curvature for metric $ds^2 = ydx^2 + xdy^2$
$begingroup$
I'm trying to solve problem from Zee's Einstein gravity in nutshell:
find curvature for metric $ds^2 = ydx^2 + xdy^2$
moreover, in the corresponding chapter curvature is derived from the term of the expansion of the metric near some point : $g_{munu}(x) = g_{munu}(0) + ... + B_{munu, rhosigma}x^rho x^sigma + ...$
so that curvature is some combination of $B_{munu, rhosigma}$
Is it possible to find curvature in this case without calculating and differentiating christoffel symbols, summing over indices of curvature tensor and etc.?
differential-geometry metric-spaces curvature
edited Dec 4 '18 at 7:49
Alex Ravsky
40.2k32282
asked Oct 21 '18 at 12:18
AntonAnton
11
$endgroup$
add a comment |
$begingroup$
I'm trying to solve problem from Zee's Einstein gravity in nutshell:
find curvature for metric $ds^2 = ydx^2 + xdy^2$
moreover, in the corresponding chapter curvature is derived from the term of the expansion of the metric near some point : $g_{munu}(x) = g_{munu}(0) + ... + B_{munu, rhosigma}x^rho x^sigma + ...$
so that curvature is some combination of $B_{munu, rhosigma}$
Is it possible to find curvature in this case without calculating and differentiating christoffel symbols, summing over indices of curvature tensor and etc.?
differential-geometry metric-spaces curvature
edited Dec 4 '18 at 7:49
Alex Ravsky
40.2k32282
asked Oct 21 '18 at 12:18
AntonAnton
11
$endgroup$
add a comment |
$begingroup$
I'm trying to solve problem from Zee's Einstein gravity in nutshell:
find curvature for metric $ds^2 = ydx^2 + xdy^2$
moreover, in the corresponding chapter curvature is derived from the term of the expansion of the metric near some point : $g_{munu}(x) = g_{munu}(0) + ... + B_{munu, rhosigma}x^rho x^sigma + ...$
so that curvature is some combination of $B_{munu, rhosigma}$
Is it possible to find curvature in this case without calculating and differentiating christoffel symbols, summing over indices of curvature tensor and etc.?
differential-geometry metric-spaces curvature
edited Dec 4 '18 at 7:49
Alex Ravsky
40.2k32282
asked Oct 21 '18 at 12:18
AntonAnton
11
$endgroup$
I'm trying to solve problem from Zee's Einstein gravity in nutshell:
find curvature for metric $ds^2 = ydx^2 + xdy^2$
moreover, in the corresponding chapter curvature is derived from the term of the expansion of the metric near some point : $g_{munu}(x) = g_{munu}(0) + ... + B_{munu, rhosigma}x^rho x^sigma + ...$
so that curvature is some combination of $B_{munu, rhosigma}$
Is it possible to find curvature in this case without calculating and differentiating christoffel symbols, summing over indices of curvature tensor and etc.?
differential-geometry metric-spaces curvature
differential-geometry metric-spaces curvature
edited Dec 4 '18 at 7:49
Alex Ravsky
40.2k32282
asked Oct 21 '18 at 12:18
AntonAnton
11
edited Dec 4 '18 at 7:49
Alex Ravsky
40.2k32282
asked Oct 21 '18 at 12:18
AntonAnton
11
edited Dec 4 '18 at 7:49
Alex Ravsky
40.2k32282
edited Dec 4 '18 at 7:49
Alex Ravsky
40.2k32282
edited Dec 4 '18 at 7:49
Alex Ravsky
40.2k32282
40.2k32282
asked Oct 21 '18 at 12:18
AntonAnton
11
asked Oct 21 '18 at 12:18
AntonAnton
11
asked Oct 21 '18 at 12:18
AntonAnton
11
11
add a comment |
add a comment |
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