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#1




The Earth warps the space around it but...
According to the last night's NOVA show on black holes, the Earth warps the space around it and this is why satellites are in orbit around it. The satellites are not turning, they are following the curve of space warped by the Earth.
I get this perfectly because this is what Einstein said about gravity. In another way this makes no sense because a beam of light would also follow the same path as the satellite. I don't think that a light shone parallel to the surface of the Earth would follow a path around the Earth. Explain. 
#2




The light is moving a lot faster. If you increased the satellite's speed, it would also not follow the same path anymore. If you increased its speed enough, the satellite would head off to deep space in a notquitestraight line, as with the light.

#3




Sufficiently massive gravity sources WILL bend light rays around them. This is predicted by relativity and has been seen visually. But the only things massive enough to actually trap light into an orbit are black holes, which is why things within the event horizon aren't visible.

#4




Earth's gravity does bend light, it just doesn't bend it very much. But we've seen light bent enough by galaxies, for example. See Gravitational lens for details.



#5




Quote:
Last edited by John Mace; 01112018 at 02:04 PM. 
#6




I generally dislike the "indentation in a rubber sheet" analogy, but to illustrate the relationship between orbit and speed, i.e. the amount of bending that a moving object experiences as it moves through curved space, the analogy works quite intuitively.
https://en.wikipedia.org/wiki/Gravit...ce_analogy.svg Imagine rolling a marble toward the curved depression, offcenter from earth. I think you get better intuition if you imagine a shallow depression and fairly slow speed. A slowly rolled marble will get bent sharply and just turn and crash into the earth. As you increase the speed of the marble it will bend less, and a fast enough marble will get deflected a little but make it back out of the depression to continue out the other side on a new trajectory. In between, there is a "sweet spot" where the ball is deflected to curve in a perfect circle around the earth momentarily. Of course, there's a lot friction here, so the momentary circular path will quickly decay as it slows, and it will spiral into the earth. But absent friction, it would continue to orbit in a circle. [Technically, this is not quite correct, something approaching in a straight line from afar would go into an elliptical orbit if anything, but let that slide.] The important point is that although the curvature is the same for every object, the amount of deflection that results from that curvature depends on the object's speed. That's why light is only deflected a tiny amount. It's also why, at a given distance from earth, there's a unique speed for an orbit. Orbital speed is where the object constantly "falls in" just the right amount to make a circle. Last edited by Riemann; 01112018 at 02:10 PM. 
#7




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gravity cannot be described by the curvature of space as the trajectory of a (test) particle depends on its speed If gravity where the curvature of space we would expect a particle to follow the curvature of space independently of its speed. However Einstein did not model gravity as the curvature of space, he modelled it as the curvature of spacetime. In spacetime the speed of particle affects its wordline (its path in spacetime) and gravity can be modelled as particles following the curvature of spacetime. 
#8




The key is that objects don't follow "straight lines" (more technically, geodesics, or shortest paths) through space. They follow geodesics through spacetime.
Take, for instance, the path of the Earth around the Sun. The shortest path from right here, right now, to the opposite side of the Sun, six months from now, is around a half(nearly)circle, and so that's the path the Earth takes from hereandnow to thereandthen. On the other hand, the shortest path from hereandnow to the opposite side of Earth's orbit, sixteen minutes from now, is what looks to us like a straight line, and that's the path that light takes (or would take, if that pesky Sun weren't in the way blocking it So say instead that it's the path a neutrino would take). 
#9




Reading this thread made it seem like there would have to be a maximal speed that a given object can orbit the earth (or any other body). For if an orbiting object speeds up, its orbit will grow larger, but then the gravitational force decreases and eventually a speed will be reached at which point it escapes the body. That doesn't mean that it no longer feels the gravity, but its path ceases to close.



#10




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#11




Syntax error: Missing closing ')'

#12




Thank you. Do you have experience with Lots of Irritating Silly Parentheses?

#13




Uh oh ... mischief with the AI ... random statement to follow ...

#14




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over every mistake. You just keep on trying till you run out of cake. 


#15




... the machines have taken over ...

#16




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http://www.netfunny.com/rhf/jokes/90q2/lispcode.html 
#17




On a related theme  getting one's head around the implications of frame dragging. I guess if we stick with the rubber sheet, in addition to the sheet taking on a curve, it has a slight rotational shear in it as well. So there is a near infinitesimal change of angle of the geodesic as you move radially to the Earth. And I would guess a very tiny difference in the path taken by an object depending upon its direction relative to the Earth's rotation  something that could be modelled as the Earth's rotation dragging on the object.

#18




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On the one hand explanations of why the trajectory of small objects changes in the proximity of large objects tend to ignore speed and force while focusing on how the large objects wrap the spacetime fabric. On the other hand the trajectory of a particle depends on its speed. If particles don't rip through spacetime, then there is something about this proces of spacetime wrapping that needs more explaining. 
#19




I’m back to clarify my befuddlement. I really miss Isaac Asimov and his ability to explain complex concepts in simple statements.
In Newton’s model, space is distinct from body and time passes uniformly, where space and time are flat dimensions. In Einstein’s model, space and time become intertwined, and spacetime gets pushed, pulled, stretched and warped by matter: “matter tells spacetime how to curve, and curved spacetime tells matter how to move.” Visual explanations in nowadays mass media do not use the idea of gravitational force but that of ‘tracks’ in spacetime caused by large objects: light seems to bend near the sun due to the curbs and twists in the fabric of spacetime. Of course these visual representations only show the space. How does time change things? When we use Minkowski’s diagram, a satellite’s trajectory around Earth turns from an ellipse into an elliptical spiral. But when Earth wraps the spacetime frame around it, this spacetime wrapping is constant in time no matter how minute. Either the ‘tracks’ one can see in visual explanations of nowadays mass media are real and light has to follow them when passing by our planet or these ‘tracks’ is just a way of putting things and in fact the force of gravity and the speed of light are all that matters. 


#20




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I cannot get how light can move against the fabric of space. If the road is curved, the car can only follow the road no matter how fast it is going. There is no "off road" in space. Some have tried to explain this in other posts but I think my brain is too small to understand. 
#21




If you're looking for an easytovisualize picture which incorporates both the curvature and spacetime, in a way that your brain has evolved to be able to grasp, then I'm afraid you're out of luck. Most humans are only easily able to visualize three dimensions: It is possible to visualize more, but it's very difficult to train yourself to do it, and it might be necessary to start the training when you're very young and your mind is still flexible. So to start with, you need at least two dimensions of space for the orbit itself. You could get away with one, but that means you're going to need to find some other way of representing an orbit, and you'll have to train yourself on that, too (though this is probably a bit easier than training yourself to visualize more dimensions, it's still likely to take a few years before it's really intuitive). Then, the way that we usually think of curvature is extrinsically, which means we need another dimension to show that. We could instead think of curvature intrinsically, but now you need some other way to represent curvature on our visualization. And then you need some way to represent time, and we're out of dimensions to do it with.

#22




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All of this holds for an object in orbit around a planet. (1) Stationary: fall straight down to the planet. (2) Moving too slow: follow a curved path that crashes into the planet. (3) Moving at just the right speed*: follow a curved path that meets itself after a full circuit around. (4) Moving too fast: follow a curved path that never bends enough to actually loop back around; escape the planet. Light is in category (4) for earth. _{*or range of speeds in the case of elliptical orbits} 
#23




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The point is that the rubber sheet analogy is actually quite a superficial one. Last edited by Asympotically fat; 01122018 at 05:05 PM. 
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#25




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To add to the above, I'll reiterate a core point that is lost in many explanations that is also missed here. The problem is that all of the simple visualisations  either rubber sheets, or diagrams of space that is distorted implicitly (or sometimes explicitly) imply that the geodesics are curved space, and thus there is a "road" to follow in space. As noted a few times above this is not so. The geodesics are in space and time  spacetime. The idea of following a curved spacetime geodesic does not get you a single road in space. You can perhaps think of the curvature in space as depending upon how long it takes you to traverse that part of space. In the limiting case  something travelling at the speed of light  you get one extreme of the paths to be taken in space, but at slower speeds the curvature of space and time means the path in space seen is different, and we can do things like go into orbit. The path in spacetime is the same for all objects. Perhaps remembering that all objects traverse spacetime at the same speed  c  the speed of causality  would help. Light travels at c, it does not travel in time, all of its speed is only in space, and so it doesn't see any curvature in time, only space. A still object in space travels at c as well, but since it is not travelling in space it must travel at c in time. So it only sees curvature in time. (Which is why gravity affects the passage of time). Anything travelling at slower speeds than c in space is travelling in time as well as space, and they see a mix of the curvatures in time and space that depends upon their speed. There is no one curvature of space, only spacetime. 
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