Cosmology 2002_20031 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

Cosmology 2002_20032 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:  s 2 = r 2 (  ) 2 + r 2 Sin 2  (  ) 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:

Cosmology 2002_20033 Metric 3 D curved space The proper distance between neighbouring points is:  s 2 = f(r) (  r) 2 + r 2 (  ) 2 + r 2 Sin 2  (  ) 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 K=const. 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.

Cosmology 2002_20034 The equatorial surface can be written as:  s 2 (  =  /2 )= f(r) (  r) 2 + r 2 (  ) 2 and the metric is: g 11 = f(r) g 22 = r 2 The Gauss Curvature [The student check]:

Cosmology 2002_20035 Must be valid also for a FLAT space A flat space has no curvature, that is: K=0 And r becomes the Euclidean distance, in this case we must have:  s 2 =  r 2 +r 2  2 And with K = 0 we have: f(r) = 1/C we have  s 2 = 1/C  r 2 +r 2  2 For the two expressions to be the same we must have: C=1 That the value of the function I wrote as f(r) must be: f(r)=1/(1-K r 2 )

Cosmology 2002_20036 Finally I write the metric as:

Cosmology 2002_20037 We make it general by: 1.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 / R 2 (t) ; k= +1, 0, -1 2.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:  r 2 =  2 R 2 (t) and K r 2 = K  2 R 2 (t) = k  2

Cosmology 2002_20038 I finally have

Cosmology 2002_20039 Area and Volumes – Space Component

Cosmology 2002_  =const Equivalent to R=const in 2D embedded in 3D

Cosmology 2002_ Euclidean – k=0

Cosmology 2002_ K = -1 Flat Minkowski Space