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Ok, I've looked a bit into your Posts. The problem is, I suck at math (i.e. differential geometry).
So please tell me if I got the basics right (your post #6):
The Minkowski basis vetors are [itex]\partial_t,\partial_r,\partial_\theta,\partial_\phi[/itex], with length [itex]1,1,r,r \, \sin(\theta)[/itex], respectively. (Q: is it ok to think of [itex]\partial_t[/itex] as implicitly acting on the interval s? i.e. change in interval per change in coordinate value?) So we construct an orthonormal basis at each event with scaled vectors
[tex]
\begin{array}{rcl}
\vec{e}_1 & = & \partial_t\\
\vec{e}_2 & = & \partial_r \\
\vec{e}_3 & = & \frac{1}{r} \, \partial_\theta \\
\vec{e}_4 & = & \frac{1}{r \, \sin(\theta)} \, \partial_\phi
\end{array}
[/tex]
(Q: This is no longer a coordinate basis? Is there some explanation for dummies as to what "nonholonomic" means in this context? If not, don't bother, I'll come back to it later)
Boosting with v=r/t yields the Milne frame.
Correct so far?
Two corrections for your subsequent paragraph:
The frame is defined for r<t, and the Hubble 'constant' scales as 1/t.
Now for something completely different:
Thanks
So please tell me if I got the basics right (your post #6):
The Milne frame is
[tex]
\begin{array}{rcl}
\vec{e}_1 & = & \frac{t}{\sqrt{t^2-r^2}} \, \partial_t + \frac{r}{\sqrt{t^2-r^2}} \, \partial_r \\
\vec{e}_2 & = & \frac{r}{\sqrt{t^2-r^2}} \, \partial_t + \frac{t}{\sqrt{t^2-r^2}} \, \partial_r \\
\vec{e}_3 & = & \frac{1}{r} \, \partial_\theta \\
\vec{e}_4 & = & \frac{1}{r \, \sin(\theta)} \, \partial_\phi
\end{array}
[/tex]
The Minkowski basis vetors are [itex]\partial_t,\partial_r,\partial_\theta,\partial_\phi[/itex], with length [itex]1,1,r,r \, \sin(\theta)[/itex], respectively. (Q: is it ok to think of [itex]\partial_t[/itex] as implicitly acting on the interval s? i.e. change in interval per change in coordinate value?) So we construct an orthonormal basis at each event with scaled vectors
[tex]
\begin{array}{rcl}
\vec{e}_1 & = & \partial_t\\
\vec{e}_2 & = & \partial_r \\
\vec{e}_3 & = & \frac{1}{r} \, \partial_\theta \\
\vec{e}_4 & = & \frac{1}{r \, \sin(\theta)} \, \partial_\phi
\end{array}
[/tex]
(Q: This is no longer a coordinate basis? Is there some explanation for dummies as to what "nonholonomic" means in this context? If not, don't bother, I'll come back to it later)
Boosting with v=r/t yields the Milne frame.
Correct so far?
Two corrections for your subsequent paragraph:
The frame is defined for r<t, and the Hubble 'constant' scales as 1/t.
Now for something completely different:
You would disagree with de Sitter space looking like an "inside out black hole"?#5 said:debunk the notion that FRW dusts (or any reasonably accurate cosmological model) can be considered as "an inside out black hole" [sic],
Thanks
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