Tag: curvature of space

A Matter of Perspective

On By rolcottIn Astronomy: LIGO, Physics: Black Holes and Relativity, UncategorizedLeave a comment

As I stepped off the escalator by the luggage carousel a hand came down heavy on my shoulder.

“Keep movin’, I gotchur bag.”

That’s Vinnie, always the surprises.  I didn’t bother to ask how he knew which flight I came in on.  What came next was also no surprise.

“You owe me for the pizza.  Now about that kinetic energy –”

“Hold that thought ’til we get to my office where I can draw diagrams.”

We got my car out of the lot, drove to the Acme Building and took the elevator to 12.

As my computer booted up I asked, “When we talked about potential energy, did we ever mention inertial frames?”

“Come to think of it, no, we didn’t.  How come?”

“Because they’ve got nothing to do with potential energy.  Gravitational and electrical potentials are all about intensity at one location in space relative to other locations in space.  The potentials are static so long as the configuration is static.  If something in the region changes, like maybe a mass moves or the charge on one object increases, then the potential field adjusts to suit.”

“Right, kinetic energy’s got to do with things that move, like its name says.  I get that.  But how does it play into LIGO?”

“Let’s stick with our spacecraft example for a bit.  I’ve been out of town for a while, so a quick review’s in order.  Objects that travel in straight lines and constant speed with respect to each other share the same inertial frame.  Masses wrinkle the shape of space.  The paths light rays take are always the shortest possible paths, so we say a light ray shows us what a straight line is.

“In our story, we’re flying a pair of space shuttles using identical speed settings along different light-ray navigation beams.  Suddenly you encounter a region of space that’s compressed, maybe by a nearby mass or maybe by a passing gravitational wave.

“That compressed space separates our inertial frames.  In your inertial frame there’s no effect — you’re still following your nav beam and the miles per second you measure hasn’t changed.  However, from my inertial frame you’ve slowed down because the space you’re traveling through is compressed relative to mine.  Does all that ring a bell?”

“Pretty much the way I remember it. Now what?”shuttle-escape-framed

“Do you remember the formula for kinetic energy?”

“Give me a sec… mass times the square of the velocity.”

“Uh-huh.  Mind you, ‘velocity’ is the combination of speed and direction but velocity-squared is just a number.  So, your kinetic energy depends in a nice, simple way on speed.  What happened to your kinetic energy when you encountered that gravity well?”

“Ah, now I see where you’re going.  In my frame my speed doesn’t change so I don’t gain or lose kinetic energy.  In your frame you see me slow down so you figure me as losing kinetic energy.”

“But the Conservation of Energy rule holds across the Universe.  Where’d your kinetic energy go?”

“Does your frame see me gaining potential energy somehow that I don’t see in mine?”

“Nice try, but that’s not it.  We’ve already seen that potential energy doesn’t depend on frames.  What made our frames diverge in the first place?”

“That gravity field curving the space I’d flown into.  Hey, action-reaction!  If the curved space slowed me down, did I speed it up?”

“Now we’re getting there.  No, you didn’t speed up space, ’cause space doesn’t work that way — the miles don’t go anywhere.  But your kinetic energy (that I can see and you can’t) did act to change the spatial curvature (that I can see and you can’t).  I suspect the curvature flattened out, but the math to check that is beyond me.”

“Lemme think…  Right, so back to my original question — what I wasn’t getting was how I could lose both kinetic energy AND potential energy flying into that compressed space.  Lessee if I got this right.  We both see I lost potential energy ’cause I’ve got less than back in flat space.  But only you see that my kinetic energy changed the curvature that only you see.  Good?”

“Good.”

(sound of footsteps)

(sound of door)

“Don’t mention it.”

~~ Rich Olcott

Throwing a Summertime curve

On By rolcottIn Cosmology, Mathematics: Dimensions, Physics: Black Holes and Relativity, UncategorizedLeave a comment

All cats are gray in the dark, and all lines are straight in one-dimensional space.  Sure, you can look at a garden hose and see curves (and kinks, dammit), but a short-sighted snail crawling along on it knows only forward and backward.  Without some 2D notion of sideways, the poor thing has no way to sense or cope with curvature.

Up here in 3D-land we can readily see the hose’s curved path through all three dimensions.  We can also see that the snail’s shell has two distinct curvatures in 3D-space — the tube has an oval cross-section and also spirals perpendicular to that.

But Einstein said that our 3D-space itself can have curvature.  Does mass somehow bend space through some extra dimension?  Can a gravity well be a funnel to … somewhere else?

No and no.  Mathematicians have come up with a dozen technically different kinds of curvature to fit different situations.  Most have to do with extrinsic non-straightness, apparent only from a higher dimension.  That’s us looking at the hose in 3D.

Einstein’s work centered on intrinsic curvature, dependent only upon properties that can be measured within an object’s “natural” set of dimensions.Torus curvature

On a surface, for instance, you could draw a triangle using three straight lines.  If the figure’s interior angles sum up to exactly 180°, you’ve got a flat plane, zero intrinsic curvature.  On a sphere (“straight line” = “arc from a great circle”) or the outside rim of a doughnut, the sum is greater than 180° and the curvature is positive.
Circle curvatures
If there’s zero curvature and positive curvature, there’s gotta be negative curvature, right?  Right — you’ll get less-than-180° triangles on a Pringles chip or on the inside rim of a doughnut.

Some surfaces don’t have intersecting straight lines, but you can still classify their curvature by using a different criterion.  Visualize our snail gliding along the biggest “circle” he/she/it (with snails it’s complicated) can get to while tethered by a thread pinned to a point on the surface. Divide the circle’s circumference by the length of the thread.  If the ratio’s equal to 2π then the snail’s on flat ground.  If the ratio is bigger than 2π,  the critter’s on a saddle surface (negative curvature). If it’s smaller, then he/she/it has found positive curvature.

In a sense, we’re comparing the length of a periphery and a measure of what’s inside it.  That’s the sense in which Einsteinian space is curved — there are regions in which the area inside a circle (or the volume inside a sphere) is greater than or less than what would be expected from the size of its boundary.

Here’s an example.  The upper panel’s dotted grid represents a simple flat space being traversed by a “disk.”  See how the disk’s location has no effect on its size or shape.  As a result, dividing its circumference by its radius always gives you 2π.Curvature 3

In the bottom panel I’ve transformed* the picture to represent space in the neighborhood of a black hole (the gray circle is its Event Horizon) as seen from a distance.  Close-up, every row of dots would appear straight.  However, from afar the disk’s apparent size and shape depend on where it is relative to the BH.

By the way, the disk is NOT “falling” into the BH.  This is about the shape of space itself — there’s no gravitational attraction or distortion by tidal spaghettification.

Visually, the disk appears to ooze down one of those famous 3D parabolic funnels.  But it doesn’t — all of this activity takes place within the BH’s equatorial plane, a completely 2D place.  The equations generate that visual effect by distorting space and changing the local distance scale near our massive object.  This particular distortion generates positive curvature — at 90% through the video, the disk’s C/r ratio is about 2% less than 2π.

As I tell Museum visitors, “miles are shorter near a black hole.”

~~ Rich Olcott

* – If you’re interested, here are the technical details.  A Schwarzchild BH, distances as multiples of the EH radius.  The disk (diameter 2.0) is depicted at successive time-free points in the BH equatorial plane.  The calculation uses Flamm’s paraboloid to convert each grid point’s local (r,φ) coordinates to (w,φ) to represent the spatial configuration as seen from r>>w.

The Shape of π and The Universe

On By rolcottIn Cosmology, Uncategorized3 Comments
pi
This square pi are rounded.

There’s no better way to celebrate 3/14/16 than chatting about how π is a mess but it’s connected to the shape of the Universe, all  while enjoying a nice piece of pie.  I’ll have a slice of that Neil Gaiman Country Apple, please.

The ancient Greeks didn’t quite know what to do about π.  For the Pythagoreans it transgressed a basic tenet of their religious faith — all numbers are supposed to be  integers or at least ratios of integers.  Alas for the faithful, π misbehaves.  The ratio of the circumference of a circle to its diameter just refuses to match the ratio of any pair of integers.

The best Archimedes could do about 250 BCE was determine that π is somewhere between 22/7 (0.04% too high) and 223/71 (0.024% too low).  These days we know of many different ways to calculate π exactly.  It’s just that each of them would take an infinite number of steps to come to a final result.  Nobody’s willing to wait that long, much less ante up the funding for that much computer time.  After all, most engineers are happy with 3.1416.

pi digitsNonetheless, mathematicians and cryptographers have forged ahead, calculating π to more than a trillion digits.  Here for your enjoyment are the 99 digits that come after digit million….

Why cryptographers ?  No-one has yet been able to prove it, but mathematicians are pretty sure that π’s digits are perfectly random.  If you’re given a starting sequence of decimal digits in π, you’ll be completely unable to predict which of the ten possible digits will be the next one.  Cryptographers love random numbers and they’re in π for the picking.

Another π-problem the Greeks gave us was in Euclid’s Geometry.  Euclid did a great job of demonstrating Geometry as an axiomatic system.  He built his system so well that everyone used it for millennia.  The problem was in his Fifth Postulate.  It claimed that parallel lines never meet, or equivalently, that the angles in every triangle add up to 180o.

Neither “fact” is necessarily true and Euclid knew that — he’d even written a treatise (Phaenomena) that used spherical geometry for astronomical calculations.  On our sweetly spherical Earth, a narwhale can swim a mile straight south from the North Pole, turn left and swim straight east for a mile, then turn left again and swim north a mile to get back to the Pole.  That’s a 90o+90o+90o=270o triangle no problem.  Euclid’s 180o rule works only on a flat plane.

cap areaBack to π.  The Greeks knew that the circumference of a circle (c) divided by its diameter (d) is π.  Furthermore they knew that a circle’s area divided by the square of its radius (r) is also π. Euclid was too smart to try calculating the area of the visible sky in his astronomical work.  He had two reasons — he didn’t know the radius of the horizon, and he didn’t know the height of the sky.  Later geometers worked out the area of such a spherical cap.  I was pleased to learn that π is the ratio of the cap’s area to the square of its chord, s2=r2+h2.

The Greeks never had to worry about that formula while figuring our how many tiles to buy for a circular temple floor.  The Earth’s curvature is so small that h is negligible relative to r.  Plain old πr2 works just fine.

CurvaturesAstrophysicists and cosmologists look at much bigger figures, ones so large that curvature has to be figured in.  There are three possibilities

  • Positive curvature, which you get when there’s more growth at the center than at the edges (balloons and waistlines)
  • Zero curvature, flatness, where things expand at the same rate everywhere
  • Negative curvature, which you get when most of the growth is at the edges (curly-leaf lettuce or a pleated skirt)

Near as the astronomers can measure, the overall curvature of the Universe is at most 10-120.  That positive but miniscule value surprised everyone because on theoretical grounds they’d expected a large positive value.  In 1980 Alan Guth explained the flatness by proposing his Inflationary Universe theory.  Dark energy may well  figure into what’s happening, but that’s another story.

Oh, that was tasty pie.

~~ Rich Olcott