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Mass causes space time curvature 
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#1
Dec112, 12:29 PM

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If I take it by literally meaning: Mass causes space time to curve. A rubber sheet where the mass is there, it causes the dent, the curvature.
So it means the greater the momentum, the greater the curve or the dent. Now if we have a very big mass, I mean to say big in terms of size, the greater will be the curve and smaller the size, smaller the mass. Is that so? In that case where are the areas that would have a bigger curve on space time rather than smaller one?  Shounak 


#2
Dec112, 06:18 PM

P: 356

Space time curvature is measured by a tensor.
More qualitatively, the curvature at a given event in space time is measured by how much a vector changes if you take it around an infitesimal loop at that event It is difficult to answer your question, the stress energy tensor causes curvature. The most well known solution is that of a non rotating non charged sphere, spacetime deviates more and more from the flat minkowski space time the closer you approach the sphere, if you want a picture to wrap your mind around, http://en.wikipedia.org/wiki/File:Flamm.jpg this might help. 


#3
Dec312, 02:16 AM

P: 268

Yes, thank you for the reply.
I went through the stressenergy tensor and understood the components. Here energymomentum tensor the energy and momentum is conserved right? My question is: The bigger the <size> of the mass, the bigger the dent(curve) in the space time. Sun, moon etc. if we imagine them as mass spheres, does the curvature is caused due to the volume of the planets? May be it is a very wrong question but still.........................  Shounak 


#4
Dec312, 06:25 AM

P: 14

Mass causes space time curvature
I'm probably not qualified to answer your question technically (I don't study physics), but I would think that the geometric size of the object is irrelevant to the curvature it causes. Using the rubbermembrane analogy, an inflated balloon would cause much less curvature than a small iron ball as it has less mass. I'd argue the same thing applies to the fabric of spacetime, as black holes are tiny objects with incredible mass that still manage to warp spacetime to a degree that even light can't escape it on the inside of the eventhorizon.



#5
Dec312, 07:44 AM

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#6
Dec312, 09:25 AM

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The image it conjures is that of a planet as a little ball rolling on the membrane, and it's the inward force of the sloped surface that causes the planet to go around and around in an orbit. It implies, moreover, that the curvature of the surface represents the curvature of spacetime (which it doesn't), and that this in turn represents gravitational attraction (which it doesn't). Flamm's paraboloid is a slice through the spacetime at constant t, whereas a planetary orbit is not at constant t; so the properties of this surface have little to do with planetary motion. Orbital motion, in fact, is caused by "curvature in time", i.e. the g_{00} component of the metric tensor. In linearized theory, g_{00} = 1  2V where V is the Newtonian potential, and it's the effect of a varying V that enters the geodesic equation and causes them to spiral around the origin. 


#7
Dec312, 09:43 AM

P: 356

It does provide a rare chance to visualize something in a theory as abstract as GR.
However, the radial component of the metric tensor in the schwarschild metric makes a non negligible contribution to the r^3 correction term for orbital trajectories. 


#8
Dec312, 11:34 AM

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#9
Dec312, 12:31 PM

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The general public generally regards physics as magic, does it matter?



#10
Dec312, 01:51 PM

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#11
Dec312, 02:32 PM

Emeritus
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I think Flamm's paraboloid depicts spatial curvature, and not spacetime curvature. If it was going to represent spacetime curvature, one of the embedded coordinates would have to be timelike. But both embedded coordinates are spacelike, at least in the exterior region.



#12
Dec312, 10:59 PM

P: 268

Well, lot of thanks to Bill. At least we come to know that Flamn's paraboloid is not what is shown in the equations. Well, I would like to extend my special thanks to Bill to point out what it actually means. Yes, this is the problem with popular science and try to depict anything on a common way. But indeed for understanding General Relativity and what is actually a curvature of space time, we need an image in order to visualize what POSSIBLY can be.
I would request Bill to enlighten us kindly in a simple language (which I know it is difficult) what is actually a space time curvature. I also have one question to Nugatory . As you have told :The only thing I'd add is that the it's not just the mass that matters but how concentrated it is." In the stress energy tensor component there are components like 'energy density' & 'momentum density'. Does that actually mean "HOW MUCH ENERGY" & "HOW MUCH MOMENTUM" it shows? I mean to say what you told is the 'concentration'? 


#13
Dec412, 02:40 PM

P: 5,632

yes to the last two questions above. extended, low density mass, can't cause all that much curvature of spacetime....analogous to the siutation where you pass thru a hollow cylinder to the center of the earth where gravitational attraction diminishes.....
But visualizing how gravitational potential curves spacetime is beyond me. have no notes with me but I may have a spacetime curvature illustration reference and will post next week if i can remember.... This is not spacetime curvature caused by earth: http://simple.wikipedia.org/wiki/Spacetime despite the illustration explanation...could not find an illustration ..... Explaining 'curvature' is not so easy....you can start via the Riemann curvature tensor and you'll find further references like Ricci...etc,etc....andthere is extrinsic (embedded) curvature and intrinsic curvature of the Riemannian manifold which Einstein utilized. If you are moving at constant speed but experiencing acceleration you are in curved space or time or both..... 


#14
Dec512, 03:07 AM

P: 268

Thanks Naty. So the intensity of momentum and energy determines HOW much the curve. But it is really interesting that determining curvature is so difficult. But I would really like to know just a little bit about intrinsic curvature. What does it mean?
One more question, sun,moon, other planets are also mass which causes the curvature in the spacetime manifold. So, is it bigger the mass (bigger the size of the planet) the bigger the curvature in that area of space? Thanks,  Shounak 


#15
Dec512, 07:50 AM

P: 5,632

the standard interpretation of curvature takes Ricci curvature as a represntation of
a volume element change and the Weyl component as a measure of tidal forces.....Exactly why those have been so widely adopted is 'above my paygrade'...but likely it is because they offer valid physical insight. For insights into curvature, try here http://en.wikipedia.org/wiki/Riemann_curvature_tensor and http://en.wikipedia.org/wiki/Riemann...trical_meaning How the different measures of curvature compare and the advantages and disadvantages of different formulations is something i have not gotten into.... but apparently Einstein's utilization of existing mathematics from Riemann's work has pretty well stood the test of time. There have been discussions in these forums of some 'upgrades' that have been investigated but I did not follow those closely. Riemann's curvature tensor is but one way to express curvature of a Riemann manifold: http://en.wikipedia.org/wiki/Curvatu...nian_manifolds 


#16
Dec512, 09:28 AM

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BUT  This is a vague and handwavy sort of description of a clear and precise mathematical concept. If you want real understanding, you have to move beyond these vague analogies (remember what I said about the rubber membrane in the post you quoted from) and look at the math. 


#17
Dec512, 09:43 AM

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#18
Dec612, 12:43 PM

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Here is a nice discussion of relativity...
and some comments on curvature here: http://math.ucr.edu/home/baez/einstein/node2.html 


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