Hamiltonian Flow Fields: How motion may be real but time an illusion
Hamiltonian Flow Fields define motion without a concept of time
Seeing all the talk about one Hamilton has made me think of another, not Alexander but William, not revolution but discovery of the principles of energy and motion in classical mechanics.
Classical mechanics is the study of how matter and energy moves. Since the time of Newton, motion has been presented as the way in which objects change their position and velocity with respect to time. For Newton and later contributors from Lagrange and Laplace to Hamilton and Maxwell, time was built into the universe by the Watchmaker God. It was the medium through which motion occurred.
Einstein objected to time having a special role in mechanics. He argued that physical law should have no special directions in space or time. Instead, they ought to be generally covariant. Einstein later showed that time is like a dimension and that our perception of time is not universal but dependent on our state of motion. Quantities that we thought had strict dependencies on time turned out to covary when their motion changed.
In a way Einstein eliminated much of the role time had in defining motion for, in his theory of relativity, it became more like any other direction in which an object or field could morph and change. Time lost much of its specialness.
As scientists started to use Einstein’s theory in more practical ways, particularly for the study of the universe as a whole, however, the need for a theory of time became more and more apparent. Attempts to develop theories of motion led to the development of time based cosmologies such as the expanding Robertson-Walker universe. Others developed the theory of geometrodynamics, the study of how spacetime geometry changes in time. Wheeler and DeWitt created their famous quantum gravity equation for how the quantum wavefunction of the universe evolves. All of these ideas screamed the primacy of time.
Yet Einstein’s original objections to time remained: how can there be a special direction in spacetime? And if there is then why does it exist? Ockham’s razor would suggest that all theories should explain why one direction is preferred over another. We can’t go back to Newton’s day. Relativity shows us in countless ways that time cannot be as unique as it appears.
One way out of this dilemma is to show how dynamics itself creates time. This approach suggests that rather than motion being change in time, time is motion and change. That is, we have our ideas about time backwards. Time does not really exist, change does. And change does not require time. All it requires are two things: energy and something called a phase space.
Energy and phase space are both critical pieces of classical dynamics going back centuries. One way to represent these two ideas is through something called a Hamiltonian.
A Hamiltonian is Energy in Phase Space
A Hamiltonian, simply put, is an expression for the energy of a dynamical system. It is named after William Rowan Hamilton who developed his theories on dynamics in the 1830s.
Every Hamiltonian depends upon variables that represent positions in phase space. These are usually represented with the variable names p and q, where p is momentum (velocity times mass for constant mass objects) and q is a coordinate, often position. If there are many, many particles or entities in the Hamiltonian, we put a little subscript on the p’s and q’s.
Hamiltonians can also depend directly on time. If it does, you know it doesn’t represent a “closed” system. A time dependence indicates that the system is exchanging energy with something else not represented. An example might be a small, hot piece of metal thrown into a cold lake. The Hamiltonian could only represent the metal and not the lake, even though the metal is losing energy to the lake.
Once you have the Hamiltonian of a system, it is very easy to derive the equations of motion for that system. You just take the derivative of the Hamiltonian with respect to your p’s and your q’s. You then get flow of the p’s and q’s in time.
Hamilton’s equations look like this:
The H is the Hamiltonian. If you aren’t familiar with (or don’t remember) calculus, that’s okay. A derivative is just a change of one quantity with respect to another. In this case, it says that the change in p (momentum) with time is equal to the negative of the change of the Hamiltonian with respect to q (your coordinate). Likewise, the change in your coordinate q with respect to time is the same as the change of your Hamiltonian with respect to your momenta p.
That means there is a very close relationship between the energy of an object, H, where it is, q, and how fast (and hard) it’s going, p. It’s all wrapped up in motion which is what mechanics is.
In my previous article I showed a number of animations of how springs look in phase space, the space of all momenta and positions of those springs. I computed those animations by solving Hamilton’s equations (or more correctly I had Python do it for me). Given an initial position and momentum for the spring, I could easily compute where it would be and how fast it would be moving (and in what direction) at any time. Here’s an example for a perfect (ideal) spring:
Hamiltonian Flow (Vector) Fields
Solving Hamilton’s equations is one thing, but what about the equations themselves? It turns out that Hamilton’s equations define what is called a flow field or vector field. The beauty of the Hamiltonian Vector Field (HVF) is that we don’t even have to solve Hamilton’s equations to see what it looks like. Instead, it is defined by Hamilton’s equations themselves. For the spring oscillator above, here is a picture of its flow field:
The flow field is an example of a vector field. That is, at every point in the phase space, (q,p) there is a vector (something with direction and magnitude) (dq/dt,dp/dt) that defines the motion of an object with that Hamiltonian if it were to end up at that point in phase space.
Unlike a single solution to the Hamiltonian equations, the flow field tells you something about all solutions. Just pick any point in phase space and your system will follow the flow field.
A damped spring (one that slows down because of friction) looks like this:
In this case the flow field spirals into the center which is a fixed point for that system. All the solutions head to that central point.
The flow field represents constant energy routes. For the spring, here is a plot of the energy over phase space:
Trajectories in phase space for a closed system follow “level sets”, which are cuts of the three dimensional energy plot at constant energy. They look like this:
Because of the conservation of energy, a spring cannot move from one level set to another unless it becomes an “open system” and allows energy to exchange with something external.
Thus, the flow field defines a direction that keeps a path in phase space on the level set.
Flow Fields and the Nature of Time
The flow theory of the nature of time actually comes out of the idea of a flow field. It says that rather than time being the fundamental medium through which systems move through phase space, it is the system of the universe as a whole that defines a flow through phase space and that flow defines time itself. Thus, we have the whole idea of time backward. We think that motion happen in time, but in fact time happens in motion.
If you go back to Hamilton’s equations you can see, in fact, that time only appears on the left hand sides of the equations. As long as the Hamiltonian is for a closed system, as the universe is, then you do not need time at all to define the Hamiltonian or the flow field that it generated. Therefore, the flow field doesn’t equate to motion in time, it defines what time itself is.
If your Hamiltonian is generally covariant, that is, it doesn’t represent time as special, it can still create time from its own flow.
The universe is just evolving over a level set, following a flow, and that flow is what we perceive as time.