- #1
chinared
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Need to find the Ricci scalar curvature of this metric:
ds2 = e2a(z)(dx2 + dy2) + dz2 − e2b(z)dt2I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor:
<The Christoffel connection> Here a'(z) denotes the first derivative of a(z) respect to z.
[itex]\Gamma\stackrel{x}{xz}[/itex]=[itex]\Gamma\stackrel{x}{zx}[/itex]=a'(z)
[itex]\Gamma\stackrel{y}{yz}[/itex]=[itex]\Gamma\stackrel{y}{zy}[/itex]=a'(z)
[itex]\Gamma\stackrel{z}{tt}[/itex]=b'(z)e2b(z)
[itex]\Gamma\stackrel{z}{xx}[/itex]=[itex]\Gamma\stackrel{z}{yy}[/itex]=-a'(z)e2a(z)
[itex]\Gamma\stackrel{t}{tz}[/itex]=[itex]\Gamma\stackrel{t}{zt}[/itex]=b'(z)
[itex]\Gamma\stackrel{}{either}[/itex]=0
<The Riemann curvature tensor>
[itex]R\stackrel{x}{zxz}[/itex]=[itex]R\stackrel{y}{zyz}[/itex]=-a''(z)-[a'(z)]2
[itex]R\stackrel{z}{tzt}[/itex]=b''(z)+[b'(z)]2
I tried to find the Ricci scalar curvature(R) from current result, but it gave a function depend on z. Is there any problem in my calculation?
Thanks for answering this question~!
ds2 = e2a(z)(dx2 + dy2) + dz2 − e2b(z)dt2I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor:
<The Christoffel connection> Here a'(z) denotes the first derivative of a(z) respect to z.
[itex]\Gamma\stackrel{x}{xz}[/itex]=[itex]\Gamma\stackrel{x}{zx}[/itex]=a'(z)
[itex]\Gamma\stackrel{y}{yz}[/itex]=[itex]\Gamma\stackrel{y}{zy}[/itex]=a'(z)
[itex]\Gamma\stackrel{z}{tt}[/itex]=b'(z)e2b(z)
[itex]\Gamma\stackrel{z}{xx}[/itex]=[itex]\Gamma\stackrel{z}{yy}[/itex]=-a'(z)e2a(z)
[itex]\Gamma\stackrel{t}{tz}[/itex]=[itex]\Gamma\stackrel{t}{zt}[/itex]=b'(z)
[itex]\Gamma\stackrel{}{either}[/itex]=0
<The Riemann curvature tensor>
[itex]R\stackrel{x}{zxz}[/itex]=[itex]R\stackrel{y}{zyz}[/itex]=-a''(z)-[a'(z)]2
[itex]R\stackrel{z}{tzt}[/itex]=b''(z)+[b'(z)]2
I tried to find the Ricci scalar curvature(R) from current result, but it gave a function depend on z. Is there any problem in my calculation?
Thanks for answering this question~!