To an outsider, physics and mathematics might appear to be almost identical disciplines. Particularly at the frontiers of theoretical physics, where a very deep knowledge of extraordinarily advanced mathematics is required to grasp even cutting-edge physics from a century ago — curved four-dimensional spacetimes and probabilistic wavefunctions among them — it’s clear that predictive mathematical models are at the core of science. Since physics is at the fundamental core of the entire scientific endeavor, it’s very clear that there’s a close relationship between mathematics and all of science.
Yes, mathematics has been incredibly successful at
describing the Universe that we inhabit. And yes, many mathematical advances
have led to the exploration of new physical possibilities that have relied on
those very advances to provide a mathematical foundation. But there’s an extraordinary
difference between physics and mathematics that one of the simplest questions
we can ask will illustrate:
What is the square root of 4?
I bet you think you know the answer, and in all honesty, you probably do: it’s 2, right?
I can’t blame you for that answer, and it’s not
exactly wrong. But there’s much more to the story, as you’re about to find out.
A ball in mid-bounce has its past and future trajectories determined by the laws of physics, but time will only flow into the future for us. While Newton’s laws of motion are the same whether you run the clock forward or backward in time, not all of the rules of physics behave identically if you run the clock forward or backward, indicating a violation of time-reversal (T) symmetry where it occurs.(Credit: MichaelMaggs Edit by Richard Bartz/Wikimedia Commons)
Take a look at the above time-lapse image of a bouncing ball. One look at this tells you a simple, straightforward story.
- The ball starts off on the left side of the image, where it’s clearly been dropped with some speed while also moving to the right.
- The ball bounces while continuing to move to the right, accelerating downward due to gravity, reaching a maximum height and then falling back down to the floor again.
- That collision with the floor robs the ball of some of its kinetic energy, but it still bounces upward, continuing to rise (but to a lesser height than after the previous bounce) and move to the right, while gravity accelerates it back down toward the floor.
- And, if we were continue to monitor this ball, we’d find that it would move to the right, while continuing on in a series of bounces, with each successive bounce taking it to a lesser and lesser height until it ceased bouncing altogether, remaining on the floor and rolling until it comes to rest.
This is, quite reasonably, the story you’d tell yourself of what’s going on.
But why, may I ask, would you tell yourself that
story rather than the opposite: that the ball begins on the right side, moving
leftward, and that it gains energy, height, and speed after each successive
“bounce” on the floor?
In Newtonian (or Einsteinian) mechanics, a system will evolve over time according to completely deterministic equations, which should mean that if you can know the initial conditions (like positions and momenta) for everything in your system, you should be able to evolve it, with no errors, arbitrarily forward in time. In practice, due to the inability to know the initial conditions to truly arbitrary precisions, this is not true.(Credit: ESO/M. Parsa/L. Calçada)
The only answer you’d likely be able to give, and you may find it dissatisfying even as you give it, is your experience with the actual world. Basketballs, when they bounce, lose a percentage of their initial (kinetic) energy upon striking the floor; you’d have to have a specially prepared system designed to “kick” the ball to higher (kinetic) energies to successfully engineer the alternate possibility. It’s your knowledge of physical reality, and your assumption that what you’re observing is aligned with your experiences, that lead you to that conclusion.
Similarly, look at the diagram, above, that shows three stars all orbiting around a central mass: a supermassive black hole. If this were a movie, instead of a diagram, you could imagine that all three stars are moving clockwise, that two move clockwise while one moves counterclockwise, that one moves clockwise and two move counterclockwise, or that all three move counterclockwise.
But now, ask yourself this: how would you know whether the movie were running forward in time or backward in time? In the case of gravity — just as in the case of electromagnetism or the strong nuclear force — you’d have no way of knowing. For these forces, the laws of physics are time symmetric: the same forward in time as they are backward in time.
Individual protons and neutrons may be colorless entities, but the quarks within them are colored. Gluons can not only be exchanged between the individual gluons within a proton or neutron, but in combinations between protons and neutrons, leading to nuclear binding. However, every single exchange must obey the full suite of quantum rules, and these strong force interaction are time-reversal symmetric: you cannot tell whether the animated movie here is shown moving forward or backward in time.
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