REVIEW
An explanation of the Newman-Janis Algorithm
General Relativity and Quantum Cosmology · Submitted 1 Jul 1998 · 8 references
Pith's verdict: A genuine no-go result — the NJA cannot build rotating perfect-fluid interiors, and Kerr-Newman is the only type-D Einstein-Maxwell output — with proofs that are sound in outline but leave one load-bearing algebraic step unshown.
T0 · AI review
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Referee report
The paper
The paper claims
The Newman-Janis algorithm's success is forced by consistency, not luck: within the family of metrics it can produce, only the standard complexification yields Kerr or Kerr-Newman.
The reading
What carries the argument
The load-bearing object is the Boyer-Lindquist-normalized two-function family of metrics the algorithm can reach, with $e^{2λ} = Σ/(j(r) + a^2χ^2)$ and $e^{2Φ} = Σ(j(r) + a^2χ^2)/(k(r) + a^2χ^2)^2$. Every result is a statement about which choices of $j$ and $k$ survive a physical condition. The conditions are read off by expanding curvature quantities in powers of $χ = cos θ$ and using the independence of $r$ and $χ$: the mixed Einstein component $G_{rχ}$ for the perfect-fluid test, the isotropic-pressure combination $G_{rr}/g_{rr} - G_{χχ}/g_{χχ}$, the Newman-Penrose scalar $Ψ_1$ written as a sum of coefficients $K_m(r)$ for the algebraic-special test, and the Ricci scalar $R$ written as a sum of $J_m(r)$ for the Einstein-Maxwell test. Setting each coefficient to zero in turn pins the free functions.
Extended reading notes
Core claim
The paper's central claim is that the Newman-Janis algorithm is not ad hoc. The authors recast it as a five-step procedure and identify the one genuinely ambiguous step: the complex extension of the seed metric's coefficients before the complex coordinate shift $x^ρ → x^ρ + iγ^ρ$. Instead of guessing that extension, they carry two undetermined functions of $r$ and $θ$ through steps 3 and 4 and impose step 5, the requirement that the output admit Boyer-Lindquist coordinates via $u = t + F(r)$, $φ = ψ + G(r)$. This reduces the output to a two-function family, written in terms of $j(r)$ and $k(r)$ with $Σ = r^2 + a^2χ^2$, $χ = cos θ$. Three theorems then follow by direct computation. First, the mixed Einstein component $G_{rχ}$ vanishes only if $k(r) = r^2$, and the isotropic-pressure condition then forces $j(r) = r^2 + d_1 r$, which is Kerr with $d_1 = 2m$: the only perfect fluid the algorithm generates is vacuum. Second, $Ψ_0$ vanishes identically and $Ψ_1 = 0$ forces $k(r) = r^2 + c_1(r + c_1/4)$, so every algebraically special output is Petrov type D. Third, adding a vanishing Ricci scalar forces $c_1 = 0$ and $j(r) = r^2 + d_1 r + d_0$, i.e. Kerr-Newman with $d_1 = 2m$ and $d_0 = Q^2$. A stated corollary is that the Drake-Turolla metrics, which match smoothly to Kerr, cannot be rotating perfect-fluid interiors of Kerr: they may be sensible fluids only in the non-rotating limit.
Load-bearing premise
The whole analysis lives inside the assumption that the algorithm's output can always be put into Boyer-Lindquist form by shifting only the time and angle coordinates by functions of $r$ alone, so any output that needs a more general coordinate transformation is excluded from the family before the uniqueness theorems are proved.
Editorial extensions
If this is right
- The choice of complexification in the standard algorithm, which differed between the $2m/r$ and $Q^2/r^2$ terms, is not a convention but the only one compatible with the output being a perfect fluid, algebraically special, or an Einstein-Maxwell solution within the Boyer-Lindquist family.
- Any proposal to generate rotating perfect-fluid interiors of Kerr by applying the algorithm to a static spherically symmetric seed is ruled out, because the only perfect fluid produced is the vacuum Kerr metric itself.
- The Drake-Turolla family of metrics that match smoothly to Kerr cannot serve as rotating perfect-fluid sources of Kerr, even where they behave as sensible perfect fluids at zero rotation.
- Within the family considered, the Kerr-Newman metric is the unique Petrov type D solution of the Einstein-Maxwell equations reachable by the algorithm, so the algorithm's reach into charged rotating black holes is exactly one solution wide.
Reading between the lines
- The argument converts an existence question into a uniqueness question: the algorithm is 'explained' not by finding the mechanism that makes it work but by showing that the surrounding family of candidate outputs contains only one physically acceptable member. That style of explanation is worth distinguishing from a derivation of why the complex shift $x → x + ia cos θ$ is the right one in the fir
- If the uniqueness is really driven by the Boyer-Lindquist ansatz rather than by the physics, a natural test is to repeat the analysis with a wider class of coordinate transformations, for instance allowing $F$ and $G$ to depend on $θ$ as well as $r$, and see whether non-vacuum perfect fluids appear. A positive result there would not refute the theorems as stated but would shrink what they explain.
- The three conditions form a nested hierarchy — perfect fluid implies vacuum Kerr, algebraic special implies type D, vanishing Ricci scalar implies Kerr-Newman — which suggests the classification is really a single statement about how few free functions survive each physical constraint. That structure may be reusable for other solution-generating transformations whose ambiguity is currently resolve
Original abstract
After the original discovery of the Kerr metric, Newman and Janis showed that this solution could be ``derived'' by making an elementary complex transformation to the Schwarzschild solution. The same method was then used to obtain a new stationary axisymmetric solution to Einstein's field equations now known as the Kerr-newman metric, representing a rotating massive charged black hole. However no clear reason has ever been given as to why the Newman-Janis algorithm works, many physicist considering it to be an ad hoc procedure or ``fluke'' and not worthy of further investigation. Contrary to this belief this paper shows why the Newman-Janis algorithm is successful in obtaining the Kerr-Newman metric by removing some of the ambiguities present in the original derivation. Finally we show that the only perfect fluid generated by the Newman-Janis algorithm is the (vacuum) Kerr metric and that the only Petrov typed D solution to the Einstein-Maxwell equations is the Kerr-Newman metric.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
After the original discovery of the Kerr metric, Newman and Janis showed that this solution could be ``derived'' by making an elementary complex transformation to the Schwarzschild solution. The same method was then used to obtain a new stationary axisymmetric solution to Einstein's field equations now known as the Kerr-newman metric, representing a rotating massive charged black hole. However no clear reason has ever been given as to why the Newman-Janis algorithm works, many physicist considering it to be an ad hoc procedure or ``fluke'' and not worthy of further investigation. Contrary to this belief this paper shows why the Newman-Janis algorithm is successful in obtaining the Kerr-Newman metric by removing some of the ambiguities present in the original derivation. Finally we show that the only perfect fluid generated by the Newman-Janis algorithm is the (vacuum) Kerr metric and that the only Petrov typed D solution to the Einstein-Maxwell equations is the Kerr-Newman metric.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | After the original discovery of the Kerr metric, Newman and Janis showed that this solution could be ``derived'' by making an elementary complex transformation to the Schwarzschild solution. | background 1.00 | 0.03 | 0.03 | asserted 0.62 |
| s1 | The same method was then used to obtain a new stationary axisymmetric solution to Einstein's field equations now known as the Kerr-newman metric, representing a rotating massive charged black hole. | background 1.00 | 0.05 | 0.04 | asserted 1.00 |
| s2 | However no clear reason has ever been given as to why the Newman-Janis algorithm works, many physicist considering it to be an ad hoc procedure or ``fluke'' and not worthy of further investigation. | gap 0.97 | 0.04 | 0.04 | asserted 0.99 |
| s3 | Contrary to this belief this paper shows why the Newman-Janis algorithm is successful in obtaining the Kerr-Newman metric by removing some of the ambiguities present in the original derivation. | contribution 0.99 | 0.94 | 0.71 | asserted 1.00 |
| s4 | Finally we show that the only perfect fluid generated by the Newman-Janis algorithm is the (vacuum) Kerr metric and that the only Petrov typed D solution to the Einstein-Maxwell equations is the Kerr-Newman metric. | result 0.99 | 0.93 | 0.31 | asserted 1.00 |
What was examined, and what was not
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References
How the literature uses it
34 arXiv papers cite this in 51 sentences
Papers citing it · 31 sentences without a confident label
Builds on it 1 paper
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Motivated by these considerations, in the present work we extend the static black hole solutions with primary scalar hair found in [ref] to rotating configurations by employing a Newman-Janis-type algorithm, following a procedure commonly adopted in the literature [ref] .
Phenomenological Rotating Extension of Black Holes with Primary Scalar Hair: Shadow Signatures in Beyond Horndeski Gravity · Introduction · 0.95
Uses its method or tool 9 papers
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The line element ([ref]) coincides with the Schwarzschild one, $m(r)=M$, and there is no difficulty in following the procedure in [ref] .
Charged rotating noncommutative black holes · New prescription for the complexification · 0.83
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The regular rotating black holes (RRBH) can be obtained from the spherically symmetric ones by making use of the Newman-Janis algorithm (for more details see [ref] ).
On the analytic extension of regular rotating black holes · Introduction · 0.90
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The construction of rotating regular black holes involves [ref] the use of the NJA [ref] , an algebraic method for transforming a static and spherically symmetric black hole solution into a rotating and axially symmetric one.
Preliminary analyses on dynamics and thermodynamics of rotating regular black holes · General form of metrics · 0.84
Compares with it 1 paper
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We find [equation] (which agrees with [ref] , see proof of Theorem 2) and [equation] but [equation] where $χ:=cosθ$.
A novel family of rotating black hole mimickers · Algebraic properties and surface gravity · 0.95
Cites it as support 1 paper
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Additionally, Drake and Szekeres find the only perfect fluid system generated by their method is the vacuum Kerr solution [ref] .
Physical interpretation of Newman-Janis rotating systems. I. A unique family of Kerr-Schild systems · Introduction · 0.97
Cites it as background 7 papers
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A few tricks have been put forward in an attempt to include angular momentum into a general static solution, the most renowned one being the Newman-Janis algorithm [ref] (see [ref] or [ref] for a comprehensive description) that was successful in deriving the Kerr solution using the Schwarzschild metric as its seed.
Generating Rotating Spacetime in Ricci-Based Gravity: Naked Singularity as a Black Hole Mimicker · Introduction · 0.90
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In contrast, most of the approaches used to derive exact rotating solutions, besides relying on symmetry properties, were methodic [ref] - [ref] and [ref] - [ref] or partly methodic relying on some ad hoc hypotheses [ref] - [ref] .
From static to rotating to conformal static solutions: Rotating imperfect fluid wormholes with(out) electric or magnetic field · Introduction · 0.81
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[See Ref. [ref] for a recent review on the algorithm, and Refs. [ref] for further insights on the method and its limitations.
Quasinormal modes-shadow correspondence for rotating regular black holes · Rotating regular black holes · 0.89
Discussion
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