take?
The Scholium problem as presented here is clearly extremely difficult, we can however simplify the problem by noting that we do not need to determine ω(τ ), A(τ ) and N (τ ) for all τ , rather it suffices to determine the instantaneous values of these quantities. This has the advantage that we can treat the problem algebraically as opposed to working with sets of non-linear coupled ordinary differential equations. The Scholium problem therefore consists of uniquely determining the instantaneous values of ω, ω0, A, N , N0, ri, ri0, r00i, mi, plus any additional unknown constants which enter into the
potential V , for example Newton’s gravitational constant G.
9.4.1
An algebraic approach to the Scholium problem
Let us illustrate some of the complications which arise when pursuing the algebraic approach. We can firstly eliminate some of the unknowns by fixing the units for mass, distance and time. This is allowed because the data {θi, ϕi} is dimensionless.
However, even with this slight simplification we still have more unknowns (4n + 9) than the number of independent equations (3n). Therefore the equations of motion alone do not provide a solution to the Scholium problem. The reason for this is because the Newtonian equations of motion only predict the subsequent evolution of the system provided the initial positons and velocities are specified, as well as the masses and any additional constants. It is these constants which make the Scholium problem underdetermined.
One simple way to proceed would be to introduce additional equation which the data {θi, ϕi} should satisfy. This could be done by taking additional derivatives of
5Note the clear distinction between this problem and the n-body problem. The n-body problem is
the problem of finding the solution xi(t) to Newton’s equations for all t given the initial data, which
9.4 Solving the Scholium Problem: What does it take? 117
the Newtonian equations of motion. With each derivative we get an additional 3n independent equations, whereas only an additional n + 7 unknowns: n unknowns for derivatives of ri and 7 for additional derivatives of ω, A and N . In particular if we
take d derivatives we have 3n(d + 1) equations and 4n + 9 + d(n + 7) unknowns. It is easy to check that we need a minimum of 4 particles in order to be able to solve the Scholium problem, in which case we would need to take at least 13 derivatives. On the other hand for only one additional derivative d = 1, we need the number of particles n ≥ 16.
Clearly in full generality the Scholium problem is not tractable analytically, one could of course simplify this problem by allowing for additional observables. For exam- ple, Barbour has suggested in [120] to take the distances rij between all the particles
as directly observable. But as we have mentioned it is difficult to independently verify the accuracy of, for example our rulers without assuming (at least in practice) the properties that one is attempting to determine. We will leave the problem of how one can solve the Scholium problem for future analysis and turn to the implications for different types of solutions.
9.4.2
The implications of different solutions
There are three situations we could find our selves in when facing the Scholium problem. That is, we could have no solution, multiple solutions and lastly a unique solution. Let us discuss these in a little more detail highlighting in particular the implications they have on the foundation of Newtonian mechanics.
If there is no solution to the Scholium problem, i.e. there is not a set of values for all the unknowns such that all equations are simultaneously satisfied, then we must conclude that Newtonian mechanics is fundamentally flawed, in particular there is no operational way in which to test this theory. This could of course happen since the Newtonian equations of motion are underdetermined in the unknown parameters, and if indeed this were the case it would be a remarkable coincidence that Newtonian mechanics has been so successful. One possible reason for why this might be the case, may be because the physical world is fundamentally relational and as a result any properties of absolute space and time are empirically inaccessible. This could therefore provide evidence for favouring relational theories over absolute ones, that is theories whose dynamical variables are relational in nature. For example, theories where the dynamical variables are simply the distances between all the planets [121].
If there are multiple solutions to the Scholium problem then it may be that New- tonian mechanics is as a physical theory ill-defined, it cannot make unique predictions. This of course depends somewhat on the degeneracy of the solution. It may be that certain parameters are uniquely determined whilst others are not. In such a case one should study the degeneracy and determine what predictions if any Newtonian me- chanics can make. For example it may be that only the relational quantities remain invariant whilst the absolute quantities such as ω, A and N are dependent on the particular solution. This again may suggest that the fundamental nature of the world is relational, however if this were the case we would be right back where we started,
namely what is an accurate clock? This form of a solution is probably the hardest to test since it may not be clear given a particular solution how one can generate another. If however, this were the case there would at least be a explanation for the remarkable success of Newtonian mechanics.
Finally, it may be that the Scholium problem has a unique solution. This would be the ideal case all unknowns will have unique values and hence it will be possible to define not only an accurate clock which marches in step with absolute time, but also the relationship between an arbitrary observers frame and an inertial frame of reference. Newtonian mechanics will as a theory be on solid ground. It would also be difficult to motivate a relational theory given both the success and simplicity of Newtonian mechanics. A theory based in absolute space which although cannot be directly observed its presence can, given the equations of motion, be deduced.