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        riddles >> medium >> Cavorite
        
(Message started by: Icarus on Jan 20^th, 2008, 8:19pm)

Title: Cavorite
Post by Icarus on Jan 20^th, 2008, 8:19pm

I recently obtained a collection of several stories by H.G. Wells, and finally have the opportunity to read The First Men in the Moon. In this story, the characters are able to travel into space by means of a newly created alloy called "Cavorite", which is able to block all gravitational attraction.

In the story, cavorite is accidentally first created as a horizontal sheet. Wells describes that the air above the sheet suddenly became weightless and is pushed up the pressure of the surrounding air, which then rushes in to fill the void. The air rushing in also suddenly becomes weightless and is pushed up as well in a great fountain spewing air into space. Fortunately, the resulting maelstrom catches the sheet itself up and hurls it into space, thus narrowly avoiding the complete removal of Earth's atmosphere.

While a good story, this doesn't strike me as good science, even by the standards of the late 19th century when Wells wrote it.

So my question is: If such a material were possible, what really would happen if it were created in a horizontal sheet of a few meters width and length? Assume that the sheet is fastened down and unmoving.

Title: Re: Cavorite
Post by HiddenHeart on Jan 21^st, 2008, 1:08am

If the sheet is fastened down and unmoving and if any gravitational wave cannot pass through it :P then these might happen:

1. At the out-most sphere the air particles just on the sheet would stay where it is! because the minute fraction of gravitational wave that reaches to it through the sheet doesn't count there. The gravitational drug on the particle towards the earth at this point is an integration of the whole earths attraction to it except the portion that is blocked by the sheet which is ignorable.

2. A particle just on the sheet might have some problem. But it would not be very much because at the Earths surface level each air particle already has a high kinetic motion due to their temperature. The only things that keep it there is the pressure(i boldly use this term) on it from other particles. Which depends mainly on the depth of the particle measured from the outer hemisphere (hhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rho.gifg).

3. Now if we don't talk about air particles and have something solid. Say a person is standing on it. He might feel lack of gravity. so due to the centrifugal force he will be thrown up. But he cannot go much higher, i guess just a few meters will be his range. After that he will be pulled aside and eventually pulled down due to the Earths asymmetry & attraction.

4. If the person is not already standing on the sheet. He cannot actually stand on that!! I guess when he try his feet will be pulled aside. Almost(but not exactly) as the same pole of magnet repulses aside when we try to put them in contact.

5. I guess the sheet itself need not be fastened to anything. Because it is attracted to the earth by itself and as it cannot create a substantial change in the pressure of the air at its space side so nothing will happen!

OK, i think i have already talked too much ::)
so, i am taking a break. :P

Title: Re: Cavorite
Post by towr on Jan 21^st, 2008, 1:59am

I would guess air above the sheet would spill down from the 'gravity mountain' to the surrounding gravity sink of the earth.
Of course surrounding air has quite a bit of kinetic energy and would happily dart across the sheet (in nice straight lines since above it it isn't effected by gravity to curve it's path).

Oh, what the heck do I know; can't someone just simulate what'll happen?

Title: Re: Cavorite
Post by Grimbal on Jan 21^st, 2008, 2:04am

I have heard of that story before, but I never thought of what would "actually" happen. I just dismissed it as impossible. It seems you could create free energy out of it. Or couldn't you?

The idea that it affects the air in a column would mean that gravity acts vertically only. For instance, if gravity comes from the center of gravity. But that is not the case.

As HiddenHeart mentions, the effect would only be proportional to the part of Earth hidden by the shield. Directly on the sheet, the effect would be to cancel gravity. At a few meters above, the effect would be much smaller already. At some altitude, it would be negligible. Earth wouldn't loose any air.

From the side, as soon as it shields a bit of the Earth, it would have a sideways repulsive effect. It would be difficult to manage to get on it. Unless you approach it from below.

The effect on air would be to suck air from below and from the side of the plate and push it up in all directions, not only up. It would create a circulation of air like when a fan is pointed upwards. How wild, I don't know. But I think it is correct that the sheet would be sucked up by the wind it creates.

Anyway, it should shield itself from gravity. It would be weightless. It would definitely fly up in the air.

One question remains, it is to how much amounts the gravity effect of half of the universe (besides the nearby planets and stars). Is it almost zero due to the distance, or does it just feel like that because of the other half canceling the effect? That could make a big difference.

Title: Re: Cavorite
Post by Icarus on Jan 21^st, 2008, 6:47am

My thoughts were pretty much the same. I considered the perpetual motion aspect, but I think instead that the air would reach a new equilibrium.

I think that Wells knew that the gravitational attraction between two bodies acts as if they were point masses at their center of gravities, but mis-interpreted this as meaning that if you blocked the direct line between the centers, all gravitational attraction would be blocked. So he thought the entire column of air above the sheet would be weightless. As HiddenHeart has described, only the air close to the plate would be seriously affected. For air far away, only the mass of earth occluded by the plate does not attract the air. And the farther away the air is, the smaller the occlusion. At most, it would cause a small hump in the upper atmosphere. And for any reasonable size sheet, this hump would be negligible in size.

Suppose the plate was a disk of 1 meter radius. Can you describe how g changes with height along the vertical line through the center of the plate?

Title: Re: Cavorite
Post by HiddenHeart on Jan 21^st, 2008, 7:47am

on 01/21/08 at 06:47:47, Icarus wrote:

Suppose the plate was a disk of 1 meter radius. Can you describe how g changes with height along the vertical line through the center of the plate?

You also have to specify the position of the plate. that is its vertical distance from the Earth's surface.

Title: Re: Cavorite
Post by Icarus on Jan 21^st, 2008, 8:17am

On the surface of the earth, of course. Sea level, if you want something more exact.

Title: Re: Cavorite
Post by Grimbal on Jan 21^st, 2008, 8:22am

I think the air would not stabilize.

Or it depends how it works.

If the cavorite only shields the straight-line action of gravity, then you could imagine a pipe going up from the center of the plate, then horizontally away from the plate, down to ground level and horizontally back to the center of the plate. It seems to me that air in that pipe would be pushed around.

The air would stabilize if cavorite works in a way compatible with the existence of a gravitational potential function. It would probably break less of Gravity if it behaves like that. In that case, since cavorite reduces the energy necessary to lift something from the top of the plate to outer space, the gravitational potential must be quite high on top of the plate. That would mean there must be a strong force preventing anything to get on the plate from the side or through a hole in the middle.

Or maybe cavorite just has negative mass. Something near a negative mass is pushed away. And indeed, the potential is higher there. So if the plate is a really massive negative mass, it would probably provide some lift to someone standing on top.

Of course, just getting near the plate would use up as much energy as what you save on your way to space.

And immense forces would be needed to keep so close to Earth something that has enough negative mass to compete with Earth's mass.

PS: I am not sure whether "high potential" actually goes with higher altitude or lower altitude. It is just a matter of changing a sign.

Title: Re: Cavorite
Post by Icarus on Jan 21^st, 2008, 8:42am

Wells, reflecting the ideas of his time, considered gravity to be some sort of radiation (a word that did not have the same connotations then that it does today). Cavorite is supposed to be impervious to all radiations. Therefore, it blocks the straight-line action of gravity.

But consider your pipe out in space, with long horizontal segments. Now put a very dense mass under one of the vertical ends. Doesn't this accomplish the same thing without the need for fantastic materials?

Title: Re: Cavorite
Post by Grimbal on Jan 21^st, 2008, 9:20am

With a normal mass the air would be attracted by the mass in the bottom horizontal segment, while in the case of cavorite, having as only effect that it shields from gravity, it shouldn't be affected.

Title: Re: Cavorite
Post by rmsgrey on Jan 21^st, 2008, 9:54am

An interesting question is what happens to the "blocked" gravity - does it get diverted, or does it get absorbed somehow?

What are the energetic consequences of soaking up gravity?

Title: Re: Cavorite
Post by SMQ on Jan 21^st, 2008, 10:14am

on 01/21/08 at 06:47:47, Icarus wrote:

Suppose the plate was a disk of 1 meter radius. Can you describe how g changes with height along the vertical line through the center of the plate?

Probably, but it's more fun to plot it. :P

http://www.dwarfrune.com/~smq/wu/cavorite.png

The plot is to scale, showing the force of gravity above the plate as a percentage of full force. As the plot clearly shows, the effect of the plate is very minor (less than a few percent disruption) at more than a few meters distance.

Also, reasoning by analogy, I don't think cavorite technology would enable the construction of a perpetual motion machine. If a gravitational field is analogous to an electrical field, cavorite is the equivalent of a gravitational conductor -- a cavorite box is alanogous to a Farady cage. You can't construct a perpetual motion machine from charged particles and an electrical field -- even with conductors -- so I don't think you can construct one from massive particles and a gravity field either.

--SMQ

Title: Re: Cavorite
Post by Michael_Dagg on Jan 21^st, 2008, 7:57pm

> In this story, the characters are able to travel into space by means
> of a newly created alloy called "Cavorite", which is able to block all
> gravitational attraction.

Mathematically, gravity works almost identically to electromagnetism
(except that the latter is signed -- only opposites attract). So you
can get a sense of how Cavorite would affect life if you can arrange
an experiment in which electromagnetic fields are neutralized. Isn't
that possible by wrapping the objects in big silver spheres? (I thought
there was something about the electrons redistributing themselves
around the sphere so as to leave no magnetic field on the inside.)

In saying this I guess I reveal my preference for understanding gravity:
rather than trying to understand the forces (something requiring vectors)
I think primarily of the potential energy field (a scalar-valued function
on space); "gravity pulls" because things move to a position of lower
potential energy. So in this framework, what exactly does this
neutralizing sheet do? Presumably the intention is that the potential
energy field on one side of the sheet is constant (so there is no
"gravitational force", i.e. gradient, to push anything anywhere).
But is it somehow one-sided? I mean, how does the sheet determine
which side of the sheet will feel no gravity? And how exactly does it
operate near the edges? I mean, in force terms, is there a sudden
change in the gravitational tug when the edge of the sheet is between
the two centers of mass? Is center of mass even relevant? (Newton _invented
calculus_ in order to prove that a homogeneous sphere exerts the same
gravitational force as a massive point located at the center of mass,
but the existence of this sheet would require a whole new integration,
I guess, and it's not clear to me what the result would be.

> Wells describes that the air above
> the sheet suddenly became weightless and is pushed up the pressure of the
> surrounding air

Implicit here is some kind of assumption about what happens around the
edges of the sheet. I'm not really sure what is being assumed... But yes,
the general principle is OK I guess: in the absence of gravity, the
Brownian motion of the air molecules would over time make them try to
achieve a uniform density, not the density-decreases-with-altitude result
of gravity. Of course with all of outer space to fill, this would mean
a net migration upward. Not exactly a whoosh, I think: for example, as
the high-altitude gravity-free air molecules become more numerous, the
air pressure up there would increase, pushing _in all directions_ (because
of the random nature of molecular collisions); that means the air molecules
that get up high will tend to push outward against the (low-pressure)
high-altitude gravity-constrained air molecules. So you'd get a sort of
fountain of air molecules migrating up, then out, then down, then back
inward over the sheet; a fraction continue to move up when the others
start moving out, so yes, there is a net upward movement.

I think it would look exactly like the effect of putting a hot plate
where the sheet is: the warm air rising, some of it pushing outward,
then cooling and falling. Have you seen those "how do hurricanes start?"
illustrations?

(continued)

Title: Re: Cavorite
Post by Michael_Dagg on Jan 21^st, 2008, 7:58pm

> Fortunately, the resulting maelstrom catches the sheet itself up and
> hurls it into space,

Hm, is the sheet itself weightless? Or only the things on top of it?
(And what if we put two sheets on top of each other?)
(Come to think of it, what happens if we carefully slice the sheet
into two layers and throw away the top one --- is the force of gravity
merely lessened? If not, does that mean an infinitesimally thin layer
is sufficient? and going the other way, if we need a certain thickness
to block gravity, what happens if we have a greater thickness -- is
gravity actually reverse? I suppose it would more likely be true that
the stuff filters gravity like water filters sunlight: each increase in
thickness removes a fixed _fraction_ of the stuff, so that there is
never a thickness that removes everything, rather, we just get an
exponential decrease in light, or in gravity, as we pile on more layers
of impeding substances.)

(Oh and another thing: how do they store this sheet? I mean, does it
just sit there and then get turned on? Or is it always repelling
gravity and somehow you choose to slide the anti-gravity field into
your work station instead of the storage closet where, I suppose,
all the air has already left the building?)

A stiff but unconstrained sheet (think of a pie tin) would be moved
in a chaotic way by the buffetting of the air molecules, raising lots
of questions about the effect of the substance when placed at an angle
to the line going through the center of the earth, that is, what does
orientation do? A flexible sheet (think of a blanket) would make
things even more complicated. For simplicity of modeling I guess I
would propose first to look at the case of a stiff and constrained sheet
(think of a car lift in a mechanic's garage). I guess it is true that
the sheet itself would be lifted, but only on the assumption that the
sheet itself is also free from gravity; otherwise it would stop rising
when the (high-altitude) air pressure under it is less than the force
of gravity on the sheet.

> thus narrowly avoiding the complete removal of Earth's atmosphere.

I think there would be little danger of this unless the sheet was
continent-sized, or unless the sheet actually _repels_ air masses
opposite to the usual direction of gravitational pull.

> So my question is: If such a material were possible, what really would
> happen if it were created in a horizontal
> sheet of a few meters width and length? Assume that the sheet is fastened
> down and unmoving.

I'm not inclined to work it out but I think you could model this
effectively (making reasonable assumptions about the behaviour of this
substance) by imagining a cloud of ions or plasma or something like that,
floating out in space beyond the pull of earth's gravity, and
constrained to a box which has one side electrically charged to attract
the ions -- but not the whole side of the box, but rather the shape of
the magic sheet would be removed. (So for example it could be a
cylindrical box with one end replaced by an annular plate which is
given the opposite charge from the ions; you can start the experiment
with the center of the annulus filled by another charged plate, then
remove this central plug to see the effect of introducing the
anti-gravity sheet.)

The key is that the plasma has to be a gas if you want to model the
behaviour of the atmosphere, because in the absence of the molecular
motion in a gas, I guess what would happen when you interrupt the pull of
gravity would be: nothing. (Astronauts' tools don't "run away" in space, they
just stay put. A tiny amount of molecular motion at all temperatures about
absolute zero will make some energetic molecules run away, i.e. that solid
steel hammer will slowly boil away, but the speed with which it happens
is negligible.)

If you work out the statistical mechanics of it all, I guess the
chamber would get to an equilibrium with more of the plasma at the
end of the cylinder that has the charged plate (the "low-altitude" end)
and less on the other end (the "high-altitude end"). When the central
plug is turned off and only the annular plate is pulling the ions,
you'd get the whole fountain display I described above, eventually
settling into a pressure distribution which is more uniform between
the two ends of the cylinder (but never entirely so).

The plasma model is a little inaccurate though: those charged ions
will tend to repel each other electrostatically, whereas air molecules
don't (in fact there is a miniscule _attraction_ between them because
they tug on each other gravitationally). I guess I expect the differences
to be minor.

My 2 cents.

Title: Re: Cavorite
Post by SMQ on Jan 22^nd, 2008, 6:26am

on 01/21/08 at 19:57:52, Michael_Dagg wrote:

Mathematically, gravity works almost identically to electromagnetism (except that the latter is signed -- only opposites attract). So you can get a sense of how Cavorite would affect life if you can arrange an experiment in which electromagnetic fields are neutralized.

Didn't I just say that? ;D

--SMQ

Title: Re: Cavorite
Post by rmsgrey on Jan 22^nd, 2008, 10:56am

For the directionality of the sheet, from the original description, as blocking all forms of radiation, the effect of the sheet is a "gravity-shadow" so tilting the sheet away from the horizontal changes the affected region in much the same way as tilting a postage stamp held below a large frosted-glass window affects the region of light-shadow.

Title: Re: Cavorite
Post by Icarus on Jan 22^nd, 2008, 6:46pm

Wells does not attempt to supply technical details such as thicknesses needed (though in the space vehicle, it is used as plating on the steel structure - so it can be quite thin), as this would only distract from his story (and make it more easily impugned). However, this much I can say:

1) The sheet works by blocking the radiation responsible for gravity (according to the best ideas of Wells' time). Therefore it is bi-directional. The potential above the plate is unaffected by the influence of any matter below the plate. The potential below the plate is unaffected by the influence of any matter above the plate. Since very little matter lies above the plate, the potential is essentially constant in the region completely shielded from the Earth. Wells' mistake was to think this region was a cylinder of constant cross-section.

For a point gravitating mass, the potential, like the force field itself, will be discontinuous at the boundary between the occluded and non-occluded regions. For a more realistic gravitating body with volume, the potential is at least C¹, as is demonstrated by SMQ's wonderful graph (good job, there)!

2) Since the plate merely blocks gravitational radiation, additional thickness would not create antigravity. While Wells does not directly say so, several comments he makes suggest that he views the cavorite as being weightless. Apparently, it blocks gravity from effecting even itself. However, the sheet is blown away by the uprush of air, not from any anti-gravity effect.

3) The maelstrom is caused by the "fact" that the air in the column above the plate is, in Wells' view, weightless, and thus no longer under 14.5 psi of pressure from the weight of the air above like the surrounding atmosphere (sorry, but I'm too lazy to look up or convert to the value in kilopascals). This pressure differential results in more air rushing in, displacing the light air, but in the process also losing the weight of the upper atmosphere, which continues the cycle. The displaced air moves upward as that is the path of least resistance. Wells does not say the Earth would permanently lose its atmosphere, but rather that it would shoot high enough that by the time it came back down, the world would be asphyxiated.

4) Wells seems to believe that this would only be a problem for a wide horizontal sheet. Smaller amounts of the material do not have this effect. (This is probably a case of ignoring a supposed problem because he didn't have an answer for it, but wanted to tell his story anyway.) The material caused the maelstrom in the story because it was first created as a horizontal sheet. This occurred at a time when they were unprepared for it because the workmen were having a disagreement over whose job it was to stoke the furnace, allowing the alloy to form prematurely as the temperature dropped.

Title: Re: Cavorite
Post by Grimbal on Jan 23^rd, 2008, 12:44am

on 01/22/08 at 18:46:32, Icarus wrote:

... under 14.5 psi of pressure ... (sorry, but I'm too lazy to look up or convert to the value in kilopascals).

Just look up "14.5 psi in kilopascal" in Google.

Title: Re: Cavorite
Post by Icarus on Jan 23^rd, 2008, 6:28pm

what part of "too lazy" didn't you understand? ::)

Title: Re: Cavorite
Post by SMQ on Jan 24^th, 2008, 4:02pm

on 01/22/08 at 10:56:24, rmsgrey wrote:

Like this? ;D
http://www.dwarfrune.com/~smq/wu/cavoritespin.gif
--SMQ

Title: Re: Cavorite
Post by HiddenHeart on Jan 25^th, 2008, 1:15am

on 01/24/08 at 16:02:31, SMQ wrote:

Like this? ;D
--SMQ

You are not "Too Lazy" i guess. :P

Great work!... Scinetific & Artistic!! :D
i'm being curious about your simulation model. i mean the formulation and the code of course. could you plz attach them! :) [to be frank i am not familier with this type of simulaton :-/. it would be really educational for us(people like me:P). i hope]
-HidenHeart

Title: Re: Cavorite
Post by SMQ on Jan 25^th, 2008, 6:55am

Trust me, you don't want to see my code. ;)

The simulation itself, though, is pretty straightforward:

Let the screen be the portion of the x-y plane from <-5.12,6370,0> to <5.12,6377.68,0>
Let the plate be a circle of radius 1 in the plane y=6371 centered on <0,6371,0>
For each angle http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif of the plate
Rotate the plate by http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif around the axis x=0,y=6371
For each pixel in the image
Determine the corresponding coordinates in the x-y plane
Initialize two vectors, s and t to 0
For each of 1000 random points in a sphere of radius 6370 centered at <0,0,0>
Let v be the vector from the screen coordinates to the random point
Add v/|v|³ (a vector in the direction of v of magnitude 1/|v|²) to t
If v does not intersect the circle of the plate, Add v/|v|³ to s
Color the pixel according to |s|/|t|
Store the resulting image
Compile the Sequence of images into an animation

I used FreeBASIC (http://www.freebasic.net) for the simulation, and GIF Construction Set (http://www.mindworkshop.com/alchemy/gifcon.html) to compile the animation.

--SMQ

Title: Re: Cavorite
Post by SWF on Feb 5^th, 2008, 7:51pm

on 01/21/08 at 06:47:47, Icarus wrote:

Suppose the plate was a disk of 1 meter radius. Can you describe how g changes with height along the vertical line through the center of the plate?

g'/g = 1 - 2/pi* ( atan(1/z) - z/(1+z²) )

Title: Re: Cavorite
Post by Eigenray on Feb 6^th, 2008, 6:31pm

Let the earth have radius R, the plate have radius r, and say the distance from the center of the earth is D = R + h. If the normal gravity is g = 1/D², I get that the gravity in the presence of the plate is

g' = 1/D² ( 1 - (D/R sin t)² )^3/2
= g [ 1 - (D/R)²/{1+(h/r)²} ]^3/2,

where t = arctan(r/h) < arcsin(R/D) is the angle to the edge of the plate from a point over the center. For fixed r, as h goes to infinity, h >> R, we have

g'/g -> ( 1 - (r/R)² )^3/2 ~ 1 - 3/2*(r/R)²,

a constant. However, we'd need r ~ 324 miles just to get this limit down to 99%.

Title: Re: Cavorite
Post by SWF on Feb 10^th, 2008, 8:58pm

Eigenray, your solution looks like what you would get if the material blocks gravity in a line of sight manner. This is a non-conservative force field, and as pointed out by Grimbal would permit a perpetual motion machine that generates energy.

The solution I gave (if I did it correctly) is for a material that shields gravity, and a force field that follows the laws of physics.

Title: Re: Cavorite
Post by Eigenray on Feb 10^th, 2008, 10:10pm

What exactly do you mean? I was integrating over lines of sight to get the force. If I integrate over lines of sight to get the potential, I get

V(D) = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif_t^arcsin(R/D) http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif_r1^r2 (-1/r)*[2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gifr sinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif]rdr dhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif
= -4http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif/(3D) * (1 - (D/R sin t)²)^3/2,

where r1,r2 satisfy

r² + D² - 2rD coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif = R².

So would the conservative force be V'(D)? If this is the case, then the force is actually upwards just above the plate. For a plate radius of 1 meter, a mass placed directly over the center should hover about D₀ = 267 meters off the ground. For http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rho.gif = r/R << 1, I get

D₀/R = 3(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rho.gif/3)^2/3 + 5(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rho.gif/3)^4/3 + O(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rho.gif²).

(By the way, The First Men in the Moon is out of copyright and is available at [link=http://www.gutenberg.org/etext/1013]Project Gutenberg[/link] among other places.)

Title: Re: Cavorite
Post by SWF on Feb 11^th, 2008, 7:42pm

Since a plate that blocks force in a line of sight matter is a non-conservative field, there is not a potential function for it. If the disk worked by line of force, you could attach a ferris wheel to a generator and put Cavorite under half it to generate power. It is easier to see this is the case if you imagine a dense point mass instead of the distributed mass of the earth.

To simplify the problem and because the effects were expected to be near the disk, I found the effect of the disk in a uniform gravitational field. This involves finding the potential function that results in no force normal to the disk (dV/dz=0 for points on the disk). Points outside the disk, but in the same plane have the same potential, and far from the disk dV/dz is a constant, -g. Finding V involves solving the Laplace equation.

Title: Re: Cavorite
Post by Eigenray on Feb 11^th, 2008, 9:44pm

Well, obviously it depends on what exactly it means for a material to 'block gravity'. But if you define

G(x) = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif_{{r visible from x}} -1/|r-x| dV,

then by definition F = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/nabla.gifG is a conservative force. With this definition, the plate would actually be repelling for small heights, since moving up lets you see more Earth and therefore lowers your potential energy.

To evaluate this integral, we can let x = (x,y,z), p = (coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif, sinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gifcoshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/phi.gif, sinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gifsinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/phi.gif), r = p + t(p-x), and the Jacobian works out to J = (1+t)² sinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif(x.p-1). Then

G = -4http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif/(3|x|) + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif_{http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif=0}^T http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif_{http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/phi.gif=0}^{2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif} http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif_t=0^{2(p.x-1)/|p-x|^2} J/|r-x| dtdhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/phi.gifdhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif,

where T = 1m/(1 Earth radius) (so the sheet is a spherical cap on the surface). I can't get Mathematica to cooperate right now or I'd try to plot the vector field in the xz plane. Here are some contour plots for G, when T = 10m/(1 Earth radius) ~ 1.6*10^-6. Darker is lower potential; the white area is the result of cropping the potential from above. The potential G(x,0,0) is minimized for x ~ 1.000195, which is about 1.24km off the ground.

Title: Re: Cavorite
Post by SWF on Feb 12^th, 2008, 6:33pm

on 02/11/08 at 21:44:03, Eigenray wrote:

It does not seem reasonable to think of Cavorite as "blocking" a scalar field just because distance from a point appears in the formula. Blocking a vector (the force) makes more sense to me.

In analogy to heat flow in a conductor which is also governed by the Laplace equation, the scalar field analogous to gravitational potential is the temperature, and the heat flux corresponds to gravitational force field. The thermal analog to Cavorite would be a perfect insulator of heat. Notice how in the thermal analogy you can block the flow of heat (vector field), but blocking the temperature (the scalar) does not have much meaning.

Similar analogies can also be made with fluid flow and flow of electricity, and unlike gravity these have a familiar means of "blocking" the vector field.

Title: Re: Cavorite
Post by Eigenray on Feb 12^th, 2008, 11:30pm

I suppose you're right. So if you were standing on the ground, and someone put a piece of cavorite above you, what would happen?