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Gravimetry, Relativity,and the Global Navigation Satellite Systems

— Second Lesson: Introduction to Differential Geometry —

Albert Tarantola

March 9, 2005

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antisymmetric Ti jk = - Tjik = - Tik j .

1.2 Connection

Consider the simple situation where some (arbitrary) coordinates x ≡ {xi} havebeen defined over the manifold. At a given point x0 consider the coordinatelines passing through x0 . If x is a point on any of the coordinate lines, let usdenote as γ(x) the coordinate line segment going from x0 to x . The naturalbasis (of the local tangent space) associated to the given coordinates consists ofthe n vectors {e1(x0), . . . , en(x0)} that can formally be denoted as ei(x0) =∂γ∂xi (x0) , or, dropping the index 0 ,

ei(x) =∂γ∂xi (x) . (1)

So, there is a natural basis at every point of the manifold. As it is assumedthat there exists a parallel transport on the manifold, the basis {ei(x)} can betransported from a point xi to a point xi + δxi to give a new basis, that we candenote {ei( x + δx ‖ x )} (and that, in general, is different from the local basis{ei(x + δx)} at point x + δx ). The connection is defined as the set of coefficientsΓ k

i j (that are not, in general, the components of a tensor) appearing in the de-velopment

e j( x + δx ‖ x ) = e j(x) + Γ ki j(x) ek(x) δxi + . . . . (2)

For this first order expression, we don’t need to be specific about the path fol-lowed for the parallel transport. For higher order expressions, the path followed

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Varadarajan, V.S., 1984, Lie Groups, Lie Albegras, and Their Representations,Springer-Verlag.

Weinberg, S., 1972, Gravitation and Cosmology: Principles and Applications ofthe General Theory of Relativity, John Wiley & Sons.

A Operations on a Manifold

A.1 Connection

The notion of connection has been introduced in section 1.2 in the main text.With the connection available, one may then introduce the notion of covariantderivative of a vector field9, to obtain

∇iwj = ∂iwj + Γ jis ws . (52)

This is far from being an acceptable introduction to the covariant derivative, butthis equation unambiguously fixes the notations. It follows from this expression,using the definition of dual basis, 〈 , 〉 = δi

j , that the covariant derivative of aform is given by the expression

∇i f j = ∂i f j − Γ si j fs . (53)

9Using lousy notations, equation (2) can be written ∂i e j = Γ ki j ek . When considering a vector

field w(x) , then, formally, ∂iw = ∂i (wj e j) = (∂i wj) e j + wj (∂i e j) = (∂i wj) e j + wj Γ ki j ek , i.e.,

∂iw = (∇i wk) ek where ∇i wk = ∂i wk + Γ ki j wj .

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Property 1 A line xi = xi(λ) is autoparallel if at every point alongthe line,

d2xi

dλ2 + γijk

dx j

dλ

dxk

dλ= 0 , (3)

where γijk is the symmetric part of the connection,

γijk = 1

2 (Γ ijk + Γ i

k j) . (4)

If there exists a parameter λ with respect to which a curve isautoparallel, then any other parameter µ = α λ + β (where α

and β are two constants) satisfies also the condition (3). Anysuch parameter associated to an autoparallel curve is called anaffine parameter.

1.4 Vector Tangent to an Autoparallel Line

Let be xi = xi(λ) the equation of an autoparallel line with affineparameter λ . The affine tangent vector v (associated to the au-toparallel line and to the affine parameter λ ) is defined, at anypoint along the line, by

vi(λ) =dxi

dλ(λ) . (5)

It is an element of the linear space tangent to the manifold at theconsidered point. This tangent vector depends on the particular

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affine parameter being used: when changing from the affine pa-rameter λ to another affine parameter µ = α λ +β , and definingvi = dxi/dµ , one easily arrives to the relation vi = α vi .

1.5 Parallel Transport of a Vector

Let us suppose that a vector w is transported, parallel to itself,along this autoparallel line, and denote wi(λ) the componentsof the vector in the local natural basis. As demonstrated in ap-pendix A.3, one has the

Property 2 The equation defining the parallel transport of a vector walong the autoparallel line of affine tangent vector v is

dwi

dλ+ Γ i

jk v j wk = 0 . (6)

Given an autoparallel line and a vector at any of its points, thisequation can be used to obtain the transported vector at any otherpoint along the autoparallel line.

1.6 Association Between Tangent Vectors and OrientedSegments

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= 0

= 0

+ 1= 0

= 0 + 1

V i =dxi

d

k V i = W

i = ddxi

− 0 = 1−k

− 0( )

Figure 1: In a connection manifold (that may or may not be met-ric), the association between vectors (of the linear tangent space)and oriented autoparallel segments in the manifold is made usinga canonical affine parameter.

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one can transport not only vectors, but also oriented autoparallelsegments.

vv

u

'

Figure 2: Transport of an oriented autoparallel segment along an-other one.

2 Sum of Oriented Autoparallel Segments

2.1 Definition and Basic Properties

In a sufficiently smooth manifold, take a particular point O asorigin, and consider the set of oriented autoparallel segments,having O as origin, and belonging to some finite neighborhoodof the origin3. We shall call these objects autovectors. Given twosuch autovectors u and v , define the geometric sum (or geosum)w = v⊕ u by the geometric construction exposed in figure 3, and

3On an arbitrary manifold, the geodesics leaving a point may form caustics(where the geodesics cross each other). The considered neighborhood of theorigin must be small enough as to avoid caustics.

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given two such autovectors u and v , define the geometric differ-ence (or geodifference) w = v! u by the geometric constructionexposed in figure 4.

De!nition of w = v ⊕ u ( v = w ⊖ u )

v

u

v

u

wv

uv' v'

Figure 3: Definition of the geometric sum of two autovectorsat a point O of a manifold with a parallel transport: the sumw = v⊕ u is defined through the parallel transport of v alongu . Here, v′ denotes the oriented autoparallel segment obtainedby the parallel transport of the autoparallel segment defining valong u (as v′ does not begin at the origin, it is not an ‘autovec-tor’). We may say, using a common terminology that the orientedautoparallel segments v and v′ are ‘equipollent’. The ‘autovec-tor’ w = v⊕ u is, by definition, the arc of autoparallel (uniquein a sufficiently small neighborhood of the origin) connecting theorigin O to the tip of v′ .

As the definition of the geodifference ! is essentially, the ‘de-construction’ of the geosum ⊕ , it is clear that the equation w =v⊕ u can be solved for v :

w = v⊕ u ⇐⇒ v = w! u . (7)

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De!nition of v = w ⊖ u ( w = v ⊕ u )

v' v'

www

uuu

v

Figure 4: The geometric difference v = w! u of two autovectorsis defined by the condition v = w! u ⇔ w = v⊕ u . It can beobtained through the parallel transport to the origin (along u ) ofthe oriented autoparallel segment v′ that “ goes from the tip ofu to the tip of w ”. In fact, the transport performed to obtain thedifference v = w! u is the reverse of the transport performed toobtain the sum w = v⊕ u (figure 3), and this explains why in theexpression w = v⊕ u one can always solve for v , to obtain v =w! u . This contrasts with the problem of solving w = v⊕ ufor u , that requires a different geometrical construction, whoseresult cannot be directly expressed in terms of the two operations⊕ and ! (see the example in figure 6).

Figure 5: The opposite -v of an ‘autovector’ v is the ‘autovector’ oppositeto v , and with the same absolute variation of affine parameter as v (or thesame length if the manifold is metric).

v-v

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It is obvious that there exists a neutral element 0 for the sum of au-tovectors: a segment reduced to a point. For we have, for any ‘autovec-tor’ v ,

0⊕ v = v⊕ 0 = v , (8)

The opposite of an ‘autovector’ a is the ‘autovector’ -a , that is alongthe same autoparallel line, but pointing towards the opposite direction(see figure 5). The associated tangent vectors are also mutually opposite(in the usual sense). Then, clearly,

(-v)⊕ v = v⊕ (-v) = 0 (9)

Given an ‘autovector’ v and a real number λ , the sense to be given toλ v (for any λ ∈ [-1, 1] ) is obvious, and requires no special discussion.It is then clear that for any ‘autovector’ v and any scalars λ and µ

inside some finite interval around zero,

(λ + µ) v = λ v⊕µ v , (10)

as this corresponds to translating an autoparallel line along itself.

The reader may easily construct the geometric representation that cor-responds to the two properties, valid in general,

(w⊕ v)# v = w ; (w# v)⊕ v = w . (11)

We have seen that the equation w = v⊕ u can be solved for v , togive v = w# u . A completely different situation appears when trying

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to solve w = v⊕ u in terms of u . Finding the u such that by par-allel transport of v along it one obtains w correspond to an ‘inverseproblem’ that has no explicit geometric solution. It can be solved, forinstance using some iterative algorithm, essentially a trial and (correc-tion of) error method.

Note that given w = v⊕ u , in general, u "= (-v)⊕w (see figure 6),the equality holding only in the special situation where the autovectoroperation is, in fact, a group operation (i.e., it is associative). This isobviously not the case in an arbitrary manifold.

Not only the associative property does not hold on an arbitrary mani-fold, but even simpler properties are not verified. For instance, let usintroduce the following definition: An autovector space is oppositive isfor any two autovectors u and v , one has w# v = -(v#w) . Fig-ure 7 shows that the surface of the sphere, using the parallel transportdefined by the metric, is not oppositive. Also note that, in general,

w# v "= w⊕(-v) . (12)

2.2 Linear tangent Space

One intuitively expects that a sufficiently smooth manifold accepts alinear tangent space at each of its points. The autovectors we have in-troduced all have their origin at a given point. The linear tangent space

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2.2 Linear Tangent Space

One intuitively expects that a sufficiently smooth manifold accepts alinear tangent space at each of its points. The autovectors we have in-troduced all have their origin at a given point. The linear tangent spaceat this origin point can be introduced via the relation

limλ→0

1λ(λ w⊕ λ v) = w + v , (13)

linking the geosum to the sum (and difference) in the tangent linearspace (through the consideration of the limit of vanishingly small au-tovectors).

2.3 Series Representations

We can now seek to write the following series expansion,

(w⊕ v)i = ai + bij wj + ci

j v j + dijk wj wk + ei

jk wj vk + f ijk v j vk

+ pijk! wj wk w! + qi

jk! wj wk v! + rijk! wj vk v!

+ sijk! vj vk v! + . . . ,

(14)

expressing the geometric sum (on the manifold) in terms of the sum inthe linear tangent space. We shall later see that this series relates to awell-known series arising in the study of Lie groups, called the BCHseries. Remember that the operation ⊕ is, in general, not associative.

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Therefore, using equations (23)–(24) and (21)–(22), we arrive at the prop-erty [

(w⊕ v) " (v⊕w)]i = Ti

jk wj vk + . . .[( w⊕(v⊕ u) ) " ( (w⊕ v)⊕ u )

]i = 12 Ai

jk! wj vk u! + . . . .(29)

Loosely speaking, the tensors T and A give respectively a ‘measure’of the default of commutativity and of the default of associativity of theautovector operation ⊕ .

We shall see on a manifold with constant torsion, the anassociativitytensor is identical to the Riemann tensor of the manifold (this corre-spondence explaining the factor 1/2 in the definition of A ).

From equation (26) follows that the torsion is antisymmetric in its twolower indices:

Tijk = -Ti

k j . (30)

We can now come back to the two developments (equations (15) and (19))

(w⊕ v)i = wi + vi + eijk wj vk + qi

jk! wj wk v! + rijk! wj vk v! + . . .

(w" v)i = wi − vi − eijk wj vk − qi

jk! wj wk v! − uijk! wj vk v! + . . . ,

(31)

with the uijk! given by expression (20). Using the definition of torsion

and of anassociativity (27)–(28), it is possible to see that one can express

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The expressions for the torsion and the anassociativity can thenbe obtained using equations (33). After some easy rearrange-ments, this gives

Tijk = Γ i

jk − Γ ik j ; Ai

jk! = Rijk! +∇!Ti

jk , (36)

where

Rijk! = ∂!Γ

ik j − ∂kΓ

i! j + Γ i

!s Γ sk j − Γ i

ks Γ s! j (37)

is the Riemann tensor of the manifold and where ∇!Tijk is the

covariant derivative of the torsion:

∇!Tijk = ∂!Ti

jk + Γ i!s Ts

jk − Γ s! j Ti

sk − Γ s!k Ti

js . (38)

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But in equations (32) we have obtained the expression of eijk ,

qijk! and r jk! in terms of the torsion tensor and the anassocia-

tivity tensor. Therefore, equations (32) give the covariant expres-sions of these three tensors.

2.6 Bianchi Identities

A direct computation shows that we have the following

Property 5 First Bianchi identity. At any point7 of a differentiablemanifold, the anassociativity and the torsion are linked through

∮( jk!) Ai

jk! =∮

( jk!) Tijs Ts

k! . (41)

This is an important identity. When expressing the anassociativ-ity in terms of the Riemann and the torsion (equation (40)), this isthe well-known ‘first Bianchi identity’ of a manifold.

The second Bianchi identity is obtained by taking the covariantderivative of the Riemann (as expressed in equation (37)) andmaking a circular sum:

7As any point of a differentiable manifold can be taken as origin of an au-tovector space.

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Property 6 Second Bianchi identity. At any point of a differentiablemanifold, the Riemann and the torsion are linked through

∮( jk!)∇ jRi

mk! =∮

( jk!) Rim js Ts

k! . (42)

Contrary to what happens with the first identity, no simplifica-tion occurs when using the anassociativity instead of the Rie-mann.

2.7 Contracted Bianchi Identities

Introducing the Ricci tensor Ri j and the scalar curvature R through

Ri j = Rsjis ; R = gi j Ri j , (43)

and the contracted torsion

Ti = Tsis , (44)

it follows from the Bianchi identities (41)–(42) the following twoequations, named the contracted Bianchi identities:

∇sEis = Ts

!r ( 12 Rr!

si + δi! Rs

r) ; ∇i Cijk = (Rjk − Rk j) + Ts Ts

jk ,(45)

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2.7 Contracted Bianchi Identities

Introducing the Ricci tensor Ri j and the scalar curvature R through

Ri j = Rsjis ; R = gi j Ri j , (43)

and the contracted torsionTi = Tsi

s , (44)

it follows from the Bianchi identities (41)–(42) the following two equations,named the contracted Bianchi identities:

∇sEis = Ts

!r ( 12 Rr!

si + δi! Rs

r) ; ∇i Cijk = (Rjk − Rk j) + Ts Ts

jk ,(45)

where the Einstein tensor Ei j and the Cartan tensor Cijk are defined as

Ei j = Ri j − 12 gi j R ; Ck

i j = Tki j + Ti δ j

k − Tj δik . (46)

Should the torsion be zero, then the contracted Bianchi identities degenerateinto

∇kEjk = 0 ; Ri j = Rji . (47)

The Ricci tensor is symmetric and the Einstein tensor satisfies a ‘conservationequation’.

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3 Gravitation

In General Relativity, the space-time is a four-dimensional manifold, endowedwith a metric that is locally Minkowskian. As it is customary to use Greek in-dices for the space-time coordinates, we should rewrite the contracted Bianchiidentities as follows

∇σ Eασ = Tσ

βρ ( 12 Rρβ

σα + δαβ Rσ

ρ)

∇σ Cσαβ = (Rαβ − Rβα) + Tσ Tσ

αβ ,(48)

the Einstein tensor and the Cartan tensor being expressed as

Eαβ = Rαβ − 12 gαβ R ; Cγ

αβ = Tγαβ + Tα δβ

γ − Tβ δαγ . (49)

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The matter content of the universe is represented, at each space-time point,by the stress-energy tensor tαβ and the moment-stress-energy tensor mα

βγ

(for details, see Halbwachs, 1960). While talphaβ fundamentally describesthe mass density content of the space-time, mα

βγ describes the spin densitycontent.

The fundamental postulate of gravitation theory is that the Einstein tensor isproportional to the stress-energy tensor, and that the Cartan tensor is propor-tional to the moment-stress-energy tensor:

Eαβ =8 π G

c4 tαβ ; Cγαβ =

8 π Gc4 mγ

αβ . (50)

This theory, including torsion and spin is called the Einstein-Cartan theoryof gravitation, and the two equations above are called the Einstein-Cartanequations. See Hehl (1973, 1974) for details.

If the moment-stress-energy tensor is zero, then, the torsion tensor and theCartan tensor vanish, the stress-energy tensor is symmetric, and we are leftwith the original Einstein theory of gravitation. Its fundamental equationsare

∇σ Eασ = 0 ; Eαβ =

8 π Gc4 tαβ ; tαβ = tβα .

(51)

4 References

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C Total Riemann Versus Metric Curvature

C.1 Connection, Metric Connection and Torsion

The metric postulate (that the parallel transport conserves lengths) is

∇kgi j = 0 . (100)

This gives∂kgi j − Γ s

ki gs j − Γ sk j gis = 0 , (101)

i.e.,∂kgi j = Γ jki + Γik j . (102)

The Levi-Civita connection, or metric connection is defined as

{ki j} =

12

gks (∂ig js + ∂ jgis − ∂sgi j

)(103)

(the {ki j} are also called the ‘Christoffel symbols’). Using equation (102), one

easily obtains {ki j} = Γki j + 12 ( Tk ji + Tjik + Ti jk ) , i.e.,

Γki j = {ki j} +12

Vki j +12

Tki j , (104)

whereVki j = Tik j + Tjki . (105)

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The tensor - 12 (Tki j + Vki j) is named ‘contortion’ by Hehl (1973). Note that

while Tki j is antisymmetric in its two last indices, Vki j is symmetric in them.Therefore, defining the symmetric part of the connection as

γki j ≡ 1

2(Γ k

i j + Γ kji) , (106)

gives

γki j = {k

i j} +12

Vki j , (107)

and the decomposition of Γ ki j in symmetric and antisymmetric part is

Γ ki j = γk

i j +12

Tki j . (108)

C.2 The Metric Curvature

The (total) Riemann R!i jk is defined in terms of the (total) connection Γ k

i j

by equation (89). The metric curvature, or curvature, here denoted C!i jk has

the same definition, but using the metric connection {ki j} instead of the total

connection:

C!i jk = ∂k{!

ji}− ∂ j{!ki} + {!

ks} {sji}− {!

js} {ski} (109)

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when geodesics and autoparallels coincide, the torsion T is a totally antisymmetrictensor:

Ti jk = -Tjik = -Tik j . (114)

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When the torsion is totally antisymmetric, it follows from the definition (105)that one has

Vi jk = 0 . (115)

Then,

Γ ki j = {k

i j} +12

Tki j , (116)

and

{ki j} =

12

(Γ k

i j + Γ kji)

= γki j , (117)

i.e., when autoparallels and geodesics coincide, the metric connection is thesymmetric part of the total connection.

Note: explain here that, if the torsion is totally antisymmetric, one introducesthe tensor J as

J!i jk = T!

is Tsjk + T!

js Tski + T!

ks Tsi j , (118)

i.e.,J!

i jk =∮

(i jk) T!is Ts

jk . (119)

It is easy to see that J is totally antisymmetric in its three lower indices,

J!i jk = -J!

jik = -J!ik j . (120)

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