- #1
shinobi20
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- TL;DR Summary
- I'd like to confirm my calculations of the principal and Gaussian curvature of the spatial part of the Friedmann-Robertson-Walker (FRW) metric. Specifically, if the spatial part has negative curvature.
I would like to calculate the principal and Gaussian curvature of the spatial part of the Friedmann-Robertson-Walker (FRW) metric; specifically, the negative Gaussian curvature . The FRW metric is,
For a certain time slice , the spatial part is,
If we set , then we get the metric (also called the first fundamental form) for the 3d hyperboloid in angular coordinates,
The 3d hyperboloid can be embedded in 4d Minkowski space. Let be the Minkowski metric and be the 3d hyperboloid metric in angular coordinates. The 3d hyperboloid in coordinates is described by the equation,
The parameterization is,
Given the point on the 3d hyperboloid, the tangent vectors are,
The normal vector on the 3d hyperboloid can be calculated as,
The normalized normal vector is,
We also need the derivative of the tangent vectors . The second fundamental form can be calculated as,
The principal curvatures for are the eigenvalues of the matrix ,
The Gaussian curvature is .
Questions:
(1) Are my calculations correct?
(2) Are the answers for the principal curvatures correct?
(3) I'm wondering about . As I know the Gaussian curvature should be for , but only for . Can anyone help clarify and explain more about the principal curvatures of the 3d hyperboloid?
(4) Another thing I'm unsure of is the sign of the normal vector, if I remove the minus sign then becomes negative and becomes positive while maintaining the same magnitude.
For a certain time slice
If we set
The 3d hyperboloid can be embedded in 4d Minkowski space. Let
The parameterization is,
Given the point
The normal vector
The normalized normal vector
We also need the derivative of the tangent vectors
The principal curvatures
The Gaussian curvature is
Questions:
(1) Are my calculations correct?
(2) Are the answers for the principal curvatures correct?
(3) I'm wondering about
(4) Another thing I'm unsure of is the sign of the normal vector, if I remove the minus sign then