Search for “fluid in equilibrium” here. It claims that T^{11} is positive for ordinary positive pressure.
First we describe a metric for a 2D space with a uniform positive curvature in a disk of radius tr, and flat outside the disk. We use polar coördinates, (r, θ) = (x^{1}, x^{2}).
g_{ij} = |
| . |
For r ≥ tr, f(r) =(((sin tr)−(cos tr)tr)+(cos tr)r)^{2}
Perhaps f(r) = (sinh r)^{2} gives us uniform negative curvature.
Γ_{k ij}= ½(−∂g_{ij}/∂x^{k} +∂g_{ik}/∂x^{j} +∂g_{kj}/∂x^{i})
∂g_{ij}/∂x^{k} is 0 except for ∂g_{22}/∂x^{1} which is f'(r).
Γ_{k ij} is 0 except for
Γ_{1 22} = −½f'(r)
and Γ_{2 12} = Γ_{2 21} = ½f'(r).
Note that the two formulae for f agree at the splice point.
For r < tr the space is isometric to a slice of a 3D of sphere radius 1.
For r > tr the space is isometric to a truncated cone where R^{𝜌}_{σμν} = 0.
Γ^{k}_{ij} = g^{kn}Γ_{n ij}.
g^{ij} = |
| . |
R^{𝜌}_{σμν} = ∂_{μ}Γ^{𝜌}_{νσ} − ∂_{ν}Γ^{𝜌}_{μσ} + Γ^{𝜌}_{μλ}Γ^{λ}_{νσ} − Γ^{𝜌}_{νλ}Γ^{λ}_{μσ}
See this code with this map:
here | tr | r | g_{ij} | g^{ij} | Γ^{k}_{ij} | ∂_{μ}Γ^{𝜌}_{νσ} | R^{𝜌}_{σμν} | R_{ijkl} | R_{ij} | R | G_{ij} |
there | tr | r | gl | gu | gam | delgam | rie | rr | rt | bigr | bigg |
∂_{μ}Γ^{𝜌}_{νσ} is zero except for
∂_{1}Γ^{1}_{22} = −½f''(r)
and ∂_{1}Γ^{2}_{12} =
∂_{1}Γ^{2}_{21} =
½(f''(r)/f(r) − (f'(r)/f(r))^{2}).
(In each case the factor with λ=1 is 0.)
R_{ijkl} = g_{in}R^{n}_{jkl}.
R_{ij} = R^{k}_{ikj}.
R = g^{ij}R_{ij}.
G_{jk} = R_{jk} − ½g_{jk}R.