[The following is a guest post from Bjoern Malte Schaefer (see his last guest post here). Bjoern is still one of the curators of the Cosmology Question of the Week blog, which is also still worth checking out. Enjoy!]
The aim of theoretical physics is a mathematical description of the processes taking place in Nature. Science is empirical, meaning that its predictions need to comply with experimental results, but other categories are very important (but not decisive): Theoreticians look for elegance, consistency and simplicity in their descriptions, they aim for abstraction and unification, and look for reduction of the laws of Nature to a few fundamental principles and at the same time for analogies in the description of different phenomena. The subject of this article is to show how these aspects are realized in classical physics, although many of the arguments apply to relativistic physics as well - or only show their true meaning in this context. I do apologize for some of the mathematics, and I promise to keep it as compact as possible.
Formulation of physical laws with differential equations
Physical laws are formulated with differential equations, which relate the rate of change of a quantity to others, for instance the rate of change of the position with time to the velocity. This rate of change is called a derivative. The solution to these equations usually involve an initial value of a quantity under consideration, and compute the value at each instant in solving the equation. The fact that the laws of physics are formulated with differential equations is very advantageous because they separate the problems of evolution of physical systems from the choice of initial conditions for the evolution. Using differential equation for e.g. deriving the motion of planets leads to the abstraction to what forces planets are subjected and how they move under these forces. It predicts naturally the orbits of planets without fixing a priori the orbits themselves, as for example Johannes Kepler might have thought.
Let’s discuss a straightforward example: The motion of a body under the action of a force in Newtonian dynamics. Newton formulated an equation of motion for this problem, which stipulates that the acceleration of a body is equal to the force acting on it, divided by the mass of the body. If, in addition, the acceleration is defined as the rate of change of velocity with time and the velocity as the rate of change of position with time, we get the usual form of Newton’s equation of motion: The second derivative of position is equal to the force divided by mass: This is the prototype of a differential equation. It does not fix the trajectory of the body (the position of the object as a function of time) but leaves that open as a solution to the differential equation under specified initial conditions (the position of the object at the starting time).
Already in Newton’s equation of motion there are two very interesting details. Firstly, the solution to the equation without any force is found to be one with a constant velocity, or with a linearly increasing coordinate, which is known as inertial motion. And secondly, the equation of motion is a second order differential equation, because of the double time derivative. This has the important consequence that motion is invariant if time moved backwards instead of forwards.
A generalisation to this idea is the classical description of gravity. In a very similar way, the gravitational potential is linked through a second-order differential equation to the source of that field, i.e. a central mass. How would this work by analogy? In the mechanics example above, the source of motion was given by the force and both were linked by the second derivatives. Here, the second derivatives of the potential are linked to the sourcing mass again by a second order differential equation, which in this context is called the Poisson-equation, named after the mathematician Denis Poisson.
Would this idea work in any number of dimensions? It turns out that one needs at least three dimensions to have a field linked to the source by a second-order differential equation, if the field is required to vanish at large distances from the source and if the field is symmetric around its source, which are all very sensible requirements. Surely the gravitational field generated by a point mass would be the same in every direction and the attracting effect of the gravitational field should decrease with increasing distance.
Is there an analogy to the forward-backward-symmetry of Newton’s equation of motion? The field equation is invariant if one interchanges the coordinates by their mirror image, therefore, Nature does not distinguish between left and right in fields, and not between forwards and backwards in motion. These are called invariances, in particular the invariance of the laws under time-reversal and parity-inversion. And finally, there’s an analogy to inertial motion, because no gravitational field is sourced in the absence of a massive object. The mass is the origin of the field in the same way as force is the reason for motion.
Joseph Louis Lagrange discovered a new way of formulating physical laws, which is very attractive from a physical point of view and which is easily generalizable to all fields of physics. How it works can be seen in a very nice analogy, which is Fermat’s principle for the propagation of light in optics. Clearly, light rays follow paths that are determined by the laws of refraction, and computing a light path using Snell’s law is very similar to using Newton’s equation of motion: At each instant one computes the rate of change of direction, which is dictated by the refractive index of the medium in the same way as the rate of change of velocity is given by the force (divided by mass). But Fermat formulated this very differently: Among all possible paths leading from the initial to the final point light chooses the fastest path. This formulation sounds weird and immediately poses a number of questions: How would the light know? Does it try out these paths? How would the light ray compare different paths? It is apparent that Fermat’s formulation is conceptually not easy to understand but one can show that it leads to exactly the right equation of motion for the light ray.
Lagrange’s idea was to construct an abstract function in analogy to the travel time of the light ray, and to measure a quantity called action. Starting from his action he could find a physically correct equation of motion by constructing a path that minimizes the action, in complete analogy to Fermat’s principle. Lagrange found out that if one starts in his abstract function with squares of first derivatives of the dynamical quantities, they would automatically lead to second order equations of motions, so the basic parity and time-reversal symmetries are fulfilled. In addition he discovered, that if he based his abstract function on quantities that are identical to all observers, he could incorporate a relativity principle and make a true statement about a physical system independent from the choice of an observer.
The formulation of the laws of physics with differential equations is very attractive because it allows to describe different solutions that might exist for a physical problem. For the motion of the planets around the Sun there is a universal mathematical description, and the planetary orbits themselves only differ by choosing different initial conditions for the differential equation. There is, however, yet another feature present in the equation of motion or the field equation, which is related to Lagrange’s abstract description.
Clearly, any description of a process must be independent if the length-, time- and mass-scales involved are changed: This feature is referred to as universality or mechanical similarity, because it allows to map solutions to the equation of motion onto others. For instance, the orbit of Mercury would be a scaled version of the orbit of Neptune, the orbits can be mapped onto each other by a redefinition of the length- and time-scales involved. This was considered be an essential property of the laws of physics, because it implies that problems fall into certain universality classes and that there is no limit of validity of the solution. Coming back to the problem of the motion of objects in gravitational fields one finds Kepler’s third law, which states that whatever the orbit of a planet, the ratio between the third power of the orbital radius divided by the orbital time squared is always a constant. It is completely sufficient to solve the problem of an orbiting planet in principle, the orbits of other planets do not even require solving the differential equation again (with different initial conditions), but all possible solutions follow from a simple scaling operation. A more comical example are astronauts walking on the surface of the moon with much smaller gravity: their movements appears to be in slow motion, but speeding up the playback would show them to move perfectly normal.
The last question is of course what the true meaning of Lagrange’s abstract function should be: It is very successful in deriving physically viable equations of motion and field equations, but before the advent of relativity it was unclear how it should be interpreted: It turns out that the Lagrange-function of moving objects is the proper time and that the Lagrange-function of the gravitational field is the spacetime curvature. Objects move along trajectories that minimize the proper time elapsing on a clock moving with that object, and the gravitational field is determined as the minimal curvature compatible with a source of the field. These interpretations require that spacetime has at least four dimensions (instead of three), and they lead to viable second-order differential equations respecting time-reversal and parity-invariance. Both quantities, proper time and curvature, are invariant under changes of the reference frame, so relativity is respected, and are invariant under choosing new coordinates - this is in fact the expression of universality. And one has learned one additional thing, which must appear beautiful to everybody: The laws of Nature are geometric, a very complicated, position dependent geometry, whose properties are defined through differential equations. The lines of least proper time are straight in spacetime in the absence of a force, and considering gravitational fields in cosmology it is even the case that the expansion of space is constant an empty universe, both as a reflection of inertial motion. But there is one new phenomenon: Gravitational fields do not vanish at large distances as Newton thought, rather, they start increasing at distances above 10^25 meters, where gravity becomes repulsive under the action of the cosmological constant, and this feature brakes scale invariance.
The formulation of the laws of Nature led physicists to a geometric description of physical processes in the form of differential equations, and variational principles are a very elegant way of formulating the origin of equations of motion and field equations. The true meaning of the variational principles only became apparent with the advent of relativity. It is even the case that other forces, like electromagnetism, the strong and the weak nuclear force have a analogous description, involving an abstract geometry on their own. Finally, it was realised by Richard Feynman that the way in which Nature realizes variational principles was through the wave-particle duality of quantum mechanics - but that is really the topic of another article.