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
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
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
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
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
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
In the affine parameterization, the expansion of the congruence is calculated to be
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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