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The Kiselev black hole is neither perfect fluid, nor is it quintessence

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The Kiselev black hole's matter is anisotropic, not a perfect fluid or quintessence.

desk verdict A correct, clearly-written correction of a widespread mislabeling; the algebra is simple and right, and the paper is worth publishing despite being limited in scope. read the letter →

arxiv 1908.11058 v1 pith:6IWE4DBI submitted 2019-08-29 gr-qc astro-ph.COhep-th

Matt Visser (Victoria University of Wellington) This is my paper · ORCID
classification gr-qcastro-ph.COhep-th MSC 83C5783C5583F05 PACS 04.70.-s04.40.-b95.36.+x
keywords Kiselevblackholeperfectfluidquintessencepressureanisotropystress-energytensorRastallgravityequationofstate
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper corrects a widespread mislabeling in the black-hole literature. The Kiselev black hole, a popular toy model, is routinely described as a perfect-fluid or quintessence spacetime. Direct computation shows its stress-energy tensor has radial pressure equal to minus the energy density and a different tangential pressure whenever the equation-of-state parameter w differs from -1. Consequently the spacetime is anisotropic and cannot be a perfect fluid, and because quintessence in standard cosmology means a scalar field with timelike gradient whose stress-energy is a perfect fluid, the Kiselev matter is not quintessence either. The correction matters because hundreds of follow-up papers build on these labels.

What carries the argument

The load-bearing object is the stress-energy tensor of the Kiselev spacetime in an orthonormal frame, Eqs. (2.1)-(2.2), together with the pressure-anisotropy identity Δ = (p_r - p_t)/((p_r + 2p_t)/3) = -3(1+w)/(2w). This identity is position-independent and nonzero for all w ≠ -1, which directly rules out an isotropic, perfect-fluid interpretation. The same identity, with w replaced by an effective position-dependent w_effective, extends to multi-component generalizations, and it also governs what happens under Rastallization.

What would settle it

Recompute the Einstein tensor of the Kiselev metric in an orthonormal frame: if for any w ≠ -1 with K ≠ 0 the radial and tangential pressures were found equal, the non-perfect-fluid claim would fail. The paper itself supplies the calculation, so the test is to verify Eq. (2.2) independently.

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Extended reading notes

Core claim

The central discovery is that the Kiselev spacetime, with metric $ds^{2}$ = -(1 - 2m/r - K/$r^{{1+3w}}$) $dt^{2}$ + $dr^{2}$/(1 - 2m/r - K/$r^{{1+3w}}$) + $r^{2}$ $dΩ_2^{2}$, has a stress-energy tensor whose orthonormal-frame components are ρ = -p_r = -3Kw/(8π $r^{{3(1+w)}}$) and p_t = -3Kw(1+3w)/(16π $r^{{3(1+w)}}$). The radial and tangential pressures are unequal for every w ≠ -1, so the matter is anisotropic. The relative pressure anisotropy Δ = (p_r - p_t)/((p_r + 2p_t)/3) = -3(1+w)/(2w) is a nonzero constant unless w = -1, the Schwarzschild-(anti)-de Sitter case. The paper also shows that the same anisotropic structure persists in two-component and N-component generalizations, with the anisotropy becoming position-dependent, and that Rastallizing the stress-energy tensor changes only book-keeping, not the geometry or the anisotropy.

Load-bearing premise

The argument assumes the standard general-relativity definition of a perfect fluid as having isotropic pressure in its rest frame, and the standard cosmological definition of quintessence as a scalar field with timelike gradient whose stress-energy is a perfect fluid; adopt looser definitions and the verbal claims weaken, though the algebra is unchanged.

Editorial extensions

If this is right

  • The special cases w = 0 (Schwarzschild), w = 1/3 (Reissner-Nordström), and w = -1 (Schwarzschild-(anti)-de Sitter) must be treated separately; only w = -1 gives an isotropic stress-energy tensor.
  • Follow-up papers that model the Kiselev spacetime as a perfect fluid are making an algebraic error and need to be reinterpreted or revised.
  • Multi-component generalized Kiselev metrics remain anisotropic with position-dependent Δ, so the 'not a perfect fluid' conclusion is robust under that generalization.
  • Rastallization does not change the spacetime geometry nor the pressure anisotropy; it is merely a parameter redefinition of the stress-energy tensor.
  • Since the standard cosmological quintessence is a perfect-fluid scalar field, results about Kiselev matter cannot be directly transferred to quintessence dark-energy models.

Reading between the lines

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

  • If the Kiselev matter is not quintessence, then dark-energy phenomenology that borrows its equation of state should not be conflated with scalar-field quintessence, because the two have different perturbation and causal properties.
  • The same orthonormal-frame pressure test could be applied to other static spherically symmetric 'exotic matter' metrics to expose mislabeled perfect fluids in the literature.
  • The N-component generalization gives a template for engineering static spherically symmetric spacetimes with prescribed anisotropy profiles by tuning the component weights w_i, an extension the paper does not explicitly pursue.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The manuscript analyzes the Kiselev black-hole metric, ds^2 = -(1-2m/r - K/r^{1+3w})dt^2 + dr^2/(1-2m/r - K/r^{1+3w}) + r^2 dOmega^2, and shows from the Einstein equations that the matter sector in an orthonormal frame satisfies rho = -p_r = -3Kw/[8pi r^{3(1+w)}] and p_t = -3Kw(1+3w)/[16pi r^{3(1+w)}]. The tangential and radial pressures are unequal for generic w, so the paper concludes that the Kiselev spacetime is not a perfect-fluid spacetime and that the word 'quintessence' is used in a nonstandard sense in this context. The analysis is extended to multi-component Kiselev metrics and to Rastall gravity, where the 'Rastallization' transformation is shown to be a mere redefinition of the stress-energy tensor.

Significance. This is a short, clearly written comment whose main value is correctional rather than exploratory. The central calculation is elementary and correct: equations (2.1)-(2.4) transparently display the pressure anisotropy, and the frame-independence of the anisotropy follows from the orthonormal-frame eigenvalue statement. The extension to two-component and N-component models is competently done, and the Rastall section usefully emphasizes that the transformation is a bookkeeping redefinition, not a modification of the geometry. The paper makes no new physical predictions, but for a comment/note whose purpose is terminological clarity that is appropriate. The main strengths are the self-contained, hand-checkable derivation and the explicit formulas that make the central claim easy to verify.

minor comments (4)
  1. [Section 2, after Eq. (2.4)] The statement 'for w != -1 we have both p_t/p_r != 1 and Delta != 0' is not correct for w = 0, where rho = p_r = p_t = 0 and both ratios are undefined; the same caveat applies to the abstract's 'unless w = -1', so the authors should explicitly exclude w = 0 or state that the claim concerns the non-vacuum matter sector.
  2. [Eq. (2.4)] The ratio p_t/p_r should be displayed as -(1+3w)/2; as typeset it can be misread as -1 + 3w/2, which would be a different and incorrect expression.
  3. [Introduction, first paragraph] The statement 'w = 1/3 corresponds to Reissner-Nordstroem' should mention the sign convention K = -Q^2 at that point, since the equality holds only with this sign choice (as noted later in the text).
  4. [Abstract and Section 1] The citation count 'over 200 direct and indirect citations as of 2019' is not independently verifiable from the manuscript; consider softening to 'approximately' or removing the precise count.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Kiselev stress-energy anisotropy is derived from the metric by direct computation.

full rationale

The derivation chain is self-contained. Starting from the Kiselev metric (1.1), the paper computes the orthonormal-frame Einstein tensor (2.1), which immediately gives ρ = −p_r and p_t = −3Kw(1+3w)/(16πr^{3(1+w)}). Because p_r ≠ p_t for w ≠ −1, the stress-energy is not isotropic, and the relative anisotropy Δ of Eq. (2.4) is a direct algebraic consequence, not an independently fitted quantity. The 'not quintessence' conclusion is likewise a definitional corollary: standard cosmological quintessence is a scalar field with timelike gradient, carrying a perfect-fluid stress-energy, as documented by external refs [5–10]; an anisotropic Type-I stress-energy therefore cannot be quintessence under that standard usage. The Rastall section is auxiliary; its redefinition formulas (5.1)–(5.10) are exhibited algebraically, and the central perfect-fluid/quintessence claim does not depend on the self-cited ref [14]. No parameter is fitted and no prediction is equivalent to an input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new objects, parameters, or entities. Its conclusions rest on standard Einstein equations and on conventional definitions of perfect fluid and quintessence, both explicitly anchored in the cited literature.

assumptions (4)
  • standard math Einstein's equations G_{μν} = 8π T_{μν} relate the metric to the stress-energy tensor.
    Used in Section 2 to compute density and pressures from the Einstein tensor.
  • domain assumption A perfect fluid is defined by isotropic pressure in its rest frame (p_r = p_t).
    Section 2 relies on this to conclude the anisotropic stress-energy is not a perfect fluid.
  • domain assumption Quintessence in cosmology is a scalar field with timelike gradient whose stress-energy is a zero-vorticity perfect fluid.
    Section 2 cites refs [5-10] for this usage and uses it to argue the Kiselev matter is not quintessence.
  • domain assumption The metric forms in Eqs. (3.1) and (4.1) are the relevant generalizations of the Kiselev spacetime.
    These are the objects of study for the generalized analysis; they are asserted as starting points.

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Pith. "Pith review of The Kiselev black hole is neither perfect fluid, nor is it quintessence." pith.science (2026). https://pith.science/paper/6IWE4DBI

@misc{pith2026190811058,
  author       = {Pith},
  title        = {Pith review of: The Kiselev black hole is neither perfect fluid, nor is it quintessence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6IWE4DBI}},
  note         = {Machine review of arXiv:1908.11058}
}
abstract

The Kiselev black hole spacetime, \[ ds^2 = - \left(1-{2m\over r} - {K\over r^{1+3w}} \right) dt^2 + {dr^2\over1-{2m\over r} - {K\over r^{1+3w}}} + r^2 \,d\Omega_2^2, \] is an extremely popular toy model, with over 200 direct and indirect citations as of 2019. Unfortunately, despite repeated assertions to the contrary, this is not a perfect fluid spacetime. The relative pressure anisotropy and average pressure are easily calculated to satisfy \[ \Delta = {\Delta p\over \bar p} = {p_r - p_t \over {1\over3} (p_r+2p_t)} =- {3(1+w)\over 2 w}; \qquad\qquad {\bar p\over \rho} = {{1\over3} (p_r + 2p_t)\over \rho} = w. \] The relative pressure anisotropy $\Delta$ is generally a non-zero constant, (unless $w=-1$, corresponding to Schwarzschild-(anti)-de Sitter spacetime). Kiselev's original paper was very careful to point this out in the calculation, but then in the discussion made a somewhat unfortunate choice of terminology which has (with very limited exceptions) been copied into the subsequent literature. Perhaps worse, Kiselev's use of the word "quintessence" does not match the standard usage in the cosmology community, leading to another level of unfortunate and unnecessary confusion. Very few of the subsequent follow-up papers get these points right, so a brief explicit comment is warranted.

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

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