Deriving the line element in homogenous isotropic space

TheMan112
Messages
42
Reaction score
1
If the Ricci-scalar R is constant for a given spatial hypersurface, then the curvature of that region should be homogenous and isotropic, right?

A homogenous and isotropic hypersurface (disregarding time) has by definition the following line element (due to spherical symmetry):

d\sigma^2 = a^2 \left(\frac{1}{1-kr^2} dr^2 + r^2(d \theta^2 + sin^2(\theta) d \Phi^2) \right)

Where k = -1, 0 or +1 and a is constant.

Why \frac{1}{1-kr^2} dr^2 ?

This is apparently very important as the value of k determines the evolution of the universe, but I don't know how to come to this line element.
 
Physics news on Phys.org
Thread 'Can this experiment break Lorentz symmetry?'

Thread 'Can this experiment break Lorentz symmetry?'

1. The Big Idea: According to Einstein’s relativity, all motion is relative. You can’t tell if you’re moving at a constant velocity without looking outside. But what if there is a universal “rest frame” (like the old idea of the “ether”)? This experiment tries to find out by looking for tiny, directional differences in how objects move inside a sealed box. 2. How It Works: The Two-Stage Process Imagine a perfectly isolated spacecraft (our lab) moving through space at some unknown speed V...
View full post »

Thread '1-Way Speed of Light'

Does the speed of light change in a gravitational field depending on whether the direction of travel is parallel to the field, or perpendicular to the field? And is it the same in both directions at each orientation? This question could be answered experimentally to some degree of accuracy. Experiment design: Place two identical clocks A and B on the circumference of a wheel at opposite ends of the diameter of length L. The wheel is positioned upright, i.e., perpendicular to the ground...
View full post »

Thread 'A sufficient condition for integrability of equation ##\nabla g=0##'

In Philippe G. Ciarlet's book 'An introduction to differential geometry', He gives the integrability conditions of the differential equations like this: $$ \partial_{i} F_{lj}=L^p_{ij} F_{lp},\,\,\,F_{ij}(x_0)=F^0_{ij}. $$ The integrability conditions for the existence of a global solution ##F_{lj}## is: $$ R^i_{jkl}\equiv\partial_k L^i_{jl}-\partial_l L^i_{jk}+L^h_{jl} L^i_{hk}-L^h_{jk} L^i_{hl}=0 $$ Then from the equation: $$\nabla_b e_a= \Gamma^c_{ab} e_c$$ Using cartesian basis ## e_I...
View full post »

Similar threads

I Schwarzschild spacetime in Kruskal coordinates
Replies
43
Views
3K
I Line element in Kerr coordinates at singularity
Replies
20
Views
1K
I Can Spherical Symmetry Be Achieved Without Varying Line Element?
Replies
14
Views
2K
I Eddington-Finkelstein coordinates for Schwarzschild metric
Replies
9
Views
3K
A Vanishing of an integral of a divergence over a closed surface
Replies
1
Views
995
I Inverting the metric coefficients in the Schwarzschild line element
Replies
32
Views
3K
I Principal and Gaussian curvature of the FRW metric
Replies
8
Views
1K
I Spatial homogeneity condition for a free particle Lagrangian
Replies
48
Views
3K
I Coord Transform in de Sitter Space: Phys Significance &Linearity?
Replies
5
Views
2K
I When time is not orthogonal to phi, that means there is a preferred direction in space? How?
Replies
14
Views
1K

Hot Threads

Recent Insights

Back
Top