Is There Gravity At The Center Of A Perfect Sphere?

[johnferk]

Thought experiment -

 

Consider a hypothetical object which is a perfect sphere of some radius "X" and some gravitational field "Y" when measured at the surface of the sphere. For the simpliest case, let's assume the matter contained in the sphere is homogeneously distributed throughout the sphere and there is no density differential as you proceed from the surface to the center of the sphere.The gravitational field at any point on the surface of the sphere will always be "Y" since the mass producing the field will be the same relative to the observer regardless of where the observer locates on the sphere's surface (i.e., the observer will always have 1 radii of mass between his/her position and the center of the sphere.

 

If the observer now travels 1/4 radii toward the center of the sphere, he/she has 1/4 radii of mass above them and 3/4 radii of mass to the center. Since gravity by definition is an "attractive force" and there is mass both above and below the observer, the observer is now being "attracted" both "upwards" and "downwards." One would predict the observer would experience a gravitational field less than that felt on the surface of the sphere since some of the "upwards" attraction would negate a portion of the "downward" attraction.

 

Now carry the scenario to the exact center of the perfect sphere. At that point, the observer now has 1 radii of mass in all directions. Gravitational fields should cancel and there should be "zero" gravity.

 

Obviously this is not what we see in real space time.

 

What am I missing here? :rolleyes:

[sigurdV]

Thought experiment -

 

Consider a hypothetical object which is a perfect sphere of some radius "X" and some gravitational field "Y" when measured at the surface of the sphere. For the simpliest case, let's assume the matter contained in the sphere is homogeneously distributed throughout the sphere and there is no density differential as you proceed from the surface to the center of the sphere.The gravitational field at any point on the surface of the sphere will always be "Y" since the mass producing the field will be the same relative to the observer regardless of where the observer locates on the sphere's surface (i.e., the observer will always have 1 radii of mass between his/her position and the center of the sphere.

 

If the observer now travels 1/4 radii toward the center of the sphere, he/she has 1/4 radii of mass above them and 3/4 radii of mass to the center. Since gravity by definition is an "attractive force" and there is mass both above and below the observer, the observer is now being "attracted" both "upwards" and "downwards." One would predict the observer would experience a gravitational field less than that felt on the surface of the sphere since some of the "upwards" attraction would negate a portion of the "downward" attraction.

 

Now carry the scenario to the exact center of the perfect sphere. At that point, the observer now has 1 radii of mass in all directions. Gravitational fields should cancel and there should be "zero" gravity.

 

Obviously this is not what we see in real space time.

 

What am I missing here? :rolleyes:

Im not sure u miss anything.

 

One reason we dont see, is that it is difficult to travel to ,say, the gravitional center of the earth. Its safer to calculate so lets wait and see them calculators arrive :)

[johnferk]

I ask the question as a thought experiment about black holes.Because they are "infinity dense" they could be considered homogeneous. Hence, at the center of the BH (and I am not talking about the event horizon, rather the black hole itself), you would carry the arguement to its extreme and there would be no gravitational field at the core of the BH.

[CraigD]

Welcome to hypography, jwkref :)

 

Thought experiment -

 

Consider a hypothetical object which is a perfect sphere of some radius "X" and some gravitational field "Y" when measured at the surface of the sphere

…

Obviously this is not what we see in real space time.

 

What am I missing here? :rolleyes:

I don’t think you’re not missing anything. What you’re describing is known as the shell theorem. It’s one of the early important results derived using calculus, in the early 1700s, by Newton.

 

Where you appear to err is in you statement “Obviously this is not what we see in real space time.” Almost certainly, this is what we would actually see, if we had access to holes more than a small fraction of distance to the center of the Earth deep, or in something easier to tunnel through, like though a decent size rubble-pile asteroid.

 

What’s more interesting, I think, than how the shell theorem applies to solid spheres (which don’t have to be the same density throughout, only within each shell), is how it applies to hollow ones. Consider the following “what if” thought experiment:

 

What if the Earth were a thin, hollow sphere of the same mass as it actually is?

 

Were this the case, conditions on the outside of the sphere would be as they actually are. On the inside, however, objects would be weightless, regardless of their distance from the inside surface.

[sigurdV]

Welcome to hypography, jwkref :)

 

What if the Earth were a thin, hollow sphere of the same mass as it actually is?

 

Were this the case, conditions on the outside of the sphere would be as they actually are. On the inside, however, objects would be weightless, regardless of their distance from the inside surface.

 

Hi CraigD

 

Dont remember who wrote it, but in an addendum he showed how negative gravity arises when a sphere like that was diminished... Interesting reading... A repeat would be nice. (Maybe Andrei Linde)

[johnferk]

Thanks much for the response(s). I don't feel so bad now that I know Newton considered the question in the 1700's.

 

I'm a tad behind on my reading of the literature :P

 

Cheers,

jkref

[jeremyb]

I ask the question as a thought experiment about black holes.Because they are "infinity dense" they could be considered homogeneous. Hence, at the center of the BH (and I am not talking about the event horizon, rather the black hole itself), you would carry the arguement to its extreme and there would be no gravitational field at the core of the BH.

 

 

has it been proven that a black hole is indeed a sphere? strange things happen under extreme conditions that seem to defy the laws of physics.

 

one thing to consider, if a black hole is a sphere, the exact center of it would be a point so small, no instrument could could get you to the exact spot, kinda like absolute zero (cold), you can go way way down there, but you will never reach absolute zero.

 

though, dont take my word on anything, my schooling ended with my HS diploma :D

[Cyberia]

A sphere is the smallest shape that a mass can attain due to gravity.

 

"Infinite" is one of those words that exist in mathsworld but not the real world. The singularity before the big bang was said to be infinitely dense and gravity did not exist in it.

 

There is also the point that black holes rotate, suggesting that what is inside rotates. Infinite density would be a one dimensional dot which cannot rotate since you would need at least two dimensions to do that.

 

We have no evidence that fundamental particles have any smaller forms and if this is so, then at the center of a black hole would be a perfect sphere of these particles. Note: Neutrons exist in neutron stars with an escape velocity of 2/3 c and they are fairly complex.

[maddog]

has it been proven that a black hole is indeed a sphere? strange things happen under extreme conditions that seem to defy the laws of physics.

one thing to consider, if a black hole is a sphere, the exact center of it would be a point so small, no instrument could could get you to the exact spot, kinda like absolute zero (cold), you can go way way down there, but you will never reach absolute zero.

though, dont take my word on anything, my schooling ended with my HS diploma :D

For a simple answer yes from a distance a black hole would appear sphere-like of course only what you could see

which is from radiation falling in and not the hole itself. With all black holes there is an event horizon.

This is EH as it is know is a barrier that prevent all matter from descended below can not reach

back out. Even light can not escape. The danger here is in trying to the physics we know out here

on matter that already fallen below the Event Horizon. A real-based coordinate system no longer

applies. Your analogy of temperature is misleading though somewhat accurate as the effect

here is also asymptotic. Where it gets misleading is like with temperature the end point has

no real world significance. In this case it is like evaluating 0/0 which is totally useless.

 

The other comment made elsewhere in this thread is when the Black Hole is rotating,

called the Kerr Metric. The black hole behaves a lot like the Earth when rotating. It flattens

out (Event Horizon) a little becoming an Oblate Sphereoid (pudgy at the equator, squished

at the poles).

 

BTW, I first learned about Black Holes when I was in High School, a Junior I believe.

 

maddog

[maddog]

Thanks much for the response(s). I don't feel so bad now that I know Newton considered the question in the 1700's.

I'm a tad behind on my reading of the literature :P

jkref,

 

Some college level physics texts (majors) have this as a very hard homework problem as

to how Newton solved the calculation of the gravity for a spherical body like the earth.

It was by working out the gravity contribution of each shell and integrating over the shell.

 

I had to do this in my Freshman Physics course. It was very hard!

 

maddog

[Eudoxus]

Black holes have no volume. They are an infinitely small point surrounded by a conceptual finite sphere of space, the event horizon. A point has no center, it has no circumference, it has no radius, it has no volume. There's no center for there to be zero net gravitational force at. It's just a point, a point at which Einstein's field equations yield a big ??? with asymptotes approaching infinity on either side.

Edited March 24, 2012 by Eudoxus

[CraigD]

Black holes have no volume. They are an infinitely small point surrounded by a conceptual finite sphere of space, the event horizon.

A serious, if pedagogical, question, Eudoxus: how can we actually know anything about the arrangement of anything within the event horizon of a black hole?

 

A point has no center, it has no circumference, it has no radius, it has no volume. There's no center for there to be zero net gravitational force at. It's just a point, a point at which Einstein's field equations yield a big ??? with asymptotes approaching infinity on either side.

Next question: As we know that General Relativity, a “extended” classical theory, in the sense that it does not include quantum mechanics, does not accurately describe very or infinitely small volumes, such as a zero-radius sphere or zero-thickness disc, are we justified in accepting solutions to its equations that describe these as the volumes containing all of the mass in a black hole?

 

In several hypography posts over the year, such as this one, I’ve pointed out that, based on intuitive consideration of how low the average density inside the event horizon of a black hole can be, the macroscopic structure within it might be very complex (and that our entire universe may be an example of a complex structure inside a black hole). In this post in the same thread, I linked to a paper exploring this possibility with more, though not completely, rigorous physics.

 

Black holes, it seems to me, are really much more complicated than their idealized GR solutions suggest.

[Cyberia]

 

The other comment made elsewhere in this thread is when the Black Hole is rotating,

called the Kerr Metric. The black hole behaves a lot like the Earth when rotating. It flattens

out (Event Horizon) a little becoming an Oblate Sphereoid (pudgy at the equator, squished

at the poles).

 

maddog

 

The centre of a black hole would be a perfect sphere. The event horizon, ie: the gravity from it producing a point where even light cannot escape called the event horizon should be an even distance from this sphere. So a sphere also. Like all large objects in space, the black hole spins and mass heading towards it falls towards it's equator, it's place of fastest spin. This could give an illusion of greater diameter.

[CraigD]

The other comment made elsewhere in this thread is when the Black Hole is rotating,

called the Kerr Metric.

The centre of a black hole would be a perfect sphere.

Sexton, I believe by “centre of a black hole”, you mean “singularity predicted by General Relativity to exist at the center of a black hole”. Please let me know if I’m misinterpreting you.

 

While my mathematical physics is too weak to follow or derive myself the Kerr Metric for a rotating, uncharged black hole, the explanations I’ve read by people who can state that the singularity described by this exact solution is not a zero volume sphere (a point) but a zero volume torus (a circular line) usually called a ring singularity.

 

I think it’s important to recognize that none of the exact solutions for black holes given by General Relativity are thought by most physicist to be physically real descriptions of what’s actually inside them. The consensus, I believe, is that quantum effects dominate at small scales inside BHs, just as they do outside them, but since there’s no proven quantum mechanical solution for gravity, nobody can more than guess at what it’s physically like.

 

The event horizon, ie: the gravity from it producing a point where even light cannot escape called the event horizon should be an even distance from this sphere. So a sphere also.

Correct. The Kerr metric doesn’t still gives a perfectly spherical event horizon, just like the Schwarzschild metric for a non-rotating BH.

 

A difference in the two solutions is that the Kerr describes a oblate spheroid outside of its spherical event horizon at every point except the poles of its rotation called the ergosphere. Outside of the EH but within the ergosphere, it’s theoretically possible to get physical work from the BH. Leading to the prediction that rotating black holes eventually become (nearly) non-rotating ones.

 

Like all large objects in space, the black hole spins and mass heading towards it falls towards it's equator, it's place of fastest spin. This could give an illusion of greater diameter.

I’m aware of no effect that causes small bodies to fall other than toward the center of mass of large bodies, other than frame dragging. While frame dragging is theoretically predicted to be detectable around smaller bodies like planets, its effect is only predicted to be dramatic around very massive, compact ones, like black holes and neutron stars.

 

One might get the impression that gravity attracts small bodies more to points on the equators of large bodies than toward their poles and other points because more collisions are observed near their equators, but this, I think, is because the stars and planets axes of rotation tend to be close to perpendicular to the plane of their ecliptic, where most of the systems matter is.

[7DSUSYstrings]

Now carry the scenario to the exact center of the perfect sphere. At that point, the observer now has 1 radii of mass in all directions. Gravitational fields should cancel and there should be "zero" gravity.

 

Obviously this is not what we see in real space time.

 

What am I missing here?

 

 

It may not be what you are missing, but what we all are missing; Clear pieces of a puzzle that didn't originate on this planet. Up till 2006, I held to the mass attraction theory, but the theory of loops of gravity, or at least at short range, could be seen to contradict Occam's Razor and, in doing so, the conservation of energy. Gravitons are thought to be a non-massive particle with weak force, but infinite range (Lederman-Schramm at FermiLab). That would, as the inverse of infinitesimality, require infinity to balance. To originate from the center of a mass requires more energy than to simply be bombarded by it. For one of our favorite examples, Einstein vascilated on this till he died, meaning a repuslsive force and it is understandable because it is difficult to solve the conflict with gravitational loops and classic physics. The problem exists in the need for a balance. With one direction in a mean free path and the other omitted, there is no conflict. Such a flow would have to be inbound. With an inbound force, energy is conserved and your query is solved.

[maddog]

While my mathematical physics is too weak to follow or derive myself the Kerr Metric for a rotating, uncharged black hole, the explanations I’ve read by people who can state that the singularity described by this exact solution is not a zero volume sphere (a point) but a zero volume torus (a circular line) usually called a ring singularity.

 

I think it’s important to recognize that none of the exact solutions for black holes given by General Relativity are thought by most physicist to be physically real descriptions of what’s actually inside them. The consensus, I believe, is that quantum effects dominate at small scales inside BHs, just as they do outside them, but since there’s no proven quantum mechanical solution for gravity, nobody can more than guess at what it’s physically like.

CraigD,

 

This is my understanding of the Kerr Metric also. When rotating the Singularity does not resolve to a point.

 

maddog

[7DSUSYstrings]

 

Now this is the variation of a singularity, I've dubbed a convergent toroid field. A field like this, characteristic of a planetary magnetosphere, might be generated by:

 

1. Long wave-particle quantum loops

 

or

 

2. Omnidirectional, graviton propagation, mean free path influx.

 

By the first I'm suggesting a wave originating from zero point energy that expands in both mass and amplitude, then returns, as in classic periodic motion, to zero. By the second I'm suggesting the same zero point origin, yet differing in the absence of the wave, it being replaced by bursts of gravitons that project omnidirectionally, decelerate and return to zero. In both cases accretion is a byproduct of complex intersections in the return phase. If the universe, as a collection of intertial vectors, has a uniform rotation, the geometry would be envisioned as the torus rotating around the singularity, thus the individual points involved would trace out a hypotrochoid.

 

There's probably no direct references to the large scale concept, but the individual elements could possibly be referenced. One comment I might make, is one should keep in mind is the topic question is certainly to extract speculation from the Janus of the mainstream philosophies of gravity's cause.