This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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' (2sphere), we
can let u1 = 6, U2 : qﬁ, so that d52 = r2 d29 + 7‘2 sin2 6 d2q5. (19.4)
Then
r2 0
(9n) =  (195)
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 2sphere 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 torsionfree con—
nections and the class of metriccompatible connections. The analytical
conditions of these are given as follows. A connection mg is said to be torsionfree
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, (metriccompatibility) . (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 torsionfree. 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
LeviCivita connection is given in terms of the Riemannian metric by (19.12) The curvature matrix of twoforms 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 __ PEP2k 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  (1915) — 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. ...
View
Full Document
 Spring '08
 LAM
 mechanics

Click to edit the document details