general relativity

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The sun is the same in a relative way…but not as much as the CMB

Meet the new day; same as the old day.  Or at least, in everything but the trivial details, every new day is indistinguishable from the previous oodles of days.  It’s just a full rotation of the Earth around its axis, such that when it’s done, the same points on Earth are lined up with the same…

…well, okay, here there’s some variation depending on how you’re measuring the day.  There are solar days (our usual measure), which are reckoned when the same line of latitude points “directly” at the sun again.  But, of course, the Earth is moving relative to the sun, and at a noticeable rate:  it goes roughly one 365.25th of the way around the sun in one day, and so it takes slightly longer to get back to the original orientation toward the sun than it would otherwise.

Then there are sidereal days, in which we reset when the same point on Earth is aligned with the same relatively distant stars.  Such distance means the angular change to those stars caused by Earth’s motion is much smaller than the angular change to the sun.  So those day lengths are, I believe, more consistent, but they are actually slightly less than 24 hours long.

I suppose, at our current level of technology and understanding, the most consistent day measure would be Earth’s rotation relative to specific spots on the cosmic microwave background radiation, which would basically be as far away as anything we can measure.

The Milky way is moving slightly relative to the CMB though.  I think this is called proper motion*, but I may be misremembering that term.  Still, that rate of motion is tiny relative to the scales involved.  Although, come to think of it, since the Earth is moving within and along with the Milky Way, it’s possible that the Earth’s motion relative to the other, relatively distant stars within the galaxy may be smaller than its motion relative to the CMB.

I don’t know whether or not that is the case.  If anyone out there knows off the top of your head, I would be delighted if you would let me know.  But don’t bother Googling it, or asking some LLM about it; I can do that for myself if we’re going to resort to such tactics.

Speaking of relative motion and such, I did something relatively new yesterday afternoon with respect to reading.  Allow me to set the stage**.

I recently started dabbling (very inconsistently) in the online courses at MIT (and other universities) on their YouTube channel(s).  I started with Quantum Mechanics and then also started the General Relativity course, since I have some personal curiosities related to that subject that I’ve yet to see answered.

In the first lecture, the GR professor went over the main texts they would be using.  Well, I already have two of them, so I figured I would get the third.  I ordered it in paperback (the hardcover was ridiculously expensive, whereas the softcover was about the same price as any good-sized trade paperback).  After it arrived, it sat rather forlornly on my desk for some weeks.

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Now, as I’ve said, I’ve been having a hard time reading fiction or nonfiction lately.  But a softcover textbook is no heavier than my lapcom would be, unlike the two hardcover GR texts I own.  So I brought the text General Relativity, by Wald, with me when I left the office yesterday.

The cool thing is, I actually read from it while waiting for the train and then while on the train.  I completed the preface/introduction and chapter one!  It’s not a long chapter, of course, and it mostly reviewed concepts of special relativity and so on***, which I know relatively (har) well.

Still, I feel reasonably pleased to have done this, and I hope to continue the process.  Maybe once I get this book under my belt, I’ll feel more inclined to do similarly with Sean Carroll’s Spacetime and Geometry, which is hardcover but not too terribly heavy.  Gravitation, on the other hand, is a book comparable in scale to Harrison’s, the “bible” of internal medicine.  It’s not a book one tends to carry around (though it provides its own rather pointed example of the effects of gravitation).

Anyway, with any luck, maybe I’ll inaugurate a more useful commuter reading schedule.  And maybe I’ll achieve enough skill at GR that I might be able to explore my perennial question.  At least, I should be able to learn more stuff, and that’s always a good thing.

I hope you all have a good day.

*Actually, no, in astronomy “proper motion” has to do with angular movement with respect to more distant stars, or something along those lines.  I don’t think it’s a great term for that.  The motion of a galaxy relative to the CMB is apparently called the “peculiar motion”, which seems like a rather peculiar choice to me.

**I don’t know how you could stop me even if you so wished.

***It has one practice problem, which is a version of the classic barn door paradox aka the ladder paradox, but I didn’t work the problem.  I know the basic answer already; it involves the relativity of simultaneity in special relativity, and the author asks us to draw a spacetime diagram demonstrating the basic issue.  I had no resources to do this on the train.

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Noisy events on the horizon of my attention span

It’s Tuesday, isn’t it?  Well, I guess it may not be Tuesday when you’re reading this, but it’s Tuesday as I’m writing it.  It’s the second day in the latest of a seemingly endless stream of utterly pointless “work weeks”.

Welcome to our world.  Welcome to our world.  Welcome to our world of noise.

That’s a paraphrase of the song that was (and may still be) sung by the dancing animatronic puppets in the main front area of the big F.A.O. Schwartz store that sits just by the southeast corner of Central Park in Manhattan.  I’m not sure why I felt like including it there, but it definitely expresses the sentiment I have that nearly everything in the universe is effectively “noise” in the information theoretic sense.  At the very least, the signal-to-noise ratio in the world is vanishingly tiny.

It’s not zero, mind you.  There’s some info hiding in all the nonsense.

Of course, whether something is signal or noise depends very much on what signal you’re seeking.  If you’re trying to detect gravitational waves, then nearly everything else around is “noise” in the sense that it is not evidence of gravitational waves, and is just going to make that evidence harder to find.  But if you’re an ornithologist, then at least some of that seeming noise might be the birdsong “signal” of a rarely seen species there in Louisiana, which I think is where the first LIGO observatory was constructed*.

And, of course, if you’re a seismologist, what you consider a significant signal would very much be noise to the LIGO people.  If there were a gravitational wave strong enough to be seismically significant, it would have to be from a very close and catastrophically violent event.

We don’t expect there to be such a thing any time soon.  And apart from such events, gravitational waves are so relatively weak‒gravity being by far the weakest of the “forces” of nature‒that so far they can only be detected from things like black hole and/or neutron star mergers, which are ridiculously violent events.

Incidentally, apparently recent observations of one such merger has given confirmatory evidence for Stephen Hawking’s black hole horizon theorem**.  That states that when two black holes merge, the (surface) area of the new, combined event horizon must be at least as large as the two prior event horizon areas combined.

In this, as in other things, black holes and their horizons act very much like the 2nd Law of Thermodynamics, and that is consistent with the Bekenstein-Hawking thesis that the entropy of a black hole is proportional to the area of the event horizon, as measured in square Planck lengths.  Indeed, the maximum entropy‒the maximum information‒of any given region of space is that which would be encoded upon an event horizon that would hypothetically enclose such a space.

As for the volume of a black hole within the event horizon…well, that’s harder to quantify.  The apparent radius, as judged from the sphere of the event horizon‒the Schwarzschild radius for a non-rotating black hole‒is almost certainly much smaller than the radius that would be perceived by someone within the horizon, for spacetime is very distorted there.  Indeed, I suspect that, at least by some measures, the volume within a black hole‒or at the very least the radius from the “center” to the horizon‒is infinite, with the “singularity” actually stretching down away forever.

Of course, an asymptotically infinite well of that sort need not always have infinite volume.  There is, for instance, the counter-example of “Gabriel’s Horn”, a shape made by rotating a truncated function (y = 1/x for x ≥ 1) around the x-axis.  This shape has infinite surface area, but it has a finite volume(!).  So you could fill it with paint, but you could never finish painting the inner and outer surface.  Weird, huh?

Of course, the dimensionality of things within a black hole’s event horizon is probably at least one step higher than things in the Gabriel’s Horn comparison, so the finite/infinite comparisons may not translate.

I’d like to be able to do a better job working that out with more than my intuition; that’s one reason why I own no fewer than four fairly serious books on General Relativity.

That’s not the only reason, of course.  I would also like to try to solve what happens to a space ship that accelerates near enough to the speed of light that its relativistic mass and relativistic length contraction puts it below its own Schwarzschild radius (at least in the direction of motion).  Also, how would that figuring be changed if the ship were rotating around the axis of its motion***?

Unfortunately, I rarely have the mental energy to put into pursuing adequate mastery of the mathematics of GR, and so I can (so far) just try to visualize and “simulate” the spacetime effects in my imagination.  That’s fine as a starting place, but even Einstein had to master the mathematics of non-Euclidean geometry and matrices and tensors before he could make General Relativity mathematically rigorous.

It’s almost certainly a pipe dream that I will ever get to that level of expertise.  My chronic pain and chronic depression (dysthymia) combined with the effects of my ASD (level 2****, apparently) and the effort that’s required for me to act “normal” enough to get along just really wear me out mentally.  It’s frustrating.  I have a stack of pertinent texts above my desk at work, where I hope they will entice me.  I even have a copy of my old Thomas and Finney college calculus text there too, so I can do some reviewing in that.

If only I were able to spend some time without pain and to get a good night’s sleep once in a while, I might even make progress.  I suspect that such things are not in the cards, however.

I would love to be dealt The Magician (in Tarot cards) but I fear that I am just The Fool.  Oh, well, that’s all just metaphorical, anyway.  It’s possible to predict the future, of course, but it is difficult, and it’s very unlikely that any set of cards‒however cool they may be‒is the way to do it.

*I remembered correctly.  It is in Louisiana.

**The theorem, being a theorem, is mathematically rigorous, but the question remains whether it describes the way our universe actually works.  That is always a matter of credences rather than “proof” in the mathematical sense.  In the real world, probabilities may come vanishingly close to zero or to one, but they never quite reach them.

***In Special Relativity, when something is traveling around a circle at a significant fraction of the speed of light, length contraction has the effect of “shrinking” the circle from the “point of view” of that which is moving at that speed.

****”Requiring substantial support” according to the official definition.  I do not have such support.