Stationary and Axisymmetric Solutions of Higher-Dimensional General Relativity

Troels Harmark

REVIEW

Stationary and Axisymmetric Solutions of Higher-Dimensional General Relativity

Troels Harmark

High Energy Physics - Theory · Submitted 18 Aug 2004 · 25 references

Pith's verdict: A correct and useful toolkit paper: the canonical form and rod-direction classification hold up, with a modest scope overreach in the abstract and a real gap in the sigma-model citations.

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arXiv IDhep-th/0408141 (version 2)

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CitationTroels Harmark. "Stationary and Axisymmetric Solutions of Higher-Dimensional General Relativity." arXiv:hep-th/0408141 (2004). https://pith.science/paper/C2OIGQC2

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Editor's note

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The thing to know: the canonical-form derivation is right, and the rod-direction generalization is the part that actually earns the paper its place. Section 2 does the work honestly — general block-diagonal ansatz, Ricci tensor computed in Appendix D with the conformal factor kept general, Eq. (D.8) forcing ∂_r Λ = 0, then the reduction to (2.12)-(2.13) with the residual R_rr + R_zz checked for consistency. Both sanity checks go through: setting D=4 with the Papapetrou substitution reproduces (1.2)-(1.3), and the diagonal case reproduces Emparan-Reall. The examples in Sections 5 and 6 are not decorative; the rod structures of Kerr, Myers-Perry (both one and two angular momenta) and the black ring are read off the explicit metrics and the angular velocities come out matching the horizons. That is real, checkable work, and the (α, Ω) parameterizations are a genuinely compact way to state those solutions. The paper is also honest where it should be: the uniqueness-from-rod-structure claim in §3.2 is explicitly labelled speculation, and the existence question is left open rather than asserted. Good practice, worth noting since not everyone does it. Two soft spots. First, the abstract's scope. Theorem 2.1 carries Condition (1) as a hypothesis, and the paper argues it holds when one Killing vector is an angle — which covers the asymptotically flat cases it cares about, but not the general class the abstract's 'for such solutions' implies. This is a scope gap, not an error; the theorem is stated correctly and the derivation is fine within its hypotheses. A sentence in the abstract restricting to solutions where Condition (1) holds would fix it. Second, and I think more substantive: Eq. (2.16) is the standard chiral equation of the SL(n,R)/SO(n) sigma-model reduction, and the paper cites neither Belinsky-Zakharov nor Maison nor Breitenlohner-Maison. The canonical form with r² = |det G| and the rod classification are new; the matrix equation is not, and the paper reads as if it were. That should be fixed before publication. Neither of these is load-bearing for the results. Who should read it: anyone working on higher-dimensional black holes, Kaluza-Klein bubbles, or solution-generating techniques — the rod bookkeeping is the practical payoff. Send it out. The referee should ask for the sigma-model citations and a tightened abstract, and otherwise this is a straightforward report.

Referee report

Major objections (3)

[Abstract, sentence s2; §2.1] The abstract states the canonical form is derived 'for such solutions', where such solutions are defined in the abstract's first sentence as having D-2 commuting Killing vector fields. The derivation in §2.1, however, requires Condition (1) of Theorem 2.1 (the wedge V_{[mu1}^{(1)}...V_{mu_{D-2}}^{(D-2)} nabla_{nu} V_{rho]}^{(i)} vanishes at at least one point for each i), which is assumed rather than proved for the general class. The text's justification ('Condition (1) can for example be argued to hold if one of the Killing vector fields is an angle, since then it is zero on the axis of rotation') covers asymptotically flat solutions with a rotational Killing vector but not configurations in which all D-2 Killing vectors are everywhere non-vanishing. The theorem is correctly stated with the hypothesis; the concern is that the abstract's phrasing is broader than what is proved. I recommend the abstract and the opening of §2.1 state the hypothesis explicitly, or that the authors argue Condition (1) holds for the class they intend to cover.
[§3.2, discussion of existence and uniqueness] The claim that a solution is completely determined by its rod-structure (parameters a_1 < ... < a_N and directions v^{(k)}) is presented as a speculation ('We now speculate that this statement can be generalized... we claim in detail that...'). This is a load-bearing interpretive claim for the rod-structure program, since the rod data are used throughout Sections 5 and 6 to characterize the solutions. The authors should either state it clearly as a conjecture with the evidence that supports it, or supply an argument (for instance an ellipticity/uniqueness argument for the boundary-value problem defined by the rod data). As written, the reader cannot tell whether the rod-structure parameterizations of Kerr and Myers-Perry in Sections 5.1 and 5.2 are uniqueness statements or merely consistency observations.
[§3.1, Eq. (3.5) and the discussion of conical singularities] The argument that a space-like rod requires the associated coordinate to be periodic with period 2 pi c, and that a time-like rod corresponds to a horizon with temperature T = 1/(2 pi c), is stated at leading order in r without a careful treatment of the subleading terms in the expansion of G near r = 0. Since the period and temperature assignments are used to fix the periods of phi and psi in the black ring analysis of §6.1 (Eqs. 6.12, 6.14, 6.15), the derivation should be made explicit: state the order to which the expansion is carried and confirm that the subleading corrections do not shift the period. This is a checkable point and would strengthen the rod-structure analysis considerably.

Minor objections (9)

[§2.1, Eq. (2.9)] The step from Eq. (D.8) to partial_r Lambda = 0 uses that the left-hand side sum G^{ij} R_ij vanishes for a Ricci-flat solution; this is correct but the text would read more clearly if the vanishing of the sum were stated explicitly rather than left implicit.
[§2.1, after Eq. (2.11)] The assumption that det(G_ij) is non-constant is stated and the constant case deferred to Appendix A, but the abstract does not mention this restriction. A brief clause in the abstract would make the scope of the main result accurate.
[Appendix A, Eq. (A.5)] The pp-wave example with G_11 = -1 - H, G_22 = 1 - H, G_12 = -H is a useful illustration, but the text should note explicitly that this class is not covered by the main canonical form and explain in one sentence why the reduction to (2.16) fails there (the r-coordinate is not a good coordinate).
[§4.2, Eq. (4.8)] The asymptotic expansion of G_22 is obtained using (4.2) and (4.3); the text should state which combination of the trace constraint and the Laplace equation is used, since the reader cannot immediately reconstruct the coefficient of the 1/sqrt(r^2+z^2) term.
[§5.1, Eq. (5.6)] The expression for G_11 in prolate spheroidal coordinates should be checked against the Boyer-Lindquist form (5.1) in the limit a -> 0; a one-line check would help the reader confirm the normalization conventions.
[§6.1, Eq. (6.7)] G_22 is given only implicitly through Eq. (6.9). Since the black ring metric is one of the two worked examples, it would be helpful to give G_22 explicitly in terms of R_1, R_2, R_3, as is done for G_11, G_12, G_33.
[§6.1, 'Getting the Myers-Perry black hole from the general black ring solution'] The limit c -> 1 with the rescalings (6.17)-(6.19) is described as 'easy to see' that one recovers (5.27). Given that this is a nontrivial check of the canonical-form machinery, a few intermediate steps (in particular how the R_i scale) would be valuable.
[Appendix H, Eq. (H.14)] The three solutions for (q, beta) are stated without derivation; a brief indication of how they are obtained from Eq. (H.13) would make the appendix self-contained.
[References] The reference list is appropriate and complete for the four-dimensional background; no additions are required.

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The paper

The paper claims

The paper establishes a canonical form for D-dimensional stationary, axisymmetric vacuum spacetimes in which the Einstein equations reduce to a single matrix equation on flat three-space.

The reading

The paper asks whether the Einstein equations for stationary, axisymmetric vacuum spacetimes in $D$ dimensions can be organized around a single matrix-valued field instead of a coupled system of metric components. It shows that, under an integrability condition on the $D-2$ commuting Killing vectors, the metric can be put in a canonical form built from a $(D-2)�times(D-2)$ symmetric matrix $G$ and one conformal factor, with $r^2=|�det G|$. The vacuum equations then reduce to $G^{-1}�nabla^2G=(G^{-1}�nabla G)^2$ on an auxiliary flat three-dimensional space, while the conformal factor is fixed by an integrable first-order system. This matters because it turns a hard geometric PDE problem into a matrix equation whose sources are thin rods on a line, and it recovers the four-dimensional Papapetrou form and the higher-dimensional Weyl solutions as special cases. The paper then reads the five-dimensional Myers-Perry black hole and the Emparan-Reall black ring through this lens and identifies their rod structures.

What carries the argument

The load-bearing object is the matrix field $G_{ij}(r,z)$, a $(D-2)�times(D-2)$ symmetric matrix whose determinant fixes the coordinate $r$. The equation $G^{-1}�nabla^2G=(G^{-1}�nabla G)^2$ is the axisymmetric reduction of the vacuum Einstein equations; the remaining metric function $�nu$ is fixed by an integrable pair of first-order equations. At $r=0$ the matrix loses rank, and the intervals where exactly one eigenvalue vanishes are the rods. Each rod carries a direction vector in the $(D-2)$-dimensional space spanned by the Killing vectors; timelike rods correspond to horizons, while spacelike rods correspond to compact directions or rotation axes. This rod data is the paper's proposed organizing principle for stationary axisymmetric solutions.

Extended reading notes

Core claim

The central claim is that any $D$-dimensional Ricci-flat metric with $D-2$ commuting Killing vector fields—subject to the integrability condition of Theorem 2.1 and excluding the constant-determinant subclass treated separately—can be written as $ds^2=�sum_{i,j=1}^{D-2}G_{ij}dx^idx^j+e^{2�nu}(dr^2+dz^2)$ with $r^2=|�det G|$. In these coordinates the Einstein equations for $G$ become $G^{-1}�nabla^2G=(G^{-1}�nabla G)^2$ on flat three-space, and $�nu$ is obtained by integrating a first-order system. The paper verifies that this reduces to Papapetrou's four-dimensional form and to generalized Weyl solutions when the Killing vectors are orthogonal, then applies it to the five-dimensional Myers-Perry black hole and the Emparan-Reall black ring, including their rod structures.

Load-bearing premise

The derivation assumes the integrability condition of Theorem 2.1—especially Condition (1), that a certain antisymmetrized tensor built from the Killing vectors vanishes at at least one point—and the paper argues this holds when one Killing vector is a rotation angle but does not prove it for the full stationary axisymmetric class it claims to cover.

Editorial extensions

If this is right

Solving a $D$-dimensional stationary axisymmetric vacuum problem reduces to solving one matrix equation for $G$ and then integrating for $�nu$, rather than attacking the full Einstein tensor.
The rod endpoints and direction vectors give a compact bookkeeping device for horizons, rotation axes, and compact directions; finite timelike rods encode horizon topology and angular velocity.
Asymptotic falloff of $G$ in four and five dimensions yields mass and angular momentum directly, as the paper demonstrates for Kerr, Myers-Perry, and the black ring.
The formalism puts five-dimensional black holes with $S^3$ horizons and black rings with $S^2�times S^1$ horizons into one language, making their shared and differing rod structures explicit.
If the paper's conjecture that rod structure determines the solution is correct, stationary axisymmetric vacuum solutions can be classified by interval data plus direction vectors rather than by solving the full field equations case by case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The matrix equation is structurally similar to an axisymmetric harmonic-map equation, which suggests that solution-generating techniques from two-dimensional sigma models might be portable to this higher-dimensional setting and could produce new black-ring solutions with two angular momenta.
The rod-direction vectors may support a combinatorial search for five-dimensional stationary axisymmetric solutions: one could enumerate admissible rod configurations with two independent angular momenta and then check regularity and asymptotic flatness.
The paper's regularity argument—more than one vanishing eigenvalue of $G$ at $r=0$ produces a curvature singularity—could be used as a selection rule to discard entire classes of rod data before solving the matrix equation.
Because the canonical form is coordinate-based, it may be a natural starting point for adapting four-dimensional uniqueness arguments to higher-dimensional black holes, though the existence of black rings shows any such theorem must be class-restricted.

Original abstract

We study stationary and axisymmetric solutions of General Relativity, i.e. pure gravity, in four or higher dimensions. D-dimensional stationary and axisymmetric solutions are defined as having D-2 commuting Killing vector fields. We derive a canonical form of the metric for such solutions that effectively reduces the Einstein equations to a differential equation on an axisymmetric D-2 by D-2 matrix field living in three-dimensional flat space (apart from a subclass of solutions that instead reduce to a set of equations on a D-2 by D-2 matrix field living in two-dimensional flat space). This generalizes the Papapetrou form of the metric for stationary and axisymmetric solutions in four dimensions, and furthermore generalizes the work on Weyl solutions in four and higher dimensions. We analyze then the sources for the solutions, which are in the form of thin rods along a line in the three-dimensional flat space that the matrix field can be seen to live in. As examples of stationary and axisymmetric solutions, we study the five-dimensional rotating black hole and the rotating black ring, write the metrics in the canonical form and analyze the structure of the rods for each solution.

The compiled record

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.49 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.13 · compares with prior results 0.92 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We study stationary and axisymmetric solutions of General Relativity, i.e. pure gravity, in four or higher dimensions. D-dimensional stationary and axisymmetric solutions are defined as having D-2 commuting Killing vector fields. We derive a canonical form of the metric for such solutions that effectively reduces the Einstein equations to a differential equation on an axisymmetric D-2 by D-2 matrix field living in three-dimensional flat space (apart from a subclass of solutions that instead reduce to a set of equations on a D-2 by D-2 matrix field living in two-dimensional flat space). This generalizes the Papapetrou form of the metric for stationary and axisymmetric solutions in four dimensions, and furthermore generalizes the work on Weyl solutions in four and higher dimensions. We analyze then the sources for the solutions, which are in the form of thin rods along a line in the three-dimensional flat space that the matrix field can be seen to live in. As examples of stationary and axisymmetric solutions, we study the five-dimensional rotating black hole and the rotating black ring, write the metrics in the canonical form and analyze the structure of the rods for each solution.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study stationary and axisymmetric solutions of General Relativity, i.e. pure gravity, in four or higher dimensions.	contribution 0.64	0.85	0.05	asserted 0.95
s1	D-dimensional stationary and axisymmetric solutions are defined as having D-2 commuting Killing vector fields.	background 0.93	0.16	0.05	asserted 1.00
s2	We derive a canonical form of the metric for such solutions that effectively reduces the Einstein equations to a differential equation on an axisymmetric D-2 by D-2 matrix field living in three-dimensional flat space (apart from a subclass of solutions that instead reduce to a set of equations on a D-2 by D-2 matrix field living in two-dimensional flat space).	contribution 0.46	0.96	0.55	asserted 1.00
s3	This generalizes the Papapetrou form of the metric for stationary and axisymmetric solutions in four dimensions, and furthermore generalizes the work on Weyl solutions in four and higher dimensions.	interpretation 0.51	0.74	0.17	asserted 1.00
s4	We analyze then the sources for the solutions, which are in the form of thin rods along a line in the three-dimensional flat space that the matrix field can be seen to live in.	contribution 0.72	0.88	0.05	asserted 1.00
s5	As examples of stationary and axisymmetric solutions, we study the five-dimensional rotating black hole and the rotating black ring, write the metrics in the canonical form and analyze the structure of the rods for each solution.	contribution 0.92	0.95	0.10	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 1.00.

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sources read

arXiv listing abstract v2

not examined

full text
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references and citing sentences
relations to other papers

levels run

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References

Works cited (25)

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Forward citations

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How later papers used it

87 later papers cite it

Evidence: substantial. 82 of the citing papers share no author with it, and 30 of those carry a confident label from Pith's citation classifier.

Pith read 30 of the 232 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

4 later papers by other authors are listed below: 4 used its method or tool. 2 more papers by the same authors are listed: 1 built on it and 1 cited it as support.

Builds on it 1 of 5 papers with this label · 1 by the same authors

By the same authors

Showed that the set of invariants for black hole space-times originally introduced with the rod structure in this paper can be naturally extended to space-times with a cosmological constant for any number of dimensions and commuting Killing vector fields.

Domain Structure of Black Hole Space-Times with a Cosmological Constant (2011) (same authors) · 0.98 · paraphrased: the licence does not allow quoting

Uses its method or tool 4 of 26 papers with this label

Used the standard techniques of this paper to read off the seed metric from the seed rod configuration.

The Black Di-Ring: An Inverse Scattering Construction (2007) · 1.00 · paraphrased: the licence does not allow quoting
Used this paper's rod description of general relativity with commuting isometries to engineer a spacetime that is asymptotically globally Minkowski.

Geometric Engineering 5d Black Holes with Rod Diagrams (2008) · 0.99 · paraphrased: the licence does not allow quoting
Followed the procedure of this paper and appendix A.2 of another work to establish that the metric in Weyl coordinates is identical to the metric.

Inverse Scattering Construction of a Dipole Black Ring (2011) · 0.96 · paraphrased: the licence does not allow quoting
Following the procedure given in this paper, deduced that the rod structure of the general solution is described by five turning points dividing the z-axis into six rods.

Static charged double-black rings in five dimensions (2011) · 1.00 · paraphrased: the licence does not allow quoting

Cites it as support 1 paper · 1 by the same authors

By the same authors

Said that this paper's proposal that the rod structure characterizes asymptotically flat five-dimensional vacuum black holes with two rotational Killing vector fields is supported by another work's uniqueness theorem.

Domain Structure of Black Hole Space-Times (2009) (same authors) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

86 arXiv papers cite this in 228 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 151 sentences without a confident label

Builds on it 5 papers

This way we show that the set of invariants for black hole space-times, originally introduced with the rod structure in [ref] , can be naturally extended to a set of invariants for black hole space-times with a cosmological constant for any number of space-time dimensions and any number of commuting Killing vector fields.

Domain Structure of Black Hole Space-Times with a Cosmological Constant · Introduction · 0.98
The domain structure generalizes the so-called rod structure proposed in [ref] as the set of invariants characterizing asymptotically flat black holes in five dimensions which are solutions of vacuum Einstein equations and possess two rotational Killing vector fields.

Domain Structure of Black Hole Space-Times · Introduction and summary · 0.84
This statement still holds if $G(u)$ is a quartic polynomial [ref] .

Gravitational multi-soliton solutions on flat space · C-metric-like coordinates · 0.80

Uses its method or tool 26 papers

Using the standard techniques of [ref] , we can read off the seed metric from the seed rod configuration: ringseedG_0=diag{-μ_1μ_4/μ_3μ_6,ρ^2μ_3μ_6/μ_2μ_5μ_7, μ_2μ_5μ_7/μ_1μ_4}.

The Black Di-Ring: An Inverse Scattering Construction · Seed Solution and Solitonic Transformations · 1.00
Given the rod structure, the results of Ref. [ref] may be used to obtain the corresponding solution of Einstein's equation from a 3-dimensional matrix valued Poisson type equation sourced by the rods.

Geometric Engineering 5d Black Holes with Rod Diagrams · Building a lens space with rods · 0.99
Following the procedure given in [ref] , one deduces that the rod structure is described by four turning points that divide the $z$-axis into five rods: * For $z<a_4$ one has a semi-infinite spacelike with normalized direction l_1=√(2√(2)/K_0)(0,2ACR,1). * For $a_4<z<a_3$ one has a finite timelike rod that corresponds to the first black hole horizon.

Charged black holes on the Taub-Bolt instanton · The rod structure · 0.98

Compares with it 8 papers

A hint on how to do this can be found by comparing our final rod diagram with the black ring rod diagram [ref] : we set $a_2=a_3$ and $a_1=a_3$.

The Black Di-Ring: An Inverse Scattering Construction · Parameter Counting · 0.98
It is important to emphasize that unlike in the original definition of the rod-structure in [ref] the domain structure is defined without using any knowledge of how the black hole space-time solves the Einstein equations.

Domain Structure of Black Hole Space-Times with a Cosmological Constant · Introduction · 0.86
On one hand, the stronger version of the rod structure used in this paper is able to distinguish the Eguchi–Hanson and Taub-bolt instantons, whereas the traditional (Harmark) rod structure [ref] is unable to do so.

Rod-structure classification of gravitational instantons with U(1)xU(1) isometry · Introduction · 0.98

Cites it as support 1 paper

The proposal of [ref] that the rod structure provides a characterization of asymptotically flat black holes in five dimensions which are solutions of vacuum Einstein equations and possess two rotational Killing vector fields is supported by the uniqueness theorem of [ref] which states that a black hole space-time with a single event horizon is unique given the rod structure and the asymptotic charges.

Domain Structure of Black Hole Space-Times · Introduction and summary · 0.98

Cites it as background 17 papers

In 5d the boundary segments which do not correspond to the horizon are specified by a length and a 2d vector which determines which linear combination of the rotational Killing fields $m_i$ vanishes – this is referred to as the interval structure (also known as rod data) [ref] (see also [ref] ).

On the uniqueness of extremal vacuum black holes · Non-extremal case · 0.85
Black hole space-times are characterised by their domain structure [ref] .

Uniqueness of Black Holes with Bubbles in Minimal Supergravity · Domain structure of black hole space-times · 0.91
A classification of such solutions was provided by Harmark in Ref. [ref] .

Geometric Engineering 5d Black Holes with Rod Diagrams · Rod structures · 0.83

Discussion

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