In this and the following sections we will construct a number of bitensors, tensorial functions of two points in spacetime. The first is , to which we refer as the “base point”, and to which we assign indices , , etc. The second is , to which we refer as the “field point”, and to which we assign indices , , etc. We assume that belongs to , the normal convex neighbourhood of ; this is the set of points that are linked to by a unique geodesic. The geodesic that links to is described by relations in which is an affine parameter that ranges from to ; we have and . To an arbitrary point on the geodesic we assign indices , , etc. The vector is tangent to the geodesic, and it obeys the geodesic equation . The situation is illustrated in Figure 5.

Synge’s world function is a scalar function of the base point and the field point . It is defined by

and the integral is evaluated on the geodesic that links to . You may notice that is invariant under a constant rescaling of the affine parameter, , where and are constants.By virtue of the geodesic equation, the quantity is constant on the geodesic. The world function is therefore numerically equal to . If the geodesic is timelike, then can be set equal to the proper time , which implies that and . If the geodesic is spacelike, then can be set equal to the proper distance , which implies that and . If the geodesic is null, then . Quite generally, therefore, the world function is half the squared geodesic distance between the points and .

In flat spacetime, the geodesic linking to is a straight line, and in Lorentzian coordinates.

The world function can be differentiated with respect to either argument. We let be its partial derivative with respect to , and its partial derivative with respect to . It is clear that behaves as a dual vector with respect to tensorial operations carried out at , but as a scalar with respect to operations carried out . Similarly, is a scalar at but a dual vector at .

We let be the covariant derivative of with respect to ; this is a rank-2 tensor at and a scalar at . Because is a scalar at , we have that this tensor is symmetric: . Similarly, we let be the partial derivative of with respect to ; this is a dual vector both at and . We can also define to be the partial derivative of with respect to . Because partial derivatives commute, these bitensors are equal: . Finally, we let be the covariant derivative of with respect to ; this is a symmetric rank-2 tensor at and a scalar at .

The notation is easily extended to any number of derivatives. For example, we let , which is a rank-3 tensor at and a dual vector at . This bitensor is symmetric in the pair of indices and , but not in the pairs and , nor and . Because is here an ordinary partial derivative with respect to , the bitensor is symmetric in any pair of indices involving . The ordering of the primed index relative to the unprimed indices is therefore irrelevant: The same bitensor can be written as or or , making sure that the ordering of the unprimed indices is not altered.

More generally, we can show that derivatives of any bitensor satisfy the property

in which “” stands for any combination of primed and unprimed indices. We start by establishing the symmetry of with respect to the pair and . This is most easily done by adopting Fermi normal coordinates (see Section 3.2) adapted to the geodesic , and setting the connection to zero both at and . In these coordinates, the bitensor is the partial derivative of with respect to , and is obtained by taking an additional partial derivative with respect to . These two operations commute, and follows as a bitensorial identity. Equation (54) then follows by further differentiation with respect to either or .The message of Equation (54), when applied to derivatives of the world function, is that while the ordering of the primed and unprimed indices relative to themselves is important, their ordering with respect to each other is arbitrary. For example, .

We can compute by examining how varies when the field point moves. We let the new field point be , and is the corresponding variation of the world function. We let be the unique geodesic that links to ; it is described by relations , in which the affine parameter is scaled in such a way that it runs from to also on the new geodesic. We note that and .

Working to first order in the variations, Equation (53) implies

where , an overdot indicates differentiation with respect to , and the metric and its derivatives are evaluated on . Integrating the first term by parts gives

The integral vanishes because satisfies the geodesic equation. The boundary term at is zero because the variation vanishes there. We are left with , or

in which the metric and the tangent vector are both evaluated at . Apart from a factor , we see that is equal to the geodesic’s tangent vector at . If in Equation (55) we replace by a generic point on , and if we correspondingly replace by , we obtain ; we therefore see that is a rescaled tangent vector on the geodesic.A virtually identical calculation reveals how varies under a change of base point . Here the variation of the geodesic is such that and , and we obtain . This shows that

in which the metric and the tangent vector are both evaluated at . Apart from a factor , we see that is minus the geodesic’s tangent vector at .It is interesting to compute the norm of . According to Equation (55) we have . According to Equation (53), this is equal to . We have obtained

and similarly These important relations will be the starting point of many computations to be described below.We note that in flat spacetime, and in Lorentzian coordinates. From this it follows that , and finally, .

If the base point is kept fixed, can be considered to be an ordinary scalar function of . According to Equation (57), this function is a solution to the nonlinear differential equation . Suppose that we are presented with such a scalar field. What can we say about it?

An additional differentiation of the defining equation reveals that the vector satisfies

which is the geodesic equation in a non-affine parameterization. The vector field is therefore tangent to a congruence of geodesics. The geodesics are timelike where , they are spacelike where , and they are null where . Here, for concreteness, we shall consider only the timelike subset of the congruence.The vector

is a normalized tangent vector that satisfies the geodesic equation in affine-parameter form: . The parameter is then proper time . If denotes the original parameterization of the geodesics, we have that , and we see that the original parameterization is singular at .In the affine parameterization, the expansion of the congruence is calculated to be

where is the expansion in the original parameterization ( is the congruence’s cross-sectional volume). While is well behaved in the limit (we shall see below that ), we have that . This means that the point at which is a caustic of the congruence: All geodesics emanate from this point.These considerations, which all follow from a postulated relation , are clearly compatible with our preceding explicit construction of the world function.

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