A Morphology of Physical Descriptions

The hierarchy of conceptual frameworks we deploy to usefully explain phenomena, from the most  self-evident but restrictive to the myriad of the allowable reads as:

1. Axioms; 
2. (Theories of) Principle; 
3. (Constructivist) Laws;
4. Solutions (to the differential Field Equations that encode the laws).

As an example, we deploy the guiding principle that all objects imbued with the same "qualities" are indistinguishable from each other. Such a viewpoint has unified many an apparently disparate phenomena.

While Newton's (Laws) assumed implicitly the equality of inertia and gravitational mass, Einstein's General theory of Relativity (GR) made this presumption explicit by geometrising their equivalence. In effect Einstein, identified away the gravitational field as merely a kinematical illusion. At least that is in the locale of stronger gravitational fields, where a freely falling frame can always be identified rendering the gravitational field illusory. The gravitational field has been "socially constructed" and we need only consider as usefully "real" the inertial acceleration field "construct" sourced from the local inertial property of the matter. Such constructivism generally asserts that it is necessary to find (or "construct") a (mathematical) object to prove that it exists. Here it is not just semantics to say that we have deconstructed the gravitational field, supplanting it with the inertial acceleration one. That certain stuff has the quality that it resists a change to its motion in the spirit of Galileo or in Newton's reworking within his First law, is thus self-evidently, tautologically Axiomatised: Inertia is the quality that massive stuff possesses if nothing else (like charge, hypercharge, strangeness..).

The fibre bundle verbein formulation of GR in which a tetrad is tagged with Lorenzian, locally Minkowskian indices (of the inertial reference frame) as well as general co-ordinate ones (of the "inertial acceleration" field) makes the "Equivalence Principle" explicit (quote intended as a nod to the slight of hand being played out). Anyway for our purposes indistinguishability principle is more usually referred to as the Symmetry Principle.

That GR aspires to embrace Mach's Principle but ultimately comes up short rendering its Field Equations as mere purveyors of its solutions as Laws of motion.

Solutions are variously described with an increasing nod to their profundity as trajectories, equations of motion, or if delivered by a Principle as "On-Shell" solutions.  Field Equations as in the case of Maxwell to the left (written in Helmholtz form) can be described in either differential (at a 'point') or Integral form. The latter emphasises the full generality of the form of the equations. That integrals deliver general families of plausible solutions, the resulting trajectory of a system being governed by its particular initial conditions: to be fed exogenously as part of the solving process.

We live in a universe of accelerating matter whose equations of motion represent the (only) possible path that our universe could have taken given the initial condition (Big Bang) primal state and these laws running their course.We view our present epoch as the end of one trajectory, one chosen course from the myriad of otherwise pencilled-in trajectories that were initially possible. The Natural Laws (of GR and Quantum Mechanics) that guide the universe along its particular course delivering the peculiarity of the universe today are to be viewed as exogenously delivered. Themselves derived as solutions to some higher guiding principle. As we shall see it is the Stationary Action Principle that is the meta-principle from whence the field equations are derived.  That an Action principle comprises a Lagrangian object; the difference between the Kinetic and Potential energies of a system renders its definition just about in within the constructivist's scope.

In the following, we will try to unpack the status of assumptions and prescriptions that give
rise to useful explanations of the aggregate behaviour of our observed universe (which we model as an interacting system of material objects undergoing relative accelerated motions in a local inertial field rather rather (say) than due to their mutual attraction).

Newton’s Laws of trajectories on an Axiomatised Euclidean Geometry

Newton’s laws are a set of dynamical laws that only allow for motions in which particles react instantaneously to a change in the relative positioning of each other (and by extension react to any particle in the whole of the universe).

That such a law prescribes the motion means that in a hierarchical sense we view these laws as more fundamental than the trajectory the universe itself traces under its guidance. Omnipresent or omnipotent, either way Newton's laws are viewed as immutable and unchanging through a universal time and space. The laws stand beyond the acts and stage that is the unfolding universe.

We take as given (as Axiomatically the case), the infinitely non-converging parallel line geometry of Euclid to describe, by positional measurement, the motion of particles in terms of arcs and lines. We say that the kinematical description of the vector calculus of changes has itself been unchanging.

As such, the prescription is twofold immutable: the timeless laws prescribe the allowable trajectories from amongst the available omnipresent set of geometrical conic sections that emanate from a particle point. 

Our hierarchy is clear: the geometry as Axiomatic, is self-evidently the most presumptuous aspect of Newton's  model, Newton's Law play out on this fixed arena and its solutions are subservient derived trajectories.

Symmetry in the Laws not of the solutions

We need to be careful in distinguishing between a fundamental law and one of its solution. We need to also be careful to create a hierarchy "laws" based on their conceptual unifying character. To be sure, solutions arising from laws need not themselves look symmetric. 

The law dictates the trajectory; the resulting trajectory merely respects the law. There need be no intrinsic symmetry in the resulting trajectory, in the sense for example, that the family of trajectories from the central force law theories of Newton and Coulomb are not especially circular, merely but variations on conic sections as above. So in a two-body orbiting problem, we observe an elliptic trajectory around the common centre of mass of the two circulating bodies.

We may choose to be an observer fixed on the larger body of the system of two bodies and then observe that we are on a focus of an ellipse with the orbiting body tracing an ellipse around our centre of mass: quite a symmetrical pattern but not the most symmetrical of conic sections that we could envisage.

Rather than truly symmetrical circular trajectories being carved out in space, Nature codifies spherical symmetry at the level of the guiding law. The encoding of symmetry is announced mathematically by representing objects characterised as indistinguishable from each-other with respect to some quality (color, charge, "inertialness") as field or algebra-valued tensor fields.  Newton’s and Coulomb’s guiding inverse square laws (of mass and charge influence at a distance) are laws whose potentials satisfy Laplace's second order differential equation. They are tensor laws describing central conservative forces; vector derivatives of a potential that act in a way that does not distinguish any directionality. The laws inherit the isotropic nature of space with which the forces permeate in being spherically symmetric.

Einstein’s Laws and dynamic Kinematics of Riemannian Geometry

In Einstein's universe, the geometry of space is no longer axiomatically assumed as flat let alone hyperbolic. Rather the Geometry of the stage (of space) itself is demoted to but a solution of the field equations of General relativity.

The space as merely a solution is now viewed

 as having an "historical" character in the sense that the geometries that preside at any moment for an inertial observer are an echo of its local past history. To the right are the principal components of the Kerr metric tensor describing the external space-time intervals resulting from a rotating Black hole.

Parenthetically, it may be helpful to think in this context of the Einstein's inertial geometrisation of a non-linear gravity, in the sense that it remembers its past, as a Martingale: a stochastic process in which the conditional expectation of the next metic interval value is given a prior (value) that is just its current value. The dynamical playing field of rulers and stop-clocks, in themselves being path dependent are a product of the Field equations. The kinematical framework of tensor calculus evolves. Einstein's revolution of the invariant was to demote an (Euclid's fifth -parallel) axiom to the status of but a 'manifold of solutions' to a governing law.

Stationarity as the Unifying Principle of all Action

We have seen how through the evolution of ideas that we must take care to define what is the reach of our laws. [Smolin, Problem with Physics], in his critique of String theory cites as one of the five problems for a Theory of Principle to resolve as the merger of General Relativity and quantum theory into one overarching theory of quantum gravity. Such a Theory of Principle with its universal description of nature is to be distinguished from a lesser constructivist theory. 

For the latter, the archetype we return to is electromagnetic field theory; merely a description of  phenomenon in terms of models or equations. It is a theory of more limited scope in that it can co-exist consistently with other such constructivist theories such as theory of dark matter with which it plainly does not interact.

In contrast the Action Principle has the required unifying character. There are an infinite set of possible paths between two points in space and the guiding hand does not insist on the simple simplicity of linearity of lines or symmetry of circles to traverse that interval. Rather, that the law guides the system to a particular trajectory from its possible set choosing that solution that solves a minimisation problem. The chosen trajectory is in a sense the efficient solution to the law.

The Natural Law will state that the system of interacting particles from its initial configuration to its end configuration will minimise the difference between the system's Kinetic Energy, T and it's potential energy, V.

The latter energy form is of gravitational (or electromagnetic if particles in system are additionally charged) type. Either way the laws prescribe energy due to the relative positioning of the system of particles (with respect to each other) and energy due to their relative speeds with respect to a distant idealised inertial observer outside the system. The law, given the status of Principle: the "Action Principle", dictates this need for the distribution of total energy between these two types to be optimally constrained. The constraint is such that the chosen trajectory is the one in which the sum of these kinetic and positional energies is the minimum. In evolving through time, the path taken at every step is the one that minimises this.

Axioms by Rights Principles as Directives

 In looking at the progressive structure of beliefs that sit at the heart of Theoretical Physics we see that Nature’s adherence to fundamental tenets sometimes mirrors the best of man’s: from those Rights we hold so implicit and inalienable (so as to be presumed to be as god given) to those we see merely as Federal Directives that require enactment at local state level. The European self-evident "Human Rights" are the Axioms of Euclidean Geometry and Grassmanian Algebra and the likes while European "Directives" such as freedom to travel across borders are Physics’ founding Principles. Laws have Solutions that are akin to the various visa entry levels of the nation states. As with the European project there has been, and will continue to be, a need for an overhaul of various tranches of this hierarchy as more abuses and inconsistencies of treatment by participants become apparent.