Notation
A tensor T with n covariant and m contravariant parts
b1b2···bm contravariant parts(vectors) Ta1a2···an
8 / 35 contravariant parts(vectors)
ωa ωab ωabc
v a v ab v abc
Abstract Index Notation
A tensor T with n covariant and m contravariant parts
b1b2···bm Ta1a2···an covariant parts(forms)
8 / 35 Abstract Index Notation
A tensor T with n covariant and m contravariant parts
b1b2···bm contravariant parts(vectors) Ta1a2···an covariant parts(forms)
ωa ωab ωabc
v a v ab v abc
8 / 35 Tensor product - different indicies, indicies determine the order:
(~v ⊗ ~w =) v aw b = w bv a 6= v bw a (= ~w ⊗~v)
Contraction - same index:
a ~v · ω = v ωa
Contractions between tensors - indicies determine the contraction:
c a c a c c (~v · T)b d = v Tab d 6= v Tba d = (~v · T)b d
Index Notation, Tensor Products and Contractions
9 / 35 Contraction - same index:
a ~v · ω = v ωa
Contractions between tensors - indicies determine the contraction:
c a c a c c (~v · T)b d = v Tab d 6= v Tba d = (~v · T)b d
Index Notation, Tensor Products and Contractions
Tensor product - different indicies, indicies determine the order:
(~v ⊗ ~w =) v aw b = w bv a 6= v bw a (= ~w ⊗~v)
9 / 35 Contractions between tensors - indicies determine the contraction:
c a c a c c (~v · T)b d = v Tab d 6= v Tba d = (~v · T)b d
Index Notation, Tensor Products and Contractions
Tensor product - different indicies, indicies determine the order:
(~v ⊗ ~w =) v aw b = w bv a 6= v bw a (= ~w ⊗~v)
Contraction - same index:
a ~v · ω = v ωa
9 / 35 Index Notation, Tensor Products and Contractions
Tensor product - different indicies, indicies determine the order:
(~v ⊗ ~w =) v aw b = w bv a 6= v bw a (= ~w ⊗~v)
Contraction - same index:
a ~v · ω = v ωa
Contractions between tensors - indicies determine the contraction:
c a c a c c (~v · T)b d = v Tab d 6= v Tba d = (~v · T)b d
9 / 35 = =
a = v ωab = ~v · ω
· = = 6
a ~v · ω = v ωa
· = = 9
~ = 1 ab ~v · ω 2! v ωab
·
~v · ω
Tensor Algebra: Contractions of co and contra parts
10 / 35 = =
a = v ωab = ~v · ω
· = = 9
~ = 1 ab ~v · ω 2! v ωab
·
~v · ω
Tensor Algebra: Contractions of co and contra parts
· = = 6
a ~v · ω = v ωa
10 / 35 = =
a = v ωab = ~v · ω
·
~v · ω
Tensor Algebra: Contractions of co and contra parts
· = = 6
a ~v · ω = v ωa
· = = 9
~ = 1 ab ~v · ω 2! v ωab
10 / 35 = =
a = v ωab = ~v · ω
Tensor Algebra: Contractions of co and contra parts
· = = 6
a ~v · ω = v ωa
· = = 9
~ = 1 ab ~v · ω 2! v ωab
·
~v · ω
10 / 35 =
= ~v · ω
Tensor Algebra: Contractions of co and contra parts
· = = 6
a ~v · ω = v ωa
· = = 9
~ = 1 ab ~v · ω 2! v ωab
· =
a ~v · ω = v ωab
10 / 35 Tensor Algebra: Contractions of co and contra parts
· = = 6
a ~v · ω = v ωa
· = = 9
~ = 1 ab ~v · ω 2! v ωab
· = =
a ~v · ω = v ωab = ~v · ω
10 / 35 ∧ = =
a b a?b b a (v ∧ w)ab v ∧ w = v w − v w =
∧ = ? =
ωa ∧ σb = ωaσb − ωbσa = (ω ∧ σ)ab
Tensor Algebra: Exterior Product
11 / 35 =
ab v a ∧ w b = v aw b − v bw a = (v ∧ w)
=
ωa ∧ σb = ωaσb − ωbσa = (ω ∧ σ)ab
Tensor Algebra: Exterior Product
∧ = ?
∧ = ?
11 / 35 a b a?b b a (v ∧ w)ab v ∧ w = v w − v w =
? =
ωa ∧ σb = ωaσb − ωbσa = (ω ∧ σ)ab
Tensor Algebra: Exterior Product
∧ = =
∧ =
11 / 35 a b a?b b a (v ∧ w)ab v ∧ w = v w − v w = ?
ωa ∧ σb = ωaσb − ωbσa = (ω ∧ σ)ab
Tensor Algebra: Exterior Product
∧ = =
∧ = =
11 / 35 ? ?
Tensor Algebra: Exterior Product
∧ = =
ab v a ∧ w b = v aw b − v bw a = (v ∧ w)
∧ = =
ωa ∧ σb = ωaσb − ωbσa = (ω ∧ σ)ab
11 / 35 ωa ∧ σbc = ωaσbc + ωbσca + ωcσab = (ω ∧ σ)abc
Tensor Algebra: Exterior Product
k-form ∧ l-form = (k + l)-form
∧ = =
12 / 35 Tensor Algebra: Exterior Product
k-form ∧ l-form = (k + l)-form
∧ = =
ωa ∧ σbc = ωaσbc + ωbσca + ωcσab = (ω ∧ σ)abc
12 / 35 3 2 1
Tensor Fields: Scalar and Covector Fields
13 / 35 3 2 1
Tensor Fields: Scalar and Covector Fields
13 / 35 Tensor Fields: Scalar and Covector Fields
3 2 1
13 / 35 Space Compression
∇ Incompatible ∇
Example: Gradient field ∇φ
Space Compression
∇~ ∇~
14 / 35 Space Compression
∇ ∇
Example: Gradient field ∇φ
Space Compression
∇~ Incompatible ∇~
14 / 35 Incompatible
Space Compression
∇~ ∇~
Example: Gradient field ∇φ
Space Compression
∇ ∇
14 / 35 ∇∧
∇∧ =
∇ ∧ ω ∇∧ =
∇ ∧ ω ∇ ∧ ∇∧ = 0
Tensor Calculus: Exterior Differentiation ∇∧ (=d)
15 / 35 =
∇ ∧ ω
=
∇ ∧ ω ∇ ∧ ∇∧ = 0
Tensor Calculus: Exterior Differentiation ∇∧ (=d)
∇∧
∇∧
∇∧
15 / 35 ∇ ∧ ω
∇ ∧ ω ∇ ∧ ∇∧ = 0
Tensor Calculus: Exterior Differentiation ∇∧ (=d)
∇∧
∇∧ =
∇∧ =
15 / 35 =
=
∇ ∧ ω ∇ ∧ ∇∧ = 0
∇∧
Tensor Calculus: Exterior Differentiation ∇∧ (=d)
∇∧
∇ ∧ ω 15 / 35 ∇∧ =
∇ ∧ ω
=
∇ ∧ ω ∇ ∧ ∇∧ = 0
∇∧
Tensor Calculus: Exterior Differentiation ∇∧ (=d) ∇∧
15 / 35
∇∧ ∇ ∧ ω
∇ ∧ ω ∇ ∧ ∇∧ = 0
Tensor Calculus: Exterior Differentiation ∇∧ (=d)
∇∧
∇∧ =
∇∧ =
15 / 35 =
∇ ∧ ω
=
∇ ∧ ∇∧ = 0
∇∧
∇∧
Tensor Calculus: Exterior Differentiation ∇∧ (=d) ∇∧
∇ ∧ ω
15 / 35 =
∇ ∧ ω
=
∇ ∧ ω ∇ ∧ ∇∧ = 0
∇∧
∇∧
Tensor Calculus: Exterior Differentiation ∇∧ (=d) ∇∧
15 / 35 ∇ ∧ ω
∇ ∧ ω
Tensor Calculus: Exterior Differentiation ∇∧ (=d)
∇∧
∇∧ =
∇∧ =
∇ ∧ ∇∧ = 0
15 / 35 Z ω = 2 L
Z ω = 3 S
Tensor Calculus: Integration
Integration of k-form fields over oriented k-dimensional surfaces:
16 / 35 Tensor Calculus: Integration
Integration of k-form fields over oriented k-dimensional surfaces:
Z ω = 2 L
Z ω = 3 S
16 / 35 0+1+1−1+1−1+1 = 2
+1−1+1+0+1−1+1 = 2
Z Z ∇ ∧ k ω = k ω Sk+1 ∂Sk
Stokes’ Theorem
Z ∇ ∧ ω = S Z ω = ∂S
17 / 35 +1+1−1+1−1+1 = 2
+1+0+1−1+1 = 2
Z Z ∇ ∧ k ω = k ω Sk+1 ∂Sk
Stokes’ Theorem
+1
Z −1 ∇ ∧ ω = 0 S Z ω = +1−1 ∂S
17 / 35 +1−1+1−1+1 = 2
+0+1−1+1 = 2
Z Z ∇ ∧ k ω = k ω Sk+1 ∂Sk
Stokes’ Theorem
+1
+1 Z ∇ ∧ ω = 0+1 S Z ω = +1−1+1 ∂S
17 / 35 +1−1+1 = 2
+1−1+1 = 2
Z Z ∇ ∧ k ω = k ω Sk+1 ∂Sk
Stokes’ Theorem
+1 Z ∇ ∧ ω = 0+1+1−1 −1 S Z ω = +1−1+1+0 ∂S
17 / 35 +1 = 2
+1 = 2
Z Z ∇ ∧ k ω = k ω Sk+1 ∂Sk
Stokes’ Theorem
Z ∇ ∧ ω = 0+1+1−1+1−1 S Z ω = +1−1+1+0+1−1 −1 +1 ∂S
−1 +1
17 / 35 Z Z ∇ ∧ k ω = k ω Sk+1 ∂Sk
Stokes’ Theorem
Z ∇ ∧ ω = 0+1+1−1+1−1+1 = 2 S Z ω = +1−1+1+0+1−1+1 = 2 +1 ∂S
+1
17 / 35 Stokes’ Theorem
Z ∇ ∧ ω = 0+1+1−1+1−1+1 = 2 S Z ω = +1−1+1+0+1−1+1 = 2 ∂S Z Z ∇ ∧ k ω = k ω Sk+1 ∂Sk
17 / 35 the volume form n
a1···an and the volume element n~ -1 1 · n~ -1= ( -1)a1···an = 1 n n! a1···an provide the unit density, volume and the space orientation. The volume of the manifold M: Z V (M) = n M
Volume Form and Volume Element
On the n dimensional orientable manifold,
18 / 35 The volume of the manifold M: Z V (M) = n M
Volume Form and Volume Element
On the n dimensional orientable manifold,
the volume form n
a1···an and the volume element n~ -1 1 · n~ -1= ( -1)a1···an = 1 n n! a1···an provide the unit density, volume and the space orientation.
18 / 35 Volume Form and Volume Element
On the n dimensional orientable manifold,
the volume form n
a1···an and the volume element n~ -1 1 · n~ -1= ( -1)a1···an = 1 n n! a1···an provide the unit density, volume and the space orientation. The volume of the manifold M: Z V (M) = n M
18 / 35 k~ · n k-vector n -1 (n − k)-form (n−k) · ~
Duality by Volume
19 / 35 k~ · n k-vector n -1 (n − k)-form (n−k) · ~
Duality by Volume
19 / 35 Duality by Volume
k~ · n k-vector n -1 (n − k)-form (n−k) · ~
19 / 35 (n−k(+1)(kn−−-form1)k)-vector-form
“Divergence” ∇ · ∇ · ∇ · ~ φ ~v ~v ~v
k~ n -1 · n k · ~
φ ω ω ω ∇ ∧ ∇ ∧ ∇ ∧
Divergence of Multivectors
For a k-vector k~v
k k n -1 ∇ · ~v := ∇ ∧ ( ~v · n) · ~
20 / 35 (n−k(+1)k−-form1)-vector
“Divergence” ∇ · ∇ · ∇ · ~ φ ~v ~v ~v
k~ n -1 · n k · ~
φ ω ω ω ∇ ∧ ∇ ∧ ∇ ∧
Divergence of Multivectors
For a k-vector k~v (n−k)-form k k n -1 ∇ · ~v := ∇ ∧ ( ~v · n) · ~
20 / 35 ((kn−−1)k)-vector-form
“Divergence” ∇ · ∇ · ∇ · ~ φ ~v ~v ~v
k~ n -1 · n k · ~
φ ω ω ω ∇ ∧ ∇ ∧ ∇ ∧
Divergence of Multivectors
For a k-vector k~v (n−k+1)-form k k n -1 ∇ · ~v := ∇ ∧ ( ~v · n) · ~
20 / 35 (n−k+1)(n−-formk)-form
“Divergence” ∇ · ∇ · ∇ · ~ φ ~v ~v ~v
k~ n -1 · n k · ~
φ ω ω ω ∇ ∧ ∇ ∧ ∇ ∧
Divergence of Multivectors
For a k-vector k~v (k−1)-vector k k n -1 ∇ · ~v := ∇ ∧ ( ~v · n) · ~
20 / 35 (n−k(+1)(kn−−-form1)k)-vector-form
∇ · ∇ · ∇ · ~ φ ~v ~v ~v
k~ n -1 · n k · ~
φ ω ω ω ∇ ∧ ∇ ∧ ∇ ∧
Divergence of Multivectors
For a multivector
n -1 ∇ · := ∇ ∧ ( · n) · ~ “Divergence”
20 / 35 (n−k(+1)(kn−−-form1)k)-vector-form
“Divergence”
Divergence of Multivectors
For a multivector
n -1 ∇ · := ∇ ∧ ( · n) · ~
∇ · ∇ · ∇ · ~ φ ~v ~v ~v
k~ n -1 · n k · ~
φ ω ω ω ∇ ∧ ∇ ∧ ∇ ∧
20 / 35 The metric tensor gab:
a b ~v · ~w = gabv v
and the inverse metric tensor g ab:
ab ω · σ = g ωaσb
provide inner products(lengths and angles).
Metric tensors raise or lower indicies:
Geometric Nature: Metric Tensor
21 / 35 Metric tensors raise or lower indicies:
Geometric Nature: Metric Tensor
The metric tensor gab:
a b ~v · ~w = gabv v
and the inverse metric tensor g ab:
ab ω · σ = g ωaσb
provide inner products(lengths and angles).
21 / 35 Geometric Nature: Metric Tensor
The metric tensor gab:
a b ~v · ~w = gabv v
and the inverse metric tensor g ab:
ab ω · σ = g ωaσb
provide inner products(lengths and angles).
Metric tensors raise or lower indicies:
e gbeTa cd
21 / 35 Geometric Nature: Metric Tensor
The metric tensor gab:
a b ~v · ~w = gabv v
and the inverse metric tensor g ab:
ab ω · σ = g ωaσb
provide inner products(lengths and angles).
Metric tensors raise or lower indicies:
Tabcd
21 / 35 Geometric Nature: Metric Tensor
The metric tensor gab:
a b ~v · ~w = gabv v
and the inverse metric tensor g ab:
ab ω · σ = g ωaσb
provide inner products(lengths and angles).
Metric tensors raise or lower indicies:
be g Tabcd
21 / 35 Geometric Nature: Metric Tensor
The metric tensor gab:
a b ~v · ~w = gabv v
and the inverse metric tensor g ab:
ab ω · σ = g ωaσb
provide inner products(lengths and angles).
Metric tensors raise or lower indicies:
e Ta cd
21 / 35 The musical isomorphisms for k-forms and k-vectors:
k V a1···ak a1b1 akbk ]: ωa1···ak 7→ ω := g ··· g ωb1···bk
k V a1···ak b1···bk [: v 7→ va1···ak := ga1b1 ··· gakbk v
Musical Isomorphisms ], [
The musical isomorphisms (lowering and raising indicies):
a b [: v 7→ va := gabv a ab ]: ωa 7→ ω := g ωb Musical Isomorphisms ], [
The musical isomorphisms (lowering and raising indicies):
a b [: v 7→ va := gabv a ab ]: ωa 7→ ω := g ωb
The musical isomorphisms for k-forms and k-vectors:
k V a1···ak a1b1 akbk ]: ωa1···ak 7→ ω := g ··· g ωb1···bk
k V a1···ak b1···bk [: v 7→ va1···ak := ga1b1 ··· gakbk v ]
Musical Isomorphisms ], [
If ω has length 1, then
2 ab |ω| = ω · ω = g ωaωb = 1
a ab hence ]ω · ω =( ]ω) ωa = g ωbωa = 1
23 / 35 Musical Isomorphisms ], [
If ω has length 1, then
2 ab |ω| = ω · ω = g ωaωb = 1
a ab hence ]ω · ω =( ]ω) ωa = g ωbωa = 1
]
23 / 35 Musical Isomorphisms ], [
If ω has length2, then
2 ab |ω| = ω · ω = g ωaωb =4
a ab hence ]ω · ω =( ]ω) ωa = g ωbωa =4
]
23 / 35 1 a1b1 akbk (∗ω)ak+1···an = k! g ··· g ωb1···bk a1···akak+1···an 1 a1···ak = k! ω a1···akak+1···ak and similarly for multivectors:
k ∗(k~v) = (V[)(k~v) · n~ -1
Hodge Dual ∗
Duality between k-form and (n − k)-form:
k V ∗(k ω) =( ])(k ω) · n
24 / 35 and similarly for multivectors:
k ∗(k~v) = (V[)(k~v) · n~ -1
Hodge Dual ∗
Duality between k-form and (n − k)-form:
k V ∗(k ω) =( ])(k ω) · n
1 a1b1 akbk (∗ω)ak+1···an = k! g ··· g ωb1···bk a1···akak+1···an 1 a1···ak = k! ω a1···akak+1···ak
24 / 35 and similarly for multivectors:
k ∗(k~v) = (V[)(k~v) · n~ -1
Hodge Dual ∗
Duality between k-form and (n − k)-form:
k V ∗(k ω) =( ])(k ω) · n
1 a1b1 akbk (∗ω)ak+1···an = k! g ··· g ωb1···bk a1···akak+1···an 1 a1···ak = k! ω a1···akak+1···ak
24 / 35 and similarly for multivectors:
k ∗(k~v) = (V[)(k~v) · n~ -1
Hodge Dual ∗
Duality between k-form and (n − k)-form:
k V ∗(k ω) =( ])(k ω) · n
1 a1b1 akbk (∗ω)ak+1···an = k! g ··· g ωb1···bk a1···akak+1···an 1 a1···ak = k! ω a1···akak+1···ak
24 / 35 Hodge Dual ∗
Duality between k-form and (n − k)-form:
k V ∗(k ω) =( ])(k ω) · n
1 a1b1 akbk (∗ω)ak+1···an = k! g ··· g ωb1···bk a1···akak+1···an 1 a1···ak = k! ω a1···akak+1···ak and similarly for multivectors:
k ∗(k~v) = (V[)(k~v) · n~ -1
24 / 35 ~ ~ n -1 ∇φ = ]∇ ∇~∧ φ ×~v = (∇ ∧ [~v) · n~ -1∇ ·~v = ∇ ∧ (~v · n) · ~
~v ∗ ~v 2 ] V]
~ φ ∗ ~v 3 V] k~ n -1 · n 1 k · ~ 3 V ∗ [ φ ω
2 [ V ∗ [ ω ω
Dualities in Euclidean space R3
25 / 35 ~ ~ n -1 ∇φ = ]∇ ∇~∧ φ ×~v = (∇ ∧ [~v) · n~ -1∇ ·~v = ∇ ∧ (~v · n) · ~
Dualities in Euclidean space R3
~v ∗ ~v 2 ] V]
~ φ ∗ ~v 3 V] k~ n -1 · n 1 k · ~ 3 V ∗ [ φ ω
2 [ V ∗ [ ω ω
25 / 35 ~ n -1 ∇~ ×~v = (∇ ∧ [~v) · n~ -1∇ ·~v = ∇ ∧ (~v · n) · ~
Dualities in Euclidean space R3
~v ∗ ~v 2 ] V]
~ φ ∗ ~v 3 V] k~ n -1 · n 1 k · ~ 3 V ∗ [ φ ω
2 [ V ∗ [ ω ω ∇~ φ = ]∇ ∧ φ
25 / 35 ~ ~ n -1 ∇φ = ]∇ ∧ φ ∇ ·~v = ∇ ∧ (~v · n) · ~
Dualities in Euclidean space R3
~v ∗ ~v 2 ] V]
~ φ ∗ ~v 3 V] k~ n -1 · n 1 k · ~ 3 V ∗ [ φ ω
2 [ V ∗ [ ω ω ∇~ ×~v = (∇ ∧ [~v) · n~ -1
25 / 35 ~ ∇φ = ]∇ ∇~∧ φ ×~v = (∇ ∧ [~v) · n~ -1
Dualities in Euclidean space R3
~v ∗ ~v 2 ] V]
~ φ ∗ ~v 3 V] k~ n -1 · n 1 k · ~ 3 V ∗ [ φ ω
2 [ V ∗ [ ω ω ~ n -1 ∇ ·~v = ∇ ∧ (~v · n) · ~
25 / 35 k V n -1 ] k · ~ ∗ l l ~v k k~ V [ · n ~ ~v ~v ~ φ ~v
φ ω
ω ω
ω
Dualities in Dimension Four
26 / 35 Dualities in Dimension Four
k V n -1 ] k · ~ ∗ l l ~v k k~ V [ · n ~ ~v ~v ~ φ ~v
φ ω
ω ω
ω
26 / 35 The directional derivative along the curve x(t) of a tensor field T: ~ a ··· (x˙ · ∇)T =x ˙ ∇aT···
a b b e.g., the geodesic equationx ˙ ∇ax˙ = 0 . ~ 0 ~ ℘(T )xδt − T (x(0)) ≈ “ Parallel transport of ~T ” ~T (x(0)) (~x˙ δt·∇)~T along the curve x from x(δt) to x(0) ~ 0 ℘(T )xδt ~T (x(δt))
x(0) δt x(δt)
Covariant Derivative on Tensor Fields The metric provides the connection operator ∇ Rules: Linear, Leibniz’s rule and
∇agbc = 0 ∇ab1···bn = 0
27 / 35 ~ 0 ~ ℘(T )xδt − T (x(0)) ≈ “ Parallel transport of ~T ” ~T (x(0)) (~x˙ δt·∇)~T along the curve x from x(δt) to x(0) ~ 0 ℘(T )xδt ~T (x(δt))
x(0) δt x(δt)
Covariant Derivative on Tensor Fields The metric provides the connection operator ∇ Rules: Linear, Leibniz’s rule and
∇agbc = 0 ∇ab1···bn = 0 The directional derivative along the curve x(t) of a tensor field T: ~ a ··· (x˙ · ∇)T =x ˙ ∇aT···
a b b e.g., the geodesic equationx ˙ ∇ax˙ = 0 .
27 / 35 Covariant Derivative on Tensor Fields The metric provides the connection operator ∇ Rules: Linear, Leibniz’s rule and
∇agbc = 0 ∇ab1···bn = 0 The directional derivative along the curve x(t) of a tensor field T: ~ a ··· (x˙ · ∇)T =x ˙ ∇aT···
a b b e.g., the geodesic equationx ˙ ∇ax˙ = 0 . ~ 0 ~ ℘(T )xδt − T (x(0)) ≈ “ Parallel transport of ~T ” ~T (x(0)) (~x˙ δt·∇)~T along the curve x from x(δt) to x(0) ~ 0 ℘(T )xδt ~T (x(δt))
x(0) δt x(δt)
27 / 35 “The curvature tensor”
Hence the operator ∇ does not commute:
d Rabc ωd = (∇a∇b − ∇b∇a)ωc
Curvature Tensor
Parallels transports do not commute on curved spaces:
28 / 35 “The curvature tensor”
Hence the operator ∇ does not commute:
d Rabc ωd = (∇a∇b − ∇b∇a)ωc
Curvature Tensor
Parallels transports do not commute on curved spaces:
28 / 35 “The curvature tensor”
Hence the operator ∇ does not commute:
d Rabc ωd = (∇a∇b − ∇b∇a)ωc
Curvature Tensor
Parallels transports do not commute on curved spaces:
28 / 35 “The curvature tensor”
Hence the operator ∇ does not commute:
d Rabc ωd = (∇a∇b − ∇b∇a)ωc
Curvature Tensor
Parallels transports do not commute on curved spaces:
28 / 35 “The curvature tensor”
Curvature Tensor
Parallels transports do not commute on curved spaces:
Hence the operator ∇ does not commute:
d Rabc ωd = (∇a∇b − ∇b∇a)ωc
28 / 35 Curvature Tensor
Parallels transports do not commute on curved spaces:
Hence the operator ∇ does not commute:
d Rabc ωd = (∇a∇b − ∇b∇a)ωc “The curvature tensor”
28 / 35 The electric field E and magnetic flux density B
E = −∇ ∧ φ − A˙ B = ∇ ∧ A
The electric displacement field D and the magnetic field H
D = ∗E + P H = ∗B − M
where P is the polarization and M is the magnetic dipole density.
Maxwell’s Equations
φ: the electric potential, A: the vector potential
29 / 35 The electric displacement field D and the magnetic field H
D = ∗E + P H = ∗B − M
where P is the polarization and M is the magnetic dipole density.
Maxwell’s Equations
φ: the electric potential, A: the vector potential The electric field E and magnetic flux density B
E = −∇ ∧ φ − A˙ B = ∇ ∧ A
29 / 35 Maxwell’s Equations
φ: the electric potential, A: the vector potential The electric field E and magnetic flux density B
E = −∇ ∧ φ − A˙ B = ∇ ∧ A
The electric displacement field D and the magnetic field H
D = ∗E + P H = ∗B − M
where P is the polarization and M is the magnetic dipole density.
29 / 35 - +
- +
+
- +
- + ∇ ∧ D = ρ
Maxwell’s Equations
Maxwell’s equations:
∇ ∧ B = 0 ∇ ∧ E + B˙ = 0 ∇ ∧ D = ρ ∇ ∧ H = j + D˙
where ρ and j are the electric charge and current density, resp.
30 / 35 Maxwell’s Equations
Maxwell’s equations:
∇ ∧ B = 0 ∇ ∧ E + B˙ = 0 ∇ ∧ D = ρ ∇ ∧ H = j + D˙
where ρ and j are the electric charge and current density, resp.
- +
- +
+
- +
- + ∇ ∧ D = ρ
30 / 35 time 3 R
where the spacetime exterior differentiation ∇4∧ satisfies ∂ ∇4∧ = t ∧ + ∇∧ ∂t
Relativistic Maxwell’s Equations
The 4-dimensional spacetime: 1 time and 3 spatial dimensions. The time parametor t induces the covector:
t = ∇4 ∧ t
31 / 35 where the spacetime exterior differentiation ∇4∧ satisfies ∂ ∇4∧ = t ∧ + ∇∧ ∂t
Relativistic Maxwell’s Equations
The 4-dimensional spacetime: 1 time and 3 spatial dimensions. The time parametor t induces the covector:
t = ∇4 ∧ t
time 3 R
31 / 35 Relativistic Maxwell’s Equations
The 4-dimensional spacetime: 1 time and 3 spatial dimensions. The time parametor t induces the covector:
t = ∇4 ∧ t
time 3 R
where the spacetime exterior differentiation ∇4∧ satisfies ∂ ∇4∧ = t ∧ + ∇∧ ∂t
31 / 35 4 k ρ in R
Relativistic Maxwell’s Equations
For a time dependent k-form field k ρ(t),
time ↑
3 k ρ(t) in R
32 / 35 Relativistic Maxwell’s Equations
For a time dependent k-form field k ρ(t),
time ↑
3 4 k ρ(t) in R k ρ in R
32 / 35 Relativistic Maxwell’s equation:
∇4 ∧ F 4 = 0 ∇4 ∧ D4 = J4
Relativistic Maxwell’s Equations
Tensors in spacetime:
A4 = φt − A (Four-potential) J4 = t ∧ j − ρ (Four-current)
F 4 = ∇4 ∧ A4 = t ∧ E − B (EM field) D4 = −t ∧ H − D (EM displacement)
33 / 35 Relativistic Maxwell’s Equations
Tensors in spacetime:
A4 = φt − A (Four-potential) J4 = t ∧ j − ρ (Four-current)
F 4 = ∇4 ∧ A4 = t ∧ E − B (EM field) D4 = −t ∧ H − D (EM displacement)
Relativistic Maxwell’s equation:
∇4 ∧ F 4 = 0 ∇4 ∧ D4 = J4
33 / 35 Relativistic Maxwell’s Equations
The four-current J4 describes world lines of charges in the spacetime.
J4 =( t ∧ j) − ρ
34 / 35 This is the end of the slide. Thank you for your attention.
Particle-Antiparticle annihilation t t
Particle exists in the beginning, and remains eternally
Particle-Antiparticle creation ~x ~x
Relativistic Maxwell’s Equations
The equation ∇4 ∧ D4 = J4 implies that
∇4 ∧ J4 = ∇4 ∧ ∇4 ∧ D4 = 0
so that the charge world lines has no boundaries:
35 / 35 Particle-Antiparticle annihilation
Particle exists in the beginning, and remains eternally
Particle-Antiparticle creation
This is the end of the slide. Thank you for your attention.
Relativistic Maxwell’s Equations
The equation ∇4 ∧ D4 = J4 implies that
∇4 ∧ J4 = ∇4 ∧ ∇4 ∧ D4 = 0
so that the charge world lines has no boundaries:
t t
~x ~x
35 / 35 Particle-Antiparticle annihilation
Particle-Antiparticle creation
This is the end of the slide. Thank you for your attention.
Relativistic Maxwell’s Equations
The equation ∇4 ∧ D4 = J4 implies that
∇4 ∧ J4 = ∇4 ∧ ∇4 ∧ D4 = 0
so that the charge world lines has no boundaries:
t t
Particle exists in the beginning, and remains eternally
~x ~x
35 / 35 This is the end of the slide. Thank you for your attention.
Relativistic Maxwell’s Equations
The equation ∇4 ∧ D4 = J4 implies that
∇4 ∧ J4 = ∇4 ∧ ∇4 ∧ D4 = 0
so that the charge world lines has no boundaries:
Particle-Antiparticle annihilation t t
Particle exists in the beginning, and remains eternally
Particle-Antiparticle creation ~x ~x
35 / 35 Relativistic Maxwell’s Equations
The equation ∇4 ∧ D4 = J4 implies that
∇4 ∧ J4 = ∇4 ∧ ∇4 ∧ D4 = 0
so that the charge world lines has no boundaries:
Particle-Antiparticle annihilation t t
Particle exists in the beginning, and remains eternally
Particle-Antiparticle creation ~x ~x
This is the end of the slide. Thank you for your attention.
35 / 35