Formation of Singularity and Apparent Horizon for Dissipative Collapse in $f(R,T)$ Theory of Gravity

Sarbari Guha; Uttaran Ghosh

REVIEW 3 major objections 5 minor 109 references

Formation of Singularity and Apparent Horizon for Dissipative Collapse in $f(R,T)$ Theory of Gravity

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read For a radiating, spherically symmetric star collapsing in f(R,T)=R+2λT gravity, the paper derives the singularity and apparent-horizon times and shows the parameter ranges in which the horizon forms first, so the final state is a black…

desk verdict The paper's central collapse-timeline result is invalid: Eq. (66) does not solve Eq. (64), so the singularity and apparent-horizon formulas are unsupported. read the letter →

arxiv 2502.05217 v1 pith:W6KZZVAT submitted 2025-02-05 gr-qc hep-th

Uttaran Ghosh , Sarbari Guha This is my paper · ORCID

classification gr-qchep-th MSC 83C5783C7583D0583C55 PACS 04.50.Kd04.70.-s04.20.Dw

keywords f(RT)gravitygravitationalcollapseapparenthorizonnakedsingularityblackholeformationheatfluxVaidyaspacetimejunctionconditions

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 considers a spherically symmetric star of isotropic matter that loses energy by heat flux, embedded in a generalized Vaidya exterior, in the modified gravity theory f(R,T)=R+2λT. Using the f(R,T) junction conditions to match the interior and exterior field equations, it derives explicit formulas for the time ts when the collapsing radius reaches zero (the singularity) and the time tah when an apparent horizon forms. Its central claim is that when the integration constants δ, C1 and C2 obey the inequalities in Tables I–III, the apparent horizon forms before the singularity, so the final object is a black hole rather than a naked singularity. A further claim is that the coupling constant λ does not appear in ts or tah, because it cancels from the pressure-isotropy condition and the horizon condition is the same as in general relativity.

What carries the argument

The load-bearing object is the temporal function w(t) in the sectional radius C(r,t)=r B0(r) w(t). The pressure-isotropy condition reduces the field equations to a single second-order differential equation for w, separating radial terms collected in D(r) from time terms. Solving that equation gives w(t) explicitly for D(r)≠0 and for D(r)=0; w=0 marks the singularity, and the null-surface condition Ċ²/A²=C'²/B², written as ẇ²=δ²w², marks the apparent horizon. The step that fixes the radial dependence is the assertion, from Appendix A, that because w² is a function of t alone, each term in the three-term sum (108) is itself a function of t alone, forcing H and d/c to be constants and yielding A0²=C4 r²B0².

What would settle it

Choose interior metric coefficients A0(r) and B0(r) for which H(r) and d(r)/c(r) genuinely vary with r, substitute the paper's solution (66) into the expression for w², and check whether the three-term sum in equation (108) can still be independent of r for suitable integration constants; if it can, the step that forces H and d/c to be constants is false and the derived times do not follow.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that a dissipative collapse (isotropic pressure plus radial heat flux) in f(R,T)=R+2λT can be solved completely for the collapse timeline once the metric coefficients are taken separable, A=A0(r), B=B0(r), C=r B0(r) w(t), and once the ratio d/c = A0²/(r²B0²) is constant. The singularity time then depends only on the constants δ, C1, C2 and not on the radial coordinate, which by the paper's criterion excludes a naked singularity. The apparent-horizon times are obtained from the condition Ċ²/A² = C'²/B², which fixes δ and yields the power-law forms A0 ∝ $r^{{n+1}}$, B0 ∝ r^n. Combining these, equations (95), (98) and (101) give ts and the two possible values of tah − ts, and the parameter tables list which sign choices of the square roots make tah < ts, i.e. black hole formation.

Load-bearing premise

The whole argument stands on the inference that because the sum of three terms equals a quantity that depends only on time, each of the three terms must itself depend only on time; that conclusion forces two functions of the radial coordinate to be constant, and the singularity and horizon times are built on it.

Editorial extensions

If this is right

When δ, C1 and C2 satisfy the tabulated constraints, the apparent horizon forms before the singularity, so an external observer never sees the singularity: the end state is a black hole.
The singularity time ts is independent of the radial coordinate, which the paper takes, following the cited no-naked-singularity criterion, to rule out a naked singularity for this class of collapse.
The f(R,T) coupling λ cancels from the pressure-isotropy equation and the horizon condition, so within this model the collapse timeline is the same as in general relativity for the same metric coefficients.
The junction conditions require the interior and exterior matter Lagrangians, and their derivatives, to match continuously across the boundary, a condition that restricts the admissible choice of matter Lagrangian.
For the D(r)=0 case, the singularity time obeys a different formula ts = (−C5 ± √y)/C4, and the apparent horizon exists only in the D(r)≠0 case.

Reading between the lines

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

If the Appendix A inference is not valid, the constancy of H and d/c — and hence the r-independence of ts on which the black-hole conclusion rests — is not established; a reader should test Eq. (108) with radially varying H(r) and d(r) before relying on the tables.
The same machinery could be applied to anisotropic pressure or shear; the paper notes the differential equation for w would acquire extra terms, so the horizon-vs-singularity ordering in Tables I–III is likely specific to the isotropic, shear-free case.
The cancellation of λ suggests that in the minimal linear f(R,T)=R+2λT model, the effect of the matter-geometry coupling may show up mainly in the boundary matching of matter Lagrangians rather than in the collapse timeline — an observable difference one could look for in specific equations of state.

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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

The main thing you should know: the paper's central result does not survive contact with its own equations. I checked the stress-test note and it is correct. With y = w^2, Eq. (64) becomes (c/2)y'' + d - D y = 0. Substituting the paper's candidate solution (66), written as y = C1/(Hk) - d/(Dk^2) - C2/(Hk^3), leaves a residual of -3d/k^2 - 4cH C2/k^3, not zero. So Eq. (66) is not a solution. That is not a missing justification; it is a wrong solution. The correct general solution, when D/c and d/c are constants, is y = d/D + C_+ e^{sqrt(2D/c)t} + C_- e^{-sqrt(2D/c)t}, with no k^{-2} or k^{-3} terms. Consequently, the singularity time ts and the apparent-horizon times (98) and (101), and every constraint in Tables I–III derived from them, are unsupported.

What the paper does well: the junction-condition analysis in Section V is a genuine extension. Matching the matter Lagrangian and its derivatives across the boundary for a heat-flux interior in f(R,T)=R+2λT with a generalized Vaidya exterior is new and clearly laid out. The observation that λ cancels from the pressure-isotropy condition and therefore does not appear in the collapse timeline is interesting, though it comes from a flawed solution. The literature coverage is thorough and the writing is mostly clear.

The soft spots are proportional and severe. Appendix A's inference that each additive term in Eq. (108) is independently r-independent is not logically valid; the left side being t-only does not force that. But even if that step were repaired, the solution form is wrong, so the whole timeline construction collapses. The GRTensor attribution without a worksheet or verification is also unsatisfying, but the direct substitution check is sufficient on its own.

Who this is for: researchers working on modified-gravity collapse who might want a worked example with dissipative heat flux. The junction-condition part could be a useful building block, but the collapse timeline results are not established. As is, I would not send this to peer review. The authors need to re-derive the w(t) solution correctly, which will change ts and tah, and then re-examine the black-hole constraints. If they fix that, the junction-condition analysis may survive as a separate contribution.

Recommendation: desk reject or return to the authors with the substitution check and Appendix A critique. Do not send it to referees until the ODE is solved correctly.

Referee Report

3 major / 5 minor

Summary. The paper studies the final state of spherically symmetric gravitational collapse of an isotropic fluid with heat flux in the f(R,T) theory with f(R,T)=R+2λT, matched to a generalized Vaidya exterior. After applying the f(R,T) junction conditions (continuity of the metric, extrinsic curvature, Ricci scalar, energy-momentum trace, and their derivatives), the authors adopt a separable interior ansatz A=A0(r), B=B0(r), C=rB0(r)w(t), reduce the pressure-isotropy condition to the single ODE (64) for w(t), and present solutions for D(r)≠0 (Eq. (66)) and D(r)=0 (Eq. (80)). Setting w=0 yields the singularity time ts (Eqs. (71), (78), (95)); the apparent-horizon condition wdot^2/w^2=δ^2 yields horizon times tah1 and tah2 (Eqs. (98)-(101)); and Tables I-III tabulate constraints on δ, C1, C2 for tah<ts, i.e., for black-hole formation. The paper concludes that the coupling λ does not affect the collapse timeline and that, under the tabulated constraints, the final singularity is hidden behind an apparent horizon.

Significance. The topic is of interest, and the paper has several sound ingredients: a systematic use of the f(R,T) junction conditions (24)-(30) with the generalized Vaidya exterior, leading to the Lagrangian-matching conditions (47) and (52); the reduction of the pressure-isotropy equation to Eq. (64); the correct treatment of the D=0 case, Eqs. (80)-(85); and the derivation of the power-law forms (90)-(91) from the apparent-horizon condition, with the internal consistency check that D(r)≠0 for those forms. The computations are first-principles: no data are fitted, and the singularity and horizon times are solved from the field equations. However, the quantitative claims of the paper (ts, tah, and the black-hole constraints in Tables I-III) all rest on Eq. (66), which does not satisfy Eq. (64), as shown in the major comments below. Because the central results are built on an incorrect solution, the manuscript cannot currently support its headline conclusions; the λ-independence of the timeline might survive a corrected derivation, but the specific times and constraints would change.

major comments (3)

[Section VI, Eqs. (64)-(66)] Eq. (66) is not a solution of Eq. (64), and this invalidates the paper's central results. Setting y=w^2, Eq. (64) becomes (c/2)y'' + d - D y = 0 with c = 1/A0^2 and d = 1/(r^2 B0^2). Substituting the claimed solution in the form y = C1/(H k) - d/(D k^2) - C2/(H k^3), with k = e^{Ht} and H^2 = 2D/c, direct differentiation gives (c/2)y'' + d - D y = d(1 - 3/k^2) - 4 c H C2/k^3, which does not vanish identically in t for generic non-zero d and C2 (for instance, at t=0 the residual is -2d - 4cHC2, which is non-zero). Hence Eq. (66) is not a solution of Eq. (64), and the singularity time (71) (equivalently (78) and (95)), the apparent-horizon times (98)-(101), and the constraints in Tables I-III, all derived from Eq. (66), are unsupported.
[Appendix A] The term-by-term independence argument leading to Eqs. (73)-(74) is logically invalid. From Eq. (108), namely w^2 = C1/(H k) - d/(D k^2) - C2/(H k^3) being a function of t only, it does not follow that each additive term is individually independent of r; r-dependence in the coefficients can cancel in the sum. The conclusions H = const and d/c = const, and hence the metric relation A0^2 = C4 r^2 B0^2 in Eq. (75), are therefore not established by the argument as written. The paper needs a proper separation-of-variables argument applied to Eq. (64) (for instance, showing that solutions w(t) exist only when D/c and d/c are constants), rather than an assertion about individual terms. Independently of this, the treatment of C1 and C2 as r-independent constants in Eq. (66) is an additional assumption that also requires justification.
[Section VI, Eqs. (66)-(71)] The attribution of Eq. (66) to the GRTensor package is not verifiable, and the failure is not a typographical slip: under the paper's own eventual assumptions H = C3 and d/c = C4 (Eqs. (73)-(74)), the general solution of Eq. (64) is y = d/D + C_+ e^{Ht} + C_- e^{-Ht}, which contains no 1/k^2 or 1/k^3 terms and cannot be reduced to Eq. (66) for any choice of C1 and C2. Sections VI and VII therefore need to be re-derived from scratch with a checkable solution of Eq. (64), after which the singularity times, the apparent-horizon times, and the parameter constraints in Tables I-III will all have to be recomputed.

minor comments (5)

[Section VIII, item 8] The statement that in the absence of shear 'the end result must necessarily be a black hole' is stronger than what Tables I-III show; several parameter ranges in those tables explicitly exclude black-hole formation, so the sentence should be rephrased to reflect the conditional nature of the results.
[Appendix A, Eq. (109)] The displayed equation H'(1+Ht)e^{Ht}=0 is not the r-derivative of C1/(H k); the correct derivative is -C1 H' e^{-Ht}(1/H + t) = 0. The conclusion H'=0 survives, but the intermediate expression should be corrected.
[Section VI, Eq. (66)] Products such as C1 H c k^2 are hard to parse and would benefit from explicit multiplication signs. Also, the ± sign in Eq. (71) should be discussed: only certain sign choices give a positive argument of the logarithm and a physically meaningful collapse time.
[Section VI, after Eq. (69)] The paper defines H = sqrt(2D/c) but never states the requirement D(r) > 0 for H to be real in the Case-I analysis; the power-law forms (90)-(91) give D(r) proportional to r^{-4}, which is consistent, but the positivity condition should be stated explicitly.
[Sections III and VIII] There are several typos: 'ithe energy-momentum tensor' in Section III should read 'the energy-momentum tensor'; 'temoral' in Section VIII should read 'temporal'; and 'Camridge' in Refs. [65] and [73] should read 'Cambridge'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the singularity and apparent-horizon times are solved from the stated f(R,T) field equations and metric ansatz, not fitted from the quantities they purport to predict; the only self-citation is peripheral.

full rationale

The central derivation is not circular. The collapse geometry is fixed by the separable ansatz (61)-(63), and the pressure-isotropy condition gives Eq. (64) for the temporal function w(t). The singularity time is then obtained by solving w(t)=0 (Eqs. (71), (78), (82)), while the apparent-horizon time follows from the null condition (86)-(87), leading to Eq. (88) and the algebraic expressions for tah - ts in Eqs. (98)-(101). No fitted data or target values enter these equations: the integration constants C1, C2, C5, C6 and the separation constant δ are free parameters, and Tables I-III merely record the algebraic inequalities under which tah < ts. The only self-citation is Ref. [56], which supplies a peripheral mass-loss formula (Eq. (102)) in the Discussion and is not load-bearing for the singularity or horizon times. The cited junction conditions [57] and the no-naked-singularity theorem [62] are external results, not self-citations. There are serious correctness concerns in the paper, notably whether Eq. (66) actually satisfies Eq. (64) and whether the Appendix A inference that each term in Eq. (108) must individually be t-dependent is valid; these are derivation errors or gaps in proof, not circularity, because the claimed predictions do not reduce by construction to the assumptions. The derivation is therefore self-contained rather than circular, and the score reflects only the presence of a minor non-load-bearing self-citation.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central derivation relies on parameter families inherited from the ansatz (δ, n, C1, C2, C5, C6, C7), plus a theory parameter λ; none are fitted to observations. The main axiomatic burden is the separability ansatz and the disputed constancy of H and d/c.

free parameters (6)

λ
Coupling constant in f(R,T)=R+2λT; introduced by the theory, required non-zero, but drops out of ts and tah.
δ
Constant defined by the apparent-horizon condition (88); later related to metric power n. Its sign and value control whether tah<ts.
n = 1 in final analysis
Real exponent in B0=C7 r^n; chosen as n=1 for the worked black-hole constraints.
C1, C2
Integration constants in the solution for w(t); their product appears in the horizon-time differences and black-hole constraints.
C5, C6
Integration constants for the D(r)=0 case; they determine ts but are not used in the main black-hole constraints.
C7
Integration constant in B0=C7 r^n and A0; scaling constant for metric coefficients.

assumptions (6)

domain assumption Field equations of f(R,T) gravity as given in Eq. (2)
The paper uses Harko et al.'s field equations without questioning them; a wrong field equation would change all results.
domain assumption Junction conditions (24)-(30) from Rosa apply to a smooth match with no thin shell
The derivation of Lm continuity rests on these junction conditions; they are cited from [57] but not re-derived.
ad hoc to paper The metric ansatz A=A0(r), B=B0(r), C=rB0(r)w(t) is admissible
This separability is assumed, not derived; it restricts the class of collapse solutions.
ad hoc to paper H and d/c are constants
The conclusion that H(r) and d(r)/c(r) are constants is required for w(t) to be time-only, but the Appendix A justification is logically flawed.
domain assumption No shear and pressure isotropy persist throughout collapse
Assumed in the matter model; the authors note shear would change the equations.
domain assumption Joshi-Goswami-Dadhich criterion applies: a singularity time independent of r implies no naked singularity
Used in Section VII to conclude black hole formation when ts is independent of r; applicability to dissipative non-dust fluids is not demonstrated.

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Pith. "Pith review of Formation of Singularity and Apparent Horizon for Dissipative Collapse in $f(R,T)$ Theory of Gravity." pith.science (2026). https://pith.science/paper/W6KZZVAT

@misc{pith2026250205217,
  author       = {Pith},
  title        = {Pith review of: Formation of Singularity and Apparent Horizon for Dissipative Collapse in $f(R,T)$ Theory of Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W6KZZVAT}},
  note         = {Machine review of arXiv:2502.05217}
}

abstract

In this paper, we consider the spherically symmetric gravitational collapse of isotropic matter undergoing dissipation in the form of heat flux, with a generalized Vaidya exterior, in the context of $f(R, T)$ gravity. Choosing $f(R, T)=R+2\lambda T$, and applying the $f(R, T)$ junction conditions on the field equations for the interior and exterior regions, we have obtained matching conditions of the matter-Lagrangian and its derivatives across the boundary. The time of formation of singularity and the time of formation of apparent horizon have been determined and constraints on the integration constants are examined for which the final singularity is hidden behind the horizon.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

109 extracted references · 70 canonical work pages

[1]

For the expression tah1 −ts from equation ( 100) to be negative for black hole formation, the following constraints must be obeyed : TABLE I: Constraints on C1 and C2 from tah1 − ts < 0 for Black Hole Formation Sign of δ Sign chosen from ± Condition for Black Hole Formation from denominator
[2]

Case I : D(r) ⁄= 0

(75) To understand the physical signiﬁcance of this condition, we consid er the radial null geodesics, which, for our interior line element are given by ds2 = 0, with θ andφ as constant. In this case, using ( 75), we ﬁnd (ignoring the constant), dt dr = ± 1 r, (76) or, t = ±ln |r|. (77) The slope of the curve tends to inﬁnity as one approaches the regu la...
[3]

δ < 0 + − ( δ2/ 64 ) < C 1C2 < 0 or C1C2 > 0
[4]

δ > 0 − Black Hole formation is not possible
[5]

δ > 0 + − ( δ2/ 64 ) < C 1C2 < 0
[6]

δ < 0 − − − ( δ2/ 64 ) < C 1C2
[7]

δ < 0 − Black Hole formation is not possible
[8]

For the expression tah2 −ts from equation ( 101) to be negative for black hole formation, the following constraints are to be obeyed : 13 TABLE II: Constraints on C1 and C2 from tah2 − ts < 0 for Black Hole Formation Sign of δ Sign chosen from ± Sign chosen from ± Condition for in numerator denominator Black Hole Formation

Show all 109 references

[9]

δ > 0 + + Black Hole formation is not possible
[10]

δ > 0 − − Black Hole formation is not possible
[11]

δ > 0 + − Black Hole formation is not possible
[12]

δ > 0 − + C1C2 > − ( δ2/ 64 )
[13]

δ < 0 + + − ( δ2/ 64 ) < C 1C2 < 0, or C1C2 > 0
[14]

Chan, Mon

R. Chan, Mon. Not. R. Astron. Soc. 288, 589 (1997)

1997
[15]

δ < 0 + − Black Hole formation is not possible
[16]

δ < 0 − + − ( δ2/ 64 ) < C 1C2 < 0 Case II : the condition z = 0
[17]

For z = 0, the expression tah1 −ts from equation ( 100) becomes zero, and black hole formation is not possible
[18]

For the expression tah2 −ts from equation ( 101) to be negative for black hole formation, the following constraints are to be obeyed : TABLE III: Conditions for Black Hole Formation from tah2 − ts < 0 Value of tah2 − ts Condition for Black Hole Formation
[19]

0 Black Hole formation is not possible
[20]

There is an additional constraint for the case z = 0 : δ and C1 should have the same sign . VIII. SUMMARY AND DISCUSSIONS To summarize, we have started with a generalized Vaidya exterior me tric, considering the exterior spacetime to the collapsing matter to be ﬁlled with a co...
[21]

Also, the derivatives of the matter Lagrang ian with respect to the coordinates of their respective spacetimes, should be continuous across the boundar y

We have shown in our analysis, just as Rosa had considered a prior i in [57] while formulating the f (R,T ) junction conditions, that the interior matter Lagrangian must match the ex terior matter Lagrangian at the boundary of the collapsing matter. Also, the derivatives of th...
[22]

If the pressure was anisotropic, then the pressure in equation (42) would be the radial pressure, since the ﬁeld equation for R11, which is the radial component of the Ricci tensor, was used to arr ive at it
[23]

In case of pressure anisotropy, we would need to ﬁnd a diﬀerent way to arrive at the solution for the temporal function w. In that case, equating the expression for q from equation ( 42) and equation ( 21) and using the equations ( 61), ( 62) and ( 63) would have provided a di...
[24]

Hence the corresponding diﬀerential equation would become more complicated if eﬀects like anisotropy and shear are involved in the collapse

Presence of shear would have further added terms involving w and its time derivative to the diﬀerential equation, since it would have appeared in the ﬁeld equation for R11 and consequently in equation ( 42). Hence the corresponding diﬀerential equation would become more compli...
[25]

The parameter λ has no eﬀect on either the singularity formation or the apparent ho rizon formation : the former because it gets cancelled out as the pressure isotropy condition is a pplied, and the latter because the condition for the formation of apparent horizon remains the...
[26]

We see that for a black hole to form as the ﬁnal state of the collap se,δ, C1 and C2 get constrained in diﬀerent ways for the possible expressions of tah −ts, obtained from the condition for apparent horizon, for non-vanis hing D(r), both when z is zero and non-zero
[27]

Tables I and II enlist the constraints that must be satisﬁed for a black hole to form when z >0 for two possible expressions of tah −ts, while Table III shows the constraints required for black hole forma tion for the z = 0 case
[28]

It has been reasoned both in GR [ 74], as well as in f (R,T ) theory [ 75] that when shearing eﬀects come into play, the matter concentra tion will be resisted by these forces

The chosen conﬁguration of the collapsing matter does not involv e shear viscosity. It has been reasoned both in GR [ 74], as well as in f (R,T ) theory [ 75] that when shearing eﬀects come into play, the matter concentra tion will be resisted by these forces. As a result, for...
[29]

Since the collapsing matter is radiating heat in the radial direction, ma ss-energy content of the collapsing body will decrease with time

In this paper, we have considered the gravitational collapse of a spherical non-rotating body where a spherical black hole will be formed as the ﬁnal state of the collapse in the cases where black hole formation is possible [ 76]. Since the collapsing matter is radiating heat ...
[30]

Glass, Phys

E.N. Glass, Phys. Lett. A 86, 351 (1981)

1981
[31]

N. O. Santos, Mon. Not. R. Astron. Soc. 216, 403 (1985)

1985
[32]

Herrera, J

L. Herrera, J. Ospino, A. Di Prisco, E. Fuenmayor, and O. T roconis, Phys. Rev. D 79, 064025 (2009)

2009
[33]

A. K. G. de Oliveira, N. O. Santos, and C. A. Kolassis, Mon. Not. R. Astron. Soc. 216, 1001 (1985)

1985
[34]

A. K. G. de Oliveira, J.A.F. de Pacheco, and N. O. Santos, M on. Not. R. Astron. Soc. 220, 405 (1986)

1986
[35]

Herrera, G

L. Herrera, G. Denmat, and N. O. Santos, Mon. Not. R. Astro n. Soc. 237, 257 (1989)

1989
[36]

R. Chan, S. Kichenassamy, G. Le Denmat, and N. O. Santos, M on. Not. R. Astron. Soc. 239, 91 (1989)

1989
[37]

R. Chan, L. Herrera, and N. O. Santos, Mon. Not. R. Astron. Soc. 265, 533 (1993)

1993
[38]

R. Chan, L. Herrera, and N. O. Santos, Mon. Not. R. Astron. Soc. 267, 637 (1994)

1994
[39]

Chan, Mon

R. Chan, Mon. Not. R. Astron. Soc. 316, 588 (2000)

2000
[40]

Herrera, and N

L. Herrera, and N. O. Santos, Phys. Rev. D, Vol. 70, 084004 (2004)

2004
[41]

Herrera, N

L. Herrera, N. O. Santos, and G. Le Denmat, Gen. Relativ. Gravit. 44, 1143 (2012)

2012
[42]

A. K. G. de Oliveira, and N. O. Santos, Astrophys. J. 312, 640 (1987)

1987
[43]

Noureen, and M

I. Noureen, and M. Zubair, Eur. Phys. J. C 75, 62 (2015)

2015
[44]

P. C. Vaidya, Nature, 171, 260 (1953)

1953
[45]

Riess et al., Astron

A.G. Riess et al., Astron. J. 116, 1009 (1998)

1998
[46]

Perlmutter et al., Astrophys.J

S. Perlmutter et al., Astrophys.J. 517 565 (1999)

1999
[47]

A. D. Felice, and S. Tsujikawa, Living Rev. Rel. 13, 3 (2010)

2010
[48]

Nojiri, and S

S. Nojiri, and S. D. Odintsov, Int. J. Geom. Meth. Mod. Ph ys. 4, 115 (2007) 16

2007
[49]

Nojiri, and S

S. Nojiri, and S. D. Odintsov, Phys. Rep. 505, 59 (2011)

2011
[50]

Capozziello, and M

S. Capozziello, and M. De Laurentis, Phys. Rep. 509, 167 (2011)

2011
[51]

Clifton, P

T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, P hys. Rep. 513, 1 (2012)

2012
[52]

Yousaf et al

studied charged perfect ﬂuid collapse and apparent horizon form ation in f (R,T ) theory. Yousaf et al. [ 53, 54] studied the inﬂuence of structure scalars obtained by orthogona l splitting of the Riemann tensor on various physical properties of the collapsing matter, such as ...
[53]

Bertolami, and M

O. Bertolami, and M. C. Sequeira, Phys. Rev. D 79, 104010 (2009)

2009
[54]

Harko, F

T. Harko, F. S. N. Lobo, G. Otalora, and E. N. Saridakis, P hys. Rev. D 89, 124036 (2014)

2014
[55]

Harko, F

T. Harko, F. S. N. Lobo, G. Otalora, and E.N. Saridakis, J CAP 12, 021 (2014)

2014
[56]

Sharif, and M

M. Sharif, and M. Zubair, JHEP 12, 079 (2013)

2013
[57]

de la Cruz-Dombriz, and A

A. de la Cruz-Dombriz, and A. Dobado, Phys. Rev. D 74, 087501 (2006)

2006
[58]

P. K. S. Dunsby, E. Elizalde, R. Goswami, S. Odintsov, an d D. Saez-Gomez, Phys. Rev. D 82, 023519 (2010)

2010
[59]

Nojiri, and S

S. Nojiri, and S. D. Odintsov, Phys. Rev. D 68, 123512 (2003)

2003
[60]

D. E. Barraco, and V. H. Hamity, Phys. Rev. D 62, 044027 (2000)

2000
[61]

Capozziello, A

S. Capozziello, A. Stabile, and A. Troisi, Class. Quant um Grav. 24 2153 (2007)

2007
[62]

Capozziello, A

S. Capozziello, A. Stabile, and A. Troisi, Class. Quant um Grav. 25 085004 (2008)

2008
[63]

Capozziello, N

S. Capozziello, N. Frusciante, and D. Vernieri, Gen. Re lativ. Gravit. 44, 1881 (2012)

2012
[64]

Multamaki, and I

T. Multamaki, and I. Vilja, Phys. Rev. D 74, 064022 (2006)

2006
[65]

Chakrabarti, and N

S. Chakrabarti, and N. Banerjee, Gen. Relativ. Gravit. 48, 57 (2016)

2016
[66]

Sharif, and H

M. Sharif, and H. R. Kausar, Astrophys. Space Sci. 331, 281 (2011)

2011
[67]

Harko, F

T. Harko, F. S. N. Lobo, S. Nojiri, and S. D. Odintsov, Phy s. Rev. D 84, 024020 (2011)

2011
[68]

P. K. Sahoo, P. H. R. S. Moraes, Parbati Sahoo, and B. K. Bi shi, Eur. Phys. J. C 78, 736 (2018)

2018
[69]

Jamil, D

M. Jamil, D. Momeni, M. Raza, and R. Myrzakulov, Eur. Phy s. J. C 72, 1999 (2012)

2012
[70]

Sharif, and M

M. Sharif, and M. Zubair, J. Phys. Soc. Jpn. 82, 064001 (2013)

2013
[71]

C. P. Singh, and V. Singh, Gen. Relativ. Gravit. 46, 1696 (2014)

2014
[72]

Shabani, and M

H. Shabani, and M. Farhoudi, Phys. Rev. D 90, 044031 (2014)

2014
[73]

P. H. R. S. Moraes, Eur. Phys. J. C 75, 168 (2015)

2015
[74]

Sharif, and M

M. Sharif, and M. Z. Gul, Mod. Phys. Lett. A 36, 2150214 (2021)

2021
[75]

P. H. R. S. Moraes, and P. K. Sahoo, Phys. Rev. D 96, 044038 (2017)

2017
[76]

P. H. R. S. Moraes, and P. K. Sahoo, Eur. Phys. J. C 77, 480 (2017)

2017
[77]

P. H. R. S. Moraes, R. A. C. Correa, and R. V. Lobato, JCAP 07, 029 (2017)

2017
[78]

Sharif, and A

M. Sharif, and A. Fatima, Eur. Phys. J. Plus 138, 721 (2023)

2023
[79]

Zaregonbadi, M

R. Zaregonbadi, M. Farhoudi, N. Riazi, Phys. Rev. D 94, 084052 (2016)

2016
[80]

M. J. Amir, S. Sattar, Int. J. Theor. Phys. 55, 1988 (2016)

2016
[81]

Abbas, R

G. Abbas, R. Ahmed, Mod. Phys. Lett. A. 34, 1950153 (2019)

2019
[82]

Yousaf, K

Z. Yousaf, K. Bamba, and M. Z. Bhatti, Phys. Rev. D 93, 064059 (2016)

2016
[83]

Yousaf, K

Z. Yousaf, K. Bamba, and M. Z. Bhatti, Phys. Rev. D 93, 124048 (2016)

2016
[84]

Yousaf, Phys

Z. Yousaf, Phys. Dark Univ. 28, 100509 (2020)

2020
[85]

Guha, and U

S. Guha, and U. Ghosh, Eur. Phys. J. Plus 136, 460 (2021)

2021
[86]

J. L. Rosa, Phys. Rev. D 103, 104069 (2021)

2021
[87]

A. Wang, Y. Wu, Gen. Relativ. Gravit, 31, 114 (1999)

1999
[88]

P. C. Vaidya, Proc. Indian Acad. Sci. (Math. Sci.) A, 33, 264 (1951)

1951
[89]

Sharif, and G

M. Sharif, and G. Abbas, Journal of the Phys. Soc. Japan 80 (2011) 104002

2011
[90]

Guha, and R

S. Guha, and R. Banerji, Int. J. Theor. Phys. 53, 2332 (2014)

2014
[91]

P. S. Joshi, R. Goswami, and N. Dadhich, Phys. Rev. D, 70, 087502 (2004)

2004
[92]

Chatterjee, A

A. Chatterjee, A. Ghosh, and S. C. Jaryal, Phys. Rev. D 102, 064048 (2020)

2020
[93]

Debnath, S

U. Debnath, S. Nath, and S. Chakraborty, Mon. Not. R. Ast ron. Soc. 369, 1961 (2006)

2006
[94]

S. W. Hawking, and G. F. R. Ellis, The Large Scale Structure Of Space-Time , Camridge Monographs on Mathematical Physics, Cambridge University Press, (1973)

1973
[95]

Darmois, ‘Les equations de la gravitation Einsteini enne,’ M´ emorial de Sciences Math´ ematiques, Fascicule XXV, ch V (1927)

G. Darmois, ‘Les equations de la gravitation Einsteini enne,’ M´ emorial de Sciences Math´ ematiques, Fascicule XXV, ch V (1927)

1927
[96]

Israel, ‘Singular Hypersurfaces and Thin Shells in G eneral Relativity,’ Nuovo Cimento 44, 1 (erratum B 49 463)(1966)

W. Israel, ‘Singular Hypersurfaces and Thin Shells in G eneral Relativity,’ Nuovo Cimento 44, 1 (erratum B 49 463)(1966)

1966
[97]

Faraoni, Phys

V. Faraoni, Phys. Rev. D 80, 124040 (2009)

2009
[98]

Deruelle, M

N. Deruelle, M. Sasaki, and Y. Sendouda, Prog. Theor. Ph ys. 119, 237 (2008)

2008
[99]

Clifton, P

T. Clifton, P. Dunsby, R. Goswami, and A. M. Nzioki, Phys . Rev. D 87, 063517 (2013)

2013
[100]

J. M. M. Senovilla, Phys. Rev. D 88, 064015 (2013)

2013
[101]

Goswami, A

R. Goswami, A. M. Nzioki, S. D. Maharaj, and S. G. Ghosh, P hys. Rev. D 90, 084011 (2014)

2014
[102]

Joshi, Gravitational Collapse and Spacetime Singularities , Camridge Monographs on Mathematical Physics, Cambridge University Press, (2007), p

Pankaj S. Joshi, Gravitational Collapse and Spacetime Singularities , Camridge Monographs on Mathematical Physics, Cambridge University Press, (2007), p. 67-68

2007
[103]

P. S. Joshi, R. Goswami, and N. Dadhich, arXiv:gr-qc/03 08012v3 13 Jan 2004

2004
[104]

Azmat, M

H. Azmat, M. Zubair, and I. Noureen, Int. J. of Mod. Phys. D, Vol. 26, No. 14 (2017) 1750181

2017
[105]

Valeri. P. Frolov, Igor. D. Novikov, Black Hole Physics - Basic Concepts and New Developments , Fundamental Theories of Physics, Kluwer Academic Publishers, (1998), p. 17

1998
[106]

Faraoni, Galaxies 2013, 1, 114-179

V. Faraoni, Galaxies 2013, 1, 114-179

2013
[107]

Jaime, and Marcelo Salgado, Clas s

Pedro Canate, Luisa G. Jaime, and Marcelo Salgado, Clas s. Quantum Grav. 33 (2016) 155005

2016
[108]

Hazarika, P

B. Hazarika, P. Phukon, arXiv:2410.00606v2 [gr-qc] 2 O ct 2024 17

2024 arXiv
[109]

Dutta, B

M. Dutta, B. Mukhopadhyay, arXiv:2411.18693v1 [gr-qc ] 27 Nov 2024 Appendix A This is the elaborate method of obtaining equations ( 73) and ( 74), by showing why the ratio d(r) c(r) is a constant both for the case when D(r) ⁄= 0 and when D(r) = 0. In each case we start with th...

2024 arXiv

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

For the expression tah1 −ts from equation ( 100) to be negative for black hole formation, the following constraints must be obeyed : TABLE I: Constraints on C1 and C2 from tah1 − ts < 0 for Black Hole Formation Sign of δ Sign chosen from ± Condition for Black Hole Formation from denominator

[2] [2]

Case I : D(r) ⁄= 0

(75) To understand the physical signiﬁcance of this condition, we consid er the radial null geodesics, which, for our interior line element are given by ds2 = 0, with θ andφ as constant. In this case, using ( 75), we ﬁnd (ignoring the constant), dt dr = ± 1 r, (76) or, t = ±ln |r|. (77) The slope of the curve tends to inﬁnity as one approaches the regu la...

[3] [3]

δ < 0 + − ( δ2/ 64 ) < C 1C2 < 0 or C1C2 > 0

[4] [4]

δ > 0 − Black Hole formation is not possible

[5] [5]

δ > 0 + − ( δ2/ 64 ) < C 1C2 < 0

[6] [6]

δ < 0 − − − ( δ2/ 64 ) < C 1C2

[7] [7]

δ < 0 − Black Hole formation is not possible

[8] [8]

For the expression tah2 −ts from equation ( 101) to be negative for black hole formation, the following constraints are to be obeyed : 13 TABLE II: Constraints on C1 and C2 from tah2 − ts < 0 for Black Hole Formation Sign of δ Sign chosen from ± Sign chosen from ± Condition for in numerator denominator Black Hole Formation