A point mass in an isotropic universe. Existence, uniqueness and basic properties

Brien C. Nolan

REVIEW

A point mass in an isotropic universe. Existence, uniqueness and basic properties

Brien C. Nolan

General Relativity and Quantum Cosmology · Submitted 12 May 1998 · 25 references

Pith's verdict: The Hawking-mass diagnostic that kills McVittie's k=-1 solution is the real contribution, and it holds up; the existence proof is sound, with §V-D the one place the interpretation outruns the derivation.

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arXiv IDgr-qc/9805041 (version 1)

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CitationBrien C. Nolan. "A point mass in an isotropic universe. Existence, uniqueness and basic properties." arXiv:gr-qc/9805041 (1998). https://pith.science/paper/SORORS6O

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Editor's note

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Worth your time, with one caveat about the abstract's framing. The real result here isn't a new metric — Nolan says outright in §I that he hasn't found a new solution, just identified which member of a known shear-free perfect-fluid class (Kramer et al., ch. 14) is the point mass in a k=-1 RW universe. The real result is the diagnostic: Hawking's renormalized quasi-local mass gives M2 = m0 h^5/w^5 for McVittie, which is m0 for k=0 and diverges for k=-1. That is a clean, decisive, citable observation, and it settles a question people had been hand-waving since 1933. The existence proof in §IV is a standard majorant-series argument — recurrence, geometric bound, generating function, radius of convergence — and it holds together. The Frobenius uniqueness step is right in substance; the bookkeeping of which coefficients vanish is slightly sloppy, but the indicial exponents p=0,1 are correct and the conclusion stands. Soft spots, in order of weight. §V-D is the one place the interpretation outruns the derivation: the conformal construction is done for the RW background, and the extension to the full solution is a single 'we simply note' sentence. The abstract lists behaviour at future null infinity among the studied properties; that is generous. §V-A's elliptic integral fixes A(t) but leaves B(t) free, and Nolan says so — so 'representation in terms of an elliptic integral is given' is an implicit quadrature, not a determination. Minor. The shear-free assumption is acknowledged as a convenience with no a priori justification, which is the right way to put it. One thing I would press if refereeing: the negative first-order pressure perturbation in the k=-1 case (§V-B) is odd, and he flags it himself. It may be genuine physics of negative spatial curvature, or a hint the matching is not quite right. Worth a referee's attention. Note that the classifier's record reports no stated limitations. That is wrong — the paper is unusually candid about its own gaps, and the record should not be read as evidence otherwise. Send it out. The Hawking-mass argument alone earns referee time.

Referee report

Major objections (3)

[§V-B, Eq. (5.3), and §IV, Eq. (4.10)] A direct integration of (3.14a) with G = -24m0(x²+4x)^{-5/2} gives y_0 ≈ -2m0x^{-3}, hence γ = e^{-β/2}(1 - 2m0e^{-β/2}x^{-3}) and e^µ = e^β(1 + 4m0e^{-β/2}e^{-3r}); the printed signs in (4.10) and (5.3) appear to be the opposite. With the corrected sign, the first-order pressure perturbation in (5.3) is -2m0e^{-β/2}e^{-3r}(β̈+2e^{-β}), which is positive under the weak energy condition, consistent with the k=0 case and with the Newtonian expectation, and removes the 'contrary to what one would expect' puzzle the author highlights. Please check the sign and the factor of two.
[§V-D] The conformal construction (5.5)-(5.8) is carried out for the RW background only; the extension to the full solution (M,g) is asserted ('we simply note that in this limit, the metric coefficients of our solution approach those of the RW background') rather than shown. Since the abstract lists behaviour at future null infinity as a studied property, the section should demonstrate that the conformal boundary of (M,g) coincides with that of the background, e.g., by showing that the exponentially small corrections do not alter the conformal factor H(u,l).
[§V-E and §I] The paper endorses McVittie's k=0 solution as satisfactorily describing a point mass in a RW universe, but §V-E identifies a curvature singularity at r=m/2 (R=2m0) in that solution. Since C1-C3 do not exclude this singularity, the criteria may not be sufficient to characterize a physical point mass; the author should discuss whether the k=0 singularity undermines the interpretation, and whether the k=-1 solution has an analogous singularity.

Minor objections (6)

[Title and abstract] There are several typographical errors: 'uniquenes s' in the title, 'occurence' in the abstract, 'Similiar' in §I, 'Raychauduri' versus 'Raychaudhuri', and 'Universtiy' in reference 7.
[§IV, Eq. (4.9)] As printed, the bracket in (4.9) appears to grow with x rather than decay, which would make γ diverge; the expression should be checked against the exact integral of (3.14a).
[§V-C] The phrase 'carrying out the same procedure' is ambiguous: for k=0 the operative condition is β̇=0, but for k=-1 the operative condition is ρ0=0, which gives e^β=(t+c)² with β̇≠0. The wording should make this distinction explicit.
[Abstract] The claim that 'existence of solutions representing more general spherical objects' is proven corresponds to the modest generalization G=O(x^{-5}) (asymptotically constant Hawking mass); the abstract could state this more precisely.
[§V-A] The elliptic-integral representation leaves B(t) undetermined, so it is not a complete determination of the solution; the abstract should clarify that the representation contains a free function of t.
[§V-E] Theorem 1 establishes existence only in a neighbourhood of x=∞, while §V-E treats the solution as global (e.g., the central singularity at R=0); the paper should state explicitly that the singularity claims are conditional on the solution extending, or prove global existence.

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The paper

The paper claims

A point mass can be embedded in an open, negatively curved Robertson–Walker universe, but McVittie's 1933 metric for that case is not the solution: its quasi-local mass diverges at infinity.

The reading

Nolan asks what it should mean, precisely, for a spacetime to describe a single point mass sitting in an expanding open universe, and answers with three conditions: spherical symmetry with a shear-free perfect fluid, metric coefficients that tend to those of a Robertson–Walker background as $r$ grows, and a Hawking quasi-local mass that is exactly a fixed positive constant $m_0$ on every metric two-sphere. He then tests McVittie's 1933 solutions against these conditions and finds a split verdict: the flat ($k=0$) case passes, because its renormalised Hawking mass is just $m_0$, but the negatively curved ($k=-1$) case fails, because the mass measured at infinity diverges. The paper's constructive half proves that a $k=-1$ solution does exist anyway, by reducing the field equations to a single nonlinear second-order equation $y'' = G(x)y^2$ with $G = O(x^{-5})$ and solving it by a convergent power series near infinity, with uniqueness among analytic solutions. What would follow if the argument holds is that the intuitive picture — one mass, one background cosmology, and a clean way to tell which is which — is mathematically available in both open cases, not just the flat one.

What carries the argument

Three pieces carry the argument. First, the Hawking quasi-local mass $M_2(S)$, defined by integrating $-Psi_2 - sigma lambda$ over a spacelike two-sphere and then taking the limit to future null infinity; it is the tool that converts a vague phrase ('represents a point mass in a Robertson–Walker universe') into a number that either stays finite or does not. Second, the reduction of the shear-free spherically symmetric perfect fluid field equations to the single equation $e^{mu/2}(mu'' - tfrac12 mu'^2 - tfrac{h'}{h}mu') = 6m_0 h^{-3}$, or in the transformed variables $y_{xx} = G(x)y^2$, which turns an existence question about Einstein's equations into an existence question about one ODE with a prescribed asymptotic value. Third, the majorant-series existence proof: writing $Y = sum_n delta^n Y_n$ and bounding each term with a geometric sequence derived from the generating function $P(X) = [1 - 2X - sqrt((1-2X)^2 - 4b_0 X)]/(2X)$, Nolan shows uniform convergence for $x$ near infinity, and a regular-singular-point analysis of the linearised equation $Gamma_{xx} = H Gamma$ rules out any second analytic solution with the required decay.

Extended reading notes

Core claim

The paper establishes that the obstruction to embedding a point mass in a $k=-1$ Robertson–Walker universe is a defect of McVittie's particular solution rather than of the problem. Using Hawking's renormalised quasi-local mass $M_2(S)$, Nolan shows that McVittie's $k=0$ metric has $M_2 = m_0$ for every metric sphere, confirming the point-mass reading, while the $k=-1$ metric gives $M_2 = m_0 h^5/w^5$, which grows without bound as $r$ (and hence the affine parameter $v$) goes to infinity. He then proves existence of a different $k=-1$ solution: with $y = e^{-mu/2}$ and $x = w^2$, the field equations reduce to $y_{xx} = G(x)y^2$ with $G(x) = -24m_0(x^2+4x)^{-5/2}$, and a majorant series argument gives a solution obeying $y = e^{-beta(t)/2} + o(1)$ as $x to infinity$, differentiable in $t$, and unique among analytic solutions. The first correction to the background is $e^{mu} = e^{beta}(1 - 4m_0 e^{-beta/2}e^{-3r}) + O(e^{-5r})$, which decays fast enough to keep the mass finite, in contrast to McVittie's $e^{-r/2}$ falloff.

Load-bearing premise

The author restricts to shear-free fluid flow purely because it makes the field equations integrable, and says outright that there is no a priori reason to set the shear to zero; without that restriction the reduction to the single equation $y_{xx} = G(x)y^2$ — and therefore the whole existence proof — has no target.

Editorial extensions

If this is right

The flat and negatively curved open cases can both be said to contain a point mass of mass $m_0$, with the mass identified intrinsically as the constant value of the Hawking quasi-local mass on metric two-spheres.
Taking $m_0 = 0$ returns the Robertson–Walker background, and taking the expansion to zero returns Schwarzschild with mass parameter $m_0$, so the solution interpolates between the two standard limits in a single one-parameter family.
The Misner–Sharp energy takes the clean form $E = tfrac{4pi}{3}R^3 rho(t) + m_0$, so the presence of the particle raises the gravitational energy by exactly its mass, and the central singularity is spacelike and trapped as in Schwarzschild.
Because the construction only needs $G(x) = O(x^{-5})$, the same existence argument covers more general spherical sources embedded in a Robertson–Walker universe, not just the exactly constant-mass case.
The background cosmology can be recovered from an arbitrary spherically symmetric shear-free perfect fluid by computing its Hawking mass: a non-constant result rules out the point-mass interpretation, and a constant one identifies the background.

Reading between the lines

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

The constant-Hawking-mass condition is a strong selection rule: it forces the energy density to be spatially homogeneous, $F(r) = 6m_0 h^{-3}$, so the paper is really characterising the unique shear-free spherical perfect fluid that is exactly a point mass in a cosmological background with no extra matter anywhere else.
The negative first-order pressure perturbation in the $k=-1$ case, $8pi p = 8pi p_0 + (ddot{beta} + 2e^{-beta})m_0 e^{-beta/2}e^{-3r} + O(e^{-5r})$, is flagged by the author as possibly an effect of negative spatial curvature; computing the next order would settle whether the central mass ever makes a net positive pressure contribution as it does in Newtonian cosmology and in the $k=0$ solution.
Because the existence proof only uses the decay rate $G = O(x^{-5})$, the same machinery should extend to any compact source whose exterior Weyl curvature falls off at least that fast, which suggests a general existence theorem for localised sources in open Robertson–Walker backgrounds.
The paper leaves the global structure — horizon, singularity, and whether the spacetime is black-hole or white-hole type — to a promised sequel, so the causal status of the central singularity is currently asserted only locally, via Hayward's criterion.

Original abstract

Criteria which a space-time must satisfy to represent a point mass embedded in an open Robertson--Walker (RW) universe are given. It is shown that McVittie's solution in the case $k=0$ satisfies these criteria, but does not in the case $k=-1$. Existence of a solution for the case $k=-1$ is proven and its representation in terms of an elliptic integral is given. The following properties of this and McVittie's $k=0$ solution are studied; uniqueness, behaviour at future null infinity, recovery of the RW and Schwarzschild limits, compliance with energy conditions and the occurence of singularities. Existence of solutions representing more general spherical objects embedded in a RW universe is also proven.

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.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,290 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.42 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.93 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Criteria which a space-time must satisfy to represent a point mass embedded in an open Robertson--Walker (RW) universe are given. It is shown that McVittie's solution in the case $k=0$ satisfies these criteria, but does not in the case $k=-1$. Existence of a solution for the case $k=-1$ is proven and its representation in terms of an elliptic integral is given. The following properties of this and McVittie's $k=0$ solution are studied; uniqueness, behaviour at future null infinity, recovery of the RW and Schwarzschild limits, compliance with energy conditions and the occurence of singularities. Existence of solutions representing more general spherical objects embedded in a RW universe is also proven.

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	Criteria which a space-time must satisfy to represent a point mass embedded in an open Robertson--Walker (RW) universe are given.	contribution 0.87	0.79	0.06	asserted 0.95
s1	It is shown that McVittie's solution in the case $k=0$ satisfies these criteria, but does not in the case $k=-1$.	result 0.95	0.83	0.16	asserted 1.00
s2	Existence of a solution for the case $k=-1$ is proven and its representation in terms of an elliptic integral is given.	result 0.67	0.94	0.26	asserted 1.00
s3	The following properties of this and McVittie's $k=0$ solution are studied; uniqueness, behaviour at future null infinity, recovery of the RW and Schwarzschild limits, compliance with energy conditions and the occurence of singularities.	contribution 0.99	0.92	0.11	asserted 0.93
s4	Existence of solutions representing more general spherical objects embedded in a RW universe is also proven.	result 0.71	0.91	0.12	asserted 1.00

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

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
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relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (25)

[1]

W. Israel, Nuovo Cimento 44 , 1 (1966)

View on Pith 1966
[2]

H. Sato, in General Relativity and Gravitation ed. B. Bertotti, F. de Felice and A. Pascolini (Reidel, Boston, 1984)

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[3]

A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, Cambridge, 1994)

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[4]

G. C. McVittie, Mon. Not. Roy. Astron. Soc. 93 , 325 (1933)

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[5]

S. W. Hawking, J. Math. Phys. 9 , 598 (1968)

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[6]

P. C. Vaidya and Y. P. Shah, Current Sci. (India) 36 , 120 (1967). P. C. Vaidya, Pramana 8 , 512 (1977). L. K. Patel and S. S. Koppar, Acta Phys. Hung. 64 , 353 (1988)

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[7]

A. Krasinski, Inhomogeneous Cosmological Models (Cambridge Universtiy Press, Cambridge, 1997)

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G. F. R. Ellis and M. Bruni, Phys. Rev. D 40 , 1804 (1989)

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A. Einstein and E. G. Straus, Rev. Mod. Phys. 17 , 120 (1945); 18 , 148 (1946)

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J. M. M. Senovilla and R. Vera, Phys. Rev. Lett. 78 , 2284 (1997)

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M. Mars, ``Axially symmetric Einstein--Straus models'', Phys. Rev. D 57 (1998)

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D. Kramer, H. Stephani, M. A. H. MacCallum and E. Herlt, Exact Solutions of Einstein's Field Equations (Cambridge University Press, Cambridge, 1980)

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P. A. Hogan, Astroph. J. 360 , 315 (1990)

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[14]

A. K. Raychaudhuri, Theoretical Cosmology (Clarendon, Oxford, 1979)

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[15]

H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, Proc. Roy. Soc. London A 269 , 21 (1962)

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[16]

S. A. Hayward, Phys. Rev. D 49 , 831 (1994)

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R. Remmert, Theory of Complex Functions Graduate Texts in Mathematics 122 (Springer-Verlag, New York, 1991)

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T. M. Apostol, Mathematical Analysis (Addison-Wesley, Reading, 1974)

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[19]

S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, 1973)

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[20]

G. F. R. Ellis, in General Relativity and Cosmology , proceedings of the XLVII Enrico Fermi Summer School, ed. R. K. Sachs (Academic Press, New York, 1971)

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[21]

E. N. Glass, J. Math. Phys. 12 , 1222 (1971)

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[22]

S. A. Hayward, Phys. Rev. D 53 , 1938 (1996)

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[23]

B. Nolan, J. Math. Phys. 34 , 178 (1993)

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[24]

M. Ferraris, M. Francaviglia and A. Spallicci, Nuovo Cim. B 111 , 1031 (1996)

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R. A. Sussmann, J. Math. Phys. 29 , 1177 (1988)

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Forward citations

Cited by 7 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Limits on the Inferred Hubble Constant Bias from a Local McVittie Gravitational Field
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Using the McVittie spacetime, the authors derive that local gravitational fields bias the Hubble constant by less than 10^-10%, rulingout this class of explications for the Hubble tension.
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A novel exact solution describes a dynamical black hole dressed with a time-dependent scalar field and immersed in an axisymmetric time-dependent electromagnetic field, where time dependence may cloak curvature singularities.
Rainbow McVittie Horizons in an Expanding Universe
gr-qc 2026-08 conditional novelty 5.0 of 10

A rainbow-deformed McVittie spacetime with one energy history reduces exactly to standard McVittie dynamics, and a linear late-time rainbow closure fitted to 32 cosmic chronometers and 1580 Pantheon+ supernovae is con...
Apparent horizon thermodynamics in an exponential $f(Q)$ gravity model
gr-qc 2026-08 conditional novelty 5.0 of 10

In an exponential f(Q) model, apparent-horizon entropy gains exponentially suppressed corrections to the area law, and the generalized second law excludes parameter values b>0.26 in the future-redshift region.
Dynamical black holes in the inflationary epoch
gr-qc 2026-03 unverdicted novelty 5.0 of 10

Only black holes with initial masses in a narrow range formed during inflation survive to the present day, reaching a maximum mass of approximately 1.043 times 10 to the minus 3 solar masses.

Show 2 more citing papers

Rotating black hole solutions with quintessential energy
gr-qc 2015-12 conditional novelty 5.0 of 10

Applying the Newman-Janis algorithm to Kiselev's spherical quintessence black hole yields a rotating family that exists for c at most 1/6, with a spin-independent static radius at sqrt(2/c).
Black Hole Persistence in Scalar Tensor Theories
gr-qc 2026-07 unverdicted novelty 4.0 of 10

A perturbative scalar-tensor construction yields an evolving horizon of size proportional to the perturbation parameter that persists through a radiation-dominated bounce but evolves asymmetrically.

How the literature uses it

34 arXiv papers cite this in 46 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 · 20 sentences without a confident label

Contests it 1 paper

The well-known McVittie solution [ref] has several shortcomings: it describes a non-accreting black hole and, beyond the special case of Schwarzschild-de Sitter, it features a spatially homogeneous energy density while the pressure is inhomogeneous [ref] .

Evolving black hole with scalar field accretion · Introduction · contest 0.84

Cites it as background 22 papers

A central mass embedded in an expanding universe is described naturally by the McVittie generalization of the FLRW spacetime [ref] .

Rainbow McVittie Horizons in an Expanding Universe · Introduction · 0.91
The metric embeds a central Schwarzschild mass in a homogeneous FLRW background without radial accretion [ref] .

Optimal Extension Regularity at the McVittie Event Horizon · Introduction · 0.93
The McVittie geometry [ref] describes a black hole in an expanding universe.[See Nolan's articles [ref] on properties of the McVittie solution and a criticism in Ref. [ref] .

Shadows of black holes at cosmological distances in the co-varying physical couplings framework · McVittie black hole in the CPC framework · 0.94

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

W. Israel, Nuovo Cimento 44 , 1 (1966)

View on Pith 1966

[2] [2]

H. Sato, in General Relativity and Gravitation ed. B. Bertotti, F. de Felice and A. Pascolini (Reidel, Boston, 1984)

View on Pith 1984

[3] [3]

A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, Cambridge, 1994)

View on Pith 1994

[4] [4]

G. C. McVittie, Mon. Not. Roy. Astron. Soc. 93 , 325 (1933)

View on Pith 1933

[5] [5]

S. W. Hawking, J. Math. Phys. 9 , 598 (1968)

View on Pith 1968

[6] [6]

P. C. Vaidya and Y. P. Shah, Current Sci. (India) 36 , 120 (1967). P. C. Vaidya, Pramana 8 , 512 (1977). L. K. Patel and S. S. Koppar, Acta Phys. Hung. 64 , 353 (1988)

View on Pith 1967

[7] [7]

A. Krasinski, Inhomogeneous Cosmological Models (Cambridge Universtiy Press, Cambridge, 1997)

View on Pith 1997

[8] [8]

G. F. R. Ellis and M. Bruni, Phys. Rev. D 40 , 1804 (1989)

View on Pith 1989

[9] [9]

A. Einstein and E. G. Straus, Rev. Mod. Phys. 17 , 120 (1945); 18 , 148 (1946)

View on Pith 1945

[10] [10]

J. M. M. Senovilla and R. Vera, Phys. Rev. Lett. 78 , 2284 (1997)

View on Pith 1997

[11] [11]

M. Mars, ``Axially symmetric Einstein--Straus models'', Phys. Rev. D 57 (1998)

View on Pith 1998

[12] [12]

D. Kramer, H. Stephani, M. A. H. MacCallum and E. Herlt, Exact Solutions of Einstein's Field Equations (Cambridge University Press, Cambridge, 1980)

View on Pith 1980

[13] [13]

P. A. Hogan, Astroph. J. 360 , 315 (1990)

View on Pith 1990

[14] [14]

A. K. Raychaudhuri, Theoretical Cosmology (Clarendon, Oxford, 1979)

View on Pith 1979

[15] [15]

H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, Proc. Roy. Soc. London A 269 , 21 (1962)

View on Pith 1962

[16] [16]

S. A. Hayward, Phys. Rev. D 49 , 831 (1994)

View on Pith 1994

[17] [17]

R. Remmert, Theory of Complex Functions Graduate Texts in Mathematics 122 (Springer-Verlag, New York, 1991)

View on Pith 1991

[18] [18]

T. M. Apostol, Mathematical Analysis (Addison-Wesley, Reading, 1974)

View on Pith 1974

[19] [19]

S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, 1973)

View on Pith 1973

[20] [20]

G. F. R. Ellis, in General Relativity and Cosmology , proceedings of the XLVII Enrico Fermi Summer School, ed. R. K. Sachs (Academic Press, New York, 1971)

View on Pith 1971

[21] [21]

E. N. Glass, J. Math. Phys. 12 , 1222 (1971)

View on Pith 1971

[22] [22]

S. A. Hayward, Phys. Rev. D 53 , 1938 (1996)

View on Pith 1996

[23] [23]

B. Nolan, J. Math. Phys. 34 , 178 (1993)

View on Pith 1993

[24] [24]

M. Ferraris, M. Francaviglia and A. Spallicci, Nuovo Cim. B 111 , 1031 (1996)

View on Pith 1996

[25] [25]

R. A. Sussmann, J. Math. Phys. 29 , 1177 (1988)

View on Pith 1988