Gravity at the Centre of the Globe

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Gravity at the Centre of the Globe

Postby Teshi » Tue Jun 12, 2007 4:27 pm UTC

So, I was reading Kim Stanley Robinson's short story Lunatics and at one point one of the characters (who may be wrong) claims that "they are getting lighter as they approach the centre of the moon" (paraphrased but quotations to show my uncertaintly.

KSR is usually on top of this kind of thing so I'm assuming that if it's wrong it's the character deliberately being simplistic and not a mistake. The character who responds to this says, "actually, I feel heavier" or something.

Anyway, so thinking about this, sorry about my very limited understanding of gravity I came up with two possible scenarios (lighter and heavier). I wonder if you could help me.

Lighter:

The mass of the earth/moon doesn't change, just your position in it. Therefore, the same object is exerting the same forces just not necessarily "down" towards the centre of the Earth. If you stood on a scale 3/4 of the way down you would weigh less "down" but you would feel some amount of weight also to your sides.

So you would feel lighter in one respect but, adding all your measureable weight together, you would weigh the same standing on the surface of the planet.

Assuming a hypothetical planet is a perfect sphere, this means that were you at the centre of the Earth you would feel exactly the same 'pull' (?) of gravity from all directions, which would amount to something around 1/2 of the Earth. It would be like being upsidedown and rightside up and lying on your front, back and sides at the same time, only weighing less.

So... presumably you would have a sensation of being weightless (floating) and being heavy (although less 'heavy'- lighter- than on the surface of the Earth) at the same time.

Heavier:

With the above, I appear to be making the assumption that gravity is directional. However, I have an image in my head of spacetime being warped by a heavy object which suggests- in my limited understanding of this image- that towards the centre of an object you actually get heavier because you are further down this 'hole' created by the object.

At the centre of this perfect globe, then, you would weigh far more than on the surface in all directions but, if I'm correctly thinking about this, you would still 'float' because you would be at the very centre of this warped 'hole'.

Thoughts:

However, this means that your overall weight (and thus the overall gravity) is in fact far more at the centre of an object with a certain mass than it is, say, buried in sand a metre below the surface of an object of the same mass, which seems a little wonky.

Time would go slightly slower at the centre of the Earth than on the surface. This is the general hypothesis with black holes, though, is it not?

Thinking about Earth not as a single entity but as a series of atoms each with their own gravitational force or bending that extends a certain distance before dropping off considerably, obviously weight would be highest at the point where the most fields of each of these individual atoms occurs which would logically be the centre of the Earth- which is the closest point in the globe to all other points.

Clearly the further you move away from the Earth's mass the less you weigh...

I'm so confused! Which is it?

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Postby Cosmologicon » Tue Jun 12, 2007 4:42 pm UTC

Quick version so's I don't get scooped:

For a uniform-density spherical mass like the Moon, your weight would decrease linearly with distance from the center. So if you went halfway to the center, you'd weigh half as much.

For the Earth in particular, since it's much more dense in the core than the crust, it's a bit different. They've worked it out, and your weight stays pretty much the same (g ~ 10 m/s^2) until you get to the core. From then on, your weight decreases linearly like on the Moon.

At the center of any spherically-symmetric object like the Earth or Moon, you'd be weightless.

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Postby evilbeanfiend » Tue Jun 12, 2007 4:42 pm UTC

assume you are a single point, the forces will all cancel, you will be weightless. assume you are a collection of points, the force on each point is still going to basically cancel so you are still weightless. you are going to float and you will not 'feel heavy'.

gravity is a force, forces have directions.


edit: if you are very big and at the centre you will presumebly feel a slight squeezing sensation as your extremities are all pulled inward by gravity. for human like objects this force will too small to notice
Last edited by evilbeanfiend on Tue Jun 12, 2007 4:48 pm UTC, edited 1 time in total.
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Postby EricH » Tue Jun 12, 2007 4:47 pm UTC

Longer version, since I already got scooped (@#$%^& ninjas.):
Definitely lighter. The first description you have is relatively accurate, in that you'll be pulled in several directions. The one difference is that you don't actually feel all those pulls--it's vector addition, so two equal 'pulls' in opposite directions cancel each other out, and you don't feel anything.
The mathematical way to come to the conclusion is to do the calculus involved in adding everything up. I'll summarize, though--inside a hollow sphere, you don't feel any gravitational pull in any direction, no matter where in the hollow you are (because if you're near, say, the north pole, there's much more mass south of you, but it's all further away than the smaller amount of mass that's north of you; in hollow sphere of uniform thickness and density, everything happens to cancel out).
So, as you travel down a shaft, toward the center of the globe, you can imagine dividing the globe into an inner, solid, sphere (consisting of everything that's closer to the center than you are), and an outer, hollow sphere (consisting of everything farther from the center than you are). The gravity you feel from the outer sphere is cancelled out, so you feel only the amount from the inner sphere. And the closer you get to the center, the smaller that inner sphere is; the gravity you feel shrinks with it.
I hope that helps, though it probably would be easier to explain with a picture...
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Postby Mighty Jalapeno » Tue Jun 12, 2007 4:54 pm UTC

This was a question I asked many years ago for a story, because (for reasons that don't need exploring at this juncture), the character lived in a dimension where the influence of matter and energy existed, without matter and energy itself, so it was a void, but with gravity. I never got a straight answer out of ANYONE, including two physicists.

I appreciate the help!

How light would the gravity be? A major plot point is he inhabits the center of this void on a tiny man-made planetoid (about a mile in diameter).

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Postby Teshi » Tue Jun 12, 2007 5:20 pm UTC

I was initially leaning towards 'lighter'/'weightless' but when I wrote up the heavier I was pretty well convinced that you'd weigh more despite the fact that that seemed very wrong.

I guess my first instinct was right, as was the character in the book.

EDIT: Thanks!

EDIT #2: Waaait a second.

I just want to clarify here. It sounds like there is[/is] weight, it's just it would cancel itself out. So are you saying I'm effectively right on both counts? You would [i]feel lighter (weightless) but be subject to "more" (or the same?) gravity?

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Postby evilbeanfiend » Tue Jun 12, 2007 5:44 pm UTC

Teshi wrote:EDIT #2: Waaait a second.

I just want to clarify here. It sounds like there is[/is] weight, it's just it would cancel itself out. So are you saying I'm effectively right on both counts? You would [i]feel lighter (weightless) but be subject to "more" (or the same?) gravity?


no the weight is 0, weight is the force exerted on an object by gravity, forces are vectors so you have to sum them as such.
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Postby Belial » Tue Jun 12, 2007 5:53 pm UTC

And since, as far as we know, gravity doesn't exist as anything other than a force, being pulled in *all* directions (like you are in the center of a planet) is effectively the same as being pulled in none (as you are in deep space.)

If we were to discover that gravity is caused by a particle or a wave or something of the sort, then yes, you'd be subject to more of *that* at the center of a planet.
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Postby Mighty Jalapeno » Tue Jun 12, 2007 5:56 pm UTC

My brain keeps thinking of the three-dimensional gravitational model, of a flat sheet pulled inwards at the center of a mass. How does that reconcile with the "no gravity at the center of a mass"? Probably a stupid question, but I was never that good at the hard sciences.

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Postby Teshi » Tue Jun 12, 2007 6:17 pm UTC

My brain keeps thinking of the three-dimensional gravitational model, of a flat sheet pulled inwards at the center of a mass.


This is what I was thinking of when I came up with the "heavier" option.

It seems to be both ways. You are subject to more forces of gravity at the bottom of the centre of the 'hole', but since they cancel each other out you feel weightless. So you are both "heavier" and completely weightless at the same time.

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Postby Mighty Jalapeno » Tue Jun 12, 2007 6:20 pm UTC

Teshi wrote:It seems to be both ways. You are subject to more forces of gravity at the bottom of the centre of the 'hole', but since they cancel each other out you feel weightless. So you are both "heavier" and completely weightless at the same time.

But in practicality (sort of), how would you be moving / reacting at the center of the earth, or, say, one mile away from it. Would you still be pulled down? Would gravity be negligible at that distance?

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Postby Cosmologicon » Tue Jun 12, 2007 6:21 pm UTC

Mighty Jalapeno wrote:My brain keeps thinking of the three-dimensional gravitational model, of a flat sheet pulled inwards at the center of a mass. How does that reconcile with the "no gravity at the center of a mass"?

The gravitational potential inside a spherical mass distribution is paraboloidal, like a bowl. If you put a ball at the very bottom of a bowl (which corresponds to the center of the mass) and release it, which way does it roll?

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Postby Mighty Jalapeno » Tue Jun 12, 2007 6:23 pm UTC

It depends. Is the bowl inside a gravitational field? If not, the ball might float away. :)

The ball at the bottom of the bowl is indicative of gravitational force holding it there, or at the very least preventing it from rolling out.

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Postby Cosmologicon » Tue Jun 12, 2007 6:38 pm UTC

You're using the word "force" in a very loose sense, and different from its well-defined meaning in physics. But ignoring that, you're right. The bottom of the bowl is a potential minimum, and indeed the structure of the gravitational field prevents it from being knocked out easily. But technically, there's no net force acting on it.

Incidentally, the story idea you mentioned happens all the time on large scales in our universe. Dark matter - which doesn't interact with normal matter except gravitationally - forms spherical haloes. Galaxies, roughly speaking, rest at the potential minima in the center of these haloes.

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Postby Mighty Jalapeno » Tue Jun 12, 2007 6:42 pm UTC

Hmmm, I might be able to use that to fudge the science in my book (which is all about travelling between dimensions, so really, the science is pretty fudged anyways!)

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Postby Teshi » Tue Jun 12, 2007 6:48 pm UTC

If you put a ball at the very bottom of a bowl (which corresponds to the center of the mass) and release it, which way does it roll?


Obviously this also works if you put the ball a little away from the centre. It will eventually come to rest at the centre.

Okay, so we have determined that the net force of gravity at the centre of a spherical object is zero, but that the forces in the centre are actually greater.

So, what about off-centre. Say you're one or two metres away from the centre. You would obviously fall towards the centre still but at a much reduced rate- your acceleration (what there would be of it)- would be less, not because there is less gravity but because you would be subject to "up" gravity as well as "down" gravity, so the net force would, although leaning to "down" be less than "up".

Right?

Also:

People in books/tv/movies always say that as people/ships/planets approach black holes they will be torn apart. I assume what they mean, then, is that the side closer to the black hole (and therefore subject to mroe gravity) gets torn away from the side further away.

Are we saying then, that assuming you could somehow appear in the centre of a black hole, you would be perfectly intact because despite the fact that the mass around you was absolutely collosal you would be in "the eye of the storm" in a manner of speaking.

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Postby MalaysianShrew » Tue Jun 12, 2007 7:14 pm UTC

No, the gravity of the hole would pull you into it condensing you and everything else it has pulled in into a single point. I don't think a human can survive being condensed into a single point.



But yeah, as you dig deeper underground, the mass above your head has gravity and is pulling you toward it ("up") while the rest of the planet is still pulling you toward itself ("down"). Subtract the forces and get the net force, which is going to be less than if all the mass of the planet were pulling you in the same direction. At the center of the planet, one half is pulling you up, one half is pulling you down, so you are effectively weightless at the center. Of course the math is much more messy because we live in 3d as opposed to the world of Dig Dug.

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Postby ArmonSore » Tue Jun 12, 2007 10:04 pm UTC

I think it's worth noting that it's not really the fact that the forces are canceling out that makes you 'feel' lighter. If two people punch you in the face(one on each cheek, each exerting the same force) the vectors clearly cancel out, but you still feel like two people just punched you in the face!

So the reason why you 'feel' lighter can't be simply because the gravitational forces are canceling out(though it is a true statement). The truth of the matter is that nobody ever 'feels' gravity(at least I haven't). What you 'feel' is the normal force of the ground pushing you up to cancel with the force of gravity pulling you down. The reason why you 'feel lighter' when you delve deeper into the ground is because the ground must exert less normal force on your body in order to cancel the net gravitational forces(So yeah, It's due to the canceling out of gravity, but we still shouldn't forget that less normal force is why you feel lighter!.)

P.S. I think the only reason why we don't 'feel' gravity is because it acts uniformly for our bodies (i.e. g is constant over the tiny height of our bodies). So I believe that you would feel a black hole's gravity as you get sucked in, because the forces are significantly different for parts of your body closest to the hole and parts farthest.
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Postby Alan » Wed Jun 13, 2007 12:04 am UTC

D&D had a hollow Earth where the inhabitants on the inner surface had gravity caused by the shell mass. But, I think there was some magic high-density mass in the shell that allowed full gravity even though the planet was hollow.

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Postby NathanielK » Wed Jun 13, 2007 12:09 am UTC

Alan wrote:D&D had a hollow Earth where the inhabitants on the inner surface had gravity caused by the shell mass. But, I think there was some magic high-density mass in the shell that allowed full gravity even though the planet was hollow.
It doesn't matter how massive the shell is; assuming it is spherically symmetric, the gravitation force inside it is zero. I would suggest that D&D may not be the best place to look for scientifically accurate world-building.
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Postby Belial » Wed Jun 13, 2007 12:19 am UTC

NathanielK wrote:
Alan wrote:D&D had a hollow Earth where the inhabitants on the inner surface had gravity caused by the shell mass. But, I think there was some magic high-density mass in the shell that allowed full gravity even though the planet was hollow.
It doesn't matter how massive the shell is; assuming it is spherically symmetric, the gravitation force inside it is zero. I would suggest that D&D may not be the best place to look for scientifically accurate world-building.


You're...uhh...wrong.

If the ground "below" (toward the outside of the sphere from you) you is massive, and the ground "above" you (on the other side of the sphere) is sufficiently far from you that its gravitational effect on you is less than the part you're standing on, then there would be gravity pulling you toward the outside of the sphere.
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Postby SpitValve » Wed Jun 13, 2007 12:39 am UTC

Belial wrote:If the ground "below" (toward the outside of the sphere from you) you is massive, and the ground "above" you (on the other side of the sphere) is sufficiently far from you that its gravitational effect on you is less than the part you're standing on, then there would be gravity pulling you toward the outside of the sphere.


Sorry I have to disagree with you there...

It actually turns out that for a hollow sphere, you do indeed feel no gravity if you're inside it. If you're outside it you feel gravity as if it came from the centre of the sphere.

There's a geometric argument for why there's no gravity within a sphere, something to do with gravity scaling as 1/r^2 and surface scaling as r^2.

I think it's something like:

Take a double-cone with small angle dtheta with you at the vertex. The cone will intersect the sphere you're inside at two circles. We consider the gravitational attraction from these circles. If the gravity is equal for any direction, then there's no net gravity.

The distance to one circle is r1, the distance to the other is r2. The radii of the circles are r1*dthetha and r2*dtheta, and the area of the circles are pi*(r1*dtheta)^2 and pi*(r1*dtheta)^2. So the mass of each is proportional to that. But because gravity goes as 1/r^2, the gravity due to each circle is the same. They're in opposite directions, so it cancels.

This is a rough argument from memory: I would appreciate people filling in the gaps :)


Edit: Actually I might be a little confused: are people talking about living between concentric spheres? Then your gravity would be towards the centre sphere.

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Postby Vaniver » Wed Jun 13, 2007 1:36 am UTC

SpitValve, that's pretty much it.
I mostly post over at LessWrong now.

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Postby Belial » Wed Jun 13, 2007 2:40 am UTC

Hmm. But the force from above doesn't adequately cancel the force from below, as it were. It would seem to still pull down.

Also, I'm assuming this world would be spinning at a pretty brisk pace. Which probably helps.
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Postby chrisb33 » Wed Jun 13, 2007 3:27 am UTC

The forces actually do cancel exactly, as SpitValve outlined. The area "underneath" you that pulls you down is much smaller than the area "above" you that pulls you up, and these two effects balance each other.

Newton proved this in his Principia:
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Postby SpitValve » Wed Jun 13, 2007 3:38 am UTC

psh, that's not even in Latin

You're right though :)

Edit: I'm going to forgive you not putting your first post in the intro thread because you're supporting my argument.

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Postby Shadowfish » Wed Jun 13, 2007 3:43 am UTC

Since no one has mentioned it yet, what spitvalve was talking about is called gauss's law

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Postby ArmonSore » Wed Jun 13, 2007 4:03 am UTC

Or Newton's Shell Theorem!
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Postby SpitValve » Wed Jun 13, 2007 4:24 am UTC

Gauss' law is most useful when there's extreme symmetry... here Newton's geometrical way is nicer I reckon. No calculus involved!

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Postby skeptical scientist » Wed Jun 13, 2007 4:48 am UTC

Yeah, but it's less than rigorous. You should really do it with calculus, although this makes a good hand-waving argument for why it's true.
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Postby SpitValve » Wed Jun 13, 2007 4:56 am UTC

skeptical scientist wrote:Yeah, but it's less than rigorous. You should really do it with calculus, although this makes a good hand-waving argument for why it's true.


Fair enough.

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Postby evilbeanfiend » Wed Jun 13, 2007 8:48 am UTC

the d&d world could work if the shell didn't have uniform density or was spinning really fast, but then it wouldn't work uniformly over the inside of the shell. the other alternative is they just got a wizard to make it work using magic.
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Postby SpitValve » Wed Jun 13, 2007 10:07 am UTC

evilbeanfiend wrote:the d&d world could work if the shell didn't have uniform density or was spinning really fast, but then it wouldn't work uniformly over the inside of the shell. the other alternative is they just got a wizard to make it work using magic.


Would rotation make a difference? I suppose if you were looking at it GR-styles it would, but it'd have to be damn fast...

If one side was denser than the other, then the denser side would be "down". But wizardry sounds easier :)

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Postby evilbeanfiend » Wed Jun 13, 2007 10:13 am UTC

SpitValve wrote:Would rotation make a difference? ...


the equator would be down, due to it no-longer being an inertial frame you feel a centrifugal force away from the axis of rotation, and given there are no other forces acting on you it doesn't have to be that quick a rotation.
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Postby Cosmologicon » Wed Jun 13, 2007 3:30 pm UTC

The problem with rotating a Dyson sphere to make artificial gravity is that it doesn't work at the poles, so all your atmosphere drains off there. What you can do is wall off a habitable zone about the Equator with enormous walls to keep the atmosphere in. But then, you don't need all that empty space off the Equator, so you might as well lose that.

Then you've got Ringworld.

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Postby skeptical scientist » Wed Jun 13, 2007 3:41 pm UTC

You don't need walls. The incline of the spherical shell provides walls.

And just because the area away from the equator isn't living space doesn't make it wasted. It still serves to capture the energy radiated by the sun. Remember, the reason you build Dyson spheres is not for living space, it's in order to capture the total energy output of your star. As you say, a ringworld would probably give you plenty of living space. Considering that a reasonably high population density on Earth would exhaust the carrying capacity long before it would exhaust the living space, one would need the entire energy output of a sun before one actually inhabited more than a fraction of the area of a Dyson sphere.
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Postby evilbeanfiend » Wed Jun 13, 2007 4:34 pm UTC

if you are inside a dyson sphere then you are no longer exempt from gravity as there is a wacking great star in the middle.
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Postby Yakk » Wed Jun 13, 2007 6:29 pm UTC

Just stick an anti-gravity radiant at the center of your world. Make it shine bright enough (maybe with a day/night cycle flux) while we are at it.

Then make the shell massive enough to generate enough gravity to cancel out the radiant, and generate 1 G downward force.

Voila -- 1 G outwards inside, and 1 G inwards outside. :)

I mean, we are talking about a hollow earth model. ;)

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Postby SpitValve » Wed Jun 13, 2007 8:50 pm UTC

evilbeanfiend wrote:
SpitValve wrote:Would rotation make a difference? ...


the equator would be down, due to it no-longer being an inertial frame you feel a centrifugal force away from the axis of rotation, and given there are no other forces acting on you it doesn't have to be that quick a rotation.


I see. I was thinking about a guy floating somewhere inside the sphere, as opposed to somebody standing on the inner surface and rotating with it.

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Postby gmalivuk » Thu Jun 14, 2007 1:59 am UTC

Yakk wrote:I mean, we are talking about a hollow earth model. ;)


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(Okay, so that particular expanding earth diagram still has magma in the middle. But I think that Neal Adams, who's the first person I heard talking about it, believes the center is hollow.)
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