What’s that funnel about, really?

If you’ve ever watched or read a space opera (oh yes, you have), you know about the gravity well that a spacecraft has to climb out of when leaving a planet.  Every time I see the Museum’s gravity well model (photo below), I’m reminded of all the answers the guy gave to, “Johnny, what can you make of this?

The model’s a great visitor-attracter with those “planets” whizzing around the “Sun,” but this one exhibit really represents several distinct concepts.   For some of them it’s not quite the right shape.DMNS gravity well

The simplest concept is geometrical.  “Down” is the direction you move when gravity’s pulling on you.

HS cone
Gravitational potential energy change
for small height differences

A gravity well model for that concept would be just a straight line between you and the neighborhood’s most intense gravity source.

You learned the second concept in high school physics class.  Any object has gravitational potential energy that measures the amount of energy it would give up on falling.  Your teacher probably showed you the equation GPE = m·g·h, where m is the mass of the object, h is its height above ground level, and g is a constant you may have determined in a lab experiment.

If the width of the gravity well model at a given height represents GPE at that level, the model is a simple straight-sided cone.

Newton energy cone
Gravitational potential energy change
for large height differences
The h indicates
an approximately linear range
where the HS equation could apply.

But of course it’s not that simple.  Newton’s Law of Gravity says that the potential energy at any height r away from the planet’s center is proportional to 1/r.

Hmm… that looks different from the “proportional to h” equation.  Which is right?

Both equations are valid, but over different distance scales.  The HS teachers didn’t quite lie to you, but they didn’t give you the complete picture either.  Your classroom was about 4000 miles (21,120,000 feet) from Earth’s center, whereas the usual experiments involve height differences of at most a dozen feet.  Even the 20-foot drop from a second-story window is less than a millionth of the way down to Earth’s center.

Check my numbers:

Height h 1/(r+h)
× 108
Difference in 1/(r+h)
× 1014
0 4.734,848,484 0
20 4.734,844,001 4.48
40 4.734,839,517 8.97
60 4.734,835,033 13.45
80 4.734,830,549 17.93
100 4.734,826,066 22.42

rh lineSure enough, that’s a straight line (see the chart).  Reminds me of how Newton’s Law of Gravity is valid except at very short distances.  The HS Law of Gravity works fine for small spans but when the distances get big we have to use Newton’s equation.

We’re not done yet. That curvy funnel-shaped gravity well model could represent the force of gravity rather than its potential energy.  Newton told us that the force goes as 1/r2 so it decreases much more rapidly than the potential energy does as you get further away.  The gravity force well has a correspondingly sharper curve to it than the gravity energy well.

Newton force cone
The force of gravity
or an embedding diagram

The funnel model could also represent the total energy required to get a real spacecraft off the surface and up into space.  Depending on which sci-fi gimmickry is in play, the energy may come from a chemical or ion rocket, an electromagnetic railgun, or even a tractor beam from some mothership way up there.

No matter the technology, the theoretical energy requirement to get to a given height is the same.  In practice, however, each technology is optimal for some situations but forbiddingly inefficient in others.  Thus, each technology’s funnel  has its own shape and that shape will change depending on the setting.

In modern physics, the funnel model could also represent Einstein’s theory of how a mass “bends” the space around it.  (Take a look at this post, which is about how mass curves space by changing the local distance scale.)  Cosmologists describe the resulting “shapes” with embedding diagrams that are essentially 2D pictures of 3D (or 4D) contour plots.  The contours are closest together where space is most compressed, just as lines showing a steep hillside on a landscape contour map are close together.

The ED around a non-spinning object looks just like the force model picture above.  No surprise — gravitational force is how we we perceive spatial curvature.

~~ Rich Olcott

Advertisements

Throwing a Summertime curve

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 ,  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.