Search for “fluid in equilibrium” here. It claims that T¹¹ 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) = (x¹, x²).

gij =

1 0 0 f(r)

.

0≤r; 0≤θ≤2π; identify (r, 0) with (r, 2π). Identify all points of form (0, θ) with each other. We choose f so that we have curvature 1 out to radius tr and 0 beyond.
f(r) is roughly proportional to r². Now we choose f to meet our goal. I think that for 0≤r≤tr, f(r) = (sin r)². (This is the embedding map for a sphere.)

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/∂xⁱ)

∂g_ij/∂x^k is 0 except for ∂g₂₂/∂x¹ 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^knΓ_{n ij}.

gij =

1 0 0 1/f(r)

.

Γ^k_ij is 0 except for Γ¹₂₂ = −½f'(r) and Γ²₁₂ = Γ²₂₁ = ½f'(r)/f(r).

R^𝜌_σμν = ∂_μΓ^𝜌_νσ − ∂_νΓ^𝜌_μσ + Γ^𝜌_μλΓ^λ_νσ − Γ^𝜌_νλΓ^λ_μσ

See this code with this map:

here	tr	r	gij	gij	Γkij	∂μΓ𝜌νσ	R𝜌σμν	Rijkl	Rij	R	Gij
there	tr	r	gl	gu	gam	delgam	rie	rr	rt	bigr	bigg

Here is the output. (This helped.)

∂_μΓ^𝜌_νσ is zero except for ∂₁Γ¹₂₂ = −½f''(r)
and ∂₁Γ²₁₂ = ∂₁Γ²₂₁ = ½(f''(r)/f(r) − (f'(r)/f(r))²).
(In each case the factor with λ=1 is 0.)

R_ijkl = g_inRⁿ_jkl.
R_ij = R^k_ikj.
R = g^ijR_ij.
G_jk = R_jk − ½g_jkR.