CMChap19 - Chapter 19 Force and Curvature The general...

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Unformatted text preview: Chapter 19 Force and Curvature The general theory of relativity due to Einstein first demonstrated the deep relationship between the physical concept of force and the geometrical concept of curvature. In this chapter, we will demonstrate this relationship explicitly at the simpler level of Newtonian mechanics, by using the construct of fictitious forces which arise from the use of noninertial frames. For this purpose we need to introduce some basic facts of Riemannian geometry. We begin with the notion of a Riemannian manifold, which is a difi'er- entiable manifold equipped with a Riemannian metric G = gij duiduj , (19.1) where gij is a smooth symmetric function on a local coordinate neighborhood of the manifold with local coordinates ul, such that det(gij) 75 0. The last condition on the determinant of gij is known as the nondegeneracy of the metric. On a Riemannian manifold, the Riemannian metric specifies rules on how to calculate the distance between two points on the manifold, and the angle between two tangent vectors at the same point. Indeed one defines the element of arc length d3 by d52 = 9,5 duiduj , (19.2) so that du dgjd gij— dt (19.3) For example, on a two—dimensional spherical surface of radius 7' (2-sphere), we can let u1 = 6, U2 : qfi, so that d52 = r2 d29 + 7‘2 sin2 6 d2q5. (19.4) Then r2 0 (9n) = - (19-5) 0 7'2 sin2 9 89 90 CHAPTER 19. FORCE AND CURVATURE Since gij is nondegenerate, the inverse matrix 917 exists, such that gijgjk = 6%. Recall the notion of the connection matrix of one—forms w; defined by [c.f. (2.8)] . ale, 2 wgej , (19.6) where the “d” is a covariant derivative, and {67;} is an orthonormal frame field on the Riemannian manifold. An orthonormal frame at a point of the manifold is a an orthonormal basis of the tangent space at that point. A connection on the tangent bundle of a manifold is called an affine connec— tion. In terms of the local coordinates at, the connection matrix elements can be written as . . wg : rg, duk . (19.7) The I‘Zh are called Christoffel symbols, and are functions of the local coordi— nates 11. Let {cal} be the dual coframe field to the frame field {ei}: (el- M) = 6?, (19.8) ’L where (, ) denotes the pairing between a vector in a certain vector space and a vector in its dual space. For the 2-sphere with local coordinates (0, (15), for example, we have the following orthonormal frame field and dual coframe field: 18 18 — —‘ l: 2: . T—a—g, eg—rsineaqi’ w rdQ, w rs1n6d¢. (19.9) 61: Among all the possible affine connections on a Riemannian manifold, there are two classes which are particularly important: the class of torsion-free con— nections and the class of metric-compatible connections. The analytical conditions of these are given as follows. A connection mg is said to be torsion-free if dwi : wj /\ w; (torsion—freeness) . (19.10) A connection rug is said to be metricvcompatible (with a metric gij) if dgij = wfgkj + (12;? gig, (metric-compatibility) . (19.11) In the above two equations, the operator d is an exterior derivative and /\ stands for an exterior product. We cannot go into the technical details of these notions here, except to mention that the exterior derivative generalizes the various differential operators in vector calculus (div, grad, curl and all that), while the exterior product generalizes the vector cross product in vector algebra. The geometric (and more intuitive) contents of (19.10) and (19.11) are shown in Figs. 19.1 and 19.2, respectively. In Fig. 19.1, two infinitesimal tangent vectors at the same point (in the mani— fold) are parallelly transported, each along the infinitesimal path given by the 91 other vector. If the resulting figure is a closed parallelogram, then the connec— tion (which determines the parallel transports) is torsion-free. In Fig. 19.2, tangent vectors X and Y (at the same point in the manifold) are parallelly transported along a curve in the manifold. If the lengths of the vectors and the angle between the vectors remain invariant under the parallel transports, then the connection is metric compatible. A fundamental theorem of Riemannian geometry is the following. Given a Riemannian metric gij, there emists a unique affine connection to, called the Levi- Civita connection, that is both torsion‘free and metric compatible. The Levi-Civita connection is given in terms of the Riemannian metric by (19.12) The curvature matrix of two-forms Q is defined in terms of the connec— tion matrix to by dew—wAw. (19.13) Using the tensorial index notation and (19.7) we can write the matrix elements of the curvature matrix in terms of the Christoifel symbols as follows. i. j k j 0,. —clw,- —w,~ Awk r2: , , fi : 6871;,“ dul /\ duh __ PEP-2k dul A duh (19 14) 1 or; org , . , g : 5 (Bug T Bulk + Filrhk '— Pal-‘1”) duh /\ dul . The quantity within the parentheses on the RHS of the last equality in the above equation is known as the Riemann curvature tensor: Riki "' + Filrhk _ Filkrhl - (19-15) — auk Bill We will see now how the Riemannian curvature tensor enters in the expres- sion for the fictitious forces. Recall (15.9), but assume that in that equation the LHS dgr/dtg = 0. For example (cf. Fig. 15.1), one may have the situation where the actual force (with respect to the inertial frame {6,}) vanishes, and the origin of the moving frame {6,} moves with a constant velocity relative to the inertial frame. We then have the coupled equations giving the acceleration of the particle with respect to the moving frame: :L'— $90,. mgoitpkadt. (.) 92 CHAPTER 19. FORCE AND CURVATURE The individual terms on the RHS can be written in terms of the Christoffel symbols as follows. _ w] dul 21' j_ -z'_i' = -i a" __ _ 2:1, (p1- — 2m dt 2m Pu dt , (19 17) .5190? . d w? . d ‘ dul 2 7. ._ z__ __1_ 2 1 _ 2 __ ”3 dt "9” dt (dt 9" dt<rll (It) . . (19.18) - 6F]- du'c dul ~ dgul 49F]- duk dul - - dzul :ZIIZ 1l___+1“]' le zl___ ,1 J ' auk dt dt ll alt? auk dt dt ll cit? Thus ‘ _ ‘dul . .612”! . 31“? . dulduk '1]:_2‘21'\?_____ t J _ z zl h .7 _____ 1 1 3’ ll} zl dt 3:le dt2 IE (auk +F1lfltk> dt dt 7 (9 9) or _, . dulduk ——% J. —— 1 .2 2‘31“? dt dt ’ ( 9 O) Where the Riemann curvature tensor R?“ has been given in (19.5). Eq.(19.20) shows explicitly how the fictitious forces depend on the Levi—Civita connection and the associated Riemannian curvature. ...
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