It’s Tuesday again, just like it was last week on this day, and I’m still doing my “daily”* blog posts, since I don’t have any desire either to write fiction or even to play any guitar.  This is at least a quasi-productive way for me to use time that I would have used to write fiction, at least until the Second Law of Thermodynamics claims me at long last, and I rush—all oblivious—into its cold but comforting embrace…to poeticize idiotically a simple fact of physics and mathematics.

Tuesdays often make me think of the Beatles song, Lady Madonna, because for me, one of the most memorable lines of that song is “Tuesday afternoon is never-ending”.  This is particularly pertinent when things are slow at work in the afternoon, though I don’t think most other people regard dull days at the office in terms of songs, like I often do.  This being me, I tend to focus on dark and/or negative songs and lyrics, or at least melancholy** ones.

I rarely think of Thursdays in terms of my stockings needing mending, at least.

The notion that Tuesday afternoon is never-ending raises an almost Zeno’s Paradox type notion.  If Tuesday afternoon really were never-ending, then Wednesday would never arrive, so there would never be another day.  Although, despite it always being Tuesday afternoon, if people could nevertheless still move and act and do things, it would be useful to break time into manageable chunks for the purposes of scheduling, planning, working, sleeping, and so on.  Also, it’s never Tuesday afternoon everyplace on Earth at once, so if Tuesday afternoon in Britain were to be never-ending, then Tuesday morning in the US, Canada, Mexico, Brazil, etc. would be never-ending, and Tuesday evening for most of Europe, and of course, Tuesday night into Wednesday morning for places east of that, right up to the international date line.

And, of course, if the Earth had stopped spinning—assuming it had done so without the numerous catastrophic effects this would otherwise entail (watch this lovely video by Vsauce to see some of these discussed)—the weather patterns on Earth would be permanently changed and made horrific.

Depending on whether Earth became the equivalent of tidally locked on the sun, or if it had just stopped rotating, it would either have a permanent sun-facing side, or it would have a day as long as its year.  Then again, even a year-long day is not literally never-ending, so I guess it would be the “tidally locked” situation.  Before long, the Prime Meridian would become a very hot strip of Earth indeed!  And the International Date Line would become extremely cold.

It is tangentially interesting to think about—having mentioned Zeno’s Paradox earlier—the notion of continuously divisible time.  If time (or distance, as in Zeno’s original paradox) were infinitely divisible, à la the real number line, it would seem that one could never experience the passage of time because before you could get to Tuesday evening you would have to go halfway through Tuesday afternoon…and before you got halfway, you’d need to get a quarter of the way…and before that you’d need to get an eighth of the way…and so on.  If things are infinitely divisible, or so says the “paradox”, you should never be able to get anywhere, either in space or time, because no matter how arbitrarily close you choose two points in space to be, or two points in time, or two numbers on a number line, there are an uncountable infinity of points in between.

Calculus, of course, deals with this issue by means of taking limits as distances go to zero, and the like; it handles instantaneous and continuous rates of change quite nicely, thank you very much, while still rigorously defining functions in terms both accurate and useful.  As for reality itself, it seems to side-step the issue entirely by making space and time, in practice, not infinitely divisible at all.

The minimum distance that makes any physical sense is the Planck length, and the minimum time is the Planck time.  To say you’ve traveled half a Planck length, or that something lasted half a Planck time, is apparently saying something that has no meaning in the real world.

Of course, the Planck length and time are REALLY small:  1.6 x 10^-35 meters and about 10^-43 seconds.  So, we cannot directly measure either of them with current technology, anyway.  Not even close.  But they are real things, when it comes to quantum mechanics, with real, verifiable physical implications that have been tested and confirmed with tremendous accuracy and applicability.

One does tend to wonder, though, about spacetime itself.  According to General Relativity, gravity is not a force in the sense that electromagnetism and the strong and weak nuclear forces are forces but is instead a manifestation of the curvature of spacetime, leading objects in it to attempt to follow the closest thing to a straight line (a geodesic) in a curved, “flexible” four-dimensional structure, in the way one has to follow a great circle on the surface of the Earth to pick the “straightest” possible path between two points on the surface of a spheroid.  This really matters for airplanes, and even for ships.

But is space itself infinitely divisible?  GR*** treats it as such, but GR conflicts with Quantum Mechanics at places of small size and high mass, producing senseless results (so I’m told…I haven’t done the figuring myself, regrettably).  Spacetime certainly seems to be able to expand indefinitely, as it has done since at least what we call the Big Bang, and it continues to do so at an increasing rate even as we speak, so to speak.  That’s trivial to conceive of with things like continuous variables, real numbers, things with uncountable infinities between any two points.  Just multiply everything by two, say, and all the numbers are twice as big, and just as uncountably infinite.

But if space is discontinuous, in some sense, as implied by presumed quantum gravity, how does the expansion manifest?  Does more space pop into existence between two regions formerly separated by a mere Planck length?  We know that if you try to separate two quarks that are bound to each other, the strong force between them becomes so intense that new, formerly virtual, quarks pop into actual being between them****.  Is this what happens with spacetime itself?  As intervals get stretched, do new nuggets of spacetime appear?

We know that it’s possible to produce new, positive energy in spacetime, balanced by the “negative” energy of gravity, so there is no local violation of conservation principles*****.  Maybe spacetime spontaneously generates more spacetime, using the force of the cosmological constant, or its equivalent, to create these new bits of spacetime as it goes along.  It seems plausible, given what we know about the finite divisibility of things we’re able to confirm experimentally, and at least little bits of spacetime seem much less energetic on a per-unit basis than things like quarks or even electrons and neutrinos.

Infinite divisibility may work quite nicely in mathematics—indeed, it does—but it may not be plausible in the real, physical world.  Spacetime is real, and if it expands, then that expansion must happen at some level and be describable in principle.

None of which changes the fact that Lady Madonna is an awesome song.

*I put “scare” quotes around that, because technically, it’s not a true daily function, since even if I continue doing it for a long time, I don’t expect to write on Sundays, and probably roughly not every other Saturday, since I won’t be going to work, and I write this during my commute.

**“Melancholy” has become a rather soft kind of negativity in modern parlance, but I wonder how people would feel if they considered when using the word that it comes from the old concept of “black bile”, one of the supposed four “humours”.

***General Relativity.

****Not a violation of Conservation of Energy…they get their substance from the energy you applied trying to separate them.

*****Again, alas, I have not done the specific math myself, but the concept is straightforward and logical.  One can similarly create a new positive electric charge as long as one creates a balancing negative charge at the same time.  It happens in nuclear decay all time.