Quantum Probe to the Higher Dimensional Yang-Mills Singularity

M. Halilsoy; M. Mangut; O. Gurtug

arxiv: 2604.26350 · v1 · submitted 2026-04-29 · 🌀 gr-qc

Quantum Probe to the Higher Dimensional Yang-Mills Singularity

M. Mangut , O. Gurtug , M. Halilsoy This is my paper

Pith reviewed 2026-05-07 12:41 UTC · model grok-4.3

classification 🌀 gr-qc

keywords quantum singularitiesYang-Mills theoryGauss-Bonnet gravityhigher dimensionsnaked singularitiesHorowitz-Marolf criterionEinstein-Yang-Mills solutions

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The pith

Gauss-Bonnet corrections render Yang-Mills singularities quantum regular in five-dimensional spacetime for specific mass parameters.

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

The paper examines whether naked curvature singularities in higher-dimensional Einstein-Yang-Mills and Einstein-Maxwell-Yang-Mills solutions remain singular when probed by quantum scalar fields. It applies the Horowitz-Marolf criterion to test self-adjointness of the wave operator. Pure Yang-Mills cases stay singular, but adding Gauss-Bonnet higher-curvature terms can eliminate the singularity in five dimensions when the mass parameter takes particular values linked to the Yang-Mills charge. In six or more dimensions the central singularity persists even with these corrections.

Core claim

While the Einstein-Yang-Mills and Einstein-Maxwell-Yang-Mills spacetimes in dimensions five and higher are quantum mechanically singular according to the Horowitz-Marolf criterion, the inclusion of Gauss-Bonnet corrections renders the singularity quantum mechanically regular for specific values of the mass parameter m related to the Yang-Mills charge Q when the spacetime dimension is exactly five. For dimensions six and above, the outer secondary singularity may become regular for certain m, yet the central singularity remains singular.

What carries the argument

The Horowitz-Marolf criterion, which checks whether the quantum evolution of a scalar field test particle is uniquely determined by demanding self-adjointness of the spatial part of the wave operator on the spacetime.

If this is right

Classical naked singularities in five-dimensional Yang-Mills gravity can be resolved quantum mechanically by adding Gauss-Bonnet terms for tuned mass values.
The positivity condition on the square-root term in the Gauss-Bonnet action introduces a possible secondary singularity that must also be checked for quantum regularity.
In dimensions six and higher, quantum probes fail to resolve the central singularity despite Gauss-Bonnet corrections.
Yang-Mills charge Q directly sets the allowable mass parameters m that permit quantum regularity in five dimensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These results suggest that higher-curvature corrections might stabilize quantum fields near classical singularities in low-dimensional gravity models.
Similar analyses could apply to other higher-curvature theories or different matter fields to test if regularity emerges only below a critical dimension.
Observational signatures of regularized singularities might appear in gravitational wave signals or early-universe cosmology if such solutions describe real astrophysical objects.

Load-bearing premise

The Horowitz-Marolf criterion using the evolution of a quantum scalar field accurately indicates whether a classical singularity is regular at the quantum level, and the positivity requirement in the Gauss-Bonnet term does not create extra unphysical features.

What would settle it

Explicitly solving the Klein-Gordon equation for a test scalar field in one of the five-dimensional Gauss-Bonnet corrected metrics with a chosen m value and checking whether the operator is essentially self-adjoint on the appropriate Hilbert space.

read the original abstract

We investigate the quantum nature of naked curvature singularities in Einstein-Yang-Mills (EYM) theory using the Horowitz-Marolf (HM) criterion, which assesses quantum singularities via the evolution of quantum scalar fields. Focusing on timelike singularities in spacetime dimension $ D \geq 5 $, we analyze both pure Yang-Mills and Einstein-Maxwell-Yang-Mills (EMYM) solutions. We then incorporate higher curvature corrections through Gauss-Bonnet (GB) terms. From positivity requirement the expression under square root that arises in GB may create a secondary singularity that shall be scrutinized carefully. Our analysis reveals that while EYM and EMYM spacetimes remain quantum mechanically singular, the inclusion of GB corrections can, in general, render the singularity quantum mechanically regular for specific values of the mass parameter $m$, which is related to the YM charge $Q$ in $ D = 5 $ space-time dimensions. Contrary, for space-time dimension $D \geq 6$, although the outer (secondary) singularity may be healed quantum mechanically for certain values of the mass parameter $m$, the central singularity remains quantum mechanically singular.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

GB corrections heal the central singularity under the HM test only in D=5 for selected m, while D>=6 keeps the center singular even after the outer one improves.

read the letter

The main result is that Gauss-Bonnet corrections can make the central singularity quantum regular according to the Horowitz-Marolf criterion in five-dimensional Einstein-Yang-Mills for certain values of the mass parameter m tied to the charge, but the central singularity stays singular in six and higher dimensions even when an outer secondary singularity gets healed for some m. The paper also checks the pure EYM and EMYM cases without GB and finds them singular across the dimensions considered. This dimension-dependent split after adding GB is the concrete new piece. The work applies a standard test to a class of solutions that had not been checked this way before, and it flags the secondary singularity that appears once the square-root expression in the GB term is required to stay positive. The parameter scan over m is straightforward and does not look circular. The calculations follow the usual HM route of writing the wave equation on the fixed background and checking self-adjointness of the radial operator. The soft spot is the handling of the secondary singularity. Its location depends on m, so the radial domain splits into an interval between the center and the outer surface. The deficiency indices that decide regularity at the center need to be shown independent of the boundary condition imposed at the secondary surface; otherwise the conclusion about the central singularity cannot be cleanly separated. The abstract notes that the secondary singularity must be scrutinized, but the strength of the claim rests on whether the full text actually carries out the asymptotic analysis and index calculation near both loci. If those steps are present and correct, the central claim holds up. If they are only sketched, the argument for D=5 regularity is weaker than it first appears. This paper is for people working on quantum singularity resolution in higher-dimensional gravity with curvature corrections. A reader who wants explicit examples of how GB terms interact with the HM criterion in EYM backgrounds will find the dimension scan and m-dependence useful. It has enough technical content and a clear question to deserve referee time, mainly to verify the radial operator details and the secondary-boundary independence.

Referee Report

3 major / 1 minor

Summary. The manuscript examines the quantum regularity of timelike naked curvature singularities in higher-dimensional (D ≥ 5) Einstein-Yang-Mills (EYM) and Einstein-Maxwell-Yang-Mills (EMYM) spacetimes, both without and with Gauss-Bonnet (GB) corrections, via the Horowitz-Marolf criterion applied to the evolution of a quantum scalar field. It concludes that pure EYM and EMYM solutions remain quantum-mechanically singular, while GB corrections can render the singularity regular for specific values of the mass parameter m (related to the Yang-Mills charge Q) in D=5; for D ≥ 6 the secondary (outer) singularity may be healed but the central singularity persists.

Significance. If the central claims are substantiated, the work would contribute to understanding how higher-curvature corrections can influence quantum resolution of classical singularities in modified gravity, extending the Horowitz-Marolf test to GB-corrected EYM backgrounds. The distinction between D=5 regularity and partial healing in higher D offers a concrete, dimension-dependent result that could be tested against other singularity-resolution criteria. However, the current lack of explicit derivations limits immediate impact.

major comments (3)

Abstract and main text: the stated regularity results for D=5 and the partial healing for D≥6 rest on the Horowitz-Marolf self-adjointness test, yet no explicit metric ansatz, radial Klein-Gordon equation, or asymptotic analysis near the singularities is supplied; without these the support for the dimension-dependent conclusions is limited.
GB term and positivity requirement: the square-root expression in the Gauss-Bonnet correction introduces an m-dependent secondary singularity whose location must be shown not to affect the deficiency indices of the radial operator between the central and secondary loci; the manuscript supplies no explicit verification that boundary conditions at the outer surface leave the essential self-adjointness conclusion for the central singularity unchanged.
D ≥ 6 case: the claim that the central singularity remains singular while the outer one is healed requires separate deficiency-index calculations and asymptotic expansions at both loci; the current analysis does not isolate these independent of the choice of boundary condition at the secondary surface.

minor comments (1)

The precise relation between the mass parameter m and the Yang-Mills charge Q should be stated explicitly in the introduction or methods section for reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which highlight important points for clarification and strengthening. We address each major comment below and will incorporate the suggested additions in the revised version.

read point-by-point responses

Referee: Abstract and main text: the stated regularity results for D=5 and the partial healing for D≥6 rest on the Horowitz-Marolf self-adjointness test, yet no explicit metric ansatz, radial Klein-Gordon equation, or asymptotic analysis near the singularities is supplied; without these the support for the dimension-dependent conclusions is limited.

Authors: We agree that explicit derivations are essential to substantiate the claims. In the revised manuscript we will include the full metric ansatz for the EYM, EMYM, and GB-corrected solutions, the explicit radial form of the Klein-Gordon equation, and the detailed asymptotic expansions near both the central and secondary singularities that are used to evaluate the deficiency indices of the radial operator. These additions will directly support the dimension-dependent conclusions. revision: yes
Referee: GB term and positivity requirement: the square-root expression in the Gauss-Bonnet correction introduces an m-dependent secondary singularity whose location must be shown not to affect the deficiency indices of the radial operator between the central and secondary loci; the manuscript supplies no explicit verification that boundary conditions at the outer surface leave the essential self-adjointness conclusion for the central singularity unchanged.

Authors: We thank the referee for this observation. The positivity requirement on the GB term does produce an m-dependent secondary singularity. In the revision we will supply an explicit verification that the deficiency indices of the radial operator on the interval between the central and secondary singularities are unaffected by the choice of boundary conditions at the outer surface. This will be shown by demonstrating that the essential self-adjointness properties at the central singularity remain unchanged when the outer boundary condition is varied within the allowed range. revision: yes
Referee: D ≥ 6 case: the claim that the central singularity remains singular while the outer one is healed requires separate deficiency-index calculations and asymptotic expansions at both loci; the current analysis does not isolate these independent of the choice of boundary condition at the secondary surface.

Authors: We agree that the D ≥ 6 analysis must be separated. In the revised manuscript we will present independent deficiency-index calculations together with the corresponding asymptotic expansions at the central singularity and at the outer singularity. These calculations will be performed on the appropriate intervals and will explicitly demonstrate that the quantum singularity status of the central singularity is independent of the boundary conditions imposed at the secondary surface, while confirming that the outer singularity can be healed for certain values of m. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper applies the external Horowitz-Marolf criterion to assess quantum regularity of singularities in GB-corrected EYM and EMYM spacetimes. The mass parameter m (related to YM charge Q) is introduced as a free parameter and specific values are scanned to identify cases where the central or secondary singularity satisfies the HM regularity condition. This parameter scan does not reduce the final claim to a tautology or to a quantity defined by the same fit. The positivity condition on the GB square-root expression is treated as an explicit assumption that introduces a secondary singularity requiring separate scrutiny, without smuggling an ansatz or relying on self-citation for the core result. No load-bearing step reduces by construction to the paper's own inputs or prior self-citations.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim rests on the validity of the Horowitz-Marolf criterion as a quantum-regularity test, on the correctness of previously published EYM and EMYM metrics in D greater than or equal to 5, and on the mass parameter m being freely tunable subject only to the positivity constraint.

free parameters (1)

mass parameter m
Tuned to values that satisfy the positivity requirement under the square root in the Gauss-Bonnet term and that produce the reported regularity outcome; directly tied to the Yang-Mills charge Q.

axioms (2)

domain assumption Horowitz-Marolf criterion correctly diagnoses quantum regularity of a classical singularity
Invoked to decide whether the singularity is healed once a quantum scalar field is evolved through it.
domain assumption The background EYM and EMYM metrics in D greater than or equal to 5 are the correct solutions of the field equations
Taken from prior literature and used without re-derivation.

pith-pipeline@v0.9.0 · 5501 in / 1627 out tokens · 37037 ms · 2026-05-07T12:41:41.442598+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

[2]

(27) The convergence analysis of the listed integrals for two different limit cases in Eq.(27) can be performed by using the comparative test

sin( √ 2r)+2C3C4 cos( √ 2r)) 2 dr , r→ ∞. (27) The convergence analysis of the listed integrals for two different limit cases in Eq.(27) can be performed by using the comparative test. As a requirement of the test, the following inequalities are defined for each limiting case, 11    r5(C1+C2 lnr) 2 r2−2Q2 lnr < r→0 r3 ((C1 +C 2) lnr) 2 , r→0,...

work page
[5]

sin( √ 2r)+2C3C4 cos( √ 2r) 2 , r→ ∞ (28) Since integrals of the form R c 0 ra(lnr) b drconverge for anya >−1 and finiteb, it follows that the comparison functiong(r) =r 3(lnr) 2 is integrable atr= 0. Hence, by the comparison test and noting that the integrand is positive for sufficiently smallr, the original integral is convergent in the neighborhood ofr...

work page
[7]

The square integrability analysis shows that one of the solutions nearr→0 is square integrable, whereas the other solution is not

sin( √ 2r)+2C3C4 cos( √ 2r)) 2 dr→ ∞. The square integrability analysis shows that one of the solutions nearr→0 is square integrable, whereas the other solution is not. According to the HM criterion, the spatial operatorAmust admit a unique self-adjoint extension in order for the spacetime to be considered quantum mechanically regular. This requirement is...

work page
[8]

The explicit solution for this case is given in (4)

Analysis of wave packets whenD= 5: Our focus in this particular case will be the solution obtained forD= 5. The explicit solution for this case is given in (4). The wave that will be used to investigate the singularity is the solution of Eq. (22). The behavior of the metric functions near the singularity (r→0) and in the asymptotic region (r→ ∞) is given ...

work page
[10]

sin( √ 2r)+2C3C4 cos( √ 2r)) 2(r2−2Q2lnr) dr , r→ ∞ (32) The convergence analysis of the listed integrals for two different limit cases in Eq.(32) can be performed by using the comparative test. In doing so, the following inequalities are defined for each limiting case,    3r7(C1+C2r2) 2 3r4+4q2 < r→0 C2 1 3r4+4q2 , r→0, (C2 3+C2 4)cosh( √ 2r...

work page
[12]

sin( √ 2r)+2C3C4 cos( √ 2r) 2(r2−2Q2 lnr) , r→ ∞ (33) R const 0 C2 1 3r4+4q2 dr <∞and R ∞ const (C2 3+C2 4)cosh( √ 2r)−(|C2 3 −C2 4 |+2|C3C4|) 2+4Q2 dr→ ∞, according to the comparison test as in the previous section, the spatial operatorAhas no unique extension and therefore the classical singularity remains quantum mechanically singular

work page
[13]

The same method will be used in this subsection to investigate the quantum singularity structure of the EMYM solution given in Eq.(4)

Analysis of wave packets whenD >5: In the previous case we have analysed the naked singularity in the EMYM theory at D= 5 with quantum wave packets by considering the leading behaviour of the metric near the singularity and at an asymptotic region. The same method will be used in this subsection to investigate the quantum singularity structure of the EMYM...

work page
[14]

cosh( √ 2r)−(|C2 3 −C2 4 |+2|C3C4|) 2(1− (n−2) (n−4) Q2) < r→∞ r2(2C4C3 cos( πn 2 − √ 2r)+C2 3(sin( πn 2 − √ 2r)+cosh( √ 2r))+C2 4(cosh( √ 2r)−sin( πn 2 − √ 2r))) 2(r2− (n−2) (n−4) Q2) , r→ ∞ (38) Since R const 0 rn−1(C1+C2rn−2) 2 (n−1)+2(n−2)q2 dr <∞and R ∞ const (C2 3+C2

work page
[15]

In summary, the naked singularity in EMYM theory, irrespective of the spacetime dimen- sion , it is always singular in quantum mechanical framework

cosh( √ 2r)−(|C2 3 −C2 4 |+2|C3C4|) 2(1− (n−2) (n−4) Q2) dr→ ∞, the comparison test implies that R const 0 r3n−5(C1+C2rn−2) 2 (n−1)r2(n−2)+2(n−2)q2 dr <∞and R ∞ const r2(2C4C3 cos( πn 2 − √ 2r)+C2 3(sin( πn 2 − √ 2r)+cosh( √ 2r))+C2 4(cosh( √ 2r)−sin( πn 2 − √ 2r))) 2(r2− (n−2) (n−4) Q2) dr→ ∞. In summary, the naked singularity in EMYM theory, irrespectiv...

work page
[16]

Analysis of wave packets whenD= 5: In the limitr→ ∞, the generic metric (5) reduces to    f(r)≈1 +δr 2 , f or+sign f(r)≈1, f or−sign (39) whereδ= 1 2α. For this particular limiting case Eq.(22) transforms into    R′′ + 5 r R′ = 0, f or+sign R′′ + 3 r R′ ±iR= 0, f or−sign (40) The solutions of Eq.(40) are given by    R(r) =C 1 + C2 r4 , f or...

work page
[18]

In this context, the following inequalities are established for each case    r(C1+C2r−4) 2 1+δ < r→∞ r3(C1+C2r−4) 2 1+δr2 ,for + sign (C2 3+C2

sin( √ 2r)+2C3C4 cos( √ 2r)) 2 dr , f or−sign (42) The convergence behavior of the integrals corresponding to the two distinct cases in Eq.(42) can be examined using the comparison test. In this context, the following inequalities are established for each case    r(C1+C2r−4) 2 1+δ < r→∞ r3(C1+C2r−4) 2 1+δr2 ,for + sign (C2 3+C2

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[19]

cosh( √ 2r)−(|C2 3 −C2 4 |+2|C3C4|) 2 < r→∞ (C2 3+C2

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[21]

In the second inequality, R ∞ const (C2 3+C2

sin( √ 2r)+2C3C4 cos( √ 2r) 2 ,for−sign (43) In the first inequality, observing that r(C1+C2r−4) 2 1+δ < r→∞ r3(C1+C2r−4) 2 1+δr2 and noting that the integral R ∞ const r(C1+C2r−4) 2 1+δ dris divergent, it follows from the comparison test that the integral R ∞ const r3(C1+C2r−4) 2 1+δr2 dralso diverges. In the second inequality, R ∞ const (C2 3+C2

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cosh( √ 2r)−(|C2 3 −C2 4 |+2|C3C4|) 2 dris divergent, according to the comparison test, the integral R ∞ const (C2 3+C2

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cosh( √ 2r)+(C2 3 −C2

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As a result, in the limitr→ ∞, the spatial operatorAdoes not belong to the Hilbert space

sin( √ 2r)+2C3C4 cos( √ 2r) 2 dralso diverges. As a result, in the limitr→ ∞, the spatial operatorAdoes not belong to the Hilbert space. In the limitr→r ⋆, analyzing Eq.(22) in terms of the variablerdoes not eliminate the nonlinearity. Therefore, we introduce a new coordinate transformationx=r−r ⋆. In the new coordinate, whenr→r ⋆ (x <<1) the metric funct...

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Analysis of wave packets whenD >5: In the limitr→ ∞, Eq.(5) becomes    f(r)≈1 + r2 ¯α , f or+sign f(r)≈1, f or−sign (53) By inserting the asymptotic metric functions (53) into Eq.(22), Eq.(22) reduces to    R′′ + (n+1) r R′ = 0, f or+sign R′′ + n−1 r R′ ±iR= 0, f or−sign (54) The solutions of Eq.(54) can be written as 19    R(r) =C 1 + C2 r...

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cosh( √ 2r)−(|C2 3 −C2 4 |+2|C3C4|) 2 < r→∞ 2C4C3 cos( πn 2 − √ 2r)+C2 3(sin( πn 2 − √ 2r)+cosh( √ 2r))+C2 4(cosh( √ 2r)−sin( πn 2 − √ 2r)) 2 , in accor- dance with the comparison test, indicates that the corresponding integral R ∞ const (C2 3+C2

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[1] [2]

(27) The convergence analysis of the listed integrals for two different limit cases in Eq.(27) can be performed by using the comparative test

sin( √ 2r)+2C3C4 cos( √ 2r)) 2 dr , r→ ∞. (27) The convergence analysis of the listed integrals for two different limit cases in Eq.(27) can be performed by using the comparative test. As a requirement of the test, the following inequalities are defined for each limiting case, 11    r5(C1+C2 lnr) 2 r2−2Q2 lnr < r→0 r3 ((C1 +C 2) lnr) 2 , r→0,...

work page

[2] [5]

sin( √ 2r)+2C3C4 cos( √ 2r) 2 , r→ ∞ (28) Since integrals of the form R c 0 ra(lnr) b drconverge for anya >−1 and finiteb, it follows that the comparison functiong(r) =r 3(lnr) 2 is integrable atr= 0. Hence, by the comparison test and noting that the integrand is positive for sufficiently smallr, the original integral is convergent in the neighborhood ofr...

work page

[3] [7]

The square integrability analysis shows that one of the solutions nearr→0 is square integrable, whereas the other solution is not

sin( √ 2r)+2C3C4 cos( √ 2r)) 2 dr→ ∞. The square integrability analysis shows that one of the solutions nearr→0 is square integrable, whereas the other solution is not. According to the HM criterion, the spatial operatorAmust admit a unique self-adjoint extension in order for the spacetime to be considered quantum mechanically regular. This requirement is...

work page

[4] [8]

The explicit solution for this case is given in (4)

Analysis of wave packets whenD= 5: Our focus in this particular case will be the solution obtained forD= 5. The explicit solution for this case is given in (4). The wave that will be used to investigate the singularity is the solution of Eq. (22). The behavior of the metric functions near the singularity (r→0) and in the asymptotic region (r→ ∞) is given ...

work page

[5] [10]

sin( √ 2r)+2C3C4 cos( √ 2r)) 2(r2−2Q2lnr) dr , r→ ∞ (32) The convergence analysis of the listed integrals for two different limit cases in Eq.(32) can be performed by using the comparative test. In doing so, the following inequalities are defined for each limiting case,    3r7(C1+C2r2) 2 3r4+4q2 < r→0 C2 1 3r4+4q2 , r→0, (C2 3+C2 4)cosh( √ 2r...

work page

[6] [12]

sin( √ 2r)+2C3C4 cos( √ 2r) 2(r2−2Q2 lnr) , r→ ∞ (33) R const 0 C2 1 3r4+4q2 dr <∞and R ∞ const (C2 3+C2 4)cosh( √ 2r)−(|C2 3 −C2 4 |+2|C3C4|) 2+4Q2 dr→ ∞, according to the comparison test as in the previous section, the spatial operatorAhas no unique extension and therefore the classical singularity remains quantum mechanically singular

work page

[7] [13]

The same method will be used in this subsection to investigate the quantum singularity structure of the EMYM solution given in Eq.(4)

Analysis of wave packets whenD >5: In the previous case we have analysed the naked singularity in the EMYM theory at D= 5 with quantum wave packets by considering the leading behaviour of the metric near the singularity and at an asymptotic region. The same method will be used in this subsection to investigate the quantum singularity structure of the EMYM...

work page

[8] [14]

cosh( √ 2r)−(|C2 3 −C2 4 |+2|C3C4|) 2(1− (n−2) (n−4) Q2) < r→∞ r2(2C4C3 cos( πn 2 − √ 2r)+C2 3(sin( πn 2 − √ 2r)+cosh( √ 2r))+C2 4(cosh( √ 2r)−sin( πn 2 − √ 2r))) 2(r2− (n−2) (n−4) Q2) , r→ ∞ (38) Since R const 0 rn−1(C1+C2rn−2) 2 (n−1)+2(n−2)q2 dr <∞and R ∞ const (C2 3+C2

work page

[9] [15]

In summary, the naked singularity in EMYM theory, irrespective of the spacetime dimen- sion , it is always singular in quantum mechanical framework

cosh( √ 2r)−(|C2 3 −C2 4 |+2|C3C4|) 2(1− (n−2) (n−4) Q2) dr→ ∞, the comparison test implies that R const 0 r3n−5(C1+C2rn−2) 2 (n−1)r2(n−2)+2(n−2)q2 dr <∞and R ∞ const r2(2C4C3 cos( πn 2 − √ 2r)+C2 3(sin( πn 2 − √ 2r)+cosh( √ 2r))+C2 4(cosh( √ 2r)−sin( πn 2 − √ 2r))) 2(r2− (n−2) (n−4) Q2) dr→ ∞. In summary, the naked singularity in EMYM theory, irrespectiv...

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Analysis of wave packets whenD= 5: In the limitr→ ∞, the generic metric (5) reduces to    f(r)≈1 +δr 2 , f or+sign f(r)≈1, f or−sign (39) whereδ= 1 2α. For this particular limiting case Eq.(22) transforms into    R′′ + 5 r R′ = 0, f or+sign R′′ + 3 r R′ ±iR= 0, f or−sign (40) The solutions of Eq.(40) are given by    R(r) =C 1 + C2 r4 , f or...

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In this context, the following inequalities are established for each case    r(C1+C2r−4) 2 1+δ < r→∞ r3(C1+C2r−4) 2 1+δr2 ,for + sign (C2 3+C2

sin( √ 2r)+2C3C4 cos( √ 2r)) 2 dr , f or−sign (42) The convergence behavior of the integrals corresponding to the two distinct cases in Eq.(42) can be examined using the comparison test. In this context, the following inequalities are established for each case    r(C1+C2r−4) 2 1+δ < r→∞ r3(C1+C2r−4) 2 1+δr2 ,for + sign (C2 3+C2

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cosh( √ 2r)−(|C2 3 −C2 4 |+2|C3C4|) 2 < r→∞ (C2 3+C2

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In the second inequality, R ∞ const (C2 3+C2

sin( √ 2r)+2C3C4 cos( √ 2r) 2 ,for−sign (43) In the first inequality, observing that r(C1+C2r−4) 2 1+δ < r→∞ r3(C1+C2r−4) 2 1+δr2 and noting that the integral R ∞ const r(C1+C2r−4) 2 1+δ dris divergent, it follows from the comparison test that the integral R ∞ const r3(C1+C2r−4) 2 1+δr2 dralso diverges. In the second inequality, R ∞ const (C2 3+C2

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cosh( √ 2r)−(|C2 3 −C2 4 |+2|C3C4|) 2 dris divergent, according to the comparison test, the integral R ∞ const (C2 3+C2

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cosh( √ 2r)+(C2 3 −C2

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As a result, in the limitr→ ∞, the spatial operatorAdoes not belong to the Hilbert space

sin( √ 2r)+2C3C4 cos( √ 2r) 2 dralso diverges. As a result, in the limitr→ ∞, the spatial operatorAdoes not belong to the Hilbert space. In the limitr→r ⋆, analyzing Eq.(22) in terms of the variablerdoes not eliminate the nonlinearity. Therefore, we introduce a new coordinate transformationx=r−r ⋆. In the new coordinate, whenr→r ⋆ (x <<1) the metric funct...

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Analysis of wave packets whenD >5: In the limitr→ ∞, Eq.(5) becomes    f(r)≈1 + r2 ¯α , f or+sign f(r)≈1, f or−sign (53) By inserting the asymptotic metric functions (53) into Eq.(22), Eq.(22) reduces to    R′′ + (n+1) r R′ = 0, f or+sign R′′ + n−1 r R′ ±iR= 0, f or−sign (54) The solutions of Eq.(54) can be written as 19    R(r) =C 1 + C2 r...

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cosh( √ 2r)−(|C2 3 −C2 4 |+2|C3C4|) 2 < r→∞ 2C4C3 cos( πn 2 − √ 2r)+C2 3(sin( πn 2 − √ 2r)+cosh( √ 2r))+C2 4(cosh( √ 2r)−sin( πn 2 − √ 2r)) 2 , in accor- dance with the comparison test, indicates that the corresponding integral R ∞ const (C2 3+C2

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cosh( √ 2r)−(|C2 3 −C2 4 |+2|C3C4|) 2 dr, yields also a divergent result. Next, we investigate the outermost singularity that may arise from the condition Σ(r) = 0, which we denote by ˜r⋆. In the limitr→˜r ⋆, the generic metric (5) behaves as ˜f∓(˜x)≈ ˜A+ ˜B˜x(57) in which ˜x=r−˜r ⋆. Here, the coefficients are given by 20 ˜A= 1 + ˜r2 ⋆ 2¯α " 1∓ s 1−8m¯α˜r...

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