It’s Tuesday now, the day that Professor Coyne, aka PCC(E), over at Why Evolution Is True calls “the cruelest day”. I’m not sure the origin of that expression; as far as I can recall, his website is the first place I encountered it, but I like it.
It’s not the beginning of the week, which has a certain hectic energy at least, with everyone in a kind of recovery from their—hopefully restful—weekend. It’s not “hump day”, which many people call Wednesday, when things are starting to coast toward the end. And, of course, it’s not its counterpart: Thursday, which is a day on which anticipation of the coming weekend can energize one for the day’s work. And, quite obviously, it’s not Friday, when those who are on a 5-day-a-week schedule are effectively already beginning their weekend**. Tuesday is the day with the least to make it stand out. Which, of course, makes it stand out.
Also, as the Beatles pointed out, and as I often note, Tuesday afternoon is never-ending. And, if time were to be truly continuous and infinitely divisible, then one could effectively make Tuesday afternoon never-ending in a Zeno’s Paradox sort of way, just by subdividing the time in between each moment as each moment passed.
Or, of course, one could fall through the event horizon of a black hole. To distant observers, that fall would indeed seem to be never-ending (though before too long the image of the faller would redshift into invisibility). And for the person falling, the end would come rather quickly. Assuming that person survived the gravitational tides, according to General Relativity, time literally comes to an end in the singularity of a black hole.
Though I always picture the heart of a black hole a bit more like one of those “Gabriel’s Horn” shapes in mathematics, which has an infinite surface area but a finite volume. Of course, I don’t have the skills and expertise to work the equations of GR, but it feels to me that, if spacetime is endlessly flexible****, then there need never be a true “end” to time; it could just stretch longer and thinner always, infinite in “surface” but finite in “volume”.
I know that’s all a bit esoteric, and I’m sure my understanding is incomplete. If there are any theoretical physicists specializing in GR reading this who can help me think more clearly about black holes and singularities and why it would be necessary for time to completely end if spacetime were continuous rather than simply to stretch—making a mathematical singularity, but not literally an end—then please do let me now.
I realize that there may be concepts that can only be dealt with rigorously using the mathematics, but on the other hand, clearly the mathematics is translatable into “ordinary language” at some level, or no one would ever be able to teach it or learn it. And I have at least a bit of mathematical background, though I haven’t formally studied how to do the matrices and whatnot involved in GR. Still, Einstein himself didn’t know how to do it when he came up with the initial ideas, so he had to learn it and then work with it, but he had the ideas first.
I don’t have his brilliance, obviously—which is certainly not an insult—but if there’s a way to demonstrate why time literally ends at a singularity***** rather than simply stretching out into an endless tube, with shrinking cross-section (in higher-dimensions) but ever-expanding “area” (again, in higher dimensions), I’d like to know. I mean, according to the whole Dark Energy paradigm, the expansion of spacetime is accelerating now and there’s no theoretical limit to how much it can expand, which seems to mean, at some level, that it has infinite stretchability.
Or perhaps it would be more accurate to say that spacetime can continue to be created between any two points that are stretching apart, somewhat—but not quite—analogous to the way in which if you try to separate two bound quarks, all you do is create two new partner quarks with the energy you’ve put in to try to stretch them so now you’ve got two pairs of inseparable quarks. Neener neener neener.
Anyway, I know that Penrose and Hawking developed their singularity theorems for black holes and those are accepted by physicists and mathematicians throughout the world. They are/were brilliant people, there’s no doubt about that. But does the theorem mean that spacetime literally vanishes at some literally infinitely dense point in the middle of a black hole—which strikes me as implausible given the stretchy-stretchy nature of spacetime—or is it a singularity more in the pure mathematical sense like the function 1/x as x approaches zero?
Enquiring minds want to know.
Wow, that wasn’t at all where I thought I was going when I started this post today, but those random, drunken walks can, at times, at least lead past interesting scenery. No one would be likely to argue that a black hole doesn’t necessarily belong in a wasteland; in a sense, it is the ultimate wasteland, at least this side of the heat death of the universe. But it is interesting, topographically (and topologically, to a novice such as I), and though it would be nice to be able to enjoy such scenery with company who would appreciate it in a similar fashion to the way I do, well…one has no “right” to such a thing and no good reason to expect it. It’s lonely, but at least the wasteland has places of beauty.
And if one gets tired of walking, and/or one is curious enough to see where it leads, one can always just jump into that black hole.
*This is a slightly altered line from the Pink Floyd song Fearless, off their excellent album Meddle.
**Some of us work every other Saturday, of course, and when you have no life, like I have no life, a weekend is not something to which to look forward, except for the chance to rest one’s back. I don’t really do anything for fun, have no friends with whom I spend time, no places that I go for entertainment or for shopping or whatever. All such things are too tainted by memories of loss, and anxiety, and the feeling of not belonging on this planet. My life is more or less a wasteland. But I can’t see any way out of it (other than the obvious), and I can’t even really tell if I’m just walking in circles within it. I think I’m walking in random patterns, like a “drunkard’s walk” (though I rarely drink). And, of course, in a random walk or drunkard’s walk, one will eventually get arbitrarily far away from one’s origin point (though the average location will be the origin, interestingly), but the distance between one and the origin increases—I think, if memory serves—only logarithmically. And I suspect that the exit from the wasteland is very far away, if it exists at all (other than, as I say, the obvious). Oh, well. Life promises one thing and one thing only; anything else is just luck***.
***A footnote within a footnote, just to note the mildly amusing fact that, so far, my footnote is longer than the main text of this post.
****A big “if”, of course. It doesn’t seem to jibe with quantum mechanics, apparently, but we have no convincing theory of quantum gravity to settle the issue. I’m so frustrated.
*****Again, according to General Relativity—I know it’s thought not to be the correct picture in such extreme circumstances, because of the uncertainty principle, among other things.