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Thread: Curved spacetime and multiple planets in orbit : an image of this ?

  1. #1 Curved spacetime and multiple planets in orbit : an image of this ? 
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    Aside from the mathematics,i was wondering what the spacetime curvage would look like for this :
    (In a side view of the solar system with spacetime curvage present)

    3 planets in orbit around the sun in the same orbittal plane, with simply one very near, one medium near, and one further away from the sun.


    I am facing the problem that it is either :


    *Planets on a smooth curve , will result in some orbits under the actual orbit plane,
    such as the orbit of the planet P1 closeby.

    Simple sketch :




    *Or when placing the planets on the same horizontal plane through the sun centre,
    then i cannot apply the correct slope of the curve for each planet.
    Because close to the sun that slope should be steeper, more towards vertical, and far of planets would need a slope more flat 'to lean against', towards the horizontal.
    And given a smooth drop in g-factor away from the sun, i cannot construct the image.

    Simple sketch :





    Can anyone provide me with such an image, or am i formulating somehow a useless question and should i consider it more theoretically or differently ?


    Last edited by Noa Drake; November 22nd, 2013 at 09:08 AM.
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  3. #2  
    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by Noa Drake View Post
    Can anyone provide me with such an image, or am i formulating somehow a useless question and should i consider it more theoretically or differently ?
    The question is a good one, and - I'm afraid - not one that is easy and straightforward to answer. Space-time around each of the objects in the solar system can be visualised somewhat like this :



    The distortion will be much greater for the central body ( sun ) than it is for the planets, but the principle is the same. Ultimately, to treat this properly, you will have no choice but to learn the maths, since there really isn't any way to visualise a 4-dimensional space-time.

    Or when placing the planets on the same horizontal plane through the sun centre,in side view, then i cannot apply the correct slope of the curve for each planet.Because close to the sun that slope should be steeper, more towards vertical, and far of planets would need a slope more flat 'to lean against', towards the horizontal. And given a smooth drop in g-factor away from the sun, i cannot construct the image.
    The problem here is that curvature is not something as simple as a slope; it is not simply a single value assigned to each point. Mathematically, curvature is described by an object called the Riemann curvature tensor, which looks like a matrix of 256 elements. Fortunately, only 20 of these components are non-zero and independent in "real" space-times, but even then it is still a lot more complicated than a single number. You can imagine every event in space and time to be endowed with one such curvature tensor, which describes the geometry of space-time in and around that particular event.


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    P.S. What happens to geodesics of particles in such space-time is then described by the equation in my signature, which is called the equation of geodesic deviation.
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    Thank you Markus.

    A 3D reality would make more sense to me to apply gravity to,
    it should according to me be possible to provide a consistent visual
    of gravity in 3 dimensions or at least in 2D viewplanes topview sideview, for any number of objekts simultaneously.
    The time issue, contraction issue (clocks etc) should then become emergent from such a model.

    So i will try and persue that, after investigating the Riemann Curvature Tensor.
    Last edited by Noa Drake; November 22nd, 2013 at 09:03 AM.
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    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by Noa Drake View Post
    Thank you Markus.A 3D reality would make more sense to me to apply gravity to, so i will persue that,after investigating your explanation.
    It is 4D, actually. Gravitational fields can change, and they do so according to the laws of GR, so you will have to account for those dynamics as well. There are some very subtle issues involved in that which sharply distinguish GR from Newton, and which have recently been experimentally verified by the Gravity Probe B experiments :

    Gravity Probe B - Wikipedia, the free encyclopedia
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    Quote Originally Posted by Noa Drake View Post
    A 3D reality would make more sense to me to apply gravity to
    The gravity that we are familiar with is entirely due to what happens with the time direction, specifically the gravitational time dilation. Although the three-dimensional space surrounding a gravitational source is not flat, the curvature of the three-dimensional space can be ignored as far as the gravity of the solar system is concerned.
    There are no paradoxes in relativity, just people's misunderstandings of it.
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    Quote Originally Posted by Noa Drake View Post
    Aside from the mathematics,i was wondering what the spacetime curvage would look like for this :
    (In a side view of the solar system with spacetime curvage present)

    3 planets in orbit around the sun in the same orbittal plane, with simply one very near, one medium near, and one further away from the sun.


    I am facing the problem that it is either :


    *Planets on a smooth curve , will result in some orbits under the actual orbit plane,
    such as the orbit of the planet P1 closeby.

    Simple sketch :




    *Or when placing the planets on the same horizontal plane through the sun centre,
    then i cannot apply the correct slope of the curve for each planet.
    Because close to the sun that slope should be steeper, more towards vertical, and far of planets would need a slope more flat 'to lean against', towards the horizontal.
    And given a smooth drop in g-factor away from the sun, i cannot construct the image.

    Simple sketch :





    Can anyone provide me with such an image, or am i formulating somehow a useless question and should i consider it more theoretically or differently ?
    This is known under the name of the N-body problem. Numerical solutions exist.
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    The N-body problem :

    "predicting the individual motions, and forces on same, of a group of celestial objects interacting with each other gravitationally. etc"

    from :
    http://en.wikipedia.org/wiki/N-body_problem


    i
    s not the same problem i describe.


    My issue is about the construction itself of the gravitational field of the sun when you put multiple planets in orbit, not the behaviour between the planets.
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    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by Noa Drake View Post
    My issue is about the construction itself of the gravitational field of the sun when you put multiple planets in orbit, not the behaviour between the planets.
    The resulting global field is a superposition of the fields of the individual bodies; in Newtonian gravity this superposition is linear, whereas in GR it is highly non-linear, which is why the two theories don't give the same predictions.
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    Quote Originally Posted by Noa Drake View Post
    The N-body problem :

    "predicting the individual motions, and forces on same, of a group of celestial objects interacting with each other gravitationally. etc"

    from :
    http://en.wikipedia.org/wiki/N-body_problem


    i
    s not the same problem i describe.


    My issue is about the construction itself of the gravitational field of the sun when you put multiple planets in orbit, not the behaviour between the planets.
    You probably don't realize but they are one and the same.
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