Cylindrical gravitational impulse wave

Jean Jules Defo; Victor Kamgang Kuetche

REVIEW 3 major objections 4 minor 36 references

Cylindrical gravitational impulse wave

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper constructs cylindrical gravitational impulse waves as two-soliton solutions in Jordan-Ehlers spacetime and shows they split into explosion and implosion signals resembling observed gravitational-wave detections.

desk verdict The paper's central nonzero explosion/implosion amplitudes contradict its own Eq. (2) once the imposed ∇²ψ=0 constraint is applied; the LIGO comparison is a tuned simulation. read the letter →

arxiv 2506.10053 v1 pith:5FZQ47HQ submitted 2025-06-11 gr-qc

Jean Jules Defo , Victor Kamgang Kuetche This is my paper · ORCID

classification gr-qc MSC 83C3583C1537K40 PACS 04.30.-w04.20.Jb

keywords cylindricalgravitationalwavesJordan-Ehlersspacetimetwo-solitonsolutionsinversescatteringmethodexplosionandimplosionwaveenergydensityEinstein-RosenChandrasekhartranscendental

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

The paper aims to show that cylindrical gravitational impulse waves in the Jordan-Ehlers spacetime can be built from two-soliton solutions of Einstein's field equations, and that such a wave naturally splits into an exploding and an imploding component. It takes a two-soliton solution constructed by the inverse scattering method, imposes the additional condition $\nabla^2\psi=0$, and then decomposes the wave into amplitudes $A$ and $B$ along outgoing and incoming characteristics. The central claim is that this construction recovers the Einstein-Rosen and Chandrasekhar limiting wave forms, gives the expected gravitational and non-gravitational energy densities, and, with added noise, produces detector-like signals of the kind observed by current gravitational-wave interferometers. If correct, the work would connect exact soliton methods in classical relativity to observed black-hole merger waveforms without relying on numerical simulation.

What carries the argument

The load-bearing object is the two-soliton solution of the Einstein equations in the Jordan-Ehlers cylindrical metric, produced by the improved inverse scattering method and imported from an earlier exact construction. The reduction mechanism is the condition $\nabla^2\psi=0$, which focuses the dynamics on the cross-polarization mode and is used to fix the metric coefficients $e^{2\gamma}$ and $\omega$. The analysis then passes to null characteristics $u$ and $v$, where the amplitudes $A$ and $B$ satisfy two coupled first-order ordinary differential equations, and the combination rules $\gamma_{,\rho}=\frac{\rho}{8}(A^2+B^2)$ and $\gamma_{,t}=\frac{\rho}{8}(A^2-B^2)$ assign gravitational and non-gravitational energy densities. This machinery is what converts an abstract two-soliton metric into explosion and implosion waveforms.

What would settle it

Substitute the explicit $\psi$, $\omega$, and $e^{2\gamma}$ obtained after imposing $\nabla^2\psi=0$ back into equations (2)-(5) and evaluate the residuals on the stated ranges of $\rho$ and $t$; any nonzero residual at the claimed parameters such as $k=2$, $\theta=0$, $q=1$ would mean the plotted amplitudes are not a solution of the Einstein field equations.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is a construction: in the Jordan-Ehlers cylindrical spacetime, a two-soliton solution with complex-conjugate poles, after the condition $\nabla^2\psi=0$, yields explosion and implosion amplitudes $A = \frac{e^{2\psi}}{\rho}\,\omega_{,v}$ and $B = \frac{e^{2\psi}}{\rho}\,\omega_{,u}$ that obey $A_{,u} = \frac{A+B}{2\rho}$ and $B_{,v} = -\frac{A+B}{2\rho}$. The corresponding densities $\gamma_{,\rho} = \frac{\rho}{8}(A^2+B^2)$ and $\gamma_{,t} = \frac{\rho}{8}(A^2-B^2)$ reproduce the Einstein-Rosen wave regime for nonzero soliton parameter and reduce to Chandrasekhar's transcendental cylindrical waves, with conserved energy, when the parameter vanishes ($A=B=0$ and $\gamma_{,t\rho}=0$). With a small modulus for the soliton parameter and Gaussian noise added, the plotted profiles of $A$ and $B$ resemble the explosion and implosion signals registered by gravitational-wave detectors.

Load-bearing premise

The whole construction depends on the borrowed two-soliton solution remaining an exact solution of the Einstein equations after the extra condition $\nabla^2\psi=0$ is imposed, and the paper asserts rather than proves that this holds.

Editorial extensions

If this is right

Exact soliton solutions of the Einstein equations can reproduce the two polarization channels of gravitational waves, here with the plus mode suppressed so the cross mode carries the signal.
The exploding ($B$) and imploding ($A$) components have amplitudes that fall off as $\rho$ grows, matching the expected late-time behavior of an axisymmetric rotating collapse.
The energy-density limits unify Einstein-Rosen waves (nonzero soliton parameter) with Chandrasekhar transcendental cylindrical waves of conserved energy (vanishing parameter).
Detector-like waveforms can be generated analytically from $A$ and $B$ with added Gaussian noise, a lower-parameter route than numerical waveform construction.
A two-black-hole collision can be linked to a cylindrical two-soliton system, supporting the idea that observed merger waveforms are captured by exact solutions.

Reading between the lines

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

Beyond the paper: a direct quantitative check would be to compare the reconstructed $A$ and $B$ profiles against a specific observed merger event using an overlap integral; the paper stops at qualitative resemblance.
Beyond the paper: the same reduction should generalize to higher soliton chains, predicting families of multi-pulse explosion-implosion signals whose structure could be probed in numerical relativity.
Beyond the paper: the complex parameter $a$ acts as a tunable handle on the waveform; varying its modulus and phase should controllably shift the balance between explosion and implosion amplitudes, a prediction that does not appear in the paper.
Beyond the paper: because the paper asserts flat spacetime cannot produce such signals, one could test that claim by running the same decomposition on flat-background cylindrical waves; if chirp-like profiles persist, the Jordan-Ehlers curvature would not be the essential ingredient.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the central claim doesn't survive contact with the paper's own field equations. The authors impose ∇²ψ=0 (Section 0.3.1) on the Jordan-Ehlers system and then conclude for a≠0 that both explosion amplitude A and implosion amplitude B are nonzero. But with ∇²ψ=0, Eq. (2) reduces to ω_t² - ω_ρ² = 0, i.e. ω_u ω_v = 0 in null coordinates, which is exactly AB = 0. So the two amplitudes cannot both be nonvanishing. The plots and the a≠0 conclusion require both. This is not a subtle issue; it is an internal contradiction.

What the paper does that is not nothing: it assembles the Tomizawa-Mishima two-soliton solution, applies the Piran et al. decomposition, and shows that in limiting cases it recovers Einstein-Rosen waves and Chandrasekhar's transcendental waves. That reduction is a useful sanity check, and the reference list shows the authors know the relevant exact-solutions literature. The plotted energy densities are consistent with known behavior of cylindrical waves (γ_ρ positive, γ_t alternating). Credit for that.

The soft spots are bigger than the reader's report suggests. First, the imported two-soliton solution is never written down or verified against Eqs. (1)-(5); the paper just says applying ∇²ψ=0 determines ω and e^{2γ}. Second, the LIGO-like signals in Appendix A are generated by hand-picking k=0.02, θ=0, ρ=3001111, adding 1.01 dB Gaussian noise, and asserting visual closeness to GW150914. No actual LIGO data are used, no fitting, no quantitative comparison, no noise model. That is not a prediction; it is parameter tuning. Third, the paper claims the analytical approach is superior to Bayesian parameter estimation, which is unsupported and off.

The stress-test note is correct and decisive. Even if the imported soliton is a valid exact solution, the imposed constraint and Eq. (2) forbid AB≠0. The paper never resolves this.

Who is this for? Someone cataloging exact cylindrical soliton solutions might skim the limits, but they would need to do the verification the authors skipped. As submitted, it should not go to a serious referee; the internal contradiction is grounds for rejection without full review. I'd tell the authors to re-derive the two-soliton solution under ∇²ψ=0, check whether AB=0 destroys the explosion/implosion interpretation, and remove or heavily rework the LIGO comparison.

Referee Report

3 major / 4 minor

Summary. The paper claims to construct a cylindrical gravitational impulse wave in Jordan-Ehlers spacetime by combining Pomeransky's inverse scattering method with Piran et al.'s decomposition into explosion and implosion amplitudes A and B. The authors impose ∇²ψ=0 to isolate the cross polarization, adopt a two-soliton solution from Tomizawa and Mishima [19], plot amplitudes and energy densities, and in Appendix A add Gaussian noise to obtain waveforms asserted to resemble LIGO-Virgo signals. They state that for a=0 the solution reduces to Chandrasekhar/Einstein-Rosen waves and that for a≠0 one has A≠0 and B≠0, with non-vanishing energy densities.

Significance. If the construction were valid, the paper would connect exact cylindrical soliton solutions to the two gravitational-wave polarizations and to LIGO-like signals, while reproducing known Einstein-Rosen and Chandrasekhar limits in a unified framework. The identification of a possible route toward such a connection is a legitimate aim, and the limiting a=0 reduction serves as a useful consistency check. However, the central result is not supported: the key amplitudes are inconsistent with the field equations under the imposed constraint, and the LIGO comparison is anecdotal rather than quantitative. The manuscript also provides no machine-checked verification or reproducible code for the imported two-soliton solution.

major comments (3)

[Section 0.2, Eqs. (2), (6)-(7); Section 0.3.1; Section 0.4] The imposed condition ∇²ψ=0 makes Eq. (2) reduce to (e^{4ψ}/2ρ²)(ω_t²−ω_ρ²)=0. In the null coordinates u=(t−ρ)/2, v=(t+ρ)/2 underlying Eqs. (6)-(7), one has ω_t²−ω_ρ²=ω_uω_v, so Eq. (2) forces ω_uω_v=0. Since A=e^{2ψ}ω_v/ρ and B=e^{2ψ}ω_u/ρ, this implies AB=0 at every point. This directly contradicts the central conclusion in Section 0.4 that for a≠0 both A≠0 and B≠0, and it invalidates the simultaneous explosion and implosion amplitudes used in all subsequent figures.
[Section 0.3.1] The two-soliton solution is imported from ref. [19], but the manuscript never writes the resulting ψ, ω, and γ functions and never verifies that they satisfy the field equations (1)-(5) after imposing ∇²ψ=0. The statement that this condition 'determines the coefficients e^{2γ} and ω' is asserted rather than demonstrated. This omission is load-bearing because the compatibility of the imported solution with the imposed constraint is precisely the point at issue.
[Appendix A, Figs. A1-A2] The claim that the profiles in Figs. A1 and A2 are 'close to the signals observed by LIGO-Hanford' and 'LIGO-Livingston' is not supported by any comparison with published LIGO data, any waveform-matching statistic, or any detector-response model. The parameter choices (k=0.02, θ=0, q=1, ρ=3001111) and the 1.01 dB Gaussian noise level are selected ad hoc, and visual resemblance after adding noise does not constitute a quantitative prediction. The concluding assertion about a 'possibility of detection' is therefore much weaker than the abstract implies.

minor comments (4)

[Throughout] The text uses 'Pomeansky' for 'Pomeransky' and mixes 'LIGO-VIRGO', 'LIGO-Virgo', and 'LIGO-VIGO'; the collaboration name should be 'LIGO-Virgo' consistently.
[Fig. 1 caption] The phrase 'explosives and implosive waves' should read 'explosion and implosion waves', and the plots would benefit from axis labels and units.
[References] Reference [6] contains a typographical duplication ('Phys. Rev. 110.110, 291') and should be corrected; 'Piran and al.' should be 'Piran et al.' throughout.
[Section 0.4] The claimed reduction to Einstein-Rosen and Chandrasekhar waves for a=0 is stated without showing the limiting procedure from the two-soliton solution; a brief derivation would clarify the connection.

Circularity Check

2 steps flagged · score 6.0 of 10

The LIGO-like waveforms are constructed by hand-tuning free parameters and adding Gaussian noise, then asserted to match LIGO; the central 'prediction' reduces to its design input.

fitted input called prediction [Section 0.2 (near Eqs. (6)-(7)) and Appendix A.1, Figs. A1-A2]
"For figures A1 and A2, we have used the observables A and B to construct a signal that is very close to that of the gravitational wave as observed by the scientific team LIGO-VIRGO [21]. ... Profile B presenting an explosion signal wave with the introduction of a Gaussian noise of the order of 1.01dB. We use the following parameters: for ( k, θ, q) = (0.02, 0, 1) with ( ρ = 3001111). This feature is actually close to the signals observed by the LIGO-Hanford [21]."

The claimed closeness to LIGO is not an independent prediction of the field equations. A and B are defined in Eqs. (6)-(7) as ω,u and ω,v, and the plotted profiles are produced by selecting the free parameters (k=0.02, θ=0, q=1, ρ=3001111) and adding 1.01 dB Gaussian noise so that the curves visually resemble published LIGO chirps. No LIGO data are used as a constraint, no overlap statistic or fitting procedure is reported, and the noise level and parameters are chosen by hand. The waveform's resemblance to LIGO is therefore an input to the plotting procedure, not a derived output of the theory.
self citation load bearing [Appendix A.1, final paragraph]
"In order to generate the previous characteristics, we introduced the Gaussian noise expressions into the field equations according to the usual procedure [31]."

Reference [31] is the authors' own prior paper. The LIGO-like figures (A1 and A2) are generated by this noise-injection step, and the paper provides no external derivation or independent check for the procedure. Since the visual 'closeness' to LIGO data is the claimed outcome, the supporting citation is self-referential. This is minor on its own, however, because Gaussian noise injection is a generic technique; the main circularity is the parameter and noise tuning that makes the curves resemble LIGO chirps.

full rationale

The core mathematical input—the Jordan-Ehlers metric (Eq. (1)), the field equations (2)-(5), the Piran amplitude definitions (6)-(7), and the imported Tomizawa-Mishima two-soliton solution—is not circular by itself; importing a known exact solution and rewriting it in new amplitudes is standard practice. The 'Einstein-Rosen' and 'Chandrasekhar' limits are consequences of the imported solution and reported parameter limits, not independent derivations, but that is a completeness issue rather than circularity. The central circularity is in the LIGO-Virgo comparison: the free parameters and noise level are hand-chosen so that plots of A and B look like published LIGO chirps, and the paper then asserts that the signals are 'actually close' to LIGO-Hanford and LIGO-Livingston without fitting to or quantitatively comparing with the observed data. The 'prediction' of LIGO-like waveforms is therefore manufactured by construction. A separate concern—whether the imposed condition ∇²ψ=0 is compatible with Eq. (2) and with the claimed simultaneous nonvanishing of A and B—is a potential internal inconsistency and a correctness risk, but it is not in itself a circularity, so it is not scored here.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The construction rests on a borrowed soliton solution, a simplifying constraint (∇²ψ=0), a set of hand-chosen parameters (k, θ, q, ρ, noise level), and interpretive identifications of A, B and the energy densities. No new entity or conserved quantity is introduced.

free parameters (5)

k = |a| = 2 (Figs. 1-3); 0.02 (Figs. A1-A2)
Modulus of the complex soliton parameter a; chosen by hand to control waveform amplitude and to simplify expressions.
θ = Arg(a) = 0 (n = 0)
Argument of the complex soliton parameter; the authors select n=0 because higher θ splits the wave amplitudes.
q = 1
Positive constant from the two-soliton solution; the authors normalize it by coordinate scaling.
Radial coordinate ρ at plot points = 0, 1, 2, 3 (Figs. 1-3); 3001111 (Figs. A1-A2)
Waveforms and energy densities are evaluated at hand-selected ρ values; the LIGO-like panels use ρ=3001111.
Gaussian noise level = 1.01 dB
White Gaussian noise added to the analytical signal to make the curves resemble LIGO-Virgo detector outputs.

assumptions (5)

domain assumption The Jordan-Ehlers metric and the field equations (1)-(5) describe the gravitational field in this cylindrical-symmetric context.
The paper assumes this metric and equation set as its starting point without derivation.
domain assumption The two-soliton solution of Tomizawa and Mishima [19] is a valid exact solution applicable to the Jordan-Ehlers metric with the condition ∇²ψ=0.
The paper imports the soliton solution without rederiving it; the central construction depends on this.
ad hoc to paper The condition ∇²ψ=0 can be imposed to isolate the × polarization without loss of the relevant physics.
This constraint is introduced to simplify the equations and is not justified beyond citing ref [22]; it excludes the + mode from the analysis.
domain assumption A = (e^{2ψ}/ρ) ω,v and B = (e^{2ψ}/ρ) ω,u represent the amplitudes of implosion and explosion waves.
Adopted from Piran et al. and Weber-Wheeler; the paper assumes this physical interpretation of the mathematical amplitudes.
ad hoc to paper Adding 1.01 dB Gaussian noise to the analytical signal reproduces conditions needed for a LIGO-like detection.
This is a modeling choice made for the figures; no LIGO noise model is used and no comparison to real detector data is performed.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Cylindrical gravitational impulse wave." pith.science (2026). https://pith.science/paper/5FZQ47HQ

@misc{pith2026250610053,
  author       = {Pith},
  title        = {Pith review of: Cylindrical gravitational impulse wave},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5FZQ47HQ}},
  note         = {Machine review of arXiv:2506.10053}
}

read the original abstract

The description of gravitational waves as explosion and implosion waves as predicted by Weber and Wheeler [{\it Rev. Mod. Phys. {\bf 29} 509 (1957)}] in Einstein and Rosen spacetime, has recently been confirmed following observations by the LIGO-VIRGO scientific team [{\it Phys. Rev. Lett. {\bf 116 } 061102 (2016)}] resulting from the collision of two massive black holes. In this dynamics, we explore a new possibility in the construction of gravitational waves like explosion and implosion waves, the special case of Jordan and Ehlers spacetime, by studying the exact solutions of the Einstein field equations. For this purpose, we use the inverse scattering method of Pomeransky in association with the method of Piran et al. [{\it Phys. Rev. D {\bf 32} 3101 (1985)}] by solving the Einstein field equations in combination with the specific metric derived from Jordan and Ehlers in order to obtain a two-soliton solution with complex conjugate poles that we assimilate to the gravitational wave. Consequently, under certain conditions, we obtain the Einstein and Rosen waves and the Chandrasekhar transcendental waves.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 34 canonical work pages

[19]

Tomizawa and T

S. Tomizawa and T. Mishima, Phys. Rev. D91 124058 (2015). 16

work page 2015
[1]

Stephani, D

H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, Exact solutions of Einstein ’s Field Equations, 2nd edition (Cambridge University Press, Cambridge, England, 2003)

work page 2003
[2]

Kennefick, Phys

D. Kennefick, Phys. Today.9 43 (2005)

work page 2005
[3]

Einstein and N

A. Einstein and N. Rosen, J. Franklin Inst.223 43 (1937)

work page 1937
[4]

Weber and J

J. Weber and J. A. Wheeler, Rev. Mod. Phys.29 509 (1957)

work page 1957
[5]

Rosen, Bull.Res

N. Rosen, Bull.Res. Counc. Isr.4 328 (1954)

work page 1954
[6]

Rosen, Phys

N. Rosen, Phys. Rev. 110.110, 291 (1958)

work page 1958
[7]

Piran, P

T. Piran, P. Sarfier and R. Stark, Phys. Rev. D32 3101 (1985)

work page 1985

Show all 36 references

[8]

Halilsoy, Il Nuovo Cimento102 6 (1988)

M. Halilsoy, Il Nuovo Cimento102 6 (1988)

1988
[9]

Alekseev, Phys

A. Alekseev, Phys. Rev. D93 061501(R) (2016)

2016
[10]

Belinskii and V

V. Belinskii and V. Zakharov, Sov. Phys. JETP48 985 (1978)

1978
[11]

Jordan, J

P. Jordan, J. Ehlers, and W. Kundt, Abh. Akad. Wiss. Mainz. Math. Naturwiss; A.S. Kompaneets, Zh. Eksp. Teor. Fiz.34 953 (1958); Sov. Phys. JETP7 659 (1958)

1958
[12]

Tomimatsu, Gen

A. Tomimatsu, Gen. Relativ. Grav.21 613 (1989)

1989
[13]

Garriga and E

J. Garriga and E. Verdaguer, Phys. Rev. D36 8 (1987)

1987
[14]

A. D. Dagotto, R. J. Glleiser, and C. O. Nicasio, Classical Quantum Gravity.8 1185 (1991)

1991
[15]

V. A. Belinski and E. Verdaguer, Gravitational soliton (Cambridge University Press, Cambridge, 2001)

2001
[16]

V. A. Belinskii, Sov. Phys. JETP50 4 (1979)

1979
[17]

V. A. Belinsky and D. Fargion, IL NUOVO CIMENTO59 1 (1980)

1980
[18]

A. A. Pomeransky, Phys. Rev. D73 044004 (2006)

2006
[20]

Tomizawa and T

S. Tomizawa and T. Mishima, Phys. Rev. D90 044036 (2014)

2014
[21]

B. P. Abbott and al., LIGO-Virgo scientific collaboration, Phys. Rev. Lett.116 061102 (2016)

2016
[22]

Mischima and S

T. Mischima and S. Tomizawa, Phys. Rev.D 96 024023 (2017)

2017
[23]

Belinskii and V.E

V.A. Belinskii and V.E. Sakharov, Sov. Phys. JETP50 (1979) 985

1979
[24]

Xanthopoulos, Phys

B. Xanthopoulos, Phys. Rev. D.34, 12 (1986)

1986
[25]

F. R. Villatoro, Nonlinear Systems, Vol. 1, Understanding Complex Systems(V. Carmona et al. (eds.), (2018)

2018
[26]

B. P. Abbott and al., LIGO-Virgo scientific collaboration, Classical Quantum Gravity. 37 055002 (2020)

2020
[27]

Stark, Rend

R. Stark, Rend. Sem. Mat. Univ. Poi. Torino.51, 3 (1993)

1993
[28]

Throne, Phys

K. Throne, Phys. Rev. 138, B251 (1965)

1965
[29]

W. B. Bonor, Nature 181, 1196 (1958)

1958
[30]

Bondi, Nature 179, 1702 (1957)

H. Bondi, Nature 179, 1702 (1957)

1957
[31]

Defo and K

J. Defo and K. Kuetche, JETP. 135 3 (2022)

2022
[32]

Chandrasekhar, Proc

S. Chandrasekhar, Proc. R. Soc. London, Ser. A,408 , 209 (1986)

1986
[33]

Stachel, J

J. Stachel, J. Math. Phys.(N.Y.)7 , 1321 (1966)

1966
[34]

Thorne, arxiv.org/abs/gr-qc/9506086v1

K. Thorne, arxiv.org/abs/gr-qc/9506086v1

arXiv
[35]

Parikh, F

M. Parikh, F. Wilczek, and G. Zahariade, arXiv:2005.07211

2005 arXiv
[36]

Abramovici et

A. Abramovici et. al. Science, 256 , 325 (1992). 17

1992

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [19]

Tomizawa and T

S. Tomizawa and T. Mishima, Phys. Rev. D91 124058 (2015). 16

work page 2015

[2] [1]

Stephani, D

H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, Exact solutions of Einstein ’s Field Equations, 2nd edition (Cambridge University Press, Cambridge, England, 2003)

work page 2003

[3] [2]

Kennefick, Phys

D. Kennefick, Phys. Today.9 43 (2005)

work page 2005

[4] [3]

Einstein and N

A. Einstein and N. Rosen, J. Franklin Inst.223 43 (1937)

work page 1937

[5] [4]

Weber and J

J. Weber and J. A. Wheeler, Rev. Mod. Phys.29 509 (1957)

work page 1957

[6] [5]

Rosen, Bull.Res

N. Rosen, Bull.Res. Counc. Isr.4 328 (1954)

work page 1954

[7] [6]

Rosen, Phys

N. Rosen, Phys. Rev. 110.110, 291 (1958)

work page 1958

[8] [7]

Piran, P

T. Piran, P. Sarfier and R. Stark, Phys. Rev. D32 3101 (1985)

work page 1985