Gravimetry, Relativity, and the Global Navigation Satellite Systems — Second Lesson: Introduction to Differential Geometry — Albert Tarantola March 9, 2005 1
1 Oriented Autoparallel Segments on a Manifold 1.1 Manifold An n -dimensional manifold is a space of elements, called ‘points’, that accepts in a finite neighborhood of each of its points an n -dimensional system of con- tinuous coordinates. Grossly speaking, an n -dimensional manifold is a space that, locally, ‘looks’ like ℜ n . We are here interested in the class of smooth manifolds that may or may not be metric, but that have a prescription for the parallel transport of vectors: given a vector at a point (a vector belong- ing to the linear space tangent to the manifold at the given point), and given a line on the manifold, it is assumed that one is able to transport the vec- tor along the line ‘keeping the vector always parallel to itself’. Intuitively speaking this corresponds to the assumption that there is an ‘inertial naviga- tion system’ on the manifold, analog to that used in airplanes to keep fixed directions while navigating. The prescription for this ‘parallel transport’ is not necessarily the one that could be defined using an eventual metric (and ‘geodesic’ techniques), as the considered manifolds may have ‘torsion’. In such a manifold, there is a family of privileged lines, the ‘autoparallels’, that are obtained when constantly following a direction defined by the ‘inertial navigation system’. If the manifold is, in addition, a metric manifold, then there is a second family of privileged lines, the ‘geodesics’, that correspond to the minimum length path between any two of its points. It is well known 1 that the two types 1 See a demonstration in appendix C.3. 2
of lines coincide (the geodesics are autoparallels and vice-versa) when the torsion is totally antisymmetric T ijk = - T jik = - T ikj . 1.2 Connection Consider the simple situation where some (arbitrary) coordinates x ≡ { x i } have been defined over the manifold. At a given point x 0 consider the coor- dinate lines passing through x 0 . If x is a point on any of the coordinate lines, let us denote as γ ( x ) the coordinate line segment going from x 0 to x . The natural basis (of the local tangent space) associated to the given coordinates consists of the n vectors { e 1 ( x 0 ) , . . . , e n ( x 0 ) } that can formally be denoted as e i ( x 0 ) = ∂ γ ∂ x i ( x 0 ) , or, dropping the index 0 , e i ( x ) = ∂ γ ∂ x i ( x ) . (1) So, there is a natural basis at every point of the manifold. As it is assumed that there exists a parallel transport on the manifold, the basis { e i ( x ) } can be transported from a point x i to a point x i + δ x i to give a new basis, that we can denote { e i ( x + δ x � x ) } (and that, in general, is different from the local basis { e i ( x + δ x ) } at point x + δ x ). The connection is defined as the set of coefficients Γ kij (that are not, in general, the components of a tensor) appearing in the development ij ( x ) e k ( x ) δ x i + . . . e j ( x + δ x � x ) = e j ( x ) + Γ k . (2) For this first order expression, we don’t need to be specific about the path followed for the parallel transport. For higher order expressions, the path 3
followed matters (see for instance equation (70), corresponding to a transport along an autoparallel line). In all the rest of this book, a manifold where a connection is defined shall be named a connection manifold . 1.3 Oriented Autoparallel Segments The notion of autoparallel curve is mathematically introduced in appendix A.2. It is enough for our present needs to know the main result demonstrated there: 4
Property 1 A line x i = x i ( λ ) is autoparallel if at every point along the line, d 2 x i dx j dx k d λ 2 + γ i = 0 , (3) jk d λ d λ where γ i jk is the symmetric part of the connection, γ i 1 2 ( Γ i jk + Γ i jk = kj ) . (4) If there exists a parameter λ with respect to which a curve is autoparallel, then any other parameter µ = α λ + β (where α and β are two constants) satisfies also the condition (3). Any such parameter associated to an autopar- allel curve is called an affine parameter . 1.4 Vector Tangent to an Autoparallel Line Let be x i = x i ( λ ) the equation of an autoparallel line with affine parameter λ . The affine tangent vector v (associated to the autoparallel line and to the affine parameter λ ) is defined, at any point along the line, by v i ( λ ) = dx i d λ ( λ ) . (5) It is an element of the linear space tangent to the manifold at the considered point. This tangent vector depends on the particular affine parameter being used: when changing from the affine parameter λ to another affine parame- v i = dx i / d µ , one easily arrives to the relation ter µ = α λ + β , and defining ˜ v i = α ˜ v i . 5
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 w i ( λ ) the components of the vector in the local natural basis. As demonstrated in appendix A.3, one has the Property 2 The equation defining the parallel transport of a vector w along the autoparallel line of affine tangent vector v is dw i jk v j w k = 0 d λ + Γ i . (6) Given an autoparallel line and a vector at any of its points, this equation can be used to obtain the transported vector at any other point along the autoparallel line. 1.6 Association Between Tangent Vectors and Oriented Segments Consider again an autoparallel line x i = x i ( λ ) defined in terms of an affine parameter λ . At some point of parameter λ 0 along the curve, we can intro- dx i duce the affine tangent vector defined in equation (5), v i ( λ 0 ) = d λ ( λ 0 ) , that belongs to the linear space tangent to the manifold at point λ 0 . As already mentioned, changing the affine parameter changes the affine tangent vector. 6
We could define an association between arbitrary tangent vectors and au- toparallel segments characterized using an arbitrary affine parameter 2 , but it is much simpler to pass through the introduction of a ‘canonical’ affine parameter. Given an arbitrary vector V at a point of a manifold, and the au- toparallel line that is tangent to V (at the given point), we can select among all the affine parameters that characterize the autoparallel line, one param- eter, say λ , giving V i = dx i / d λ (i.e., such that the affine tangent vector v with respect to the parameter λ equals the given vector V ). Then, by def- inition, to the vector V is associated the oriented autoparallel segment that starts at point λ 0 (the tangency point) and ends at point λ 0 + 1 , i.e., the seg- ment whose ‘affine length’ (with respect to the canonical affine parameter λ being used) equals one. This is represented in figure 1. Let O be the point where the vector V and the autoparallel line are tangent, let P be the point along the line that the procedure just described associates to the given vector V , and let Q be the point associated to the vector W = k V . It is easy to verify (see figure 1) that for any affine parameter considered along the line, the increase in the value of the affine parameter when passing from O to point Q is k times the increase when passing from O to P . The association so defined between tangent vectors and oriented autoparallel segments is consistent with the standard association between tangent vectors and oriented geodesic segments in metric manifolds without torsion, where the autoparallel lines are the geodesics. The tangent to a geodesic x i = x i ( s ) , 2 To any point of parameter λ along the autoparallel line we can associate the vector (also belonging to the linear space tangent to the manifold at λ 0 ) V ( λ ; λ 0 ) = λ − λ 0 1 − λ 0 v ( λ 0 ) . One has V ( λ 0 ; λ 0 ) = 0 , V ( 1; λ 0 ) = v ( λ 0 ) , and the more λ is larger than λ 0 , the ‘longer’ is V ( λ ; λ 0 ) . 7
i i = dx V d = + 1 0 = 0 dx i i = d i = W k V = 0 + 1 = 0 0 = 1 ( ) − − − 0 k Figure 1: In a connection manifold (that may or may not be metric), the associ- ation between vectors (of the linear tangent space) and oriented autoparallel segments in the manifold is made using a canonical affine parameter. 8
parameterized by a metric coordinate s , is defined as v i = dx i / ds , and one has g ij v i v j = g ij ( dx i / ds ) ( dx j / ds ) = ds 2 / ds 2 = 1 , this showing that the vector tangent to a geodesic has unit length. 1.7 Transport of Oriented Autoparallel Segments Consider now two oriented autoparallel segments, u and v with common origin, as suggested in figure 2. To the segment v we can associate a vector of the tangent space, as we have just seen. This vector can be transported along u (using equation 6) until its tip. The vector there obtained can then be associated to another oriented autoparallel segment, giving the v ′ suggested in the figure. So, on a manifold with a parallel transport defined, one can transport not only vectors, but also oriented autoparallel segments. v v ' u Figure 2: Transport of an oriented autoparallel segment along another one. 2 Sum of Oriented Autoparallel Segments 9
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