D

This gives the rate of change of the separation of two infinitesimally neighbouring curves as measured in Hq. Operating again with T)jds and projecting into Hq, one finds

Changing the order of the derivatives in the first term and using (4.2), this reduces to

^"Fsi^Fs^j = -Rabc<iiZcVbV* + h\ Vb.,cxZ°+ f»fb±Zb. (4.4)

This equation, known as the deviation or Jacobi equation, gives the relative acceleration, i.e. the second time derivative of the separation, of two infinitesimally neighbouring curves as measured in Hq. We see that this depends only on the Riemann tensor if the curves are geodesies.

In Newtonian theory, the acceleration of each particle is given by the gradient of the potential O and therefore the relative acceleration of two particles with separation Za is O. abZb. Thus the Riemann tensor term Rdx-d VbVd is analogous to the Newtonian O. The effect of this 'tidal force* term can be seen, for example, by considering a sphere of particles freely falling towards the earth. Each particle moves on a straight line through the centre of the earth but those nearer the earth fall faster than those further away. This means that the sphere does not remain a sphere but is distorted into an ellipsoid with the same volume.

In order to investigate the deviation equation further we shall introduce dual orthonormal bases ej, e2, e2, e4 and e1, e2, e8, e4 of Tq and T*q at some point q on an integral curve y(s) of v, with e4 = v. One would like to propagate them along y(s) to obtain similar such bases at each point of y(s). However, if one parallelly propagates them along y(s) (i.e. so that T>/8s of each vector is zero) e4 will not remain equal to v, and ej, e2, e3 will not remain orthogonal to v, unless y(s) is a geodesic. We therefore introduce a new derivative along y(s) called the Fermi derivative ~DF/ds. This is defined for a vector field X along y(s) by:

It has the properties:

(iii) if X and Y are vector fields along y(s) such that

DFX DfY

(iv) if X is a vector field along y(s) orthogonal to V then

(This last property shows that the Fermi derivative is a natural generalization of the derivative D j8s.)

Thus, if one propagates an orthonormal basis of Tq along y(s) so that the Fermi derivative of each basis vector is zero, one obtains an orthonormal basis at each point of y(s), with E4 = V. The vectors Ej, E2, E2 may be interpreted as giving a non-rotating set of axes along y(s). These could be realized physically by small gyroscopes pointing in the direction of each vector.

The definition of the Fermi derivative along y(s) can be extended from vector fields to arbitrary tensor fields by the usual rules:

(i) T>f/8s is a linear mapping of tensor fields of type (r, s) along y(s) to tensor fields of type (r,s), which commutes with contractions;

From these rules it follows that the dual basis E1, E2, E3, E4 of T*g is also Fermi-propagated along y(s). Using Fermi derivatives, (4.3) and (4.4) may be written as:

-¿±za = V"Vd + hab V»;cxz<+ V»Vb±Z\ (4.6)

One may express these equations in terms of the Fermi-propagated dual bases. As J.Z is orthogonal to V it will have components with respect to EX) E2, E2 only. Thus it may be expressed as ZaEa where we adopt the convention that Greek indices take the values 1, 2, 3 only. Then (4.6) and (4.6) can be written in terms of ordinary derivatives:

where F°;/S are the components of Va.b for which a = a and b = ¡3. As the components Za obey the first order linear ordinary differential equation (4.7), they can be expressed in terms of their values at some point q by: Z*{s) = Aa/l(s)Z% (4.9)

where Aa/!(s) is a 3 x 3 matrix which is the unit matrix at q and satisfies

In the case of a fluid the matrix Aa/! can be regarded as representing the shape and orientation of a small element of fluid which is spherical at q. This matrix can be written as

where 0a/S is an orthogonal matrix with positive determinant and Sa/1 is a symmetric matrix. These will both be chosen to be the unit matrix at q. The matrix 0a/! may be thought of as representing the rotation that neighbouring curves have undergone with respect to the Fermi-propagated basis while Sa/3 represents the separation of these curves from y(s). The determinant of Sa/3, which equals the determinant of Aap, may be thought of as representing the three-volume of the element of the surface orthogonal to y(s) marked out by the neighbouring curves.

At q where Aap is the unit matrix, dOa/S/ds is antisymmetric and &Saplds is symmetric. Thus the rate of rotation of neighbouring curves at q is given by the antisymmetric part of while the rate of change of their Boparation from y(a) is given by the symmetric part of T^.^ and the rate of ohangc of volume ib given by the trace of We' therefore define the vorticity tensor as

the expansion tensor as

and the volume expansion as

We further define the shear tensor as the trace free part of 8ab,

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