Time Reversal

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    What would we see if we could travel backward through time, like H. G. Wellsís Time Traveler returning home from the time of the Eloi and the Morlocks? In that scenario time would elapse forward for us and elapse backward in the rest of the Universe as we see it. The second hand on my analog wristwatch will continue to rotate clockwise but outside our time warp the hands of clocks will rotate counterclockwise. Simple logic lets us figure out what would happen, of course, but what do the laws of physics have to say? First, letís specify that we are taking a Godís-eye view of things because the laws of optics would otherwise distort what we imagine seeing.

    We start where mathematical physics began, with gravity. The most simplistic statement we have says that gravity makes things fall down. Under time reversal would things fall up?

    Imagine that we could see, in reversed time, the incident that inspired Albert Einstein to conceive the equivalence principle, the foundation on which he established his theory of General Relativity. We would see a carpenter and his tools leaping out of a sandpile and rising back to the roof from which they fell; we would see them fall upward. But then we would discern that gravity is still pulling them downward. They would come up out of the sandpile at high speed, slow their rate of ascent as they rose, and come to a full stop when they reached the roof. Gravity doesnít seem to be changed by time reversal.

    We know that latter statement must stand true to Reality, that in reversed time gravity cannot become repulsive, because we know that the planets must continue to move on their orbits about the sun, albeit in the retrograde direction. In order to follow their ellipses they must be attracted to the sun in accordance with Isaac Newtonís inverse-square law. So reversing time does not reverse gravity. It also does not reverse centrifugal force, which we can deduce through simple dynamic geometry, just as Newton did.

    How does time reversal affect electromagnetism? Through the electric force bodies can either attract or repel each other. Do those roles get interchanged by time reversal?

    The simplest electric system that we have for an example consists of a single hydrogen atom, which comprises a negatively-charged electron and a positively-charged proton attracting each other. Hydrogen can last a long time: absorbed into cold planets, such as Uranus or Neptune, it can exist unchanged for billions of years. So we know that in reversed time we wonít see hydrogen atoms flinging themselves apart. The force holding the electron and proton together remains attractive.

    Likewise, we know that like charges will continue to repel each other in reversed time. If that statement did not stand true to physics, then inside stars, at all depths, protons would rush together and commit fusion at rates far greater than what we see in forward time. The Universe that we see in backward time must differ from the Universe that we see in forward time only in having its motions reversed, so again the electric force doesnít change: in reversed time like charges still repel each other.

    A similar analysis also tells us that the magnetic force doesnít reverse in reversed time, but we also get something else. The Poynting vector, described by the vector cross product of an electric field vector and a magnetic field vector, describes a flow of energy and that flow reverses when time elapses backward; thus, we infer that in reversed time either the electric field or the magnetic field, but not both, must reverse its direction.

    Maxwellís Equations must remain valid under time reversal and itís easy to see that they do. Faradayís law tells us that the curl of an electric field equals the negative time derivative of an associated magnetic induction field. The time derivative operator changes algebraic sign under time reversal, so either the electric field or the magnetic induction field, but not both, must reverse in order to keep the equation valid. Ampereís law, as modified by James C. Maxwell, tells us that the curl of a magnetic field stands in direct proportion to an electric current density and the time derivative of an associated electric field. In reversed time the charged particles constituting the electric current reverse their motions, so if the magnetic field remains unchanged by the time reversal, then electric charges must reverse their signs, positive becoming negative and vice versa, and the electric field must reverse because of the reversed sign on the time derivative. Maxwellís first equation, Gaussís law for the electric field agrees: if the electric field changes in reversed time, then so do electric charges. Unfortunately, Maxwellís Equations donít give us enough information to determine which field, the electric or the magnetic, reverses in backward time, but the equations themselves remain valid nonetheless.

    The laws of thermodynamics give us pause. The first law, conservation of energy, remains unaffected by time reversal. The amount of energy going into an interaction equals the amount of energy coming out of the interaction, regardless of the direction in which the interaction plays out. The third law, Nernstís theorem, also remains unaffected. The second law, which defines the arrow of time, gives us a different matter.

    According to the second law of thermodynamics, no system can, of itself, reduce the amount of entropy associated with it. Implicit in that statement we have the proviso that the system move forward in time. In reversed time, then, we must have the statement that no system can, of itself, increase the amount of entropy associated with it. In less abstract terms, we would have to say that heat, of itself, will not go from a hotter body to a colder body in backward time.

    In a system that consists of very-weakly interacting particles, the entropy stands in direct proportion to the natural logarithm of the number of different ways in which those particles can form a given macroscopic manifestation. There are relatively few ways in which smoke particles can be arranged into a small puff put into a jar compared to the number of ways in which those particles can be distributed evenly throughout the jarís entire volume; thus, the particles in a puff of smoke, pushed around by Brownian motion, will spread and fill the jar, thereby giving the jar and its contents an increase in their entropy. If time were to reverse its elapse, we would expect to see the diffusion process run backward and the smoke re-accumulate into a small puff. But, given the probabilistic nature of the air-smoke mixture, can we say that such a re-accumulation conforms to the laws of physics?

    The Newtonian laws of motion and the equations that express them include a tacit assumption that the geometric and dynamic quantities involved are determinate with infinite precision. If Reality did not conform to that assumption, then objects and events in the past, observed from a time-reversed perspective, would differ from the same objects and events observed from a time-forward perspective. The smoke in the jar, for example, seen by an observer traveling backward in time would not re-accumulate into a small puff. As some of the first computer models of climate showed, leading to the discovery of the butterfly effect, even the most minuscule of differences in the orientations, locations, and motions of the elements of a system lead to major differences in the systemís manifestation. Unless the reversal of time made all of the properties of all systems of particles reverse with infinite precision, history would look remarkably different from what we remember. Without infinite precision in time reversal, Brownian motion would keep the smoke in our jar widely dispersed.

    But couldnít history actually differ from what we remember? How do we know that it doesnít? How do we know that the past contains only one unique history?

    Consider the Burgess Shale, an outcrop of fossilized mud high in the Canadian Rockies in British Columbia. A little over half an eon ago it was mud at the bottom of a shallow sea near the base of a 160-meter cliff. If we were to go to that outcrop today, we might find, as an example, the impression and fossilized remains of a little creature called Hallucigenia.

    Hallucigenia was a lobopodian worm less than an inch long. It had a double row of spines on its back and seven or eight pairs of tentacle-like legs, each leg ending in a pair of tiny claws. In the seafloor mud off the coast of Laurentia 505 million years ago, one of those worms slithered across the landscape, its arms wriggling in the mud to propel the creature forward and also to search for food, such as the eggs of other creatures. Then an undersea mudslide buried it and a great many other creatures, all of which became part of the Burgess Shale. Finding the remains of that one creature in the Shale gives us solid (actually, truly solid) evidence that history is unchangably fixed; that if time were to reverse and we could see into the middle of the Cambrian Period, we would see that one particular individual worm going about its business. That analysis tells us that the laws of physics remain infinitely precise under time reversal.

    But now we come to the quantum theory, which tells us that Reality contains no such thing as infinite precision. Heisenbergís indeterminacy principle makes that fact explicit. It tells us that we face an indeterminate future as time elapses forward. We thus conceive Reality as consisting of a single, determinate past emerging from a Now that transforms the potentialities of an inchoate set of indeterminate futures into the actuality of that single, absolutely determinate past. If we turn time around, should the past become indeterminate in accordance with the laws of the quantum theory?

    The remains of an Hallucigenia say no. History must have a perfectly unique shape and, therefore, it must be absolutely determinate. That fact necessitates that the laws of physics, which control how events occur and cause each other, must also be absolutely determinate in reversed time.

    But the quantum theory emerges from the proposition that the Universe does not acknowledge any action whose value is less than Planckís constant. That proposition comes from applying the finite-value theorem to the principle of least action and appears most clearly in Heisenbergís indeterminacy principle (šxšp h).

    With each and every particle in the Universe we associate a state function. It has the mathematical form of a wave and it relates the particleís necessary properties (such as mass, electric charge, and spin) to its contingent properties (such as locus, momentum, and energy). The state function itself does not represent anything that we can measure; rather, it is a complex number whose square gives us the probability density of finding the particle at any given point of space at a given time. The particle only has a definite position when it interacts with another particle in a collision.

    Consider a collision between two particles. It occurs at a definite time and place. The two particles on their blurry trajectories have, by chance, encountered each other, exchanged energy and momentum, and flown off on new blurry trajectories. In reversed time that collision wonít occur in the same way (backwards, of course), if at all. Bornís theorem tells us that each particle has only an indefinite location on its blurry trajectory (thatís what makes the trajectory blurry), so as it travels backward in time it will not necessarily come back to the same point at the same time that the collision occurred. It has only a probability of being there-then. History must thus be indeterminate: events cannot occur in reversed time as backward copies of the same events occurring in forward time. Unless we have missed something.

    We can borrow and adapt a concept devised by David Bohm, that of the pilot wave. Indeed, the quantum theory in reverse time might actually be Bohmís implicate order. In forward time, though, we canít take all of Bohmís version of the quantum theory, because it doesnít conform to Bellís theorem, but one small piece of it will give us what we need. Let us hypothesize that whenever two particles collide, the collision produces a spherical memory wave that expands at the speed of light. As the memory wave expands, its amplitude drops quickly and it effectively ceases to affect the ghostwave described by the state function.

    In reversed time the memory wave contracts, coming to a point at the very place and time where-when the collision that created it occurred. As the wave contracts, its amplitude increases and it begins to augment the ghostwaves associated with the two particles. The surface of the wave gets smaller and it shifts the probabilities in the ghostwave, transforming what might look like a Gaussian distribution of probabilities into a Dirac Delta, which represents a one-hundred-percent probability that both particles come to the right point at the right instant. Thus, in reversed time the collision looks exactly like a temporal mirror image of the collision occurring in forward-elapsing time.

    But now we must ask the question thatís been hanging fire since I began this essay Ė is time reversal real? Does any phenomenon actually move backward in time?

    We have at least one opinion that says yes. According to Richard Feynman, antimatter consists of ordinary particles traveling backward in time. A positron, for example, carries a positive electric charge because it is a time-reversed electron. Verification of that proposition comes through Diracís Equation, which has negative-energy solutions. But negative energy denotes an absurdity, so in his 1949 paper "Theory of Positrons" Feynman interpreted the negative-energy states as positive-energy states of a particle moving backward in time. He could hardly have done otherwise: by way of Einsteinís mass-energy relation, we see that the negative-energy state corresponds to a negative inertia, which, on first impression, is absurd.

    If I attach a rope to an object with a negative inertia and then pull on the rope, the applied force, directed toward me, should make the object move away from me. The equal and oppositely directed change in the objectís linear momentum, required by Newtonís third law of motion (conservation of linear momentum), would give the object a negative velocity (a velocity directed opposite the force), but the rope prevents the object from acquiring that velocity and, thus, responding to the applied force as the Newtonian law requires. Nothing can violate a fundamental law of physics, so we must infer that negative inertia cannot and does not exist.

    As Feynman said, then, the negative-energy solutions of Diracís Equation must refer to particles moving backward in time. Under time reversal the electric charges of particles get reversed, negative charges becoming positive charges and vice versa, so now we also know that electric fields get reversed under time reversal and magnetic fields do not.

    So now we know a little more about how the laws of physics work and see a little clearer why Reality has the form that it does.

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