- #1

- 6

- 0

## Main Question or Discussion Point

Need to find the Ricci scalar curvature of this metric:

ds

I 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)e

[itex]\Gamma\stackrel{z}{xx}[/itex]=[itex]\Gamma\stackrel{z}{yy}[/itex]=-a'(z)e

[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)]

[itex]R\stackrel{z}{tzt}[/itex]=b''(z)+[b'(z)]

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~!

ds

^{2}= e^{2a(z)}(dx^{2}+ dy^{2}) + dz^{2}− e^{2b(z)}dt^{2}I 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)e

^{2b(z)}[itex]\Gamma\stackrel{z}{xx}[/itex]=[itex]\Gamma\stackrel{z}{yy}[/itex]=-a'(z)e

^{2a(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~!