- #1
gasgas
- 4
- 1
Hello, I'm having quite a lot of trouble with working through the proof of this proposition in Hawking and Ellis's "Large Scale Structure of Spacetime".
First of all, I am unsure of how the matrix element A that is in the proof can be the same matrix element as in equation 4.39 because there we had a point q where the Jacobi field was equal to zero, but later in the proof we conclude that such a point must exist so assuming it at the beginning doesn't make sense to me.
That aside, I assume that matrix A being equal to the identity at point p is just a convenient choice of coordinates we can make as the expansion then works out to be the trace of dA/dt. After that, I fail to see how any component of dA/dt being large at p implies that the singularity of the expansion occurs somewhere close to p. I can see it is true when one of the diagonal elements is large as that would make the expansion large which is inversely proportional to the maximum proper time to reach the singularity, but that makes no use of equation 4.39 and works only for diagonal elements.
Finally, I am utterly confused by the final argument of the conjugate point to a point r which is "further along" the geodesic than p. Are they looking at it as being past directed and it is the value of the derivatives of Jacobi field at r which defines the matrix A and then propagating backwards? Also, I fail to see how not having a point conjugate to r between p and r implies that the expansion at theta is positive, and how that then implies that there must be a point q conjugate to r "before" p.
Any help of alternative reading material is very appreciated.
First of all, I am unsure of how the matrix element A that is in the proof can be the same matrix element as in equation 4.39 because there we had a point q where the Jacobi field was equal to zero, but later in the proof we conclude that such a point must exist so assuming it at the beginning doesn't make sense to me.
That aside, I assume that matrix A being equal to the identity at point p is just a convenient choice of coordinates we can make as the expansion then works out to be the trace of dA/dt. After that, I fail to see how any component of dA/dt being large at p implies that the singularity of the expansion occurs somewhere close to p. I can see it is true when one of the diagonal elements is large as that would make the expansion large which is inversely proportional to the maximum proper time to reach the singularity, but that makes no use of equation 4.39 and works only for diagonal elements.
Finally, I am utterly confused by the final argument of the conjugate point to a point r which is "further along" the geodesic than p. Are they looking at it as being past directed and it is the value of the derivatives of Jacobi field at r which defines the matrix A and then propagating backwards? Also, I fail to see how not having a point conjugate to r between p and r implies that the expansion at theta is positive, and how that then implies that there must be a point q conjugate to r "before" p.
Any help of alternative reading material is very appreciated.
