**Minkowski Spacetime** (**M ^{4}**) is covered completely by the

-infinity < x < infinityThe corresponding metric is of course-infinity < y < infinity

-infinity < z < infinity

-infinity < t < infinity.

dsif we choose the positive signature.^{2}=dx+^{2}dy+^{2}dz-^{2}dt^{2}

A transformation to **polar coordinates**

x = ρ cos (φ)gives us the metricy = ρ sin (φ)

dsThe (certainly not unique) inverse transformation is^{2}=dρ+ ρ^{2}^{2}dφ+^{2}dz-^{2}dt.^{2}

ρ = (xwith^{2}+ y^{2})^{1/2}φ = cos

^{-1}(x / ρ) for y > 0, or y = 0 and x != 0, andφ = 2π - cos

^{-1}(x / ρ) for y < 0,

0 < ρ < infinity(We take the range of cos0 <= φ < 2 π.

This is actually a pair of **diffeomorphisms**, each covering one half of the x-y plane
for any value of z (and t),
excluding the origin. The two match on their boundaries (have the same value and derivatives there),
and so we treat them as a single transformation.

But because the inverse transformation is not defined on the z axis. this
**diffeomorphism** does not cover the region
ρ = 0. There, all values of φ map to the same values of
x and y. The metric is degenerate on the z axis, and of course, the inverse metric is singular there.
We will call this a **degeneracy singularity**. For the **polar coordinate chart**, the z axis exists at
ρ = 0, but there is no **diffeomorphism** between **polar**
and **Cartesian coordinates** which includes it.
From the "active" point of view, this **diffeomorphism** is a map between two manifolds
which have the topology **R ^{4} - R**.

Now consider a transformation from **Cartesian** to **spherical coordinates**:

x = r sin (θ) cos (φ)This gives us the metricy = r sin (θ) sin (φ)

z = r cos (θ).

dswith^{2}=dr+ r^{2}^{2}dθ+ r^{2}^{2}sin^{2}(θ)dφ-^{2}dt^{2}

r = (xwith^{2}+ y^{2}+ z^{2})^{1/2}θ = cos

^{-1}(z / r),

0 < r < infinityAgain, the0 < θ < π

0 <= φ < 2 π.

If we have learned from our discussions of **polar** and **spherical coordinates** in flat space, we expect to
be wary of the coordinates we use to describe the **Schwarzschild Spacetime**. Since it is asymptotically flat and
spherically symmetric, we will use coordinates which *asymptotically* are the same as **spherical
coordinates** in flat space. In these coordinates, the metric for the **Schwarzschild Spacetime** is:

dsThis metric is clearly singular at r = 0 and at r = 2M. In agreement with our topological expectations for this metric, r = 2M is the "edge" of a^{2}= r / (r - 2M)dr+ r^{2}^{2}dθ+ r^{2}^{2}sin^{2}(θ)dφ+ (2M / r - 1)^{2}dt.^{2}

The **Kruskal Coordinates** for the **Schwarzschild Spacetime** are generally described implicitly as

(r / 2M - 1) eNote that tanh^{r / 2M}= X^{2}- T^{2}t / 2M = 2 tanh

^{-1}(T / X).

Simplify[ Solve[ {(r / (2M) - 1) E^(r / (2M)) == X^2 - T^2, t / (2M) == 2 ArcTanh[T / X]}, {r, t}]]where ProductLog[x] is the solution for y in x = y e{{t -> 4 M ArcTanh[T / X], r -> 2 M (1 + ProductLog[(-T^{2}+ X^{2}) / E])}}

Using the same technique to solve for X and T, we find that the inverse transformation is

X = eAn equivalent transformation occurs if X -> -X and T -> -T.^{r / 4M}cosh (t / 4M) (r / 2M - 1)^{1/2}T = e

^{r / 4M}sinh (t / 4M) (r / 2M - 1)^{1/2}.

This **diffeomorphism** is defined throughout the region r > 2M.
As we noted above, the continuation of this transformation to values of r < 2M corresponds to
values of X^{2} - T^{2} > -1.
This tells us that the **Kruskal Coordinate Chart** covers the region between the red lines in the
X-T plane below, which includes the region covered by the original coordinate chart:

Because of the restriction -X < T < X, the **diffeomorphism** only covers the region between the blue lines
T = +- X. These lines represent r = 2M for infinite t, which indicates that the t coordinate is also
problematic for the region near r = 2M. Lines of constant t correspond to straight lines through
the origin whose slopes are between
-1 and 1, with the X axis corresponding to t = 0.
Note that the second and fourth quadrants of this diagram represent negative values of t.
Lines of constant r are hyperbolae (in green above).
From the indifference of the inverse transformation to sign, we see that the left and right half planes represent
a double cover of this **diffeomorphism**.

In these coordinates (and abbreviating the ProductLog function as **P**), the **Schwarzschild metric** is

dsThis appears to be singular at X = T (at r = 2M). However, if we parameterize the metric using the function^{2}= 16 M^{2}P((X^{2}- T^{2}) / e) / ((X^{2}- T^{2}) (1 +P((X^{2}- T^{2}) / e))) (dX-^{2}dT) +^{2}4 M^{2}(1 +P((X^{2}- T^{2}) / e))^{2}(dθ+ sin^{2}^{2}(θ)dφ).^{2}

R (X, T) = 2M (1 +the metric takes the simple (and suggestive) formP((X^{2}- T^{2}) / e))(implying that X^{2}- T^{2}= e^{R / 2M}(R / 2M - 1)),

dsExamining the limit of R (X, T) as T -> X, we find^{2}= 32 e^{-R / 2M}M^{3}/ R (dX-^{2}dT) + R^{2}^{2}(dθ+ sin^{2}^{2}(θ)dφ).^{2}

Limit[ R[X, T], T->X]The function R is equivalent to the coordinate r in the region r > 2M; we will interpret it as an analytical continuation of r to the region r <= 2M. Lines of constant R in this region are hyperbolae crossing the T axis between the red and blue lines in the diagram above. Because t is infinite on the blue lines, it cannot be continued into that region in the same fashion as R.2M

This metric has a singularity at R = 0, of which we will say more soon.
Of course the metric is only valid outside whatever interior solution
is responsible for the curvature; we still expect it to describe a region of topology
**R ^{4} - B^{4}**, which excludes the point R = 0. So we see
that the

Because it is not always obvious which manifold is described by a given metric, and because two very different metrics may in fact describe the same geometry, we are led to ask the following questions:

- Is there a way to know in advance whether a metric singularity is a
**coordinate singularity**or an**essential singularity**, which cannot be removed by a coordinate transformation (**degeneracy singularities**are thankfully obvious) ? - Given two metrics, such as the
**Schwarzschild metrics**in asymptotically**spherical coordinates**and**Kruskal Coordinates**, is there any way to determine if they are related by a**diffeomorphism**?

Any product of curvature tensors or their derivatives, such that all indices are fully contracted, is
invariant with respect to **diffeomorphisms**. As a scalar, its numerical value does not depend
in any way on the coordinate system used to measure it. We have already met one such scalar: R,
the scalar curvature. Because it is identically zero for vacuum spacetimes, which includes many
metrics of interest, we consider the simplest scalar invariant quadratic in the Riemann tensor,
the **Kretschmann Invariant**:

RFor the^{.}R = R^{a b c d}R_{a b c d }

RSo we see that while the algebraic forms of the invariants are very different, there must exist a simple variable substitution which makes the forms equivalent, if they are related by a^{.}R_{spherical}= 48 M^{2}/ r^{6}R

^{.}R_{Kruskal}= 3 / (4 M^{4}(1 +P((X^{2}- T^{2}) / e))^{6})= 48 M^{2}/ R^{6}

*Ris proportional to the intrinsic angular momentum of the spacetime. Or consider [Gass]^{.}R = ε^{a b c d}R_{a b e f}R_{c d}^{e f}

D Rwhich vanishes on the ergosphere of a Kerr black hole (in 4 dimensions). Or perhaps [Bicak]^{.}D R = D_{a}R_{b c d e}D^{a}R^{b c d e}

D D Rwhich is non-vanishing for expanding type N spacetimes. It is clear that we need to quantify the problem of selecting curvature invariants of interest.^{.}D D R^{.}D D R^{.}D D R =D

_{a}D_{b}R_{c d e f}D_{g}D_{h}R_{i j k l}D^{a}D^{b}R^{c j e l}D^{g}D^{h}R^{i d k f}

The number of independent curvature invariants of order 0 (involving no covariant derivatives) in D > 2 dimensions is

N[Haskins], while for order k there are:^{D}_{0}= ( D - 2 ) ( D - 1 ) D ( D + 3 ) / 12

NFor some dimensions of interest we have^{D}_{k}= D ( k + 1 ) ( D + k + 1 ) ! / ( 2 ( D - 2 ) ! ( k + 3 ) ! )

The 14 invariants of order 0 in 4 dimensions were first constructed [Witten] using spinor methods and later covariantly [Greenberg] in terms of the Weyl Tensor (C), the traceless portion of the Ricci Tensor (S), the Einstein Curvature Tensor (E) and the dual operator *:

D N ^{D}_{0}N ^{D}_{1}N ^{D}_{2}3 3 15 27 4 14 60 126 5 40 175 420 6 90 420 1134 7 175 882 2646 8 308 1680 5544 9 504 2970 10692 10 780 4950 19305 11 1155 7865 33033

We will be interested in curvature invariants which can be consistently compared in diverse dimensions, and unfortunately neither of the methods used to identify independent invariants generalizes well to dimensions other than 4. This leads us to two problems:

R S ^{.}S^{.}S^{.}SC ^{.}CC ^{.}E^{.}E* C ^{.}C* C ^{.}E^{.}EC ^{.}C^{.}CC ^{.}E^{.}E^{.}C^{.}E^{.}E* C ^{.}C^{.}C* C ^{.}E^{.}E^{.}C^{.}E^{.}ES ^{.}SC ^{.}E^{.}E^{.}C^{.}E^{.}E^{.}C^{.}E^{.}ES ^{.}S^{.}S* C ^{.}E^{.}E^{.}C^{.}E^{.}E^{.}C^{.}E^{.}E

- Invariants independent in one dimension may not be so in others.
- Since ε has D indices, dual invariants only exist in D mod 4 = 0 and
in D mod 2 = 0. For example, we can construct:
*R R = ε

in D mod 4 = 0 and in D mod 2 = 0 we can construct:^{a b c d}R_{a b e f}R_{c d}^{e f}*R R R R = ε

^{a b c d e f g h}R_{a b i j}R_{c d}^{i j}R_{e f k l}R_{g h}^{k l}(with a maximum of ( D ! ) ( D ( D - 1 ) / 2 )

^{( D / 4 )}products)*R *R R = ε

This severely limits the invariants we can construct from dual tensors, both because of the lack of a consistent definition from one value of D to another, and from the point of view of computability in a reasonably finite amount of time.^{a b c d}ε^{e f g h}R_{a b i j}R_{e f}^{i j}R_{c d g h}*R *R R R R = ε

^{a b c d e f}ε^{g h i j k l}R_{a b m n}R_{g h o p}R_{c d}^{m n}R_{i j}^{o p}R_{e f k l}(with a maximum of ( D ! )

^{2}products).

For vacuum solutions to Einstein's Equations, we find that

R R ^{a b c d}R^{e}_{a}^{f}_{c}R_{b e d f}R ^{a b}R_{a b}R ^{a b c d}R^{e}_{a}^{f}_{c}R_{b f d e}R ^{a b c d}R_{a b c d}R ^{a b; c}R_{a b; c}R ^{a b}R^{c}_{a}R_{b c}R ^{a b; c}R_{a c; b}R ^{a b c d}R_{a c}R_{b d}R ^{a b}_{; a}R^{c}_{b; c}R ^{a b c d}R^{e}_{a}R_{b c d e}R ^{a b c d; e}R_{a b c d; e}R ^{a b c d}R^{e f}_{a b}R_{c d e f}R ^{a b c d}_{; a}R^{e}_{b c d; e}ε

^{a b c d ...}ε^{e f g h ...}R_{a b e f}R_{c d g h}R... / ((D/2)! (8 π)^{D/2})This is theεEuler Class, which is only definable for even D. For a compact manifold, the integral of theEuler Classover the entire manifold is a topological invariant called theEuler Characteristic. Note that theEuler Classincludes additional terms for manifolds with boundary.^{a b c i ...}ε^{e f g}_{i ...}R_{b c e}^{h}R_{f g a h}= δThe following Mathematica code can be used to compute this last invariant:^{a}_{[e}δ^{b}_{f}δ^{c}_{g]}R_{b c}^{e h}R^{f g}_{a h}deltasuuulll = Table[ Table[ Table[ Table[ Table[ Table[ 0,{i, dim}], {j, dim}], {k, dim}], {l, dim}], {m, dim}], {n, dim}];deltaul[ a_, b_] := If[a == b, 1, 0];Do[ Do[ Do[ Do[ Do[ Do[ deltasuuulll[[i, j, k, l, m, n]] =

deltaul[i, l] deltaul[j, m] deltaul[k, n] + deltaul[i, m] deltaul[j, n] deltaul[k, l] +Rlluu = simpler[ raise[ raise[ Rllll, 3], 4]];deltaul[i, n] deltaul[j, l] deltaul[k, m] - deltaul[i, m] deltaul[j, l] deltaul[k, n] -

deltaul[i, l] deltaul[j, n] deltaul[k, m] - deltaul[i, n] deltaul[j, m] deltaul[k, l],

{n, dim}], {m, dim}], {l, dim}], {k, dim}], {j, dim}], {i, dim}];

Ruull = simpler[ raise[ raise[ Rllll, 1], 2]];

starRstarR = simpler[ Sum[ Sum[ Sum[ Sum[ Sum[ Sum[ Sum[

deltasuuulll[[aa, bb, cc, ee, ff, gg]] Rlluu[[bb, cc, ee, hh]] Ruull[[ff, gg, aa, hh]],{hh, dim}], {gg, dim}], {ff, dim}], {ee, dim}], {cc, dim}], {bb, dim}], {aa, dim}]];

R = 2 D Λ / (D - 2),andR

^{a b}R_{a b}= 4 D Λ^{2}/ (D - 2)^{2}

Rby contracting the field equations appropriately.^{a b}R^{c}_{a}R_{b c}=R^{a b c d}R_{a c}R_{b d}=8 D Λ

^{3}/ (D - 2)^{3}

Another set of invariants of some interest are the **Pontrjagin Classes**. These are not scalar invariants,
but rather differential forms defining a set of characteristic classes
of the tangent space manifold. The first **Pontrjagin Class** is
[Eguchi, Gilkey and Hanson]

pa 4-form defined for D >= 4, and the second is_{1}= - Tr (R ^ R) / (8 π^{2})= - (R^{a}_{b c d}R^{b}_{a e f}+ R^{a}_{b e f}R^{b}_{a c d}-R^{a}_{b e d}R^{b}_{a c f}- R^{a}_{b f d}R^{b}_{a e c}-R

^{a}_{b c e}R^{b}_{a d f}- R^{a}_{b c f}R^{b}_{a e d}) / (48 π^{2}),

pan 8-form defined for D >= 8. p_{2}= Tr (R ^ R ^ R ^ R) / (64 π^{4})(which has 8 ! / 2^{4}= 2520 terms),

The following Mathematica code can be used as a model for computing the first **Pontrjagin Class**:

Thewedge[ q1_, aa_, bb_, i_, j_, q2_, cc_, dd_, k_, l_] :=(q1[[aa, bb, i, j]] q2[[cc, dd, k, l]] + q1[[aa, bb, k, l]] q2[[cc, dd, i, j]] -q1[[aa, bb, k, j]] q2[[cc, dd, i, l]] - q1[[aa, bb, l, j]] q2[[cc, dd, k, i]] -

q1[[aa, bb, i, k]] q2[[cc, dd, j, l]] - q1[[aa, bb, i, l]] q2[[cc, dd, k, j]]) / 6

If[ (i == k) || (i == l) || (j == k) || (j == l), 0, 1];

pindex[ ii_, jj_Integer] := Block[{inds},If[ Head[ii] === Integer, If[ jj < ii, jj, jj + 1],(inds = Select[ Table[ i, {i, dim}], !MemberQ[ ii, #]&];Return[ inds[[jj]]];)]];pontrjagin[] := Block[{p1, p2},(If[ dim < 4, Return[], ];Switch[ dim,

4, If[ (p1 = simpler[ Sum[ Sum[- wedge[ rulll, aa, bb, 1, 2,rulll, bb, aa, 3, 4], {bb, dim}], {aa, dim}] / (8 Pi5, Do[ If[ (p1 = simpler[ Sum[ Sum[- wedge[^{2})]) =!= 0,Print[ "p1[1, 2, 3, 4] = ", p1], ],

rulll, aa, bb, pindex[ii, 1], pindex[ii, 2],6, Do[ Do[ If[ (p1 = simpler[ Sum[ Sum[- wedge[rulll, bb, aa, pindex[ii, 3], pindex[ii, 4]],

{bb, dim}], {aa, dim}] / (8 Pi

^{2})]) =!= 0,Print[ "p1[", pindex[ii, 1], ", ", pindex[ii, 2],

", ", pindex[ii, 3], ", ", pindex[ii, 4], "] = ", p1], ], {ii, dim}],

rulll, aa, bb, pindex[{ii, jj}, 1], pindex[{ii, jj}, 2],rulll, bb, aa, pindex[{ii, jj}, 3], pindex[{ii, jj}, 4]],

{bb, dim}], {aa, dim}] / (8 Pi

^{2})]) =!= 0,Print[ "p1[", pindex[{ii, jj}, 1], ", ",

pindex[{ii, jj}, 2], ", ", pindex[{ii, jj}, 3], ", ",

pindex[{ii, jj}, 4], "] = ", p1], ],

{jj, ii + 1, dim}], {ii, dim - 1}]];)];

For non-vacuum solutions, there exist invariants of the stress-energy tensor which can be expressed in terms of the invariants above:

TT^{a b}T_{a b}= (R^{a b}R_{a b}+ R^{2}(D / 4 - 1) - Λ R (D - 2) + D Λ^{2}) / α^{2}T

^{a b}T^{c}_{a}T_{b c}= (R^{a b}R^{c}_{a}R_{b c}- 3 R (R^{a b}R_{a b}/ 2 + Λ^{2}(D / 2 - 1)) +3 Λ R^{a b}R_{a b}- R^{3}(D / 8 - 3 / 4) + 3 Λ R^{2}(D / 4 - 1) + Λ^{3}D) / α^{3}T

^{a b; c}T_{a b; c}= (R^{a b; c}R_{a b; c}+ R^{, a}R_{, a}(D / 4 - 1)) / α^{2}T

^{a b; c}T_{a c; b}= (R^{a b; c}R_{a b; c}- R^{, a}R^{b}_{a; b}+ R^{, a}R_{, a}/ 4) / α^{2}

RFor a^{a b}_{; a}R^{c}_{b; c}= R^{, a}R^{b}_{a; b}- R^{, a}R_{, a}/ 4

Tand^{a b}= ρ v^{a}v^{b}+ P (g^{a b}+ v^{a}v^{b})T = (ρ + P) v

^{a}v_{a}+ D PT

^{a b}T_{a b}= (ρ + P)^{2}(v^{a}v_{a})^{2}+ 2 P (ρ + P) v^{a}v_{a}+ D P^{2}

TThe invariants involving the covariant derivative of T^{a b}T^{c}_{a}T_{b c}= (ρ + P)^{3}(v^{a}v_{a})^{3}+ 3 P (ρ + P)^{2}(v^{a}v_{a})^{2}+ 3 P^{2}(ρ + P) v^{a}v_{a}+ D P^{3}

Of course, we would like the invariants we examine to include those of some physical or intrinsic
geometric interest, but in all but the simplest cases it is a priori impossible to determine which those might be.
For example, by equating our two expressions for T^{a b} T_{a b}, we see that for D = 4,
Λ = 0, and a pressure-less perfect fluid, the square of the Ricci Tensor is
equal to the square of the kinetic energy density. But in most cases, we must simply take our chances and see what turns up!

The next section explores spheres in arbitrary dimensions.

- Table of Contents
- Index:

©2005, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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