VECTORS IN CURVED SPACES

6.1. MATHEMATICAL PRELIMINARIES

I will now provide an intuitive description of vectors in curved spaces and their generalization to the infinite-dimensional (also curved) spaces. The important concept of the vector and multivector, or polyvector, derivative will also be explained. The usual functional derivative is just a component description of the vector derivative in an infinite-dimensional space. My aim is to introduce the readers into those very elegant mathematical concepts and give them a feeling about their practical usefulness. To those who seek a more complete mathematical rigour I advise consulting the literature [22]. In the case in which the concepts discussed here are not found in the existing literature the interested reader is invited to undertake the work and to develop the ideas initiated here further in order to put them into a more rigorous mathematical envelope. Such a development is beyond the scope of this book which aims to point out how various pieces of physics and mathematics are starting to merge before our eyes into a beautiful coherent picture.

The relationγ^µ·γ_β =δ^µ_β implies

∂_αγ^µ·γ_β+γ^µ·∂_αγ_β = 0, (6.5) from which it follows that

Γ^µ_αβ =γ^µ·∂_αγ_β. (6.6) Multiplying the latter expression byγ_µ, summing overµand using

γ_µ(γ^µ·a) =a, (6.7)

which holds for any vectora, we obtain

∂_αγ_β = Γ^µ_αβγ_µ. (6.8) In general the connection isnot symmetric. In particular, when it is sym-metric, we have

γ^µ·(∂_αγ_β−∂_βγ_α) = 0 (6.9) and also

∂_αγ_β −∂_βγ_α= 0 (6.10)

in view of the fact that (6.9) holds for anyγ^µ. For a symmetric connection, after using (6.1), (6.2), (6.10) we find

Γ^µ_αβ = ¹₂g^µν(g_να,β+g_νβ,α−g_αβ,ν). (6.11) Performing the second derivative we have

∂_β∂_αγ^µ=−∂_βΓ^µ_ασγ^σ−Γ^µ_ασ∂_βγ^σ =−∂_βΓ^µ_ασγ^σ+ Γ^µ_ασΓ^σ_βργ^ρ (6.12) and

[∂_α, ∂_β]γ^µ=R^µ_ναβγ^ν , (6.13) where

R^µ_ναβ =∂_βΓ^µ_να−∂_αΓ^µ_νβ+ Γ^µ_βρΓ^ρ_αν−Γ^µ_αρΓ^ρ_βν (6.14) isthe curvature tensor. In general the latter tensor does not vanish and we have a curved space.

SOME ILLUSTRATIONS

Derivative of a vector. Letabe an arbitrary position-dependent vector, expanded according to

a=a^µγµ. (6.15)

Taking the derivative with respect to coordinatesx^µ we have

∂_νa=∂_νa^µγ_µ+a^µ∂_νγ_µ. (6.16) Using (6.8) and renaming the indices we obtain

∂_νa=^³∂_νa^µ+ Γ^µ_νρa^ρ´γ_µ, (6.17) or

γ^µ·∂_νa=∂_νa^µ+ Γ^µ_νρa^ρ≡D_νa^µ, (6.18) which is the well known covariant derivative. The latter derivative is the projection of∂_νaonto one of the basis vectors.

Locally inertial frame. At each point of a spaceV_N we can define a set ofN linearly independent vectorsγa,a= 1,2, ..., N, satisfying

γa·γ_b =η_ab, (6.19)

where η_ab is the Minkowski tensor. The set of vector fields γa(x) will be called theinertial or Lorentz (orthonormal) frame field. .

A coordinate basis vector can be expanded in terms of local basis vector

γ^µ=e^µ_aγ^a, (6.20)

where the expansion coefficients e^µ_a form the so called fielbein field (in 4-dimensions “fielbein” becomes “vierbein” or “tetrad”).

e^µ_a=γ^µ·γ_a. (6.21)

Also

γ_a=e^µ_aγ_µ, (6.22)

and analogous relations for the inverse vectorsγ_µ and γ^a satisfying

γ^µ·γ_ν =δ^µ_ν, (6.23)

γ^a·γ_b =δ^a_b, (6.24)

From the latter relations we find

γ^µ·γ^ν = (e^µ_aγ^a)·(e^ν_bγ^b) =e^µ_ae_ν^a=g^µν, (6.25) γ^a·γ^b = (e_µ^aγ^µ)·(e_ν^bγ^ν) =e_µ^ae^µb =η^ab. (6.26)

A vectoracan be expanded either in terms ofγ_µ orγ_a:

a=a^µγ_µ=a^µe_µ^aγ_a=a^aγ_a, a^a=a^µe_µ^a. (6.27) Differentiation gives

∂_µγ^a=ω^a_bµγ^b, (6.28) whereω^a_bµ is the connection for the orthonormal frame field γ^a. Inserting (6.20) into the relation (6.3) we obtain

∂_νγ^µ=∂_ν(e^µ_aγ^a) =∂_νe^µ_aγ^a+e^µ_a∂_νγ^a=−Γ^µ_νσγ^σ, (6.29) which, in view of (6.28), becomes

∂_νe^µa+ Γ^µ_νσe^σa+ω^ab_νe^µ_b = 0. (6.30) Because of (6.25), (6.26) we have

∂νeµa−Γ^σ_νµeσa+ω^a_bνeµb = 0. (6.31) These are the well known relations for differentiation of the fielbein field.

Geodesic equation in V_N. Let p be the momentum vector satisfying the equation of motion

dp

dτ = 0. (6.32)

Expandingp=p^µγ_µ, wherep^µ=mX˙^µ, we have

˙

p^µ+p^µγ˙_µ= ˙p^µγ_µ+p^µ∂_νγ_µX˙^ν. (6.33) Using (6.8) we obtain, after suitably renaming the indices,

( ˙p^µ+ Γ^µ_αβp^αX˙^β)γµ= 0, (6.34) which isthe geodesic equationin component notation. The equation of mo-tion (6.32) says that the vectorp does not change during the motion. This means that vectors p(τ) for all values of the parameter τ remain paral-lel amongst themselves (and, of course, retain the same magnitude square p²). After using the expansion p = p^µγ_µ we find that the change of the componentsp^µ is compensated by the change of basis vectorsγ_µ.

Geometry in a submanifoldV_n. In the previous example we considered a geodesic equation in spacetime V_N. Suppose now that a submanifold

— a surface — Vn, parametrized by ξ^a, is embedded in V_N. Let¹ e^a, a= 1,2, ..., n, be a set of tangent vectors to V_n. They can be expanded in

1Notice that the indexahas now a different meaning from that in the case of a locally inertial frame considered before.

terms of basis vectors ofV_N:

e_a=∂_aX^µγ_µ, (6.35)

where

∂_aX^µ=e_a·γ^µ (6.36)

are derivatives of the embedding functions ofV_n. They satisfy

∂aX^µ∂bXµ= (ea·γ^µ)(eb·γµ) =ea·eb =γab, (6.37) which is the expression for the induced metric ofV_n. Differentiation ofe_a gives

∂_be_a=∂_b∂_aX^µγ_µ+∂_aX^µ∂_b.γ_µ (6.38) Using

∂_bγ_µ=∂_νγ_µ∂_bX^ν (6.39) and the relation (6.8) we obtain from (6.38), after performing the inner product withe^c,

(e^c·∂bea) =∂a∂bX^µ∂^cXµ+ Γ^σ_µν∂aX^µ∂bX^ν∂^cXσ. (6.40) On the other hand, the left hand side of eq. (6.38) involves the connection ofVn:

e^c·∂_be_a= Γ^c_ba, (6.41) and so we see that eq. (6.40) is a relation between the connection ofV_nand V_N. Covariant derivative in the submanifold V_n is defined in terms of Γ^d_ba.

An arbitrary vectorP inV_N can be expanded in terms ofγ_µ:

P =Pµγ^µ. (6.42)

It can be projected onto a tangent vectore_a:

P ·e_a=P_µγ^µ·e_a=P_µ∂_aX^µ≡P_a (6.43) In particular, a vector ofVN can be itself a tangent vector of a subspace V_n. Let p be such a tangent vector. It can be expanded either in terms of γ_µ ore_a:

p=p^µγ^µ=p_ae^a, (6.44) where

p_a=p_µγ^µ·e_a=p_µ∂_aX^µ,

p_µ=p_ae^a·γ_µ=p_a∂^aX_µ. (6.45) Such symmetric relations betweenp_µandp_ahold only for a vector pwhich is tangent to Vn.

Suppose now that p is tangent to a geodesic of V_n. Its derivative with respect to an invariant parameterτ along the geodesic is

d

dτ(p^ae_a) = d

dτ (p^a∂_aX^µγ_µ)

= ( ˙p^a∂_aX^µ+∂_a∂_bX^µp^aξ˙^b+p^a∂_aX^αΓ^µ_αβX˙^β)γ_µ (6.46) where we have used eq. (6.8) and

d

dτ∂_aX^µ=∂_a∂_bX^µξ˙^b.

The above derivative, in general, does not vanish: a vector pof VN that is tangent to a geodesic in a subspaceV_n changes withτ.

Making the inner product of the left and the right side of eq. (6.46) with e^c we obtain

dp

dτ ·e^c = ˙p^c+ Γ^c_abp^aξ˙^b= 0. (6.47) Here Γ^c_ab is given by eq. (6.40). For a vector p tangent to a geodesic of V_n the right hand side of eq. (6.47) vanish.

On the other hand, starting fromp=p^µγµ and using (6.8) the left hand side of eq. (6.47) gives

dp dτ ·e^c=

µdp^µ

dτ + Γ^µ_αβp^αX˙^β

¶

∂^cX_µ= 0. (6.48) Eqs.(6.47), (6.48) explicitly show that in general a geodesic of V_n is not a geodesic ofV_N.

A warning is necessary. We have treated tangent vectorse_ato a subspace Vn as vectors in the embedding space V_N. As such they do not form a complete set of linearly independent vectors inV_N. An arbitrary vector of VN, of course, cannot be expanded in terms of ea; only a tangent vector to V_n can be expanded so. Therefore, the object ^³dp

dτ ·e^c´e_c should be distinguished from the object ^³dp

dτ ·γ^µ´γ_µ = dp/dτ. The vanishing of the former object does not imply the vanishing of the latter object.

DERIVATIVE WITH RESPECT TO A VECTOR

So far we have considered derivatives of position-dependent vectors with respect to (scalar) coordinates. We shall now consider the derivative with respect to a vector. LetF(a) be a polyvector-valued function of a vector valued argumentawhich belongs to ann-dimensional vector spaceAn. For an arbitrary vector e inAn the derivative of F in the direction e is given

by µ

e· ∂

∂a

¶

F(a) = lim

τ→0

F(a+eτ)−F(a)

τ = ∂F(a+eτ)

∂τ

¯¯

¯¯

¯τ=0

. (6.49) Forewe may choose one of the basis vectors. Expandinga=a^νe_ν, we have

µ e_µ· ∂

∂a

¶

F(a)≡ ∂F

∂a^µ = lim

τ→0

F(a^νe_ν +e_µτ)−F(a^νe_ν) τ

= lim

τ→0

F((a^ν +δ^ν_µτ)e_ν)−F(a^νe_ν)

τ . (6.50)

The above derivation holds for an arbitrary function F(a). For instance, forF(a) =a=a^νe_ν eq. (6.50) gives

∂F

∂a^µ = ∂

∂a^µ(a^νe_ν) =e_µ=δ_µ^νe_ν (6.51) For the componentsF ·e^α =a^α it is

∂a^α

∂a^µ =δµα (6.52)

The derivative in the directione_µ, as derived in (6.50), is the partial deriva-tive with respect to the componenta^µ of the vector argumenta. (See Box 6.1 for some other examples.)

In eq. (6.50) we have derived the operator e_µ· ∂

∂a ≡ ∂

∂a^µ. (6.53)

For a running indexµthese are components (or projections) of the operator

∂

∂a =e^µ µ

e_µ· ∂

∂a

¶

=e^µ ∂

∂a^µ (6.54)

which isthe derivative with respect to a vectora.

The above definitions (6.49)–(6.53) hold for any vectoraofAn. Suppose now that all those vectors are defined at a point a of an n-dimensional manifold V_n. They are said to be tangent to a point x in V_n [22]. If we allow the pointx to vary we have thus a vector fielda(x). In components it is

a(x) =a^µ(x)e_µ(x), (6.55) where a^µ(x) are arbitrary functions of x. A point x is parametrized by a set ofncoordinatesx^µ, hencea^µ(x) are functions ofx^µ. In principlea^µ(x) are arbitrary functions ofx^µ. In particular, we may choose

a^µ(x) =x^µ. (6.56)

Box 6.1: Examples of differentiation by a vector 1)Vector valued function:

F =x=x^νe_ν ; ∂F

∂x^µ =e_µ=δ_µ^νe_ν ; ∂F

∂x =e^µ∂F

∂x^µ =e^µe_µ=n.

2)Scalar valued function F =x² =x^νxν ; ∂F

∂x^µ = 2xµ; ∂F

∂x =e^µ2xµ= 2x.

3)Bivector valued function

F =b∧x= (b^αe_α)∧(x^βe_β) =b^αx^βe_α∧e_β,

∂F

∂x^µ = lim

τ→0

b^αeα∧(x^βe_β+eµτ)−b^αeα∧x^βe_β

τ =b^αe_α∧e_µ,

∂F

∂x =e^µ∂F

∂x^µ =b^αe^µ(e_α∧e_µ) =b^αe^µ·(e_α∧e_µ) +b^αe^µ∧e_α∧e_µ,

=b^α(δ_α^µe_µ−δ_µ^µe_α) =b^αe_α(1−n) =b(1−n).

Then

a(x) =x^µe_µ(x). (6.57) Under a passive coordinate transformation the components a^µ(x) change according to

a^0µ(x⁰) = ∂x^0µ

∂x^ν a^ν(x). (6.58)

This has to be accompanied by the corresponding (active) change of basis vectors,

e^0µ(x⁰) = ∂x^ν

∂x^0µe_ν(x), (6.59)

in order for a vector a(x) to remain unchanged. In the case in which the components fieldsa^µ(x) are just coordinates themselves, the transformation (6.58) reads

a^0µ(x⁰) = ∂x^0µ

∂x^ν x^ν. (6.60)

In new coordinates x^0µ the components a^0µ of a vector a(x) = x^µe_µ are, of course, not equal to the new coordinates x^0µ. The reader can check by performing some explicit transformations (e.g., from the Cartesian to spherical coordinates) that an object as defined in (6.56), (6.57) is quite

a legitimate geometrical object and has, indeed, the required properties of a vector field, even in a curved space. The set of points of a curved space can then, at least locally², be considered as a vector field, such that its components in a certain basis are coordinates. In a given space, there are infinitely many fields with such a property, one field for every possible choice of coordinates. As an illustration I provide the examples of two such fields, denotedX and X⁰:

a(x) =x^µeµ(x) =a^0µ(x⁰)e⁰_µ(x⁰) =X,

b(x) =b^µ(x)e_µ(x) =x^0µe⁰_µ(x⁰) =X⁰. (6.61) Returning to the differential operator (6.49) we can consideraas a vector field a(x) and the definition (6.49) is still valid at every point x ofV_n. In particular we can choose

a(x) =x=x^νe_ν. (6.62) Then (6.50) reads

µ eµ· ∂

∂x

¶

F(x)≡ ∂F

∂x^µ = lim

τ→0

F((x^ν+δµντ)eν)−F(x^νeν)

τ . (6.63)

This is the partial derivative of a multivector valued functionF(x) of posi-tionx. The derivative with respect to the polyvectorx is

∂

∂x =e^µ µ

e⁰_µ· ∂

∂x

¶

=e^µ ∂

∂x^µ (6.64)

Although we have denoted the derivative as∂/∂xor∂/∂a, this notation should not be understood as implying that∂/∂acan be defined as the limit of a difference quotient. The partial derivative (6.50) can be so defined, but not the derivative with respect to a vector.

DERIVATIVE WITH RESEPCT TO A POLYVECTOR

The derivative with respect to a vector can be generalized to polyvectors.

Definition (6.49) is then replaced by µ

E∗ ∂

∂A

¶

F(A) = lim

τ→0

F(A+Eτ)−F(A)

τ = ∂F(A+Eτ)

∂τ . (6.65)

2Globally this canot be true in general, since a single coordinate system cannot cover all the space.

Here F(A) is a polyvector-valued function of a polyvector A, and E is an arbitrary polyvector . The star “ * ” denotes the scalar product

A∗B=hABi0 (6.66)

of two polyvectorsA andB, wherehABi0 is the scalar part of the Clifford product AB. Let e_J be a complete set of basis vector of Clifford algebra satisfying³

e_J∗e_K =δ_JK, (6.67)

so that any polyvector can be expanded asA=A^JeJ. ForE in eq. (6.65) we may choose one of the basis vectors. Then

µ

e_K∗ ∂

∂A

¶

F(A)≡ ∂F

∂A^K = lim

τ→0

F(A^Je_J+e_Kτ)−F(A^Je_J)

τ . (6.68)

This is the partial derivative ofF with respect to the multivector compo-nentsA_K. The derivative with respect to a polyvector Ais the sum

∂F

∂A =e^J µ

e_J∗ ∂

∂A

¶

F =e^J ∂F

∂A^J. (6.69)

The polyvectorAcan be a polyvector field A(X) defined over the position polyvector fieldX which is a generalizatin of the position vector field x defined in (6.61). In particular, the field A(X) can be A(X) = X. Then (6.68), (6.69) read

µ

e_K∗ ∂

∂X

¶

F(X)≡ ∂F

∂X^K = lim

τ→0

F(X^Je_J+e_Kτ)−F(X^Je_J)

τ , (6.70)

∂F

∂X =e^J µ

e_J∗ ∂

∂X

¶

F =e^J ∂F

∂X^J (6.71)

which generalizes eqs. (6.63),6.64).

VECTORS IN AN INFINITE-DIMENSIONAL

In document The Landscape of Theoretical Physics: A Global View (Strani 187-196)