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Thread: Inverse square gravity law: what does it say about space time curve?

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    Inverse square gravity law: what does it say about space time curve?

    Take a thought experiment where a tower-block is built at the south pole, and rises to 1 light year away from the Earth.

    Each flour is ten meters above the one below.

    Each floor is covered in a sheet of aluminium, the same thickness as the aluminium on the floor below.

    Each floor can measure the weight of the aluminium on that floor.

    Would the weight of aluminium on each floor be the same as the weight on all the other floors, even at one light year distance?

    tower grav.jpg

    edit: forgetting of course, the gravitational affect of the tower itself...
    Last edited by WaxRubiks; 2019-Apr-04 at 05:23 PM.
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    Answering the title: The inverse square gravity law is Newtonian gravitation with no curved spacetime and cannot say anything about it.
    The answer for the thought experiment: Yes: your image has the area and mass of each floor increasing as distance squared in a inverse square gravity law.

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    Quote Originally Posted by Reality Check View Post
    Answering the title: The inverse square gravity law is Newtonian gravitation with no curved spacetime and cannot say anything about it.
    The answer for the thought experiment: Yes: your image has the area and mass of each floor increasing as distance squared in a inverse square gravity law.
    no, I'm sure, but is there an equivalent to the inverse sqaure law in space-time curvature theories?
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    Quote Originally Posted by WaxRubiks View Post
    no, I'm sure, but is there an equivalent to the inverse sqaure law in space-time curvature theories?
    Yes, in the limit of weak gravity, general relativity is a space-time curvature theory that is equivalent to the inverse square law. It is remarkable that two so completely different models of "what is going on" can make the same predictions in a limit like that, but it's not so unusual. Classical mechanics and quantum mechanics are also very different, yet make the same predictions in the limit of systems whose "action" is much larger than Planck's constant.
    Last edited by Ken G; 2019-Apr-05 at 01:12 PM.

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    Quote Originally Posted by Ken G View Post
    Yes, in the limit of weak gravity, general relativity is a space-time curvature theory that is equivalent to the inverse square law. It is remarkable that two so completely different models of "what is going on" can make the same predictions in a limit like that, but it's not so unusual. Classical mechanics and quantum mechanics are also very different, yet make the same predictions in the limit of systems whose "action" is much larger than Planck's constant.
    thanks.

    The thing I was thinking about is that the space-time curvature caused by a body, seems to be represented through the whole of space, regardless of how far away you are away from that body.
    Would that be a fair way to look at it?
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    It's hard to picture what the curvature is doing, but one approach that works all right is to imagine that space is very flexible and can be stretched or compressed easily, if it is told to do so. Also imagine that space is continuous and cannot be ripped, but it is dynamically stretched and compressed based on Einstein's theory. Then imagine that mass causes the space to contract into the center of the mass, such that the space around it gets "sucked in" by its need to be continuous and not rip. This is more or less the dynamical version (by which I mean, treating time as something that elapses) of spacetime curvature (where you treat time as part of a 4-d manifold). And if you take the "dynamical space" picture, rather than the "4-d spacetime" picture, you can see why the effects of a mass get smaller with distance away, much like the effect on a trampoline when you jump on the center of it gets weaker as you head away from the center.

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    Quote Originally Posted by Ken G View Post
    It's hard to picture what the curvature is doing,
    I thought that since the curvature seems to be represented through the whole of space, then maybe a piece of matter's contribution to the curvature of space could be considered like an angle....an angle in 4D space? Or something...

    curv.jpg
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    Quote Originally Posted by WaxRubiks View Post
    no, I'm sure, but is there an equivalent to the inverse sqaure law in space-time curvature theories?
    GR has no exact equivalent other than what Ken G stated - in the appropriate limits, GR deduces to Newtonian gravitation. Parameterized post-Newtonian formalism or Linearized gravity may also be relevant. These might give force laws that are an inverse square law + corrections.

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    I cannot visualize a 4D space construct, let alone Einstein's spacetime construct. My take on it is that we just need to trust Einstein's mathematical technique because it is in good agreement with observations for which Newtonian mechanics fell a bit short.

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    Quote Originally Posted by Hornblower View Post
    I cannot visualize a 4D space construct, let alone Einstein's spacetime construct. My take on it is that we just need to trust Einstein's mathematical technique because it is in good agreement with observations for which Newtonian mechanics fell a bit short.
    from my OP Newtonian analogy, it might seem that the space-time warpage by say a neutron star, is represented throughout the whole of space...I kind of visualise it like a piece of origami, ie the whole of space is somehow warped around the neutron star, not by much, not to a noticeable amount..though.
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    Quote Originally Posted by WaxRubiks View Post
    from my OP Newtonian analogy, it might seem that the space-time warpage by say a neutron star, is represented throughout the whole of space...I kind of visualise it like a piece of origami, ie the whole of space is somehow warped around the neutron star, not by much, not to a noticeable amount..though.
    This one has me stumped. As I think I understand it, Einsteinian spacetime is smoothly curved more and more sharply as we approach a massive spherical body. Origami consists of plane surfaces folded around sharp linear creases. I cannot by any stretch of the imagination see origami as an analogy to spacetime curvature. The familiar rubber sheet and heavy ball analogy makes far more sense as a visualizeable 3D construct. Can you show us, in appropriate mathematical detail, why you think origami might be a useful analogy?

    While I am at it, what is this "not to a noticeable amount"? It will be very noticeable near something as dense as a neutron star. The curvature flattens out to insignificance at great distances.

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    Quote Originally Posted by Hornblower View Post
    This one has me stumped. As I think I understand it, Einsteinian spacetime is smoothly curved more and more sharply as we approach a massive spherical body. Origami consists of plane surfaces folded around sharp linear creases. I cannot by any stretch of the imagination see origami as an analogy to spacetime curvature. The familiar rubber sheet and heavy ball analogy makes far more sense as a visualizeable 3D construct. Can you show us, in appropriate mathematical detail, why you think origami might be a useful analogy?

    While I am at it, what is this "not to a noticeable amount"? It will be very noticeable near something as dense as a neutron star. The curvature flattens out to insignificance at great distances.
    well obviously I'm no astrophysicist mathematician. My OP presents the idea that Newtonian forces never really change with distance, they are just spread out more....and then it was discussed that space-time had a similar properties as the inverse square law....so maybe space-time curvature is the same at any distance? I'm not sure obviously.
    Space-time curve would just be spread out more with distance rather than an actual curvature really changing, may be?
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    Quote Originally Posted by WaxRubiks View Post
    well obviously I'm no astrophysicist mathematician. My OP presents the idea that Newtonian forces never really change with distance, they are just spread out more....and then it was discussed that space-time had a similar properties as the inverse square law....so maybe space-time curvature is the same at any distance? I'm not sure obviously.
    That doesn't make much sense: we know the force decreases with distance. And this can be thought of being because it gets spread out over a larger area at greater distance.

    I don't think there is a simple equivalent for the way in which curvature changes with distance. (I should read Ken's answers before replying, really!) Maybe you can get an idea why not from the fact that the amplitude of gravitational waves decreases linearly with distance.

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    I guess the other way of visualising what happens is to use a different coordinate system. For example Painleve-Guillstrand (sp?) coordinates can be thought of as spatial coordinates falling towards a mass. The rate of fall decreases with distance (with, I think, an inverse square law).

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    Gullstrand. And yes, it's very useful to have a name for those coordinates, that's the way I like to think about it because it's easier to imagine dynamics than 4D.

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