For some time I have been exploring behaviors in differential geometries, one of which is the geometry of General Relativity. I have also been looking at circuits in electrical engineering and was taken with the idea of resonances. They occur in circuits that have an inductor, capacitor and a resistance. Resonances also occur in physical media such as musical instruments, bells and bridges. In all of these cases, one can “drive” the system with an oscillating force and observe the possibility of resonance by the appearance of a super strong response to what may seem like a small driving force.
Alternatively, one can look at such systems without a driving force and start it from some initial condition; one then observes the system “ringing” or resonating for some time thereafter.
We approach such systems ordinarily as being described by linear equations and think of the two cases as examples of solutions to such equations that have both homogeneous contributions (the latter “ringing” solutions) and inhomogeneous contributions from the driving force. For the inhomogeneous contributions, a wave solution at a given frequency results in a steady state solution of the same frequency with an amplitude determined by the details of the media. For the homogeneous contributions, there is no initial frequency, but typically an algebraic equation determines the resonant frequency possibilities; there are only a finite number of such solutions.
You might think the situation is different in the differential geometries that I have been considering. The equations are not linear. There is no longer the presumption that small changes in the initial conditions lead to small changes in the resultant behaviors. For example one might get “chaotic” solutions from the simplest of forms.
However, I think one can still analyze such systems in differential geometry as if they were resonant systems. The basic idea is that resonance effects should manifest whenever the various energy contributions cancel in the sense that they leave a system that looks like there are no external forces. I can put this thought into the equations for a differential geometry theory.
The left hand side is the acceleration in the theory, which depends in part on the metric field. In general the theories have isometries, symmetries that leave the metric unchanged, and these isometries lead to “electromagnetic fields” or Coriolis Forces. The first term on the right represents such a contribution. Finally there will be a host of other effects, which reflect various contributions that drive the curvature of space-time, such as inertial stresses. We think of these terms as forces that “drive” the behavior. They are the analog of forces in a circuit. If we take the analogy seriously, we speculate that resonance occurs when these forces cancel: .
What I find interesting is that the equation that remains is a homogenous equation that has discrete solutions if the assumed flow is harmonic:
The reason is that the resultant equation is an eigenvalue equation for the frequency:
The possible frequencies are based on the eigenvalues of an antisymmetric matrix ; the eigenvalues are zero or purely imaginary. Thus the solutions lead to a discrete set of real frequencies. I suggest that these are the resonant frequencies of the “circuit”.
There are a variety of interpretations, depending on the details of the differential geometry. For example, for general relativity, the antisymmetric matrix could be the Coriolis force and reflect a rotational frame. It could also reflect the electromagnetic field and the rotational field is the magnetic field. For decision process theory, the matrix can represent the payoff field. In each of these cases, one expects that there will be essentially a “free fall” solution that has a discrete set of frequency contributions. In addition, there will be solutions with any frequency, but such solutions will no longer be “free fall”. They will correspond to having a non-zero “driving force”. It is interesting that we find both types of solutions in the numerical exercises for decision process theory.
It is interesting to speculate on whether there are physical manifestations of such discrete solutions in physical processes such as transmission lines. I think the usual analysis does not indicate the possibility. The possibility might occur however if one considers transmission lines in magnetic fields or possibly with additional symmetries, such as circular transmission lines that rotate.