The Metric: The easy way

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The Metric The easy way

• Guido Chincarini
• Here we derive the Robertson Walker Metric, the
  3D Surface and Volume for different Curvatures.
  The concept is introduced w/o the need of General
  Relativity to make it simpler and east to
  understand physically.
• This part after the Curvature Lecture and before
  the Hubble law and Cosmological redshift.
• A good book for the General Relativityt is also
• Introduction to General relativity by R.Adler, M.
  Bazin and M. Schiffer, Pub. McGraw-Hill
• And the superb
• Gravitation by W. Misner, S. Thorne and J.A.
  Wheeler

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The Space

• Hypothesis
• The space is symmetric and, to be more precise
• Homogeneous and isotropic.
• If I have a surface embedded in a 3 dimensional
  space then I write
• ?s2 r2 (??)2 r2 Sin2? (??)2
• The question is how do I write a similar metric
  for a 3 dimension space embedded in a 4 dimension
  space. Obviously I will have to add a ?r which
  gives the third dimension of a 3 D curved space
  in a 4 dimension space. That is I can write the
  metric as

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Metric 3 D curved space

• The proper distance between neighbouring points
  is
• ?s2 f(r) (?r)2 r2 (??)2 r2 Sin2? (??)2
• The fact that the space is symmetric, that is
  homogeneous and isotropic means that all the
  surfaces must have the same curvature, that is
  Kconst.
• For convenience, this does not change the
  reasoning because of what we stated above
  (symmetry), we select an equatorial surface, that
  is we work on a surface for which ? ?/2.

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The equatorial surface can be written as

• ?s2 (? ?/2 ) f(r) (?r)2 r2 (??)2
• and the metric is
• g11 f(r) g22 r2
• The Gauss Curvature The student check

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Must be valid also for a FLAT space

• A flat space has no curvature, that is
• K0
• And r becomes the Euclidean distance, in this
  case we must have
• ?s2 ?r2 r2 ??2
• And with K 0 we have
• f(r) 1/C we have
• ?s2 1/C ?r2 r2 ??2
• For the two expressions to be the same we must
  have
• C1
• That the value of the function I wrote as f(r)
  must be
• f(r)1/(1-K r2)

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Finally I write the metric as
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We make it general by

• I transform the coordinates using a parameter
  R(t) which is a function of time. The proper
  distance and the curvature become adimensional.
• ? r/R(t) and K(t) k / R2 (t) k 1, 0, -1
• I add an other dimension, the time. In agreement
  with the Theory of Relativity an event is
  deifined by the space coordinates and the time
  coordinate. Therefore I have
• ? r2 ??2 R2(t) and
• K r2 K ?2 R2 (t) k ?2

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I finally have
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Area and Volumes Space Component

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?const Equivalent to Rconst in 2D embedded in 3D
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Euclidean k0

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K -1 Flat Minkowski Space