- #1
davidge
- 554
- 21
I have a question regarding the FLRW metric used for cosmological analysis in S & G Relativity.
Let the coordinates of a point in the space time be ##(t,r,\theta,\varphi)##. For constant ##t, \theta## and ##\varphi## we have the metric $$d \tau^2 = \frac{dr^2}{1 - kr^2}$$
My doubt is about this form of the metric. We know that it can describe a paraboloid, a 1-dim sphere (circle) or a plane, depending on the sign of ##k##.
If ##k = 1## or ##k = -1## we have a maximally symmetric space, correct? (Namely the circle or the paraboloid, respectively.) Now are these the only possible one dimensional spaces of max symmetry? To put it another way, that form above of the metric is the unique possible form for one dimension if it is to describe a maximally symmetric space? (By one dimension I mean of course, that we need only one coordinate (##r## in this case) to describe the space.)
Let the coordinates of a point in the space time be ##(t,r,\theta,\varphi)##. For constant ##t, \theta## and ##\varphi## we have the metric $$d \tau^2 = \frac{dr^2}{1 - kr^2}$$
My doubt is about this form of the metric. We know that it can describe a paraboloid, a 1-dim sphere (circle) or a plane, depending on the sign of ##k##.
If ##k = 1## or ##k = -1## we have a maximally symmetric space, correct? (Namely the circle or the paraboloid, respectively.) Now are these the only possible one dimensional spaces of max symmetry? To put it another way, that form above of the metric is the unique possible form for one dimension if it is to describe a maximally symmetric space? (By one dimension I mean of course, that we need only one coordinate (##r## in this case) to describe the space.)