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noz92

cause of gravity

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What is the cause of gravity. I know that it's caused by bent space-tme, but, every model we have shows space-time as a 2-D surface. If this is correct, then every boddy in the universe would be roughly the same highth, and (using the earth as an example, and assuming that the earth doesn't tilt) how could a north pole exist, if neither space nor time would exist there (although, on the geographical north pole, I'm not sure how you could set your watch :) [laghs]). So the innacuracy of this model is very clear, but I'm not sure how space-time could be bent any other way.

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The 2-D surfaces are just visual aids they do no represnt spacetime which is four dimensional (infact due to the signature of it's metric even two dimensional spacetime can never be proeperly represented in 3 dimensional approximately Euclidean space). The curvature of spacetime is something hat is very difficult to grasp, but people are already famir with the idea of two dimensional curved spaces, so it makes a good starting place.

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Gravity is spacetime-curvature. Spacetime-cuvature is cause by mass (loosely speaking). Mass is described by the stress-energy-momentum tensor (the point that stress and momentum are also included is why I said "loosely speaking" in the last sentence).

 

Curvature is a confusing term. It´s an attibute of space-time at every point. There are several equivalent definitions for it. However, it´s not nessecarily what you expect of an attribute called "curvature". For example, the surface of a cylinder is bent but not curved. Also, the definitions of curvature does not need the spacetime to be embedded in another space - it would be a bit contradictory if the universe (everything there is) was embedded in something greater. So as Aechylus said: Those pictures where a 2D surface is embedded in 3D are just visual aids.

 

Example showing curvature: Take a unit sphere (an apple, for example) and a vector (pencil). Put your vector in the point (0,-1,0) (note that vectors in GR have a point they belong to). Now rotate your pencil along the equator (z=0) to the point (0, 1, 0). Rotate it along the circle (x=0) to your original point afterwards. Compare with the original pencil´s direction. Try the same on a cylinder. Funny, eh?

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The 2d system you see is just something used to make the concept easier to understand. Curvature in 4d space is not something a human can accuratly visualise. When this is used, it is vastly easier to see the basic results of something. The representation should not be taken too literally, its just a mental aid.

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Yeah, that whole bowling-ball-and-trampolene analogy probably causes more confusion than enlightenment. The biggest misunderstanding one is apt to get out of it is that the warping of spacetime results in objects being pulled towards a heavier object because there is some all prevading gravitational field all throughout the universe that pulls all objects down (i.e. not towards the center of the heavier object), and both the light and heavy object are "sitting" on a certain 2D layer of the spacetime continuum, and the smaller object only moves because it "slides" down this layer where it's sloped. The obvious problem with that interpretation is that one must subsume gravity into the picture beforehand, explaining how the smaller object slides down a sloped spacetime plane, but then nothing's been explained about why gravity exists in the first place.

 

The best way to understand the warping of spacetime around masses is to image a universe with 1 fewer dimension than our universe has. How? First, understand what Aeschylus means by a 4 dimensional spacetime: 3 space and 1 time. Save the 1 time dimension and get rid of 1 space dimension in your imaginary universe, and then imagine the time dimension as if it were the 3rd space dimension from the 4D universe. This is OK since the whole reason we consider time to be a fourth dimension is because it seems to have all the same properties of the 3 space dimensions, and therefore its only difference may just be in how we experience it. So imagine that the 2 space dimensions compose an infinit plane. Now, since all objects in the universe must occupy all spatial dimensions, then all objects in your imaginary universe must be 2D. An object like the Sun or the Earth would be discs. Now visualize the time dimension: if you've got a flat plane for the 2 space dimensions (like the ground), then you must imagine time as a vertically oriented dimension (like lamp posts or anything extruding upwards from the ground). In effect, what you end up envisioning is a 3D universe which is still 2 space and 1 time, but indistinguishable (visually, that is) from our familiar 3D universe of spatial dimensions without taking time (4th dimension) into consideration. Now you've still got these discs floating around in this 2D plane. Let's fix that: because all objects (whether 2D or 3D or nD) prevade throughout all time, then they exist at all points of the time dimension. A rock just sitting there may only exist in a limited volume of space, but it still exists in that volume at time t1, t2, ... tn, etc. This passing of time can be visualized in your imaginary universe as multiple layers of 2D space planes stacked ontop of each other, and in each one of these planes, the discs representing objects continue to exist at their coordinates in the 2D plane (if they're not moving). So you end up getting a series of discs stacked on top of each other. This can be simplified from a multitude of discs to one long cylinder. That's right, objects like planets and other discs are actually cylinders in our 3D (2 space, 1 time) universe. So now you're ready to imagine the warping of spacetime: take your verticle time dimension and imagine that, instead of being parallel to all the cylinders, the time dimension actually bends towards the cylinders. The more massive the cilynder, the greater the bend. What happens after the time dimensions intersects with the cylinder's surface? I don't know, I guess it gets "sucked" into the cilynder never to be seen again. Finally, apply the principle of "the shortest distance between 2 points is a straight line" to all lighter objects being pulled by gravity due to the heavier objects. With warped spacetime, a "straight line" isn't so straight anymore, but objects still behave as if they were straight. That is, they simply follow the path they've always been following (through time or space). If that path so happens to be curved, they'll curve with it, almost as if they don't perceive the curvature and see it as a "straight line" instead. So you must image the cilynders of lighter objects actually bending along with the time dimension towards the heavier object. And that's that.

 

When you return to our familiar 4D universe, you can now image what's going on when a lighter objects falls towards the heavier object. As it accelerates towards the heavier object, what you're in fact seeing is its position at higher and higher points along the time dimension. What you're looking at when you see the heavier object is actually a 4D cylinder* and at each moment in time, the 3D sphere that you see is actually different slices of this 4D cylinder. The fact that time bends means that it becomes more parallel with the space dimensions, and this means that passage through time starts to look more like passage through space, and this is indeed what's going on. As the object moves through time, the time dimension being curved results in the object changing its trajectory so that it ends up moving more through space than exclusively through time. Does this help at all, or did I confuse you even more?

 

* No, I'm not suggesting that there actually ARE 4D cilynders, I'm just saying that it works as a model for understand GR.

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For example, the surface of a cylinder is bent but not curved. Also, the definitions of curvature does , eh?

 

We say a cylinder has extrinsic curvature, but no intrinsic curvature.

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yep, and no curvature in the sense of GR (or at least I wouldn´t know where extrinsic curvature plays a role in GR).

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