Are there resonant circuits in General Relativity?

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: {{{T}_{a}}=0}.

What I find interesting is that the equation that remains is a homogenous equation that has discrete solutions if the assumed flow is harmonic:

{{V}^{a}}={{\tilde{V}}^{a}}{{e}^{i\omega t}}

The reason is that the resultant equation is an eigenvalue equation for the frequency:

{{f}_{ab}}{{\tilde{V}}^{b}}=i\omega {{g}_{ab}}{{\tilde{V}}^{b}}

The possible frequencies are based on the eigenvalues of an antisymmetric matrix {{f}_{ab}}; 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.

Strategic preferences

At the heart of the discussion of strategic decision-making is how to value each agent’s strategic position. In the theory of games, this is based on the notion of utility; the assumption is that each agent values the outcome of the decision independently. I carry over that notion of valuation into decision process theory. I assume that each agent or player can measure the utility of any given strategy by assigning a numerical preference. As in game theory, the player can also measure the utility of a mixture of utilities by thinking of the choice as assigning sequence of frequencies to each pure possibility and making choices with those frequencies over a sequence of plays. In either case the preferences are idiosyncratic: they go with the player who owns those choices.

It is clear that preferences defined in this way provide a numerical position that is more than just a number for each strategy, it represents a physical attribute of the decision-making process. It is very much like a global positioning system for keeping track of positions on the earth; a fair amount of coordination is involved to relate one position to another. We get by with our GPS systems because this complexity is hidden inside our devices. Such a global positioning system is also used in theories of which decision process theory is a special case. My initial approach was to adopt the same strategy as used in such theories to define a positioning system. I have adopted what is called in the literature, harmonic coordinates to define the positions. It requires the definition of a scalar field for each strategic direction. This scalar field captures the physical characteristics of preferences associated with that direction. There are constraints on the behaviors of these scalar fields that arise from the theory that can be verified by detailed analysis of real world behaviors. The theory addresses the detailed coordination alluded to above.

Since many detailed examples are given in the white papers on this site, it may be helpful to put those numerical examples into a more general context. I suggested that a very large class of models, called stationary models in the literature of general relativity, is a useful class of models to study for decision-making. In these models, there is always a frame of reference in which the decision flows are stationary and the distance metric is independent of time. In the formal language of differential geometry, time is an isometry. I specialized to the case in which the flows are not only stationary but zero in this special frame, which I called the central co-moving frame. The detailed coordination described above however is absent in this frame.

To get an idea of why, consider an analog of a wave traveling in water. The coordination of interest is the behavior of the wave. In particular, if one generates a wave at a source, we want to know the behavior of all the subsequent ripples. There is nothing however that prevents us from viewing the wave from the perspective of an army of corks uniformly distributed and riding on the surface of the water. Each cork, from his point of view, is at rest (co-moving). The model assumption is that over time, the attributes of the water are constant for each cork (though in principle different for different corks). The cork doesn’t see anything of direct interest to us. Nevertheless, from the behavior of each cork, we gain spatial knowledge about the water. That knowledge can be used to reconstruct the ripple behavior if we add additional equations. The harmonic equations from differential geometry are just such additional equations. They take the spatial information from the co-moving frame and provide wave equations that depend on that spatial information to project the behavior of the ripples that might occur.

Software delivery schedules and standing waves

I worked on a large software development project in which the most interesting thing was why the project was delivered two years late to the customer. Both the customer and we, the vendor, knew from the outset what was required for the project. We both agreed on the work that needed to be accomplished and the time it would take to accomplish that work. About half way through the completion of the project, the customer added new requirements, which we as the vendor agreed could be done maintaining the original schedule, but with a known amount of increased effort.

We both looked at the project from a static perspective: we based our estimate of the initial effort on jobs we had done in the past that were similar. We estimated the increased efforts due to additional requirements on the same historical data. This view looks at the completion date as a random event distributed with a normal distribution having a small error. The shape of the distribution, including the standard deviation, is based on the historical data. Despite the best efforts of the development team, the total effort needed to complete the project as well as the delivery time were vastly underestimated.

I believe that the lessons learned from this project are directly related to the need to view the software delivery process as a dynamic as opposed to a static process. We in fact brought in an outside (Systems Dynamics) consultant on the project and learned the following.

  • Our detailed understanding of the project was fundamentally correct: the number of engineers needed to produce a given amount of code didn’t change after the new requirements were added.
  • Our understanding of the quality of the code produced by the engineers didn’t change based on an assessment of their skill set.
  • It was well understood that new hires would be less skilled than those with training in the areas under development. Despite this understanding, standard practice was to not take into account such details when doing cost and schedule estimates. Normally, such differences would generate small increases in costs associated with training.

Though there were many other factors, these three lessons were already sufficient to gain an understanding of why the costs and schedules were terribly out of whack. Let’s say that skilled developers would develop code that contained at most 10% errors. For argument sake, suppose that the delivery of product to the customer would allow 1% error. Suppose further that a test cycle to determine errors would take 6 months and that this was built into the original schedule. One test cycle after initial completion is sufficient to deliver a product of the requisite quality. Now imagine the situation of adding new requirements with the commensurate hiring of new personnel with less experience on the product. Because of the new hires, the initial delivery would contain many more errors, say as an example, 25% errors. So if we start with 100 units, we have 25 units that have errors after the initial pass. After the planned 6 months, we have 25/4 units with errors assuming the same quality of testing and fixing, which is not sufficient to deliver a quality product. This means an additional 6 months of testing, which gets us down to 25/16. To get below the 1 defect requirement, we now need an additional 6 months of testing, to reach the defect level of 25/64. Thus we get an additional year of development because of the change in quality of the new hires.

The actual quality was worse on the project and there were a few other factors, but the essential aspect of the story is unchanged: a dynamic look at the mechanisms demonstrates that small factors lead to unexpected and huge effects. This example illustrates how decisions propagate and impact outcomes. Decision process theory provides a theoretical foundation for such effects that is illustrated in the various models that we have detailed in our white papers. In this software example, we were misled when we assumed certain effects were static, such as the quality of the engineers. We assumed that the engineers would instantly become experts, ignoring what we equally well knew to be true that a training period was needed to make that happen; such a training period could in fact be a couple of years, well beyond the time we allowed for a development cycle.

There are examples in physics where we also make such assumptions, which don’t usually cause problems, but again can lead to incorrect results. A mechanism that is closely related to decision process theory would be that of gravity. In physics we assume that for most purposes, gravity is static. Under extreme conditions however, this is a bad assumption. If our sun were to explode, we would not feel the gravitational effect for several minutes because of the time it takes for the cause to make its effect felt. Such effects might be labeled gravity waves, despite our incorrectly labeling of gravity as a static scalar field.

In decision process theory, it is also true that causes generate effects that are separated by a finite time. There is always a propagation speed: effects are never instantaneous. In our models, in our model calculations we have focused initially on streamlines, which are paths along which the scalar fields are constants. For a picture, imagine the motion of air with smoke; the smoke provides visual evidence of the behavior of the streamline. One specific model might be of someone speaking, whose voice generates sound waves. We would capture the streamlines as displaying a global wave pattern, one that is not visible on any one streamline. Once sound waves are generated, we would see the streamlines undulate: the path of a velocity peak would propagate with a form we call a harmonic standing wave, analogous to shaking a jump rope whose other end is attached to a wall. This wave velocity is quite distinct from, and often much faster than, the media velocity.

Here is a model calculation using Mathematica of what a harmonic standing wave looks like in decision process theory for an attack-defense model:

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