Overcharging a Black Hole and Cosmic Censorship

Veronika E. Hubeny

REVIEW

Overcharging a Black Hole and Cosmic Censorship

Veronika E. Hubeny

General Relativity and Quantum Cosmology · Submitted 17 Aug 1998 · 17 references

Pith's verdict: A real new observation (near-extremal, not extremal, starting point) plus a clean shell no-go, wrapped in an abstract that promises more than the body delivers — and the body knows it.

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arXiv IDgr-qc/9808043 (version 1)

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CitationVeronika E. Hubeny. "Overcharging a Black Hole and Cosmic Censorship." arXiv:gr-qc/9808043 (1998). https://pith.science/paper/NDHA5B2F

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Editor's note

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The thing worth your time here isn't the title. It's the observation that starting near-extremal rather than exactly extremal lets a finite jump carry the hole past M² = Q², which is precisely the gap in the earlier no-go arguments (Wald 1974, Cohen–Gautreau, Needham, Semiz) that all worked at extremality or in strict test processes. That is a genuine new idea, and the ε-family in eqs. (20)–(24) is clean, explicit, and checkable by hand. The shell section is the other keeper: eq. (40) contradicts the overcharging inequalities outright, so an imploding shell cannot reach the singularity with Q+q > M+E. That's a crisp algebraic no-go, not a numerical claim, and it's the most solid result in the paper. The citation pattern is fine; Quinn–Wald is used within its stated hypotheses, and the self-force expansion is quoted, not rederived, which is appropriate. One correction to the record: the classifier says no stated limitations. That's wrong. The paper is unusually candid — section 3.2 and section 6 both say plainly that the capture margin scales as ε² while the backreaction scales as ε, so small corrections do not imply an unchanged outcome. The stress-test concern lands, and it lands on the abstract, not the algebra. Since h ~ ε and |ṙ| ~ ε, the accumulated ΔE/E is O(1) at every ε; the ε→0 limit is simultaneously the test-particle limit and the limit where backreaction decides the outcome. The paper's own numbers show it: E = 2.99e-3 against a required E∞ ≈ 0.0122. So [abstract:s0]'s 'as long as backreaction may be considered negligible' is literally true and jointly unsatisfiable within the construction. That's a framing problem, and a real one, but it isn't a mathematical error. Two smaller gaps: the shell bounce-below-horizon case rests on 'numerical calculations indicate' with nothing shown, and section 3.3 is a single parameter point with no scan, so we can't tell whether the bounce is generic. Both are minor next to the framing issue. This is a paper for anyone working on cosmic censorship or on the test-particle-versus-backreaction boundary; it's a good teaching example of a correction that is small and still decisive. Send it out.

Referee report

Significance

The paper makes a genuine and, at the time, previously overlooked observation: the extremal no-go of Wald, Cohen-Gautreau, Semiz and Needham does not automatically extend to near-extremal holes, because a finite jump can overshoot extremality where an infinitesimal one cannot. The ε-family of eqs. (20)–(24) is a clean, explicit parameterization of that observation, and the shell no-go of §4 (eq. 40) is a crisp algebraic result. The manuscript is also unusually honest about its own limitations: §3.2, §3.3 and §6 state plainly that the backreaction obstruction is not resolved and that the corrected trajectory in the worked example bounces. The derivation is self-contained, the numerical example is falsifiable, and the negative shell result is a real contribution. The main weakness is that the abstract's framing does not fully convey the body's finding that the condition 'backreaction negligible' cannot be met in the sense the outcome requires

Major objections (4)

[Abstract [abstract:s0, abstract:s1] vs. §3.2–3.3, eqs. (26)–(28), (33)] The ε-family of eqs. (20)–(24) makes the self-force terms small relative to the leading acceleration, but the quantity that decides capture is the accumulated energy loss ΔE = q∫ h f vdot/|rdot| dr of eq. (33). Since h ~ O(ε) (eq. 26) and |rdot| ~ O(ε) near the minimum (eq. 28 gives rdot²(rm) ~ O(ε²)), the integrand q h f vdot/|rdot| is ε-independent at fixed r, and the integral is cut off only at r − r+ ~ ε, giving ΔE ~ ε, i.e. ΔE/E = O(1) for every ε. The paper's own numbers confirm this: E = 2.99×10⁻³ while the 'best case' starting energy required is E∞ ≈ 0.0122 in §3.3, a factor of roughly four, not a small correction. So the ε→0 limit is simultaneously the test-particle limit and the limit in which backreaction is decisive; there is no member of this family in which the corrected trajectory converges to the test trajectory. The body acknowledges the tension in words ('the difference between the equations of motion leading to vastly different outcomes is also tiny'), but the abstract's juxtaposition of s0 and s1 reads as a positive result. I recommend revising the abstract to state explicitly that the condition in s0 cannot be met in the sense required for the outcome, and that the test-limit overcharging does not survive the first-order correction in the worked example
[§3.3, eq. (34) and the 'best case' rescue] The claim that E∞ ≈ 0.0122 would allow the particle to fall past the horizon rests on the same linearized E(r) integral that the paper itself says breaks down near the horizon, where q h f vdot/|rdot| ~ O(ε²/(r−1)²) reaches O(1). The authors note this and note that the minimum of the corrected rdot² is ≈ 6×10⁻⁹ < ε³. As it stands, the rescue is not under control: the dominant contribution to the integral arises precisely where the approximation is invalid. Either a better-controlled calculation (e.g. a resummation, or a direct numerical integration of the full self-force equation of motion) is needed, or the claim should be softened to 'the linearized estimate suggests, but does not establish, that a larger initial energy would allow capture.'
[§4, eqs. (41)–(42)] The second shell case (bounce below the horizon) is argued only by 'numerical calculations indicate that this is indeed the case,' with no scan shown. Since the abstract [abstract:s3] states that cosmic censorship is upheld for the shell, and half of that conclusion (the bounce case) rests on an unshown numerical scan, the evidence should be presented (e.g. a figure of the allowed region in (m², E, q) space at fixed M, Q) or the claim qualified to the implode-to-singularity case, which is proved analytically in eq. (40). The analytic case is clean and convincing; the numerical case needs to be visible to the reader
[§3.3, single numerical example] The only quantitative check of the corrected equation of motion is the single example with ε ≈ 0.0022, a ≈ 1.34, b ≈ 1.03, c ≈ 0.805. The paper itself notes that 'it is not clear that the situation described by our example is in fact generic in terms of the outcome.' Given that the central conclusion (unresolved) rests on this one example plus the ε-scaling argument, a small scan over (a, b, c) at fixed ε would materially strengthen the claim that the bounce is generic rather than an artifact of the chosen parameters, and would also test whether the 'best case' rescue of eq. (34) succeeds for some members of the family

Minor objections (5)

[Eq. (7)] The second equality in eq. (7) appears to have lost a factor of 1/2: the correct expression is (r+ − Q)/2, which is what the ε-expansion of eq. (22) uses (q > ε + ε²). As typeset it reads r+ − Q/2, which is off by a factor of two near extremality
[Eq. (9) vs. eq. (8)] The radical in eq. (9) should be checked against the substitution of eq. (8) into (E − qQ/r)/sqrt(g(r)); the combination 2MQEq as printed does not obviously reproduce the numerical bound quoted in §3.3, and the reader may wish to verify the algebra
[Figures 1–3] The axis labels in the source are unclear (appearing as 'r/Plus' and 'r/OverDot2'). Please ensure the published figures have legible axis labels and units, and that Figs. 2 and 3 indicate the value of r at which rdot² first becomes negative
[§3.2, eq. (17)] The identification ∇[b(G−)a]c = ∇[bRa]c should be explained; the notation Ra is not defined and the dimensional consistency with the retarded Green's function is not obvious
[§5] The 'rocket' extension is speculative and could be shortened or explicitly labeled as an aside; the N-particle discussion (N = 2, 4, 6) is qualitative and would benefit from at least one quantitative statement of the allowed parameter range

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The paper

The paper claims

A single charged particle can, in the test-particle limit, drive a near-extremal Reissner-Nordström black hole past extremality, but the paper's own self-force estimate makes the particle bounce short of the horizon, leaving the censorship

The reading

The paper asks whether a black hole's horizon can be destroyed by dropping in matter that carries more charge than mass. Wald's earlier gedanken experiment said no for an exactly extremal hole, because the particles that would overshoot are precisely the ones the hole refuses to swallow. Hubeny's move is to start slightly below extremality instead, so that the black hole can jump over the extremal bound in one finite step rather than creeping up to it. Working in the test-particle approximation, she writes down an explicit family of parameters in which a radially infalling charge passes the horizon and leaves $M+E < Q+q$, i.e. a naked singularity. She then checks the neglected self-force terms and finds they can be made small in absolute size, but the capture margin is smaller still, and her one worked example reverses the outcome. The shell version of the same experiment, where the calculation is exact by spherical symmetry, fails outright because the shell's own charge repels it before it reaches the horizon.

What carries the argument

The load-bearing object is the radial equation of motion for a charged test particle in ingoing Eddington coordinates, $dot r^2 = frac{1}{m^2}[E + qA_v(r)]^2 - g(r)$ with $A_v = -Q/r$ and $g(r) = 1 - 2M/r + Q^2/r^2$. Capture requires $dot r^2 > 0$ for all $r ge r_+$; overcharging requires $E < Q+q-M$. Combining the two gives a lower bound on the particle charge, $q > r_+ - Q^2$, which is empty at extremality and nonempty just below it. That inequality is the whole mechanism: it says the extremal no-go is a knife-edge case, not a barrier. The second piece of machinery is the epsilon-scaling ansatz that turns the three conditions into a single small parameter, letting the author display simultaneously the $O(e^2)$ capture margin, the $O(e)$ self-force corrections taken from the Quinn-Wald equation, and the $O(e)$ classical radius. For the shell case the analogous object is the junction equation $sqrt{g_{in}+dot R^2} - sqrt{g_{out}+dot R^2} = -m/R$, whose effective potential contains the shell's own self-repulsion and self-attraction terms.

Extended reading notes

Core claim

The paper's central claim is conditional and stated as such: if backreaction can be neglected, a near-extremal Reissner-Nordström black hole can be overcharged by a single charged particle. The construction is explicit. With $M=1$, $Q=1-2e^2$, and particle parameters $q=ae$, $E=ae-2be^2$, $m=ce$ for $a>1$, $1<b<a$, $c<sqrt(a^2-b^2)$, the test-particle radial equation gives $dot r^2(r_m) > 0$ at leading order, so the particle passes $r_+$, and $(M+E)-(Q+q) = -2(b-1)e^2 < 0$, so the final configuration exceeds extremality. The same scaling makes the particle's classical radius $q^2/m = a^2 e/c$ and the relative size of the Quinn-Wald self-force terms arbitrarily small as $e to 0$. The paper is explicit that this does not settle the question. The capture margin is $O(e^2)$ while the radiation-damping correction is $O(e)$, so the correction is not small compared with the margin; the single numerical example ($Q=0.99999$, $q=3times10^{-3}$, $m=1.8times10^{-3}$, $E=2.9897times10^{-3}$) shows the corrected $dot r^2$ turning negative, i.e. a bounce. A 'best case' rescue in which all radiated energy escapes requires a starting energy of about $0.0122$ and appears to allow capture, but the author notes the linearized energy integral is unreliable near the horizon, where the correction ratio grows like $e^2/(r-1)^2$ and reaches order one. The companion shell calculation is exact and negative: for an imploding charged shell, the condition $dot R^2(r_m)>0$ is algebraically incompatible with overcharging, and the less stringent condition $d dot R^2/dr(r_+)>0$ is incompatible with the physical bound $0<m^2<E^2$, so the shell bounces outside the horizon.

Load-bearing premise

The argument needs the neglected self-force terms to be negligible for the outcome, but the paper's own scaling shows the capture margin shrinks faster than the corrections do, so smallness relative to the leading terms does not deliver smallness relative to what the process has to overcome.

Editorial extensions

If this is right

The extremal no-go for charged-particle capture is not a barrier but a limiting case: any black hole even slightly below extremality admits a finite window of particle charges and energies that overshoot the bound in the test limit.
Whether cosmic censorship survives in this scenario cannot be decided by comparing the size of backreaction terms to the size of the leading terms; the relevant comparison is to the capture margin, and the paper's own scaling shows the margin loses.
For a charged imploding shell, censorship is upheld without approximation, because the shell's electromagnetic self-repulsion prevents it from crossing the horizon whenever the parameters would overcharge the hole.
Any claim that a near-extremal hole can be destroyed must specify the fate of the energy radiated during the infall, since the final mass depends on how much of that radiation is recaptured.

Reading between the lines

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

The paper's structure suggests a general lesson for gedanken experiments: an experiment whose success margin scales as one power of a small parameter higher than its own error terms is undecidable at that order, and the honest verdict is 'not resolved' rather than either 'censorship holds' or 'censorship fails'.
The contrast between the point particle and the shell points to a one-parameter family interpolating between them, a ring of $N$ symmetrically placed particles, where self-repulsion is diluted by a factor of order $N$ while self-force corrections persist; the paper reports exploratory checks for $N=2,4,6$ but does not settle the backreaction question there.
The uncharged rotating analogue, a spinning particle dropped into a near-extremal Kerr hole, removes the electromagnetic self-force that dominates the obstruction here and would be a cleaner place to look for a definite answer.
If the bounce found in the worked example is generic rather than an artifact of the linearized energy integral, then the near-horizon energy loss acts as a self-regulating mechanism that keeps the horizon intact without any appeal to a new principle.

Original abstract

We show that, contrary to a widespread belief, one can overcharge a near extremal Reissner-Nordstrom black hole by throwing in a charged particle, as long as the backreaction effects may be considered negligible. Furthermore, we find that we can make the particle's classical radius, mass, and charge, as well as the relative size of the backreaction terms arbitrarily small, by adjusting the parameters corresponding to the particle appropriately. This suggests that the question of cosmic censorship is still not wholly resolved even in this simple scenario. We contrast this with attempting to overcharge a black hole with a charged imploding shell, where we find that cosmic censorship is upheld. We also briefly comment on a number of possible extensions.

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, 4,267 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.82 · human participants 0.01 · controlled comparison 0.26 · randomized assignment 0.01 · states quantitative result 0.07 · 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.

We show that, contrary to a widespread belief, one can overcharge a near extremal Reissner-Nordstrom black hole by throwing in a charged particle, as long as the backreaction effects may be considered negligible. Furthermore, we find that we can make the particle's classical radius, mass, and charge, as well as the relative size of the backreaction terms arbitrarily small, by adjusting the parameters corresponding to the particle appropriately. This suggests that the question of cosmic censorship is still not wholly resolved even in this simple scenario. We contrast this with attempting to overcharge a black hole with a charged imploding shell, where we find that cosmic censorship is upheld. We also briefly comment on a number of possible extensions.

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	We show that, contrary to a widespread belief, one can overcharge a near extremal Reissner-Nordstrom black hole by throwing in a charged particle, as long as the backreaction effects may be considered negligible.	result 0.99	0.92	0.89	asserted 0.92
s1	Furthermore, we find that we can make the particle's classical radius, mass, and charge, as well as the relative size of the backreaction terms arbitrarily small, by adjusting the parameters corresponding to the particle appropriately.	result 0.99	0.95	0.09	asserted 0.99
s2	This suggests that the question of cosmic censorship is still not wholly resolved even in this simple scenario.	interpretation 1.00	0.31	0.48	hedged possibility 1.00
s3	We contrast this with attempting to overcharge a black hole with a charged imploding shell, where we find that cosmic censorship is upheld.	result 0.98	0.94	0.07	asserted 1.00
s4	We also briefly comment on a number of possible extensions.	future work 0.97	0.53	0.02	unclear 0.91

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

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

Figures

Figures reproduced from arXiv:gr-qc/9808043 by the authors.

References

Works cited (17)

[1]

R. Penrose, “Gravitational Collapse: the Role of Genera l Relativity,” Riv. Nuovo Cim. 1, 252 (1969)

View on Pith 1969
[2]

R. M. Wald, “Gravitational Collapse and Cosmic Censorsh ip,” gr- qc/9710068

View on Pith
[3]

C. J. S. Clarke, “A Review of Cosmic Censorship,” Class. Q uant. Grav. 10, 1375, (1993)

View on Pith 1993
[4]

T. P. Singh, “Gravitational collapse, Black Holes and Na ked Singulari- ties,” gr-qc/9805066 25

View on Pith Pith review arXiv
[5]

R. Wald, “Gedanken Experiments to Destroy a Black Hole,” Ann. Phys. 83, 548, (1974)

View on Pith 1974
[6]

J. M. Cohen & R. Gautreau, “Naked Singularities, Event Ho rizons, and Charged Particles,” Phys. Rev. D 19, 2273, (1979)

View on Pith 1979
[7]

I. Semiz, “Dyon Black Holes Do Not Violate Cosmic Censors hip,” Class. Quant. Grav. 7, 353, (1990)

View on Pith 1990
[8]

J. D. Bekenstein & C. Rosenzweig, “Stability of the Black Hole Horizon and the Landau Ghost,” Phys. Rev. D 50, 7239, (1994), gr-qc/9406024

View on Pith Pith review arXiv 1994
[9]

T. Needham, “Cosmic Censorship and Test Particles,” Phy s. Rev. D 22, 791, (1980)

View on Pith 1980
[10]

T. C. Quinn & R. M. Wald, “An Axiomatic Approach to Electr omag- netic and Gravitational Radiation Reaction of Particles in Curved Space- time,” Phys. Rev. D 56, 3381, (1997), gr-qc/9610053

View on Pith Pith review arXiv 1997
[11]

B. S. DeWitt & R. W. Brehme, “Radiation Damping in a Gravi tational Field,” Ann. Phys. 9, 220, (1960)

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[12]

J. M. Hobbs, “A Vierbein Formalism of Radiation Damping ,” Ann. Phys. 47, 141, (1968)

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[13]

K. Kuchaˇ r, private communication

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[14]

V. De La Cruz & W. Israel, “Gravitational Bounce,” Il. Nu ovo Cim. A 51, 744, (1967)

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[15]

K. Kuchaˇ r, “Charged Shells in General Relativity and T heir Gravita- tional Collapse,” Czech. J. Phys. B 18, 435, (1968)

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[16]

D. G. Boulware, “Naked Singularities, Thin Shells, and the Reissner- Nordstr¨ om Metric,” Phys. Rev. D8, 2363, (1973)

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[17]

B. Leaute & B. Linet, “Electrostatics in a Reissner-Nor dstr¨ om space- time,” Phys. Lett. A 58, 5, (1976) 26 r/Plus 1.002 r/Plus 1.004 r/Plus r 5./Multiply10-6 0.00001 0.000015 r/OverDot2 Fig. 1 Figures.nb 1 1.03 r/Plus 1.05 r/Plus 1.07 r/Plus r 0.002 0.004 0.006 0.008 0.01 r/OverDot2 Fig. 2 Figures.nb 1 0.9999992 r/Plus 0.9999993 r/Plus r 2/Multiply10-8...

View on Pith 1976

Forward citations

Cited by 12 Pith papers

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How later papers used it

76 later papers cite it

Evidence: substantial. 76 of the citing papers share no author with it, and 12 of those carry a confident label from Pith's citation classifier.

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1 later paper by other authors is listed below: 1 built on it.

Builds on it 1 of 2 papers with this label

Generalized the analysis constructed for Reissner-Nordstrom black holes by this paper.

Charged spherically symmetric black holes in the Lyra geometry and a preliminary investigation on the overcharging process (2024) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

61 arXiv papers cite this in 102 sentences

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Papers citing it · 61 sentences without a confident label

Contests it 2 papers

To reconcile these conflicting results, Sorce and Wald (SW) [ref] showed that the linear-order analysis in Hubeny's gedankenexperiment [ref] was insufficient, as the potential violation in the near-extremal expansion must be of the second order [ref] .

Weak Cosmic Censorship Conjecture in Gedanken Experiments at All Orders · Introduction · contest 0.80
This work has put all the Hubeny type of (classical) gadanken experiments [ref] into question.

Quasi-normal Modes of near-extremal black holes in Generalized spherically symmetric spacetime and Strong Cosmic Censorship Conjecture · An exotic near-extremal black hole and Creation of over-extremal spacetime, a naked singularity · contest 0.71

Builds on it 1 paper

The equations of motion for charged timelike shells in asymptotically flat black hole backgrounds were previously studied in [ref] and here we generalize them to the asymptotically AdS case.

Bouncing inside the horizon and scrambling delays · Massive charged shell · 0.99

Uses its method or tool 1 paper

Following [ref] , we define the minimum possible value of $δM$ given by the requirement that particle falls into the star which is given by

Extending the weak cosmic censorship conjecture to the charged Buchdahl star by employing the gedanken experiments · Perturbation inequalities and gedanken experiment · 0.88

Uses its data or values 1 paper

So for the given background spacetime with parameters $M=1$ and $Q=0.99999$ as stated in [ref] , we can then select the appropriate parameters of the charged particle by setting $α'=1.33902$, $β'=1.0301$, and $γ'=0.80505$.

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In contrast to the finding that neither a test charged particle nor a charged scalar field can destroy an extremal Reissner-Nordström black hole [ref] , we demonstrate that both a test particle and a charged scalar field can overcharge an extremal nonsingular ABG black hole at linear order, owing to nonlinear electrodynamic field corrections.

Overcharging a nonsingular black hole in general relativity: the nonlinear electrodynamic field effects · Conclusion and Discussion · 0.95
Notice that the restriction ([ref]) is exactly the same obtained previously for the classical gedanken experiments involving charged test bodies, see [ref] .

Challenging the weak cosmic censorship conjecture with charged quantum particles · Challenging the WCCC · 0.96
Although the back reaction effects are completely neglected, the analysis in Ref. [ref] is in sharp contrast with previous works [ref] .

Cosmic censorship in overcharging a Reissner-Nordstr\"{o}m black hole via charged particle absorption · Test particle case · 0.85

Cites it as background 26 papers

It is well known [ref] that the RN black hole can be overcharged and thus the WCCC is violated for linear accretion process while it would always be restored when non-linear order perturbations are included.

Testing the weak cosmic censorship conjecture for a Reissner-Nordstr\"{o}m-de Sitter black hole surrounded by perfect fluid dark matter · Conclusions · 0.84
We briefly review the basic idea of overcharging a RN black hole by the infall of a charged particle, giving the precise meaning of “overcharging”, based on the discussion in Ref. [ref] .

Cosmic censorship in overcharging a Reissner-Nordstr\"{o}m black hole via charged particle absorption · Overcharging a Reissner-Nordström black hole without back reaction effects · 0.97
Another avenue for violating cosmic censorship was identified by Hubeny [ref] , who noticed that a near-extremal Reissner-Nordström black hole, possessing a charge $Q$ that is almost as large as its mass $M$ (in geometrized units in which $G = c = 1$), can absorb a test particle of such charge $q$, mass $m$, and energy $E_0$ that the final configuration cannot be a black hole: the final charge $Q + q$ exceeds the final mass $M + E_0$, signalling the presumed destruction of the event horizon.

Self-force as a cosmic censor · Introduction · 0.86

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.

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