- #1
Apashanka
- 429
- 15
Our 4-D space is ##x^1,x^2,x^3 ,t##.
Our sub-manifold is defined by ##(x^1,x^2,x^3)##
Therefore for this sub-manifold to be maximally symmetric and for which the tangent vector ##\frac{∂}{∂t}(\hat t)## orthogonal to this sub-manifold
The metric becomes,
##ds^2=g(t)dt^2+f(t)(dr^2+r^2d\Omega^2)##
From the known metric for 4-D space and comparing this with above ##g(t)=-1## and for static ##f(t)=1## and for expanding ##f(t)=a(t)^2## ,is it the case ??
And can we say that the hypersurface to any t is orthogonal to the increament direction of t??
Our sub-manifold is defined by ##(x^1,x^2,x^3)##
Therefore for this sub-manifold to be maximally symmetric and for which the tangent vector ##\frac{∂}{∂t}(\hat t)## orthogonal to this sub-manifold
The metric becomes,
##ds^2=g(t)dt^2+f(t)(dr^2+r^2d\Omega^2)##
From the known metric for 4-D space and comparing this with above ##g(t)=-1## and for static ##f(t)=1## and for expanding ##f(t)=a(t)^2## ,is it the case ??
And can we say that the hypersurface to any t is orthogonal to the increament direction of t??