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2.1 Synge’s world function

2.1.1 Definition

In this and the following sections we will construct a number of bitensors, tensorial functions of two points in spacetime. The first is x', to which we refer as the “base point”, and to which we assign indices ' a, ' b, etc. The second is x, to which we refer as the “field point”, and to which we assign indices a, b, etc. We assume that x belongs to N (x'), the normal convex neighbourhood of x'; this is the set of points that are linked to x' by a unique geodesic. The geodesic b that links x to ' x is described by relations m z (c) in which c is an affine parameter that ranges from c0 to c1; we have z(c0) =_ x' and z(c1) =_ x. To an arbitrary point z on the geodesic we assign indices m, n, etc. The vector tm = dzm/dc is tangent to the geodesic, and it obeys the geodesic equation Dtm/dc = 0. The situation is illustrated in Figure  5View Image.

View Image

Figure 5: The base point ' x, the field point x, and the geodesic b that links them. The geodesic is described by parametric relations zm(c), and tm = dzm/dc is its tangent vector.
Synge’s world function is a scalar function of the base point ' x and the field point x. It is defined by
integral ' 1 c1 m n s(x, x ) = -(c1 - c0) gmn(z)t t dc, (53) 2 c0
and the integral is evaluated on the geodesic b that links x to x'. You may notice that s is invariant under a constant rescaling of the affine parameter, c --> c = ac + b, where a and b are constants.

By virtue of the geodesic equation, the quantity m n e =_ gmnt t is constant on the geodesic. The world function is therefore numerically equal to 12 e(c1 - c0)2. If the geodesic is timelike, then c can be set equal to the proper time t, which implies that e = - 1 and s = - 1(Dt )2 2. If the geodesic is spacelike, then c can be set equal to the proper distance s, which implies that e = 1 and s = 1(Ds)2 2. If the geodesic is null, then s = 0. Quite generally, therefore, the world function is half the squared geodesic distance between the points ' x and x.

In flat spacetime, the geodesic linking x to x' is a straight line, and s = 12jab(x - x')a(x - x')b in Lorentzian coordinates.

2.1.2 Differentiation of the world function

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

We let sab =_ \~/ bsa be the covariant derivative of sa with respect to x; this is a rank-2 tensor at x and a scalar at x'. Because s is a scalar at x, we have that this tensor is symmetric: sba = sab. Similarly, we let sab' =_ @b'sa = @2s/@xb'@xa be the partial derivative of sa with respect to x'; this is a dual vector both at x and x'. We can also define sa'b =_ @bsa'= @2s/@xb@xa' to be the partial derivative of sa' with respect to x. Because partial derivatives commute, these bitensors are equal: sb'a = sab'. Finally, we let sa'b' =_ \~/ b'sa' be the covariant derivative of sa' with respect to x'; this is a symmetric rank-2 tensor at x' and a scalar at x.

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

More generally, we can show that derivatives of any bitensor _O_...(x,x') satisfy the property

_O_...;ba'... = _O_...;a'b..., (54)
in which “...” stands for any combination of primed and unprimed indices. We start by establishing the symmetry of _O_...;ab' with respect to the pair a and ' b. This is most easily done by adopting Fermi normal coordinates (see Section 3.2) adapted to the geodesic b, and setting the connection to zero both at x and x'. In these coordinates, the bitensor _O_...;a is the partial derivative of _O_... with respect to xa, and _O_...;ab' is obtained by taking an additional partial derivative with respect to xb'. These two operations commute, and _O_ ' = _O_ ' ...;b a ...;ab follows as a bitensorial identity. Equation (54View Equation) then follows by further differentiation with respect to either x or x'.

The message of Equation (54View Equation), 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, '' ' ' '' '' ' sa b gde = sa bdge = sgeab d.

2.1.3 Evaluation of first derivatives

We can compute sa by examining how s varies when the field point x moves. We let the new field point be x + dx, and ds =_ s(x + dx,x')- s(x, x') is the corresponding variation of the world function. We let b + db be the unique geodesic that links x + dx to ' x; it is described by relations zm(c) + dzm(c), in which the affine parameter is scaled in such a way that it runs from c 0 to c 1 also on the new geodesic. We note that dz(c ) = dx'= _ 0 0 and dz(c1) = dx.

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

integral c1( ) ds = Dc g zmdzn + 1-g zmzn dzc dc, c0 mn 2 mn,c

where Dc = c1 - c0, an overdot indicates differentiation with respect to c, and the metric and its derivatives are evaluated on b. Integrating the first term by parts gives

integral c1 ds = Dc [gmnzm dzn]c1- Dc (gmn¨zn + Gmncznzc)dzm dc. c0 c0

The integral vanishes because zm(c) satisfies the geodesic equation. The boundary term at c0 is zero because the variation m dz vanishes there. We are left with a b ds = Dcgabt dx, or

sa(x,x') = (c1 - c0)gabtb, (55)
in which the metric and the tangent vector are both evaluated at x. Apart from a factor Dc, we see that sa(x, x') is equal to the geodesic’s tangent vector at x. If in Equation (55View Equation) we replace x by a generic point z(c) on b, and if we correspondingly replace c1 by c, we obtain sm(z, x') = (c - c0)tm; we therefore see that sm(z,x') is a rescaled tangent vector on the geodesic.

A virtually identical calculation reveals how s varies under a change of base point x'. Here the variation of the geodesic is such that dz(c ) = dx' 0 and dz(c ) = dx = 0 1, and we obtain a' b' ds = -Dcga'b't dx. This shows that

' sa'(x,x') = -(c1 - c0) ga'b'tb, (56)
in which the metric and the tangent vector are both evaluated at ' x. Apart from a factor Dc, we see that ' sa (x,x') is minus the geodesic’s tangent vector at x'.

It is interesting to compute the norm of sa. According to Equation (55View Equation) we have g sasb = (Dc)2g tatb = (Dc)2e ab ab. According to Equation (53View Equation), this is equal to 2s. We have obtained

gabsasb = 2s, (57)
and similarly
ga'b's 's '= 2s. (58) a b
These important relations will be the starting point of many computations to be described below.

We note that in flat spacetime, s = j (x - x')b a ab and s '= - j (x - x')b a ab in Lorentzian coordinates. From this it follows that sab = sa'b'= - sab'= - sa'b = jab, and finally, ab a'b' g sab = 4 = g sa'b'.

2.1.4 Congruence of geodesics emanating from x'

If the base point x' is kept fixed, s can be considered to be an ordinary scalar function of x. According to Equation (57View Equation), this function is a solution to the nonlinear differential equation 1 ab 2 g sasb = s. 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 sa =_ s;a satisfies

sa sb = sa, (59) ;b
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 s < 0, they are spacelike where s > 0, and they are null where s = 0. Here, for concreteness, we shall consider only the timelike subset of the congruence.

The vector

a sa u = ---1/2- (60) |2s |
is a normalized tangent vector that satisfies the geodesic equation in affine-parameter form: ua;bub = 0. The parameter c is then proper time t. If c* denotes the original parameterization of the geodesics, we have that * - 1/2 dc /dt = |2s |, and we see that the original parameterization is singular at s = 0.

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

* h = ---h---, h* = sa;a - 1, (61) |2s |1/2
where h* = (dV)- 1(d/dc*)(dV ) is the expansion in the original parameterization (dV is the congruence’s cross-sectional volume). While * h is well behaved in the limit s --> 0 (we shall see below that * h --> 3), we have that h --> oo. This means that the point x' at which s = 0 is a caustic of the congruence: All geodesics emanate from this point.

These considerations, which all follow from a postulated relation 1gabsasb = s 2, are clearly compatible with our preceding explicit construction of the world function.



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