Cosmological Solutions to Einstein's Equations

part of the spacelike submanifold of an AdS³

A metric for Anti-deSitter (AdS) Spacetime in n > 2 dimensions is [Hawking]

ds² = dr² + sinh(r)² dΩ_n-2² - cosh(r)² dt²

There is a degeneracy singularity at r = 0. The nonzero Christoffel Symbols for n = 3 and 4 are (with c_r = cosh(r))

Γ^r_{r r} c_r / (1 - c_r²) c_r / (1 - c_r²)
Γ^r_{c c} c_r (1 - c_r²) / (1 - c²)
Γ^r_{φ
φ} c_r (1 - c_r²) (1 - c²) c_r (1 - c_r²)
Γ^r_{t t} - c_r (1 - c_r²) - c_r (1 - c_r²)
Γ^c_{r c} - c_r / (1 - c_r²)
Γ^c_{c c} c / (1 - c²)
Γ^c_{φ
φ} c (1 - c²)
Γ^φ_{r
φ} - c_r / (1 - c_r²) - c_r / (1 - c_r²)
Γ^φ_{c
φ} - c / (1 - c²)
Γ^t_{r t} 1 / c_r 1 / c_r

The r direction is geodesic, and the r-t and spacelike hypersurfaces are geodesic. The latter indicates that the spacetime is globally hyperbolic, and can be foliated into timelike hypersurfaces. Note also that the φ direction is geodesic when c is +-1.

The nonzero components of the Riemann Tensor, which are all negative definite, for n = 3 and 4 are

R_{r c r c} - 1 / (1 - c²)
R_{r φ r φ} - 1 - (1 - c²)
R_{r t r t} - c_r² / (1 - c_r²) - c_r² / (1 - c_r²)
R_{c φ c φ} - (1 - c_r²)²
R_{c t c t} - c_r² (1 - c_r²) / (1 - c²)
R_{φ t φ t} - c_r² (1 - c_r²) - (1 - c²) c_r² (1 - c_r²)

and the Ricci Tensor is

R_{a b} = - (n - 1) g_{a b}

which implies that AdSⁿ is an Einstein Manifold with negative cosmological constant

Λ = - (n - 1) (n / 2 - 1)

The invariants we have chosen to examine are

R - n (n - 1)
R^{a b} R_{a b} n (n - 1)²
R^{a b c d} R_{a b c d} 2 n (n - 1)
R^{a b} R^c_a R_{b c} - n (n - 1)³
R^{a b c d} R_{a c} R_{b d} - n (n - 1)³
R^{a b c d} R^e_a R_{b c d e} - n (n - 1)²
R^{a b c d} R^{e f}_{a b} R_{c d e f} - 4 n (n - 1)
R^{a b c d} R^e_a^f_c R_{b e d f} - n (n - 1) (n - 2)
R^{a b c d} R^e_a^f_c R_{b f d e} - n (n - 1) (n - 3)
R^{a b; c} R_{a b; c} 0
R^{a b; c} R_{a c; b} 0
R^{a b}_{; a} R^c_{b; c} 0
R^{a b c d; e} R_{a b c d; e} 0
R^{a b c d}_{; a} R^e_{b c d; e} 0
Euler class - (-1)^{n / 2} 2 / (Area (Sⁿ))
ε^{a b c i ...} ε^{e f g}_{i ...} R_{b c e}^h R_{f g a h} - n (2 n - 4) (2 n - 2)

Except for the last, these are all exactly the negative of those for the sphere with unit radius. Hence we recognize AdSⁿ as a spacetime of constant negative curvature. The last invariant is equal to that of the unit sphere.

A metric for deSitter (dS) Spacetime in n > 2 dimensions is [Hawking]

ds² = a² cosh(t / a)² dΩ_n-1² - dt²

Hypersurfaces of constant time in the deSitter spacetime are spheres. The nonzero Christoffel Symbols for n = 3 and 4 are (with c_t = cosh(t / a))

Γ¹_{1 1} c₁ / (1 - c₁²) c₁ / (1 - c₁²)
Γ¹_{1 t} 1 / c_t 1 / c_t
Γ¹_{2 2} c₁ (1 - c₁²) / (1 - c₂²)
Γ¹_{φ
φ} c₁ (1 - c₁²) c₁ (1 - c₁²) (1 - c₂²)
Γ²_{1 2} - c₁ / (1 - c₁²)
Γ²_{2 2} c₂ / (1 - c₂²)
Γ²_{2 t} 1 / c_t
Γ²_{φ
φ} c₂ (1 - c₂²)
Γ^φ_{1
φ} - c₁ / (1 - c₁²) - c₁ / (1 - c₁²)
Γ^φ_{2
φ} - c₂ / (1 - c₂²)
Γ^φ_{φ
t} 1 / c_t 1 / c_t
Γ^t_{1 1} - c_t (1 - c_t²) / (1 - c₁²) - c_t (1 - c_t²) / (1 - c₁²)
Γ^t_{2 2} - (1 - c₁²) c_t (1 - c_t²) / (1 - c₂²)
Γ^t_{φ
φ} - (1 - c₁²) c_t (1 - c_t²) - (1 - c₁²) (1 - c₂²) c_t (1 - c_t²)
Γ^t_{t t} c_t / (1 - c_t²) c_t / (1 - c_t²)

The timelike direction is geodesic, and the timelike hypersurfaces orthogonal to the φ direction are geodesic. Also, the c₁ direction is geodesic when c_t is 1, corresponding to the minimum at t = 0 when the space is a sphere of radius a. The c₂ direction is geodesic when c₁ is +-1, the φ direction is geodesic when c_n-2 is +-1, etc.

The nonzero components of the Riemann Tensor, which are all positive definite, for n = 3 and 4 are

R_{1 2 1 2} a² c_t⁴ / (1 - c₂²)
R_{1 φ 1 φ} a² c_t⁴ a² (1 - c₂²) c_t⁴
R_{1 t 1 t} a² c_t² / ((1 - c₁²) (1 - c_t²)) a² c_t² / ((1 - c₁²) (1 - c_t²))
R_{2 φ 2 φ} a² (1 - c₁²)² c_t⁴
R_{2 t 2 t} a² (1 - c₁²) c_t² / (1 - c_t²) a² (1 - c₁²) c_t² / ((1 - c₂²) (1 - c_t²))
R_{φ t φ t} a² (1 - c₁²) (1 - c₂²) c_t² / (1 - c_t²)

The Ricci tensor and all computed curvature invariants are the same as for Sⁿ with r -> a.

FRW Cosmologies

A general metric for the Friedmann-Robertson-Walker cosmologies is [Hawking]

ds² = S²(t) (- dt² + dr² + f²(r) dΩ²)

The spacelike hypersurface for this metric is a space of constant curvature, characterized by a constant K, which determines the functions S(t) and f(r) (and the notation we will use in the following):

K = 1 0 -1

S(t) = - E (1 - cos(t)) / 3 t² E (cosh(t) - 1) / 3

f(r) = sin (r) r sinh (r)

c_r = cos (r) cosh (r)

c_t = cos (t) cosh (t)

where E is the constant energy (kinetic + potential) of the homogeneous dust filling the universe.

These metrics all have a degeneracy singularity at t=0 besides the obvious one at r=0. We will take the coordinate chart for t to be the interval (0, π) for the K=1 case, and (0, infinity) for K=0 or -1. We can use the following Mathematica code to generate the metrics:

ftab = {Sin[r], r, Sinh[r]};
Stab = {(-e / 3) (1 - Cos[t]), t^2, (e / 3) (Cosh[t] - 1)};
rtab = {cr, r, cr};
ttab = {ct, t, ct};
crtab = {ArcCos[cr], r, ArcCosh[cr]};
cttab = {ArcCos[ct], t, ArcCosh[ct]};
xp = Flatten[ {r, Table[ Symbol["t" <> ToString[i]], {i,dim - 3}], ph, t}];
gp = IdentityMatrix[dim];
gp[[dim, dim]] = -1;
fn = ftab[[K]];
Sfn = Stab[[K]];
Do[
gp[[j, j]] = fn^2 Product[ Sin[xp[[i]]]^2, {i, 2, j - 1}],
{j, 2, dim - 1}];
Do[
gp[[i, i]] = Sfn^2 * gp[[i, i]],
{i, dim}];
gp = simpler[gp];
x = Flatten[ {rtab[[K]], Table[ Symbol["c" <> ToString[i]], {i, dim - 3}], ph, ttab[[K]]}];
g = simpler[coordxform[gp, xp, Flatten[ {crtab[[K]], Table[ ArcCos[ Symbol["c" <> ToString[i]]],
{i, dim - 3}], ph, cttab[[K]]}], x]];

In 4 dimensions (henceforth notated D = 4) this yields the following metrics:

K = 1 0 -1

g_{r r} (1 - c_t)² E² / (9 (1 - c_r²)) t² (c_t - 1)² E² / (9 (c_r² - 1))
g_{c c} (1 - c_r²) (1 - c_t)² E² / (9 (1 - c²)) r² t² / (1 - c²) (c_r² - 1) (c_t - 1)² E² / (9 (1 - c²))
g_{φ φ} (1 - c²) (1 - c_r²) (1 - c_t)² E² / 9 r² (1 - c²) t² (1 - c²) (c_r² - 1) (c_t - 1)² E² / 9
g_{t t} - (1 - c_t) E² / (9 (1 + c_t)) - t² - (c_t - 1) E² / (9 (1 + c_t))

The nonzero Christoffel Symbols in D = 4 are

K = 1 0 -1

Γ^r_{r r} c_r / (1 - c_r²) - c_r / (c_r² - 1)
Γ^r_{r t} - 1 / (1 - c_t) 2 / t 1 / (c_t - 1)
Γ^r_{c c} c_r (1 - c_r²) / (1 - c²) - r / (1 - c²) - c_r (c_r² - 1) / (1 - c²)
Γ^r_{φ
φ} (1 - c²) c_r (1 - c_r²) - (1 - c²) r - (1 - c²) c_r (c_r² - 1)
Γ^c_{r c} - c_r / (1 - c_r²) 1 / r c_r / (c_r² - 1)
Γ^c_{c c} c / (1 - c²) c / (1 - c²) c / (1 - c²)
Γ^c_{c t} - 1 / (1 - c_t) 2 / t 1 / (c_t - 1)
Γ^c_{φ
φ} c (1 - c²) c (1 - c²) c (1 - c²)
Γ^φ_{r
φ} - c_r / (1 - c_r²) 1 / r c_r / (c_r² - 1)
Γ^φ_{c
φ} - c / (1 - c²) - c / (1 - c²) - c / (1 - c²)
Γ^φ_{φ
t} - 1 / (1 - c_t) 2 / t 1 / (c_t - 1)
Γ^t_{r r} - (1 + c_t) / (1 - c_r²) 2 / t (1 + c_t) / (c_r² - 1)
Γ^t_{c c} - (1 - c_r²) (1 + c_t) / (1 - c²) 2 r² / ((1 - c²) t) (c_r² - 1) (1 + c_t) / (1 - c²)
Γ^t_{φ
φ} - (1 - c²) (1 - c_r²) (1 + c_t) 2 (1 - c²) r² / t (1 - c²) (c_r² - 1) (1 + c_t)
Γ^t_{t t} - 1 / (1 - c_t²) 2 / t 1 / (c_t² - 1)

The timelike direction is geodesic, and the r-t hypersurface and the timelike hypersurface orthogonal to φ are both geodesic.

The nonzero components of the Riemann Tensor are all positive definite:

K = 1 0 -1

R_{r c r c} 2 (1 - c_t) E² / (9 (1 - c²)) 4 r² t² / (1 - c²) 2 (c_t - 1) E² / (9 (1 - c²))
R_{r φ r φ} 2 (1 - c²) (1 - c_t) E² / 9 4 (1 - c²) r² t² 2 (1 - c²) (c_t - 1) E² / 9
R_{r t r t} E² / (9 (1 - c_r²) (1 + c_t)) 2 t² E² / (9 (c_r² - 1) (1 + c_t))
R_{c φ c φ} 2 (1 - c_r²)² (1 - c_t) E² / 9 4 r⁴ t² 2 (c_r² - 1)² (c_t - 1) E² / 9
R_{c t c t} (1 - c_r²) E² / (9 (1 - c²) (1 + c_t)) 2 r² t² / (1 - c²) (c_r² - 1) E² / (9 (1 - c²) (1 + c_t))
R_{φ t φ t} (1 - c²) (1 - c_r²) E² / (9 (1 + c_t)) 2 (1 - c²) r² t² (1 - c²) (c_r² - 1) E² / (9 (1 + c_t))

as are the nonzero components of the Ricci Tensor:

K = 1 0 -1

R_{r r} 3 / ((1 - c_r²) (1 - c_t)) 6 / t² 3 / ((c_r² - 1) (c_t - 1))
R_{c c} 3 (1 - c_r²) / ((1 - c²) (1 - c_t)) 6 r² / ((1 - c²) t²) 3 (c_r² - 1) / ((1 - c²) (c_t - 1))
R_{φ φ} 3 (1 - c²) (1 - c_r²) / (1 - c_t) 6 (1 - c²) r² / t² 3 (1 - c²) (c_r² - 1) / (c_t - 1)
R_{t t} 3 / ((1 - c_t)² (1 + c_t)) 6 / t² 3 / ((c_t - 1)² (1 + c_t))

The invariants we have chosen to examine are

K = 1 0 -1

R 54 / ((1 - c_t)³ E²) 12 / t⁶ 54 / ((c_t - 1)³ E²)
R^{a b} R_{a b} 2916 / ((1 - c_t)⁶ E⁴) 144 / t¹² 2916 / ((c_t - 1)⁶ E⁴)
R^{a b c d} R_{a b c d} 4860 / ((1 - c_t)⁶ E⁴) 240 / t¹² 4860 / ((c_t - 1)⁶ E⁴)
R^{a b} R^c_a R_{b c} 39366 / ((1 - c_t)⁹ E⁶) 432 / t¹⁸ 39366 / ((c_t - 1)⁹ E⁶)
R^{a b c d} R_{a c} R_{b d} 118098 / ((1 - c_t)⁹ E⁶) 1296 / t¹⁸ 118098 / ((c_t - 1)⁹ E⁶)
R^{a b c d} R^e_a R_{b c d e} 52488 / ((1 - c_t)⁹ E⁶) 576 / t¹⁸ 52488 / ((c_t - 1)⁹ E⁶)
R^{a b c d} R^{e f}_{a b} R_{c d e f} 122472 / ((1 - c_t)⁹ E⁶) 1344 / t¹⁸ 122472 / ((c_t - 1)⁹ E⁶)
R^{a b c d} R^e_a^f_c R_{b e d f} 61236 / ((1 - c_t)⁹ E⁶) 672 / t¹⁸ 61236 / ((c_t - 1)⁹ E⁶)
R^{a b c d} R^e_a^f_c R_{b f d e} 30618 / ((1 - c_t)⁹ E⁶) 336 / t¹⁸ 30618 / ((c_t - 1)⁹ E⁶)
R^{a b; c} R_{a b; c} - 393660 (1 + c_t) / ((1 - c_t)⁹ E⁶) - 8640 / t¹⁸ - 393660 (1 + c_t) / ((c_t - 1)⁹ E⁶)
R^{a b; c} R_{a c; b} - 373977 (1 + c_t) / ((1 - c_t)⁹ E⁶) - 8208 / t¹⁸ - 373977 (1 + c_t) / ((c_t - 1)⁹ E⁶)
R^{a b}_{; a} R^c_{b; c} - 59049 (1 + c_t) / ((1 - c_t)⁹ E⁶) - 1296 / t¹⁸ - 59049 (1 + c_t) / ((c_t - 1)⁹ E⁶)
R^{a b c d; e} R_{a b c d; e} - 708588 (1 + c_t) / ((1 - c_t)⁹ E⁶) - 15552 / t¹⁸ - 708588 (1 + c_t) / ((c_t - 1)⁹ E⁶)
R^{a b c d}_{; a} R^e_{b c d; e} - 39366 (1 + c_t) / ((1 - c_t)⁹ E⁶) - 864 / t¹⁸ - 39366 (1 + c_t) / ((c_t - 1)⁹ E⁶)
Euler class 243 / (2 π²(1 - c_t)⁶ E⁴) 6 / (π²t¹²) 243 / (2 π²(c_t - 1)⁶ E⁴)
ε^{a b c i ...} ε^{e f g}_{i ...} R_{b c e}^h R_{f g a h} - 1944 / ((1 - c_t)⁶ E⁴) - 96 / t¹² - 1944 / ((c_t - 1)⁶ E⁴)

We see that for constant t, these spaces are indeed spaces of constant curvature, and that the singularity at t=0 is in fact an essential singularity. Note that the order 0 invariants are all positive, and that once again the last invariant is negative. Also note that that even though the spacelike hypersurfaces are of constant curvature, the order 1 invariants are nonzero: the invariants measure the curvature of the whole spacetime.

In general these invariants contain very little independent information, which as we have seen is indicative of the geometrical simplicity of these spacetimes. The coefficients of the invariants have the following patterns, with those for K = -1 the same as those for K = +1:

K = 1 0

R 18 (D - 1) (D - 3) 4 (D - 1) (D - 3)
R^{a b c d} R_{a b c d} 324 (D - 1) (2 D - 3) 16 (D - 1) (2 D - 3)
R^{a b c d} R^{e f}_{a b} R_{c d e f} 64 (D - 1) (4 D - 9)
R^{a b c d} R^e_a^f_c R_{b e d f} 16 (D - 1) (D - 2) (4 D - 9)
R^{a b c d} R^e_a^f_c R_{b f d e} 1458 (D - 1) (D - 3) (4 D - 9) 16 (D - 1) (D - 3) (4 D - 9)
R^{a b}_{; a} R^c_{b; c} - 6561 (D - 1)² (D - 3)² - 144 (D - 1)² (D - 3)²
R^{a b c d; e} R_{a b c d; e} - 26244 (D - 1) (4 D - 7) - 576 (D - 1) (4 D - 7)
R^{a b c d}_{; a} R^e_{b c d; e} - 6561 (D - 3)² (2 D - 2) - 288 (D - 1) (D - 3)²
ε^{a b c i ...} ε^{e f g}_{i ...} R_{b c e}^h R_{f g a h} - 324 (D - 1) (D - 2) (4 D - 15) - 32 (D - 1) (D - 2) (4 D - 15)

The next section explores black hole solutions to Einstein's Equations.

Table of Contents

Index:

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Please send comments or suggestions to the author.

Γ^r_{r r}	c_r / (1 - c_r²)	c_r / (1 - c_r²)
Γ^r_{c c}		c_r (1 - c_r²) / (1 - c²)
Γ^r_{φ φ}	c_r (1 - c_r²)	(1 - c²) c_r (1 - c_r²)
Γ^r_{t t}	- c_r (1 - c_r²)	- c_r (1 - c_r²)
Γ^c_{r c}		- c_r / (1 - c_r²)
Γ^c_{c c}		c / (1 - c²)
Γ^c_{φ φ}		c (1 - c²)
Γ^φ_{r φ}	- c_r / (1 - c_r²)	- c_r / (1 - c_r²)
Γ^φ_{c φ}		- c / (1 - c²)
Γ^t_{r t}	1 / c_r	1 / c_r

R_{r c r c}		- 1 / (1 - c²)
R_{r φ r φ}	- 1	- (1 - c²)
R_{r t r t}	- c_r² / (1 - c_r²)	- c_r² / (1 - c_r²)
R_{c φ c φ}		- (1 - c_r²)²
R_{c t c t}		- c_r² (1 - c_r²) / (1 - c²)
R_{φ t φ t}	- c_r² (1 - c_r²)	- (1 - c²) c_r² (1 - c_r²)

R	- n (n - 1)
R^{a b} R_{a b}	n (n - 1)²
R^{a b c d} R_{a b c d}	2 n (n - 1)
R^{a b} R^c_a R_{b c}	- n (n - 1)³
R^{a b c d} R_{a c} R_{b d}	- n (n - 1)³
R^{a b c d} R^e_a R_{b c d e}	- n (n - 1)²
R^{a b c d} R^{e f}_{a b} R_{c d e f}	- 4 n (n - 1)
R^{a b c d} R^e_a^f_c R_{b e d f}	- n (n - 1) (n - 2)
R^{a b c d} R^e_a^f_c R_{b f d e}	- n (n - 1) (n - 3)
R^{a b; c} R_{a b; c}	0
R^{a b; c} R_{a c; b}	0
R^{a b}_{; a} R^c_{b; c}	0
R^{a b c d; e} R_{a b c d; e}	0
R^{a b c d}_{; a} R^e_{b c d; e}	0
Euler class	- (-1)^{n / 2} 2 / (Area (Sⁿ))
ε^{a b c i ...} ε^{e f g}_{i ...} R_{b c e}^h R_{f g a h}	- n (2 n - 4) (2 n - 2)

Γ¹_{1 1}	c₁ / (1 - c₁²)	c₁ / (1 - c₁²)
Γ¹_{1 t}	1 / c_t	1 / c_t
Γ¹_{2 2}		c₁ (1 - c₁²) / (1 - c₂²)
Γ¹_{φ φ}	c₁ (1 - c₁²)	c₁ (1 - c₁²) (1 - c₂²)
Γ²_{1 2}		- c₁ / (1 - c₁²)
Γ²_{2 2}		c₂ / (1 - c₂²)
Γ²_{2 t}		1 / c_t
Γ²_{φ φ}		c₂ (1 - c₂²)
Γ^φ_{1 φ}	- c₁ / (1 - c₁²)	- c₁ / (1 - c₁²)
Γ^φ_{2 φ}		- c₂ / (1 - c₂²)
Γ^φ_{φ t}	1 / c_t	1 / c_t
Γ^t_{1 1}	- c_t (1 - c_t²) / (1 - c₁²)	- c_t (1 - c_t²) / (1 - c₁²)
Γ^t_{2 2}		- (1 - c₁²) c_t (1 - c_t²) / (1 - c₂²)
Γ^t_{φ φ}	- (1 - c₁²) c_t (1 - c_t²)	- (1 - c₁²) (1 - c₂²) c_t (1 - c_t²)
Γ^t_{t t}	c_t / (1 - c_t²)	c_t / (1 - c_t²)

R_{1 2 1 2}		a² c_t⁴ / (1 - c₂²)
R_{1 φ 1 φ}	a² c_t⁴	a² (1 - c₂²) c_t⁴
R_{1 t 1 t}	a² c_t² / ((1 - c₁²) (1 - c_t²))	a² c_t² / ((1 - c₁²) (1 - c_t²))
R_{2 φ 2 φ}		a² (1 - c₁²)² c_t⁴
R_{2 t 2 t}	a² (1 - c₁²) c_t² / (1 - c_t²)	a² (1 - c₁²) c_t² / ((1 - c₂²) (1 - c_t²))
R_{φ t φ t}		a² (1 - c₁²) (1 - c₂²) c_t² / (1 - c_t²)

K =	1	0	-1
S(t) =	- E (1 - cos(t)) / 3	t²	E (cosh(t) - 1) / 3
f(r) =	sin (r)	r	sinh (r)
c_r =	cos (r)		cosh (r)
c_t =	cos (t)		cosh (t)

K =	1	0	-1
g_{r r}	(1 - c_t)² E² / (9 (1 - c_r²))	t²	(c_t - 1)² E² / (9 (c_r² - 1))
g_{c c}	(1 - c_r²) (1 - c_t)² E² / (9 (1 - c²))	r² t² / (1 - c²)	(c_r² - 1) (c_t - 1)² E² / (9 (1 - c²))
g_{φ φ}	(1 - c²) (1 - c_r²) (1 - c_t)² E² / 9	r² (1 - c²) t²	(1 - c²) (c_r² - 1) (c_t - 1)² E² / 9
g_{t t}	- (1 - c_t) E² / (9 (1 + c_t))	- t²	- (c_t - 1) E² / (9 (1 + c_t))

K =	1	0	-1
R_{r c r c}	2 (1 - c_t) E² / (9 (1 - c²))	4 r² t² / (1 - c²)	2 (c_t - 1) E² / (9 (1 - c²))
R_{r φ r φ}	2 (1 - c²) (1 - c_t) E² / 9	4 (1 - c²) r² t²	2 (1 - c²) (c_t - 1) E² / 9
R_{r t r t}	E² / (9 (1 - c_r²) (1 + c_t))	2 t²	E² / (9 (c_r² - 1) (1 + c_t))
R_{c φ c φ}	2 (1 - c_r²)² (1 - c_t) E² / 9	4 r⁴ t²	2 (c_r² - 1)² (c_t - 1) E² / 9
R_{c t c t}	(1 - c_r²) E² / (9 (1 - c²) (1 + c_t))	2 r² t² / (1 - c²)	(c_r² - 1) E² / (9 (1 - c²) (1 + c_t))
R_{φ t φ t}	(1 - c²) (1 - c_r²) E² / (9 (1 + c_t))	2 (1 - c²) r² t²	(1 - c²) (c_r² - 1) E² / (9 (1 + c_t))

K =	1	0	-1
R_{r r}	3 / ((1 - c_r²) (1 - c_t))	6 / t²	3 / ((c_r² - 1) (c_t - 1))
R_{c c}	3 (1 - c_r²) / ((1 - c²) (1 - c_t))	6 r² / ((1 - c²) t²)	3 (c_r² - 1) / ((1 - c²) (c_t - 1))
R_{φ φ}	3 (1 - c²) (1 - c_r²) / (1 - c_t)	6 (1 - c²) r² / t²	3 (1 - c²) (c_r² - 1) / (c_t - 1)
R_{t t}	3 / ((1 - c_t)² (1 + c_t))	6 / t²	3 / ((c_t - 1)² (1 + c_t))

K =	1	0	-1
R	54 / ((1 - c_t)³ E²)	12 / t⁶	54 / ((c_t - 1)³ E²)
R^{a b} R_{a b}	2916 / ((1 - c_t)⁶ E⁴)	144 / t¹²	2916 / ((c_t - 1)⁶ E⁴)
R^{a b c d} R_{a b c d}	4860 / ((1 - c_t)⁶ E⁴)	240 / t¹²	4860 / ((c_t - 1)⁶ E⁴)
R^{a b} R^c_a R_{b c}	39366 / ((1 - c_t)⁹ E⁶)	432 / t¹⁸	39366 / ((c_t - 1)⁹ E⁶)
R^{a b c d} R_{a c} R_{b d}	118098 / ((1 - c_t)⁹ E⁶)	1296 / t¹⁸	118098 / ((c_t - 1)⁹ E⁶)
R^{a b c d} R^e_a R_{b c d e}	52488 / ((1 - c_t)⁹ E⁶)	576 / t¹⁸	52488 / ((c_t - 1)⁹ E⁶)
R^{a b c d} R^{e f}_{a b} R_{c d e f}	122472 / ((1 - c_t)⁹ E⁶)	1344 / t¹⁸	122472 / ((c_t - 1)⁹ E⁶)
R^{a b c d} R^e_a^f_c R_{b e d f}	61236 / ((1 - c_t)⁹ E⁶)	672 / t¹⁸	61236 / ((c_t - 1)⁹ E⁶)
R^{a b c d} R^e_a^f_c R_{b f d e}	30618 / ((1 - c_t)⁹ E⁶)	336 / t¹⁸	30618 / ((c_t - 1)⁹ E⁶)
R^{a b; c} R_{a b; c}	- 393660 (1 + c_t) / ((1 - c_t)⁹ E⁶)	- 8640 / t¹⁸	- 393660 (1 + c_t) / ((c_t - 1)⁹ E⁶)
R^{a b; c} R_{a c; b}	- 373977 (1 + c_t) / ((1 - c_t)⁹ E⁶)	- 8208 / t¹⁸	- 373977 (1 + c_t) / ((c_t - 1)⁹ E⁶)
R^{a b}_{; a} R^c_{b; c}	- 59049 (1 + c_t) / ((1 - c_t)⁹ E⁶)	- 1296 / t¹⁸	- 59049 (1 + c_t) / ((c_t - 1)⁹ E⁶)
R^{a b c d; e} R_{a b c d; e}	- 708588 (1 + c_t) / ((1 - c_t)⁹ E⁶)	- 15552 / t¹⁸	- 708588 (1 + c_t) / ((c_t - 1)⁹ E⁶)
R^{a b c d}_{; a} R^e_{b c d; e}	- 39366 (1 + c_t) / ((1 - c_t)⁹ E⁶)	- 864 / t¹⁸	- 39366 (1 + c_t) / ((c_t - 1)⁹ E⁶)
Euler class	243 / (2 π²(1 - c_t)⁶ E⁴)	6 / (π²t¹²)	243 / (2 π²(c_t - 1)⁶ E⁴)
ε^{a b c i ...} ε^{e f g}_{i ...} R_{b c e}^h R_{f g a h}	- 1944 / ((1 - c_t)⁶ E⁴)	- 96 / t¹²	- 1944 / ((c_t - 1)⁶ E⁴)

K =	1	0
R	18 (D - 1) (D - 3)	4 (D - 1) (D - 3)
R^{a b c d} R_{a b c d}	324 (D - 1) (2 D - 3)	16 (D - 1) (2 D - 3)
R^{a b c d} R^{e f}_{a b} R_{c d e f}		64 (D - 1) (4 D - 9)
R^{a b c d} R^e_a^f_c R_{b e d f}		16 (D - 1) (D - 2) (4 D - 9)
R^{a b c d} R^e_a^f_c R_{b f d e}	1458 (D - 1) (D - 3) (4 D - 9)	16 (D - 1) (D - 3) (4 D - 9)
R^{a b}_{; a} R^c_{b; c}	- 6561 (D - 1)² (D - 3)²	- 144 (D - 1)² (D - 3)²
R^{a b c d; e} R_{a b c d; e}	- 26244 (D - 1) (4 D - 7)	- 576 (D - 1) (4 D - 7)
R^{a b c d}_{; a} R^e_{b c d; e}	- 6561 (D - 3)² (2 D - 2)	- 288 (D - 1) (D - 3)²
ε^{a b c i ...} ε^{e f g}_{i ...} R_{b c e}^h R_{f g a h}	- 324 (D - 1) (D - 2) (4 D - 15)	- 32 (D - 1) (D - 2) (4 D - 15)