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Gravity Query


GeeKay

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If gravity is what happens when mass curves space and time, what would be the situation were a given mass remotely non-spherical - i.e. a perfectly flat bar or sheet, for example? Would there still be a localised curvature of space and time? If so, how does gravity's tidal pull fit into this arrangement?

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Well the 'curvature effect' would be uneven and somewhat reflect the shape of the object. You would, in principle, be able to deduce the shape of the object from the variation in the curvature 'field'.

 

Note the words in inverted commas are meant to be taken in a somewhat colloquial sense rather than a strict physics one.

 

However most objects on a scale sufficiently large to really exert appreciable gravity are roughly round. Even Saturn and its rings could be called round.

There are two effects that promote this.

 

Firstly, surface tension tends to create globular shapes as with raindrops.

 

Secondly, extended shapes are subject to quite large disruptive forces due to twisting and rotation. So they tend to break up.

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Thanks for the helpful clarifications. Yes, I do understand that in the 'real world' of gravitational attraction the sphere offers the least surface area for a given volume of mass. However, I'm trying to step out of the real world for a moment in order to understand more fully the relationship between gravity, geometry and matter. Change the geometric arrangement of matter, for instance, and it seems that this alters the curvature of space/time, hence gravity itself. Thus, the most extreme example in this 'thought experiment' would be to imagine how a large uniformly flat expanse of matter would affect local space/time. Without a conic section being present (such as a section of a sphere), how could there be a tidal pull? If there is no tidal pull, does this mean an absence of gravitational attraction, period?

 

My apologies for pursuing this line of thought, but I feel this is just the kind of question any astute eight-year old child would ask. At present I would be unable to answer it - that is, without tying myself up in knots in the process.

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Tides are a result of differences in gravity. Unless gravity was uniform everywhere, you would still have tides. And in the absence of tides, you could still have gravity — not having a gradient means the function is constant. It doesn't mean it's zero.

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Thanks for the helpful clarifications. Yes, I do understand that in the 'real world' of gravitational attraction the sphere offers the least surface area for a given volume of mass. However, I'm trying to step out of the real world for a moment in order to understand more fully the relationship between gravity, geometry and matter. Change the geometric arrangement of matter, for instance, and it seems that this alters the curvature of space/time, hence gravity itself. Thus, the most extreme example in this 'thought experiment' would be to imagine how a large uniformly flat expanse of matter would affect local space/time. Without a conic section being present (such as a section of a sphere), how could there be a tidal pull? If there is no tidal pull, does this mean an absence of gravitational attraction, period?

 

My apologies for pursuing this line of thought, but I feel this is just the kind of question any astute eight-year old child would ask. At present I would be unable to answer it - that is, without tying myself up in knots in the process.

 

I think in the case of a infinite plane you would end up with a uniform gravitational field - and not only uniform along the plane but towards and away from the plane (ie independent of distance from the plane).

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Nevertheless, I still wonder if a 'uniform' gravity field is possible under such conditions. It seems that Newton's field equations and General Relativity - in particular the transference of rest-energy into energy of motion with changes in altitude - require a gravitational tidal pull of some kind. This in turn would appear to be dependent upon a rotund mass curving space/time about itself. Another way would be to see it in terms of a conic section - an infalling object entering ever more constricting regions of space/time. This funnel-like geometry then is responsible for the gravitational differential, hence the tidal effect. Would this apply to a flattened geometry, however - one without a conic section? No conic section, no tidal pull? No tidal pull, therefore no G? Or would gravity behave very differently in such circumstances?

 

Ref. Einstein's Universe (esp. the chapter 'Shells of Time') by Nigel Calder: BBC publications (1979)

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Even near an almost infinite plane the space time will become curved - remember this is an intrinsic 4 dimensional curvature and not something that can be readily visualize in our 3 dimension monkey brains. No tidal pull no gravity is just plain wrong - I stand to be corrected but I think most GR calcs are done basis point-sized test particles which by their non-extended nature cannot see tidal forces.

 

What important tidal forces act on a billard ball rolling down a track - sure they are there but you can completely ignore them. Einstein uses the equivalence of gravitation (which could have tidal forces) and acceleration (which does not) as the very basis of most of Relativity. I think you are visualizing this as increased density of force lines - "ever more constricting regions of space/time" - and whilst this does explain lots of things (the inverse square law can be nicely comprehended with this notion) it is only an analogy, a heuristic that aids understanding. Einstein talks of a 4d spacetime and a geometric curvature

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imatfaal

and a geometric curvature

 

 

Remember also that we are used to seeing 2D curves on flat paper when we talk about curvature.

In that case there is only one direction available to curve in and only one plane for the radius of curvature.

 

When we go to 3D, there are two possible radii of curvature at any point on a curve and if we go to 4D spacetime then there are three, with time as one of the plane's axes.

 

This is really the province of differential geometry in higher dimensions, which I think is ajb's speciality.

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To the above contributors: many thanks for your helpful and insightful comments. I began this line of enquiry at the point where it seemed I'd run out of understanding about GR. I think I have a better grasp of it now. Heaven help my brain cells, though, should a quantum theory of gravity ever emerge!

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If gravity is what happens when mass curves space and time, what would be the situation were a given mass remotely non-spherical - i.e. a perfectly flat bar or sheet, for example? Would there still be a localised curvature of space and time? If so, how does gravity's tidal pull fit into this arrangement?

 

It is really coincidental that you post this topic, when I was thinking on similar lines. My thoughts, were given a hypothetical almost planet sized mass 'Cube' how would gravity works on it and effect the entities that exist on its surface? Make this a thought experiment, and ignore the fact that a planet sized mass, would become a orb due to its huge mass.

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