2. Differentiable Manifolds And Tensors

2.1 Definition Of A Manifold 2.2 The Sphere As A Manifold2.3 Other Examples Of Manifolds 2.4 Global Considerations2.5 Curves 2.6 Functions On M2.7 Vectors And Vector Fields 2.8 Basis Vectors And Basis Vector Fields 2.9 Fiber Bundles 2.10 Examples Of Fiber Bundles 2.11 A Deeper Look At Fiber Bundles 2.12 Vector Fields And Integral Curves 2.13 Exponentiation Of The Operator d/d2.14 Lie Brackets And Noncoordinate Bases 2.15 When Is A Basis A Coordinate Basis?

2.16 One-forms 2.17 Examples Of One-forms 2.18 The Dirac Delta Function 2.19 The Gradient And The Pictorial Representation Of A One-form 2.20 Basis One-forms And Components Of One-forms 2.21 Index Notation2.22 Tensors And Tensor Fields 2.23 Examples Of Tensors 2.24 Components Of Tensors And The Outer Product 2.25 Contraction2.26 Basis Transformations 2.27 Tensor Operations On Components 2.28 Functions And Scalars 2.29 The Metric Tensor On A Vector Space 2.30 The Metric Tensor Field On A Manifold 2.31 Special Relativity

2.1. Definition of a Manifold

n Set of all n-tuples of real numbers

1 2, , , nx x x

Definition: ( Topological ) manifold

A manifold is a set M in which every point P has an open neighborhood U that is related to some open set fU(U) of n by a continuous 1-1 onto map fU.

A ( topological ) manifold is a continuous space that looks like n locally.

Caution: Length is not yet defined.

: nU Uf U M f U R

1 2, , , nU U UP x P x P x P

xUj(P) are the coordinates of P under fU.

Dim(M) = n.

M is covered by patches of Us. Subscript U will often be omitted

: nf U M f U RLet be 1-1. (U, f ) = Chart

: ng V M g V R

U,V open

, , ,U V f U g V f U g V open

Let S U V

1, , nf S x x 1, , ng S y y

1 : n ng f R R 1 1, , , ,n nx x y y

1, ,j j ny y x x jy x 1, ,j n

( Coordinate transformation )

Charts (U, f ) & (V, g ) are Ck - related if

k j

ki

y

x

exist & are continuous i, j

Atlas = Union of all charts that cover M

M is a Ck manifold if it is covered by Ck charts.

A differentiable manifold is a Ck manifold with k > 1.

Some allowable structures on a differentiable manifold:

• Tensors

• Differential forms

• Lie derivatives

M is smooth if it is C.

M is analytic (C) if all coordinate transformations in it are analytic functions.

2.2. The Sphere As A Manifold

2 2 22 1 2 3 3 1 2 3, ,S P x x x R x x x const

Every P has a neighborhood that maps 1-1 onto an open disc in 2.

( Lengths & angles not preserved )

Spherical coordinates: 1 2,x x

2 2:f S R ,P f P

1 2 1 2, 0 , 0 2Range f x x x x

Breakdowns of f :

• North & south poles in S2 mapped to line (0, x2) & (, x2) in 2, resp.

• Points (x1, 0) & (x1, 2) correspond to same point in S2.

1 2 1 2, 0 , 0 2Range f x x x x

Both poles & semi-circle joining them ( = 0 ) not covered

2nd map needed

One choice of 2nd map:

Another spherical coordinates with its = 0 semi-circle given by { (/2 , ) | {/2 , 3/2 } } in terms of coordinates of 1st system.

Stereographic map (fails at North pole only)

2P Q R

North pole mapped to all of infinity

The 2-D annulus bounded by 2 concentric circles in 2 can be covered by a single ( not differentiable everywhere ) coordinate patch

These conclusions apply to any surface that is topologically equivalent to S2.

2.3. Other Examples Of Manifolds

Loosely, any set M that can be parameterized continuously by n parameters is an n-D manifold

Examples

(i) Set of all rotations R(,,) in E3. ( Lie group SO(3) )

(ii) Set of all boost Lorentz transformations (3-D).

(iii) Phase space of N particles ( 6N-D).

(iv) Solution space of any ( algebraic or differential ) equation

(v) Any n-D vector space is isomorphic to Rn.

(vi) A Lie group is a C-manifold that is also a group.

Rn is a Lie group wrt addition.

2.4. Global Considerations

Any n-D manifolds of the same differentiability class are locally indistinguishable.

Let f: M N

If f & f –1 are both 1–1 & C, then f is a diffeomorphism of M onto N.

M & N are diffeomorphic.

Example of diffeomorphic manifolds:

• Smooth crayon & sphere

• Tea cup & torus

2.5. Curves

:C I M R

1, 1, ,iP x C i n

Parametrized curve with parameter

2.6. Functions on M

:f M R

1, , nP f P f x x

f is differentiable if f(x1,…,xn) C1

Reminder: jx P C

Any sufficiently differentiable set of equations j j iy y x

that is locally invertible ( with finite Jacobian ) is an acceptable coordinate transformation

, 1, ,i j n

2.7. Vectors and Vector Fields Let i ix x 1, ,i n be a curve C through P on M.

1, ,i nf x f x x be a differentiable function on M.

1 , , ng f x x is a differentiable function on C. if x

1

in

ii

d g d x f

d d x

Chain rule: 1

in

ii

d d x

d d x

id x

d

d xi components of vector tangent to C

{ d xi }= infinitesimal displacement along C

d

d = vector tangent to C with components wrt basis ix

Tangent (vector) space TP(M) at P :

d d da b

d d d

1

i in

ii

d x d xa b

d d x

Finer points:

• Each d/d denotes an equivalent class of diistinct curves having the same tangent at P.

• Two vectors d/d & d/d may denote the same tangent to the same curve under different parametrization.

• Only vectors at the same point on M can be added to produce another vector. ( Vectors at different points belong to different vector spaces )

Advantages of defining vector as d/d :

• No finite distance involved works on manifolds without metric.

• Coordinate free.

• Conforms with the geometric notion that a tangent is generated by some infinitesimal displacement along the curve.

2.8. Basis Vectors And Basis Vector Fields

Let M be an n-D manifold.

• P M, TP(M) is an n-D vector space.

• Any collection of n linearly independent vectors in TP(M) is a basis for TP(M) .

• A coordinate system { x i } in a neighborhood U of P defines a coordinate basis { / x i } at all points in U.

Let ie be an arbitrary (non-coordinate) basis for TP(M) .

ii

i

V Vx

jj

j

V e PV T M

For a vector field, V i & V j are functions on M.

The vector field is differentiable if these functions are.

{ / x i } is a basis it is a linearly independent set.

This is guaranteed if { x i } are good coordinates at P.

Proof:

Let { y i } be another set of good coordinates at P.

1, ,j j ny y x x 1, ,j n is invertible

Inverse function theorem det det 0i

j

yJ

x

i.e., vectors with components given by the columns of J are linearly independent .

The j th column1 2 3

, , ,T

j j j

y y y

x x x

represents the vector

1 2

1 2

n

j j j n

y y y

x y x y x y

jx

QED

2.9.a.Bundles

Let E & B be 2 topological spaces with a continuous surjective mapping

: E B

Then ( E, B, ) is a bundle with base B.

E.g. Cartesian bundle

Ref: Choquet, § III.2

1 2 1 1, ,B B B 1 1 2 1,x x x with

Bundle is a generalization of the topological products. E.g.,

A cylinder can be described as the product of a circle S1 with a line segment I.

A Mobius strip can only be described as a bundle.

If the topological spaces –1(x) are homeomorphic to a space F xB,

then –1(x) is called a fibre Fx at x, and F is a typical fibre.

Cartesian product: , ,X Y x y x X y Y

A fibre bundle ( E, B, , G ) is a bundle ( E, B, ) with a typical fibre F,

a structural group G of homeomorphism of F onto itself,

and a covering B by a family of open sets { Uj ; jJ } such that

a) The bundle is locally trivial ( homeomorphic to a product bundle )

1:j j jU U F ,j jp p p p

2.9. Fibre Bundles

c) The induced mappings

:j k j kg U U G 1

, ,j k j x k xx g x

are continuous

These transition functions satisfy

j k k i j ig x g x g x

b) Let j kx U U then1

, , :k x j x F F

is an element of G

A vector bundle is a fibre bundle where the typicle fibre F is a vector space & the structural group G is the linear group.

The tangent bundle pp M

T M T M

is a vector bundle with F = n.

In a coordinate patch (U,x) of M, the natural basis of Tp(U) is { / xi}.

The natural coordinates of T(U) are ( x1, …, xn, / x1, … , / xn ) = ( x, v ).

A vector field is a cross section of a vector bundle.

nF R , x v x

2.10. Examples of Fiber Bundles

• Tangent bundle T(M).• Tensor field on manifold.• Particle with internal, e.g., isospin, state.• Galilean spacetime: B = time, F = 3.

– Time is a base since every point in space can be assigned the same time in Newtonian mechanics.

– Relativistic spacetime is a manifold, not a bundle.

• Frame bundle: derived from T(M) by replacing Tp(M) with the set of all of its bases , which is homomorphic to some linear group. ( see Aldrovandi, §6.5 )

• Principal fibre bundle: F G. E.g. frame bundle.

2.11. A Deeper Look At Fiber Bundles

A fibre bundle is locally trivial but not necessarily globally so.

→ generalizes product spaces

Example T(S2) :

Schutz’s discussion is faulty.

It is true that S2 has no continuous cross-section frame bundle of S2 is not trivial.

For proof: see Choquet, p.193.

However, the (Brouwer) fixed point theorem states that every homeomorphism from a closed n-ball onto itself has at least one fixed point.

It says nothing about n-spheres Sn.

Example: see Choquet, p.126.

Mobius band = Fibre bundle ( M, S1, π, G ) with typical fibre F = I .

The base S1 is covered by 2 open sets.

G = { e, σ} C2 .

2.12. Vector Fields And Integral Curves

Vector field: A rule to select V(P) from TP P M.

Integral curve of a vector field: A curve C() with tangent equal to V.

i

i jd xv x

d i i jV P v xwhere

Vector field ~ set of 1st order ODEs.

Solution exists & is unique in a region where {vi(x)} is Lipschitzian (Choquet, p.95)

Uniqueness of solution integral curves non-crossing

( except where V i = 0 i )

Congruence: the set of integral curves that fills M.

f : X → Y , where X, Y are Banach spaces, is Lipschitzian in U X if k > 0 s.t.

f x f y k x y ,x y U

W x yy x

y xV x y

r y r x

Integral curves: 2 2r x y

1 xV y

r

2 yV x

r

2 2

, 0,0x y

yV x xy Vr r

1 2, , ,, ,V x y V x y V xx

yy

y xr r

2 22 2

2 2

x yy x

r r 2 1r

V is not Lipschitzian in any neighborhood of (0,0) since 2 1r k r

, 0,0y rx

as r → 0

V is not Lipschitzian if any partial of V j does not exist.

2.13. Exponentiation of the Operator d/d

Let M be C.

If the integral curve xi() of Y = d/d is analytic, then

0

00 !

n n ii

nn

d xx

n d

00 !

n ni

nn

dx

n d

0

dide x

0

Y ie x

2.14. Lie Brackets and Noncoordinate Bases

Natural vector field basis ix

, 0i jx x

with ,i j

Let dV

d

dW

d& then

, ,d d

V Wd d

d d d d

d d d d

i j j ii j j i

i j

V W W Vx x x x

i j j ii j j i

V W W Vx x x x

(Einstein's summation notation)

j ii j j i

i j i j j i j i

W VV W W V

x x x x x x x x

j ii j

i j j i

W VV W

x x x x

i i

j jj j i

W VV W

x x x

, 0V W ,V W are noncoordinate basis vectors if

Lie bracket

involves Tp's of neighboring points

Example of a noncoordinate basis (Exercise 2.1):

cos sin

sin cos

r x y

x y

x y

x y

Coordinate grid:

xi is constant on the integral curves xj of ∂/∂xj.

Integral curves of vector fields

dV

d

dW

d&

λ need not be constant on integral curves of d/dμ

expi i

P

dx R x

d

expi i

R

dx A x

d

exp exp i

P

d dx

d d

exp expi i

P

d dx B x

d d

exp , expi i i

P

d dx B x A x

d d

2 2

2 3 2 32 2

1 11 ,1

2 2i

P

d d d dO O x

d d d d

2 3, i

P

d dx O

d d

2 3, i

PV W x O

Lie algebra: (Additional) closure under Lie bracket.

Vector fields on U M is a linear space closed under linear combinations of constant coefficients.

Lie algebra of vector fields on U M : Closed under both.

Invariances of M → Lie group (Chap 3)

Ex 2.3: Jacobi Identity

2.15. When Is A Basis A Coordinate Basis?

, 0i je e

i i

de

d

,i j

Proof for : (2-D case only)

Let be a basis for vector fields in an n-D region U M.

Then { λi } is a coordinate (holonomic) basis for U

Let { xi } be coordinates for U.

Let

, exp expi i

P

d dx x

d d

dA

d

dB

d , 0A B

Proof for : , 0i j i j j ix x x x x x

exp exp i

P

d dx

d d

Task is to show that (α,β) are good coordinates and

( /α, /β) are basis vectors.

, exp expi i

P

d dx x

d d

exp expi i

P

x d d d x

d d d

0

exp!

n n

nn

d d

d n d

1

1 1 !

n n

nn

d

n d

1

10 !

n n

nn

d

n d

expd d

d d

exp expi i

P

x d d d x

d d d

,ix

d

d

,ix

d

d

, 0

so that ( /α, /β) are basis vectors if (α,β) are good coordinates

, exp expi i

P

d dx x

d d

defines a map (α,β) → ( x1, x2 )

If the map is invertible, then (α,β) are good coordinates.

By the implicit function theorem, this inverse exists

1 2

1 2e 0d t

x x

x x

This is true since vectors A & B are linearly independent. QED

General case:

1, , expi n j ij

Px Y x j j

Y

1-Forms

Let T*x(M) be the dual to the tangent space Tx(M) at xM.

Elements of T*x(M) are called cotangent vectors, covariant vectors, covectors,

differential forms, or 1-forms.

In contrast, Elements of Tx(M) are called tangent vectors, contravariant vectors, or simply vectors.

Let X, Y be linear space over . A mapping f : X → Y is called linear if

f ax bu a f x b f u , & ,a b x u X K

The set L(X,Y) of all linear mappings from X to Y is a linear space over with

f g x f x g x

a f x a f x

( vector addition )

( scalar multiplication )

The set L(X, ) is called the dual X* of X.

V V

, ,PL T M K

**X X

a b V a V b V , & , Pa b V W T M K

one can define ,V

( contraction , not to be confused with an inner product )

Thus, V K * &P PT M V T M

i.e., a 1-form is a linear function that takes a vector to a number in .

aV b W a V b W

2.17. Examples of 1-Forms

Gradient of a function f is a 1-form denoted by d f

Matrix algebra:

Column matrix ~ vector Row matrix ~ covector

Matrix multiplication ~ contraction

Hilbert space in quantum mechanics:

Ket | ψ ~ vector Bra ψ | ~ covector

ψ | φ ~ contraction / sesquilinear product

There is no natural way to associate vectors with covectors.

Doing so requires the introduction of a metric or inner product.

2.18. The Dirac Delta Function

Let C[1,1] be the set of all C real functions on interval [1,1].

( C[1,1], + ) is a group.

( C[1,1], + ; ) is an -D vector space.

The dual 1-forms are called distributions.

E.g., the Dirac delta function δ(x-x0) is a distribution.

By definition, δ(x-x0) maps a function f(x) to a real number f(x0) :

,af f f a : 1,1a C R

by

We shall avoid the abiguous notation

a f

1,1a 1

1

dx x a f x

By introducing the inner product 1

1

f g dx f x g x

we can associate with vector g a unique 1-form G by the condition

,G f g f 1,1f C 1

1

dx f x g x

g f

1

1

,a f dx x a f x f a

Since

one may say δ(xa) is a function in C[1,1] that gives rise to the 1-form δa .

However, calling δ(xa) is a function is not mathematically correct.

The “delta function” is actually a Dirac measure (distribution of order 0).

It can be shown that the Dirac measure cannot be associated with a locally integrable function. [see Choquet, VI,B.]

2.19. The Gradient & the Pictorial Representation of a 1-Form

A 1-form field is a cross section of the cotangent bundle T*(M).

The gradient of a function f : M → is the 1-form *Pd f T M

d d fd f

d d

P

dT M

d

satisfying

Let h(x) = height at x. Then

h d h x ii

d hx

x

i

i

hx

x

V = number of ω planes crossed by V

→ 1-form is a set of parallel planes

Steepest descend & gradient vector can only be defined in the presence of a metric.

exterior deis rivthe atived

2.20. Basis 1-Forms & Components of 1-Forms

Given a basis

PV T M iiV V e

*PT M

ie for TP(M), there is a natural dual basis ie for T*P(M)

s.t. ,i i i ij j j je e e e e e

Since

i i jje V e V ewe have j i

jV e e j ijV iV i

iV e e V

Let then

iiV V e i

iV e iie V e i

i e V

where i ie ie V is arbitrary → ii e i

ie e

ie is indeed a basis for T*P(M)

ii V

Coordinate basis:i

j ji x

xd

xx

i

j

2.21. Index Notation

Coordinate bases:

V V

ii

V Vx

ix

→ vector :

Dual bases: ii dx idx → 1-

form :

Contraction: ii V

Einstein notation: implied summation if a pair of upper & lower indices are denoted by the same letter.

2.22. Tensors & Tensor Fields

A tensor F of type (NM) at P is a multi-linear function that

takes N 1-forms & M vectors to a number in , i.e.,

* *

....... ................ .........

: P PP PM factorsN factors

T M T MT M T M

F K

Multi-linear means F is linear in each of its arguments.

For a (21) tensor, we have , ;V F K

;V T K

Linearity:

, ; , ; , ;a b V a V b V F F F etc

A vector is a (10) tensor, a 1-form is a (0

1) tensor & a function is a (00) tensor.

→ ; _T is a 1-form.

_ , _ ; _F F and

Operators or transformations on a linear space are (11) tensors.

2.2.3 Examples of Tensors

Matrix algebra:

Column vector is a (10) tensor, row vector is a (0

1) tensor, & matrix is a (11) tensor.

Function space C[-1,1]:

A differential operator maps linearly a vector (function) into another vector.

→ it is a (11) tensor.

Elasticity:

Stress vector = resultant force per unit area across a surface

Let the stress vector over cube face S(k) with normal ek be F(k) , then

τ takes a 1-form into another vector → it is a (20) tensor called the stress tensor.

k i ii

k F e or F e

The normal to a plane is equivalent to a 1-form ( a set of parallel planes ) .

2.24. Components of Tensors & the Outer Product

Outer ( direct, or tensor ) product :

(NM) tensor (P

Q) tensor = (N+PM+Q) tensor

1 1

1 1, ; , ,N N

M M

i i iij j j jS e e e eS

Example: V W is a (20) tensor with ,V W p q V p W q

Caution: V W W V since ,W V p q W p V q

Reminder: * *, P Pp q T M T M = Cartesian product

The components of a (NM) tensor S are given by

so that 1 1

1 1

N M

M N

i i j jj j i iS e e e e S

2.25. Contraction

The summation of a pair of upper & lower tensor indices is called contraction.

1. Contraction of a (NM) with a (P

Q) tensor gives a (N+P1M+Q1) tensor.

2. Contraction within a (NM) tensor gives a (N1

M1) tensor.

Contraction is basis independent.

Proof for case A is rank (20) & B rank (0

2) :

Let C be the (11) tensor with components i i k

j k jC A B

→ ; i jj iV C V C i k j

k j iA B V , ,kke e VA B

, ,kke e VA B , _ ,VA B

2.26. Basis Transformations

Tensor analysis: A tensor is an object that transforms like a tensor.

Consider

' 'j

i j ie e

'i i ie e e by

''

ji j ie e

(Λ nonsingular )''

j i ji k k ' '

' 'j k ki j i with

Let i ij je e then ' '

i i kj k je e e e '

i kk j '

ij

' '' '

k i k ii j i je e Hence '

'k

j

' 'k k iie e '

'k k i

ie e

' 'j

i j ie e

' 'i iV e V

' 'k k iie e

' 'i ie

'i kk e V 'i k

k V ( contravariant to ej )

'j

j ie '

jj i ( covariant to ej )

''

j jj jV e V e V are basis independent'

'i i

i ie e

Coordinate transform: 'i ix y

Coordinate basis: i ie

x

' 'i i

ey

Dual basis: i ie dx ' 'i ie dy

' 'i ie

y

'

j

i j

x

y x

' '

jji i

x

y

'

jj ie

''

ii

j j

y

x

2

' , ' ', '' '

jj ji j j ij i

x

y y

( not true for general bases )

2.27. Tensor Operations On Components

Operations on components that result in another tensor are called tensor operations.

They are necessarily basis independent.

Examples:

1. Aij + B i

j = C ij → A + B = C

2. a Aij → a A

3. Aij B k

l = C i kj l → AB = C

4. C i kk l = D i

l ( contraction )

Equations containing only tensor operations on tensors are called tensor equations.

They are necessarily basis independent.

2.28. Functions & Scalars

Scalar = (00) tensor at a point on M. E.g., V j ωj |P.

Scalar function on M = (00) tensor field on M. E.g., V j (x) ωj (x)

V j (x) is a function on M but not a scalar function.

Note: It is possible to have a scalar function f(x) s.t. f(x) = V j (x) x.

Can’t tell whether a function is scalar by looking at its values alone.

2.29. The Metric Tensor on a Vector Space

Given a real vector space X, one can define a bilinear inner product

TT OD

: X X R ,V U V Uby

One can associate a (02) metric tensor g with by ,V U V U g

g is symmetric since is :

,i j i jg e egj igi je e j ie e

' ' ' 'k l

i j k l i jg g ' 'k l

i k l jg

Matrix notation: Tg g

Since Λ is invertible, we can set Λ = O D, where O is orthogonal & D is diagonal.

T TD O 1D O

1g DO g OD

1g DO g OD

Since g is real-symmetric, it can be digonalized with an orthogonal transform:

dg D g D O can be chosen s.t. 2d i idiag g d

Setting1

i

d i

dg

iD diag d

g is diagonal with elements 1

Canonical form of g is diag( 1, …, 1, +1, …, +1 ).

The corresponding basis is orthonormal, i.e., i j i je e

Signature of g = t = Tr g = (number of +1) (number of 1)

Metric is positive-definite if t = n ( all +1’s ).

Metric is negative-definite if t = n ( all 1’s ).

Definite metrics are called Euclidean.

Minkowski metric is indefinite.

g = I wrt the Cartesian basis for a Euclidean space.

Transformations between different Cartesian bases are orthogonal since

TI I T I O n

Canonical form of the Minkowski metric is 1, 1, 1, 1diag

Corresponding bases are called Lorentz bases.

Transformations between different Lorentz bases satisfy T

They are called Lorentz transformations. 4 3,1L O

, _V VgA metric tensor associates a unique 1-form to each vector via

iiV V e where i iV e V , iV eg j

j iV g ji jg V ,j

j iV e eg

j k ki j ig g → k i k i j

i i jg V g g V i jj V iV

ji i j

i i jj

V g V

V g V

i.e., g i j

raises & gi j lowers an index.

j i m j mik kg V VE.g

.In a metric space, tensors of ranks (N

M) , (N+1M1), (N+2

M2) , …., are all equivalent.

They’re simply called tensors of order N+M.

In a Euclidean with a orthonormal basis, the components of a vector & its dual have the same numerical value.

→ In calculations, there’s no need to distiguish them.

2.30. The Metric Tensor Field on a Manifold

A metric tensor field assigns continuously a metric tensor g to TP(M) p M.

As already stated, g must be (02), symmetric, & invertible.

A metric is a high level structure that allows the definition of distance, curvature, metric connection, parallel transport, …. (see Chap 6)

Reminder: Lie derivatives & differential forms do not require a metric.

Continuity of g(p) → canonical form of g must be a constant p M.

If one is free to choose the basis for any TP(M), g can be made canonical everywhere.

However, these bases may not all be coordinate ones (see Ex 2.14).

One exception is n with Euclidean metric & Cartesian coordinates (c.f. Ex 2.15-6).

Arc-length Δl of a curve with tangent d

Vd

is given by

2 xl x V V 2V V 2,V V g

If g is positive definite, then ,l V V g

If g is indefinite, then case Δl2 > 0 is called space-like & Δl2 < 0, time-like.

,l V V g then gives the proper distance / proper time.

Case Δl2 = 0 is called light-like.

Note that 0l 0x

2.31. Special Relativity

Minkowski spacetime = 4 with metric of signature +2.

Lorentz frames: coordinates ( x0, x1, x2, x3 ) = ( ct, x, y, z ).

Event intervals: 2 2 2 22 0 1 2 3s x x x x

are invariant under Lorentz transformations.

x x

1,1,1,1diag Metric tensor is in canonical form everywhere:

Δs2 defines a pseudo-norm for the vector 0 1 2 3, , ,x x x x x

Inner product is defined as:

V W V W

0 1 2 3, , ,V V V V V

V W

0 1 2 30 1 2 3, , , , , ,V V V V V V V V V

V W

0 1 2 3, , ,

f f f fd

x xf

x x

0 1 2 3

, , ,f f f f

x xf

x xd