Time Reversal of a Raindrop

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    It is raining lightly this morning. High above the town of Visalia, particles of water dust floating on the wind stick together into clumps. When a clump becomes too big to ride the wind, it falls as a raindrop, gathering more water dust into itself as it falls through the cloud that brought it to Central California. With myriads of other raindrops it falls into the trees and onto the yard in front of my apartment.

    On the ground next to the sidewalk leading away from my apartment some of the water has accumulated into a small, shallow puddle. Where and when a raindrop hits a puddle a ripple forms and then expands away from that point, a thin rise in the water acting out a growing circle. Where the ripple reaches the edge of the puddle the extra water that it carries above the level of the puddle slops onto the wet soil and then flows back into the puddle.

    In dynamic terms the raindrop precipitates an exchange of energy. Initially the raindrop possesses potential energy relative to ground level: it possesses that energy purely by virtue of its altitude in a gravitational field. As the drop descends, it trades potential energy (energy of position) for kinetic energy (energy of motion) and transfers some of that energy to air molecules through collisions that slow the drop’s descent.

    When the raindrop punches into the water in the puddle, it pushes the water aside, doing work against the pressure acting to hold the water in place and thereby losing the last of its kinetic energy. It accomplishes that act by exerting pressure downward on the water and the pressure, exerted uniformly in all directions, pushes the surrounding water laterally. The normal pressure of the puddle pushes the water back into the air gap behind the raindrop, driving enough water into the gap that the water rises up into a small hill.

    Pressure within the hill pushes the water evenly in all horizontal directions, sending the water out in a circular ripple. The water itself doesn’t move far, but the ripple propagates by shifting gravitational potential energy out in an ever-widening circle, making the water itself move to and fro. At the edge of the puddle the ripple slops its water onto the wet soil and then the water flows back into the puddle. The energy in the ripple, derived from the kinetic energy of the raindrop, gets turned into heat in the soil, mass movement of water turned into vibrations of the molecules in the soil.

    We know that the energy involved in this event is conserved; it’s neither created nor destroyed. The energy can only move from one object to another and from one form to another. Those movements must conform to the requirements associated with the law of entropy. That law, often called the arrow of time, applies to fluids and things, like energy, that can be analogized to fluids: it seems to give form to what we call causality and we can state it most simply by saying that fluids tend to spread out with the elapse of time.

    In our raindrop, potential energy is initially contained in the drop and gravity converts it into kinetic energy, still concentrated in the drop. When the drop hits the puddle it shudders the molecules in the standing water, thereby turning some of its kinetic energy into heat. The rest of the kinetic energy goes into the work of pushing the water aside. That energy ends up being carried by the ripple and deposited into the soil. And some small amount of the original kinetic energy is converted into heat in the water by viscosity (fluid friction).

    Because heat consists of kinetic energy manifested in the random motions of molecules that jostle each other, it spreads out as molecules collide with their neighbors. If we could measure extremely small increments of temperature, we would find that after the ripple breaks on the soil, the soil is faintly warmer. A short time later, though, the temperature of the soil goes back to matching the temperature of its surroundings: the heat has spread outward from where the ripple broke. We are no longer able to locate that energy. Aside from a minuscule rise in the level of the puddle, there is no sign that the raindrop has fallen.

    Imagine that we have the means to record that event in all of its aspects and that we have actually made that recording. When we replay the recording we can watch the particles moving in and around the raindrop and watch them exchange energy. We can watch energy moving from particle to particle in the puddle after the raindrop falls into it. And we can watch the energy spreading out and getting lost among other energies in the soil. We see everything moving in accordance with the laws of physics, as we expect.

    But suppose we run the recording backwards. We might focus our attention onto a small patch of soil and see a small amount of water run up onto it, in apparent defiance of gravity. Actually, the random motions of the water molecules result in collisions that produce a net motion of some of the water toward the soil at the edge of the puddle. At the same time the random motions of molecules in the soil produce a coordinated motion that kicks the water up into a ripple. That kick-up happens around the puddle in a way that produces a perfectly circular ripple that shrinks, propagating toward the center of the circle.

    When the ripple comes to the center of the circle, the water piles up into a narrow hill, which descends rapidly into the puddle, leaving a hole in the water as it sinks below the surface. Then the water at the bottom of the hole rushes upward and ejects a drop with such exquisitely precise force that it leaves the surface of the puddle flat and smooth and not moving.

    As the drop rises, air molecules above it move out of its way, as if guided by magic, and air molecules below it rush into the space behind it and undergo collisions that drive them upward against the drop’s underside with sufficient impulses, constituting a collective force, that push the drop upward. Driven by the air molecules, the drop reaches the cloud and particles of water dust fall away from it until the drop disappears.

    It looks as if we are traveling backward in time and we can tell because causality looks weird. The laws that govern the exchanges of momentum and energy work in reverse as they do forward, but the law of entropy doesn’t seem to work right at all. The law of entropy tells us that fluids and fluid-like entities spread out, but in our reversed recording the energy concentrates itself. We don’t actually move backward in time, so this exercise is moot and we can ignore it... or so we might think.

    In the modern theory of quantum electrodynamics physicists, led by Richard Feynman, represent antimatter as consisting of ordinary particles moving backward in time. Thus, they conceive the mutual annihilation of an electron and a proton, for example, as an electron, under the right circumstances, kicking off a pair of gamma photons and thereby getting itself kicked backward in time as the positron. Regardless of whether an observer consists of a person or an elementary particle, that observer must "see" the laws of physics making the world work properly, both forward and backward in time.

    An observer traveling backward in time would see the fundamental forces unaffected by the change in temporal perspective. Gravity remains an attractive force under time reversal and we know this because the planets would still have to be attracted to the sun in order to move on their elliptical orbits, albeit backward. The electric force between opposite charges remains attractive: if it didn’t, atoms would fly apart immediately for someone beginning to move backward in time. In fact, the only phenomenon that moves matter and radiation that is not time reversible is the increase of entropy.

    To illustrate that last statement, I offer the conduction of heat in an insulating solid, such as glass. The molecules in such a solid vibrate around their average positions within the solid. Imagine that we increase the amplitude on one molecule’s vibration, that we give it additional kinetic energy by, say, hitting it. That molecule then jostles its neighbors harder, thereby increasing the amplitudes of their vibrations: some energy gets transferred to those neighbors. Those neighbors jostle their neighbors a little harder and so on and so on. The extra energy that we gave one molecule spreads outward away from that molecule. Because of that spreading, the entropy associated with the additional energy increases, as the second law of thermodynamics requires.

    On the other hand, we know that we will never see a ring-shaped array of molecules jostling their neighbors in a way that makes energy move toward and accumulate in a single molecule at the center of the ring. The entropy associated with that energy would have to decrease spontaneously and that’s just not possible. But if we travel backward in time, that is precisely what we would see. Thermodynamics is not indifferent to the direction in which we move through time. That fact tells us that thermodynamics is the nexus where classical mechanics meets quantum mechanics.

    We know that classical Newtonian mechanics, the mechanics of the planets and of the billiard table, is perfectly time-reversible: from a video of an alien planet and its moons you would not be able to tell whether the planet and its moons were moving forward or backward in time. Quantum mechanics, on the other hand, is irreversible: it’s indeterminate in forward time, so it must be determinate in reversed time. In quantum mechanics we occupy an instant, called Now, that progresses into the future and ravels up myriad possibilities into a single, definite actuality. In reversed time Now unravels our single, definite history into a wide array of potentialities. That proposition enables us to infer something about the fundamental entity of quantum mechanics, the aleatric wave that accompanies every particle of matter and radiation.

    The aleatric wave, standing or traveling, defines, in accordance with Born’s theorem, the probabilities of a particle having a certain value of momentum and/or position. Those probabilities evolve in a way that roughly tracks the motion of the particle until the particle interacts with another particle or object and thereby "collapses the wave function", which is to say that the interaction gives the particle’s accidental properties a perfectly precise value with a probability of one. Then the particle rides another aleatric wave that originates in the interaction.

    While the probabilities rule the values of the particle’s accidental properties, the aleatric wave itself is a precisely deterministic entity. It has a specific value at each and every point in space at each and every instant of time and those values evolve in a deterministic manner in accordance with one or another of the relativistic analogues of Schrödinger’s equation (depending upon what kind of particle it guides). That wave is perfectly reversible for all time: no matter when we look at it, the wave is precisely defined and evolving and that evolution reverses when we reverse the elapse of time.

    This stands in stark contrast to the ripples in our rainpuddle: at some instant of time they dissipate and their energy is scattered among the molecules in the water and soil. The laws of classical physics don’t allow that energy to reconverge on where the ripple broke and thus to reconstitute the ripple. The ripple is only time reversible because of the quantum nature of matter. The aleatric waves involved with the particles reverse precisely and thus cause the scene to evolve in reverse.

    Each particle associates with an aleatric wave that originated in the particle’s last interaction. Moving forward in time, that wave denumerates the probabilities of the particle occupying a given point in space at a given instant. Moving backward in time, the wave narrows down into a certainty and guides the particle to the precise locus in spacetime where the wave originated, in the form of a Dirac delta.

    The aleatric waves of multiple particles are not necessarily independent of each other. The particles are pushed toward or away from each other by forcefields, primarily by the electromagnetic (gravity pulls equally on all particles, so we can ignore it for now). Those forcefields have their own aleatric fields and waves; indeed, in the quantum theory an electromagnetic field consists of an aleatric field populated by virtual photons. We can actually ignore the particles and describe them by saying that the electromagnetic aleatric field refracts the particles’ aleatric waves.

    Our raindrop is not so much a collection of random particles, then, as it is a collective aleatric field/wave carrying the particles with it. That complex aetherial knot of immateriality corresponds to what we would call the ghost of the raindrop. When the raindrop hits the puddle and merges with it, that ghost separates from its particles and continues on its way downward, not interacting with anything else. We know that must happen because it happens to single particles.

    When a single particle comes out of an interaction (a collision with another particle), an aleatric wave accompanies it. That wave encodes Heisenberg’s indeterminacy principle by denumerating the probabilities of the particle actually existing at points around the most probable location of the particle. It exists mostly in a minuscule region around the particle and moves with the particle until another collision changes the particle’s motion. Instead of collapsing and vanishing, as physicists assume, the aleatric wave continues propagating along its original path, albeit without the particle coming with it.

    I assert the truth of that latter statement because we know that nothing that affects the motions of particles can arise ex nihilo. That fact applies here because we know that certain phenomena proceed backward in time, so a time-reversed point of view is a legitimate one. In that point of view a collapsed aleatric wave would appear to spring up gratuitously. Thus we infer that the aleatric wave continues to exist after its particle gets knocked off of it. The wave may cause some minuscule perturbations in the locations of other particles as it washes over them, but such effects are undetectable.

    So the ghost of our raindrop continues to exist as a raindrop. The forces that bound the particles together in the drop have their own ghosts and thus continue to bind the ghosts (the aleatric waves) of the particles together. In reversed time this ghost would rise up out of the ground, gather its original particles into itself, and carry them back up into the sky. In that way we can have an indeterminate future and a perfectly determined past, as we expect.

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