9.3 HigherOrder Metrics 

A similar path to the same goal could also be taken in those manifolds in which the line element is expressed in a less simple way, e.g., by a fourth root of a differential expression of the fourth degree… 
Riemann, 1854 

Given three points A,B,C, let dx^{1} denote the distance between A and B, and let dx^{2} denote the distance between B and C. Can we express the distance ds between A and C in terms of dx^{1} and dx^{2}? Since dx^{1}, dx^{2}, and ds all represent distances with comensurate units, it's clear that any formula relating them must be homogeneous in these quantities, i.e., they must appear to the same power. One possibility is to assume that ds is a linear combination of dx^{1} and dx^{2} as follows 

_{} 

where g_{1} and g_{2} are constants. In a simple onedimensional manifold this would indeed be the correct formula for ds, with g_{1} = g_{2} = 1, except for the fact that it might give a negative sign for ds, contrary to the idea of an interval as a positive magnitude. To ensure the correct sign for ds, we might take the absolute value of the right hand side, which suggests that the fundamental equality actually involves the squares of the two sides of the above equation, i.e., the quantities ds, dx^{1}, dx^{2} satisfy the relation 

_{} 

where we have put g_{ij} = g_{i} g_{j}. Thus we have g_{11}g_{22}  4(g_{12})^{2} = 0, which is the condition for factorability of the expanded form as the square of a linear expression. This will be the case in a onedimensional manifold, but in more general circumstances we find that the values of the g_{ij} in the expanded form of (2) are such that the expression is not factorable into linear terms with real coefficients. In this way we arrive at the secondorder metric form, which is the basis of Riemannian geometry. 

Of course, by allowing the secondorder coefficients g_{ij} to be arbitrary, we make it possible for (ds)^{2} to be negative, analagous to the fact that ds in equation (1) could be negative, which is what prompted us to square both sides of (1), leading to equation (2). Now that (ds)^{2} can be negative, we're naturally led to consider the possibility that the fundamental relation is actually the equality of the squares of boths sides of (2). This gives 

_{} 

where the sum is evaluated for m,n,a,b each ranging from 1 to n, where n is the dimension of the manifold. Once again, having arrived at this form, we immediately dispense with the assumption of factorability, and allow general fourthorder metrics. These are nonRiemannian metrics, although Riemann actually alluded to the possibility of fourth and higher order metrics in his famous inagural dissertation. He noted that 

The line element in this more general case would not be reducible to the square root of a quadratic sum of differential expressions, and therefore in the expression for the square of the line element the deviation from flatness would be an infinitely small quantity of degree two, whereas for the former manifolds [i.e., those whose squared line elements are sums of squares] it was an infinitely small quantity of degree four. This pecularity [i.e., this quantity of the second degree] in the latter manifolds therefore might well be called the planeness in the smallest parts… 

It's clear even from his brief comments that he had given this possibility considerable thought, but he never published any extensive work on it. Finsler wrote a dissertation on this subject in 1918, so such metrics are now often called Finsler metrics. 

To visualize the effect of higher order metrics, recall that for a secondorder metric the locus of points at a fixed distance ds from the origin must be a conic, i.e., an ellipse, hyperbola, or parabola. In contrast, a fourthorder metric allows more complicated loci of equidistant points. When applied in the context of Minkowskian metrics, these higherorder forms raise some intriguing possibilities. For example, instead of a spacetime structure with a single lightlike characteristic c, we could imagine a structure with two null characteristics, c_{1} and c_{2}. Letting x and t denote the spacelike and timelike coordinates respectively, this means (ds/dt)^{4} vanishes for two values (up to sign) of dx/dt. Thus there are four roots given by ±c_{1} and ±c_{2}, and we have 

_{} 
The resulting metric is 

_{} 

The physical significance of this "metric" naturally depends on the physical meaning of the coordinates x and t. In Minkowski spacetime these represent what physical rulers and clocks measure, and we can translate these coordinates from one inertial system to another according to the Lorentz transformations while always preserving the form of the Minkowski metric with a fixed numerical value of c. The coordinates x and t are defined in such a way that c remains invariant, and this definition happily coincides with the physical measures of rulers and clocks. However, with two distinct lightlike "eigenvalues", it's no longer possible for a single family of spacetime decompositions to preserve the values of both c_{1} and c_{2}. Consequently, the metric will take the form of (3) only with respect to one particular system of xt coordinates. In any other frame of reference at least one of c_{1} and c_{2} must be different. 

Suppose that with respect to a particular inertial system of coordinates x,t the spacetime metric is given by (3) with c_{1} = 1 and c_{2} = 2. We might also suppose that c_{1} corresponds to the null surfaces of electromagnetic wave propagation, just as in Minkowski spacetime. Now, with respect to any other system of coordinates x',t' moving with speed v relative to the x,t coordinates, we can decompose the absolute intervals into space and time components such that c_{1} = 1, but then the values of the other lightlines (corresponding to ±c_{2}') must be (v + c_{2})/(1 + v c_{2}) and (v  c_{2})/(1  v c_{2}). Consequently, for states of motion far from the one in which the metric takes the special form (3), the metric will become progressively more asymmetrical. This is illustrated in the figure below, which shows contours of constant magnitude of the squared interval. 



Clearly this metric does not correspond to the observed spacetime structure, even in the symmetrical case with v = 0, because it is not Lorentzinvariant. As an alternative to this structure containing "superlight" null surfaces we might consider metrics with some finite number of "sublight" null surfaces, but the failure to exhibit even approximate Lorentzinvariance would remain. 

However, it is possible to construct infiniteorder metrics with infinitely many superlight and/or sublight null surfaces, and in so doing recover a structure that in many respect is virtually identical to Minkowski spacetime, except for a set (of spacetime trajectories) of measure zero. This can be done by generalizing (3) to include infinitely many discrete factors 

_{} 

where the values of c_{i} represent an infinite family of sublight parameters given by 

_{} 

A plot showing how this spacetime structure develops as n increases is shown below. 



This illustrates how, as the number of sublight cones goes to infinity, the structure of the manifold goes over to the usual Minkowski pseudometric, except for the discrete null sublight surfaces which are distributed throughout the interior of the future and past light cones, and which accumulate on the light cones. The sublight null surfaces become so thin that they no linger show up on these contour plots for large n, but they remain present to all orders. In the limit as n approaches infinity they become discrete null trajectories embedded in what amounts to ordinary Minkowski spacetime. To see this, notice that if none of the factors on the right hand side of (4) is exactly zero we can take the natural log of both sides to give 

_{} 

Thus the natural log of (ds)^{2} is the asymptotic average of the natural logs of the quantities (dx)^{2}  c_{i}^{2}(dt)^{2}. Since the values of c_{i} accumultate on 1, it's clear that this converges on the usual Minkowski metric (provided we are not precisely on any of the discrete sublight null surfaces). 

The preceding metric was based purely on sublight null surfaces. We could also include n superlight null surfaces along with the n sublight null surfaces, yielding an asymptotic family of metrics which, again, goes over to the usual Minkowski metric as n goes to infinity (except for the discrete null surface structure). This metric is given by the formula 

_{} 

where the values of c_{i} are generated as before. The results for various values of n are illustrated in the figure below. 



Notice that the quasi Lorentzinvariance of this metric has a subtle periodicity, because any one of the sublight null surfaces can be aligned with the time axis by a suitable choice of velocity, or the time axis can be placed "in between" two null surfaces. In a 1+1 dimensional spacetime the structure is perfectly symmetrical modulo this cycle from one null surface to the next. In other words, the set of exactly equivalent reference systems corresponds to a cycle with a period of m, which is the increment between each c_{i} and c_{i+1}. However, with more spatial dimensions the sublight null structure is subtly less symmetrical, because each null surface represents a discrete cone, which associates two of the trajectories in the xt plane as the sides of a single cone. Thus there must be an absolutely innermost cone, in the topological sense, even though that cone may be far off center, i.e., far from the selected time axis. Similarly for the superlight cones (or spheres), there would be a single state of motion with respect to which all of those null surfaces would be spherically symmetrical. Only the accumulation shell, i.e., the actual lightcone itself, would be spherically symmetrical with respect to all states of motion. 
