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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.

    Summery:
    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'.

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


    A
    ...\
    ....\___________________
    .....B............................C

    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;

    a2=b2+c2-2bc*CosA

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

    Haths
     
  2. jcsd
  3. Feb 11, 2009 #2

    Mentz114

    User Avatar
    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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