The Metric: The easy way

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_2003

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: Cosmology 2002_2003

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 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_2003

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]: Cosmology 2002_2003

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: 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: C=1 That the value of the function I wrote as f(r) must be: f(r)=1/(1-K r2) Cosmology 2002_2003

Finally I write the metric as: Cosmology 2002_2003

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 Cosmology 2002_2003

I finally have Cosmology 2002_2003

Area and Volumes – Space Component Cosmology 2002_2003

=const Equivalent to R=const in 2D embedded in 3D Cosmology 2002_2003

Euclidean – k=0 Cosmology 2002_2003

K = -1 Flat Minkowski Space Cosmology 2002_2003

The Metric: The easy way