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RModule
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Hello,
I'm a mathematics student specializing in (semi-)Riemannian geometry and relating this to relativity. My main reference is O'neill's "Semi-Riemanninan Geometry with Applications to Relativity".
This book defines the Robertson-Walker spacetime as follows:
Let [tex]S[/tex] be a connected three-dimensional Riemanninan manifold of constant curvature [tex]k = -1,0,1[/tex]. Let [tex]f>0[/tex] be a smooth function on an open interval [tex]I[/tex] in [tex]R_1^1[/tex] Then the warped product [tex]M(k,f) = I \times_f S[/tex] is called a RW spacetime. Explicitly [tex]M(k,f)[/tex] is the manifold [tex]I\times S[/tex] with line element [tex]-dt^2+ f^2(t)d\sigma^2[/tex] where [tex]d\sigma^2[/tex] is the line element of [tex]S[/tex].
Is this a non-standard definition? From what I can see on wiki, the manifold should be simply connected. Further, does this definition imply that [tex]M[/tex] is orientable?
Because, from this definition [tex]\mathbb{R}P^3 \times I[/tex] would be a RW spacetime, even though it is not simply connected. Which leads me to the following; [tex]S^3[/tex] and [tex]\mathbb{R}P^3[/tex] are locally "the same" but globally quite different. What relativistic implications would not being simply connected have on the spacetime? I guess you essentially could see the same stars from two(or more) different directions. Are there any other important differences?
Thank you.
I'm a mathematics student specializing in (semi-)Riemannian geometry and relating this to relativity. My main reference is O'neill's "Semi-Riemanninan Geometry with Applications to Relativity".
This book defines the Robertson-Walker spacetime as follows:
Let [tex]S[/tex] be a connected three-dimensional Riemanninan manifold of constant curvature [tex]k = -1,0,1[/tex]. Let [tex]f>0[/tex] be a smooth function on an open interval [tex]I[/tex] in [tex]R_1^1[/tex] Then the warped product [tex]M(k,f) = I \times_f S[/tex] is called a RW spacetime. Explicitly [tex]M(k,f)[/tex] is the manifold [tex]I\times S[/tex] with line element [tex]-dt^2+ f^2(t)d\sigma^2[/tex] where [tex]d\sigma^2[/tex] is the line element of [tex]S[/tex].
Is this a non-standard definition? From what I can see on wiki, the manifold should be simply connected. Further, does this definition imply that [tex]M[/tex] is orientable?
Because, from this definition [tex]\mathbb{R}P^3 \times I[/tex] would be a RW spacetime, even though it is not simply connected. Which leads me to the following; [tex]S^3[/tex] and [tex]\mathbb{R}P^3[/tex] are locally "the same" but globally quite different. What relativistic implications would not being simply connected have on the spacetime? I guess you essentially could see the same stars from two(or more) different directions. Are there any other important differences?
Thank you.
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