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Mathamatical Description of the Cosmological Principle

  1. Feb 11, 2009 #1
    I think this belongs here, while not relativistic, is is about objects moving relative to one another.

    This stems from the description of 'Dark Energy' in a mathamatical context to show that you can place a 'center of the universe' at any point in space, and the accelaration vector between two bodies is inderpendant of the position vectors to this 'arbitary center of the universe'.

    The presumption is that there exsits a 'center of the universe', and that from it we can draw a diagram


    Where A is our 'center of the universe', B is one body, C is the other body. There is a radial distances rB and rC respectfully, and the magnitude of radial accelaration from this 'center of the universe' is proportional to these distances.

    From B, an oberserver notes that C is accelarating away from him at some accelaration a'C and reversely, from C an observer notes that B is accelarating away from him at some accelaration a'B

    How do I show that no matter where I place A, the magnitude of accelaration is not dependant on the 'center of the universe'?

    I thought you could describe a'C and a'B as;

    a'C = rCK*cos( 1/2 n ) - rBK
    a'B = rBK*cos( 1/2 n ) - rCK

    Where K is the constant of proportionality and n is the angle between the bodies. Using the geometry between them. I guess that the cosine formula is now somehow incorperated into this;


    Because there are simmilarities in both equations, but I'm not sure where to go from here in the proof...

  2. jcsd
  3. Feb 11, 2009 #2


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    Gold Member

    You can model the principles of isotropy and homogeneity by setting up a line element that defines the distance measure in the cosmos by all observers. By ansatz we say that every observer will be able to construct a spherically symmetric coordinate system around themselves. Spherically symmetric implies no preferred direction (isotropy) and all observers can do this ( homogeneity).

    The line element is one of the Robertson-Walker-Lemaitre varieties and the simplest form looks like this

    [tex]ds^2 = c^2dt^2-a(t)^2d\sigma^2[/tex]

    see Wiki on 'Robertson-Walker metric'.
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