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REVIEW 3 major objections 4 minor 80 references

Anisotropic gravastar as horizonless regular black hole spacetime and its images illuminated by thin accretion disk

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A single equation of state describes both an anisotropic gravastar and a regular black hole, and the two look nearly identical under a thin accretion disk.

desk verdict Nice model-building and image work, but the advertised smooth transition to the Hayward RBH limit is not supported by the cutoff equation as written. read the letter →

arxiv 2411.12358 v1 pith:MLPWG6HO submitted 2024-11-19 gr-qc

M. F. Fauzi , H. S. Ramadhan , A. Sulaksono This is my paper · ORCID
classification gr-qc PACS 04.70.-s04.40.-b
keywords gravastarregularblackholehorizonlesscompactobjectHaywardmetricanisotropicpressurethinaccretiondiskshadowimagelightring
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 tries to establish that a horizonless ultracompact object and a regular black hole are not separate models but two phases of one equation of state. For sub-extremal parameters the construction is an anisotropic gravastar with continuous pressure—a de Sitter core, a positive-pressure crust, and a Schwarzschild exterior—and at the extremal limit it becomes the Hayward regular black hole with $p=-\epsilon$. The authors then ray-trace optically thin accretion disks around the object and compare the images with the horizonless Hayward spacetime and with the thin-shell gravastar. They find that with emitting matter outside the star, the inner light-ring structure matches the horizonless regular black hole and the quantum horizonless compact object, while differing from thin-shell gravastars; when emitting matter is placed inside, strong gravitational redshifts create a dark center and the resemblance flips toward the thin-shell gravastar.

What carries the argument

The load-bearing object is the continuous-pressure ansatz $p(r)=-\epsilon(r)\,[1-g(r)(R/r)(m(r)/m(R))\,\Theta(x-1)^2]$ built on the Hayward mass profile $m(r)=x\alpha_c\ell r^3/(2(r^3+\ell^3))$, together with the radius rule $R=R_c/x$ and the sigmoidal cut-off $\Theta(x-1)$. This single formula guarantees the de Sitter core $p(0)=-\epsilon(0)$, zero pressure at the surface $p(R)=0$, regularity $p'(0)=0$, and, through the cut-off, exactly the de Sitter equation of state $p=-\epsilon$ for $x\ge1$. The transverse pressure is fixed by the TOV equation, and $e^{\nu}$ is obtained by backward integration matched to Schwarzschild at $R$; two choices of $g(r)$ (constant, and $[1+\omega(1-r/R)]^{-1}$) are used to show which gravastar features are robust. This machinery carries the argument because it turns the proposed connection between regular black holes and horizonless stars into an explicit two-branch spacetime that can be ray-traced.

What would settle it

Recompute the axial GLM2 image with an absorption coefficient proportional to the interior energy density $\epsilon(r)$: if the central intensity vanishes or the inner ring disappears for $x=0.98$, the transparency assumption, not the spacetime geometry, is the real source of the reported image.

Watch

Extended reading notes

Core claim

The central claim is that a single matter ansatz can describe both a regular black hole and its horizonless counterpart. The pressure profile $p(r)=-\epsilon(r)\,[1-g(r)(R/r)(m(r)/m(R))\,\Theta(x-1)^2]$, with the star radius $R=R_c/x$ and a sigmoidal cut-off $\Theta(x-1)$, gives $p(0)=-\epsilon(0)$, $p(R)=0$, and $p'(0)=0$, and it forces the de Sitter equation of state $p=-\epsilon$ as $x\to1$. Solving the TOV equation with this pressure, and matching the surface to a Schwarzschild exterior, produces an interior $e^{\nu}$ that is nearly zero near the center, so the spacetime develops strong redshifts of the same kind as the quantum horizonless compact object. In images, the GLM1 (ISCO) disk profile yields four major light rings whose positions barely change with compactness, matching the horizonless Hayward configuration; the GLM2 (center) profile yields a dark central region from strong redshift, flipping the similarity toward the thin-shell gravastar and away from horizonless Hayward.

Load-bearing premise

The load-bearing premise is that light rays traverse the star's interior without being absorbed or scattered; if the interior matter interacts with photons, the predicted dark center and inner light-ring pattern no longer follow.

Editorial extensions

If this is right

  • For $x>x_{\rm ps}\approx0.85324$ the star surface lies inside the Schwarzschild photon sphere, so the object produces photon spheres and light rings despite having no horizon.
  • With an ISCO-type disk, the anisotropic gravastar's light-ring pattern is nearly independent of compactness, unlike the horizonless Hayward spacetime and unlike the three-ring pattern of thin-shell gravastars.
  • With a center-emitting disk, the gravastar image has a dark center produced by strong redshift, while the horizonless Hayward image is brightest at the center; this is the cleanest distinction between the two horizonless models.
  • At realistic telescope resolution with the disk outside the object, inclined images of the gravastar and the horizonless Hayward spacetime are nearly indistinguishable, and both differ from a black hole mainly by a slightly brighter center and a resolved inner light ring.
  • The relativistic Doppler asymmetry is weaker for the Schwarzschild exterior of the gravastar than for the Hayward spacetimes, offering a possible but resolution-limited discriminator.

Reading between the lines

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

  • If the transparency assumption fails, the GLM2 dark-center signature may not survive; a natural next step is to repeat the ray tracing with a frequency-dependent opacity derived from the same $\epsilon$ and $p$ profiles.
  • The exponential suppression $e^{\nu}(0)\sim \exp(-1/y)$ suggests a parameter correspondence between $y=1-x$ and the interaction parameter of the quantum horizonless compact object; comparing quasinormal-mode spectra across that map would sharpen the claimed equivalence.
  • The paper notes but does not analyze stability; since horizonless ultracompact spacetimes must contain stable photon orbits, the model may be prone to the light-ring instability, which would limit how long such an object could exist even if its images match.
  • A concrete observational extension is to test the GLM2 prediction with very long baseline interferometry at horizon-scale resolution: a resolved dark center with an inner light ring, rather than a sharp shadow, would favor a transparent horizonless object of this type.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper constructs a phenomenological spacetime model intended to interpolate between an anisotropic gravastar and a Hayward regular black hole. The setup fixes a Hayward-type density profile truncated at a pressure-defined surface R=Rc/x, where x=α/αc parametrizes the mass: x<1 corresponds to a star configuration and x≥1 to a Hayward regular black hole. The radial pressure is given by the ansatz p(r)=-ϵ(r)[1-g(r)(R/r)(m(r)/m(R))Θ(x−1)^n], which by construction yields a de Sitter core p(0)=-ϵ(0), p'(0)=0, and (for the ideal step cutoff of Eq. (29)) zero pressure at r=R. A sigmoidal cutoff Θ is then introduced with the stated purpose of making the pressure approach p=-ϵ as x→1. The interior g_{tt} is obtained by integrating the TOV equation backward from the matched Schwarzschild surface, giving a strongly redshifted center that the authors compare to the QHCO of Chen and Yokokura. The authors then integrate null geodesics through the combined spacetime, assume an optically thin interior, and generate thin-disk images (GLM1 and GLM2 emission profiles; axial and inclined observers with Doppler shifts and Gaussian filtering) for their gravastar, the horizonless Hayward spacetime, and the Hayward black hole. They report QHCO-like inner light rings for exterior emission, a distinct inner-ring structure relative to thin-shell gravastars, and a redshift-induced dark center for interior emission.

Significance. The paper addresses a timely question—whether a single phenomenological model can connect regular black holes and horizonless compact objects—and its image pipeline is concrete and carefully executed. Credit is due for the explicit construction of the equation-of-state ansatz, the transparent TOV integration scheme, the comparison set (Hayward, thin-shell gravastar, QHCO), and for producing falsifiable predictions: the exp(-1/y) central-redshift scaling of Fig. 11, the QHCO-like inner light ring structure for exterior emission, and the redshift-induced central darkening for interior emission. The authors are also appropriately explicit about their caveats (transparent interior, no stability analysis, no transition dynamics). As it stands, however, the central mechanism advertised in the abstract—a model that 'corresponds to regular black hole spacetime in the appropriate limit'—is not realized by the printed equations: the smooth cutoff of Eqs. (30)–(31) does not deliver p=-ϵ at x=1 nor a zero-pressure vacuum boundary, and Eq. (33) is displayed with the wrong exponent sign and a missing boundary factor.

major comments (3)
  1. [III.A, Eqs. (29)–(31)] The smooth cutoff does not realize the two limits claimed in the text. With the sigmoid (31), Θ(0)=1/2 at x=1, so Eq. (30) gives R=2Rc rather than an indefinite radius, and at the nominal surface r=R one obtains p(R)=-ϵ(R)(1-Θ(1)^n)=-(1-2^{-n})ϵ(R), i.e., -3ϵ(R)/4 for n=2. This is neither zero (so R is not a pressure-defined vacuum boundary, and the exterior Schwarzschild matching of Eq. (32) requires a surface layer) nor equal to -ϵ(R) (so the de Sitter equation of state is not recovered). The accompanying sentence, 'the cut-off function ... does not vanish if n>1 ... ensures that the term inside the brackets ... becomes zero,' is backwards: the bracket is 1-X with X∝Θ^n, and p=-ϵ requires X→0, i.e., Θ→0, whereas the sigmoid gives Θ^n=2^{-n}>0 at x=1. Consequently the x≥1 Hayward branch is imposed by declaration rather than reached as a limit of Eq. (30), and the abstract's 'corresponds to regular black hole spacetime in the appropriate limit' is unsupported as written. The issue is not merely formal: for x values inside the transition zone used in the images (x=0.98 lies beyond x_t in Fig. 10), p(r)<0 on r<R and p(R)<0, so the pressure is discontinuous at the surface and the object is not a continuous-pressure gravastar there. The authors should either modify Θ so that Θ(1)=0 (treating the hard cutoff (29) as the definition and the sigmoid as a κ→0 regularization), or explicitly restrict the claimed correspondence to that limit and revise the abstract accordingly.
  2. [III.A, Eq. (33)] The displayed interior metric component is inconsistent with Eq. (9). From Eq. (9), ν'=2(4πr̃³p+m)/(r̃(r̃-2m)), so the correct solution matched to the Schwarzschild exterior is e^{ν(r)}=e^{ν(R)}exp(-2∫_r^R [4πr̃³p(r̃)+m(r̃)]/[r̃(r̃-2m(r̃))] dr̃). As printed, exp(2∫_r^R ...) equals e^{ν(R)-ν(r)} and omits the boundary factor e^{ν(R)}; this is load-bearing because e^ν enters the photon effective potential (46) and therefore every image in Sec. IV. The prose description of backward integration with the matching condition e^{ν(R)}=1-2m(R)/R suggests the numerical integration was performed correctly, but the published formula must be fixed.
  3. [IV.C, and conclusions] Both headline image results—the QHCO-like inner light rings for exterior emission and the central darkening for GLM2 interior emission—depend on the assumption that photons traverse the star's interior without any interaction with the medium, which has nonzero ϵ, p, and p_t. The authors flag this caveat in the conclusions, and I do not regard it as fatal, but it is load-bearing for one of the two main claims, and it is physically nontrivial given that the interior is a material medium rather than vacuum. A quantitative supporting argument (e.g., an order-of-magnitude optical-depth estimate or a statement of the maximal opacity compatible with the reported images) would materially strengthen the paper; at minimum, the abstract's summary of the image results should carry the same transparency caveat that Sec. IV.C states.
minor comments (4)
  1. [III.A] The sentence 'The integration of Eq. (16) is carried numerically' should refer to the master equation for the metric function ν (Eq. (33)), since Eq. (16) is the closed-form Hayward metric component; the equation numbers in this paragraph should be checked.
  2. [Throughout] There are many typographical errors and garbled figure labels that should be cleaned up before publication; examples include 'Mazur and Motolla' (p. 3), 'Tolman Oppenheimer Volkof' (p. 5), 'Schwarzshcild' (p. 12) and 'Schwarzchild' (p. 13), 'Lorenz invariant' (p. 24), 'surpresses' (p. 19), 'emmision' (p. 26), and the axis labels '5=0:05' (Fig. 3), 'p=0(0)' (Figs. 4 and 7), and 'X=m(R)' (Figs. 17–24), which presumably should read κ=0.05, p/ϵ(0), and X/m(R).
  3. [IV.B.1 and IV.C.3] The symbol σ is used both for the width of the GLM emission profile and for the standard deviation of the Gaussian filter applied to the inclined images; these should be denoted by different symbols to avoid confusion.
  4. [IV.C.1 and Fig. 12] The effective-potential and image comparisons at fixed x compare models with different total masses (m_G(R)=0.7383 versus m_H=0.8063 at x=x_ps, as the paper itself notes); a caption sentence emphasizing that m(R) differs between the models would prevent an over-literal reading of the comparison.

Circularity Check

2 steps flagged · score 6.0 of 10

The gravastar/RBH correspondence is encoded in the pressure ansatz and step function; the image computations are genuine forward results, not fitted.

  1. self definitional [Sec. III A, Eq. (28) and the paragraph following it]
    "With this particular pressure profile, it can be seen that at r = R, the last term becomes unity, leading to the pressure becoming zero, thus defining the star radius. At the center (r = 0), the local compactness is zero, resulting in p = −ϵ, indicating that we have a de Sitter core as an anisotropic gravastar requires. It can also be observed that the condition p′(r = 0) = 0 is always satisfied."

    The fulfillment of the anisotropic-gravastar conditions is not derived from a microphysical EoS or from the TOV equations; it is written into the functional form of p(r). With g(R)=1, the factor R m(r)/(r m(R)) equals unity at r=R, so p(R)=0 by construction. Likewise, m(r)/r=O(r^2) near r=0 makes p(0)=-ϵ and p'(0)=0 identities for any admissible g(r). The paper's statement that the model meets the gravastar requirements is therefore a restatement of the ansatz, not a consequence obtained from the field equations.

  2. self definitional [Sec. III A, Eqs. (29)-(30) and the following sentence]
    "Thus, we introduce a cut-off function Θ(x − 1) ... to modify the radius and the radial pressure profile as follows: R = Rc/x [Θ(x−1)]−1, p(r) = −ϵ(r)(1 − g(r) R/r m(r)/m(R) Θ(x−1)n) ... As x → 1, the radius becomes indefinite, and the cut-off function in the pressure profile does not vanish if n > 1. This ensures that the term inside the brackets in the pressure profile becomes zero as x → 1, resulting in the de Sitter EoS p = −ϵ."

    The advertised transition to the regular-black-hole limit is imposed by the definition of Θ rather than produced by the dynamics. For x>1, Θ=0 forces the correction term to vanish and R to diverge, so p=-ϵ and the Hayward form hold by construction. The paper explicitly says the cutoff is introduced 'to modify' the radius and pressure profile so that this happens. Thus the abstract's central claim that the object 'corresponds to regular black hole spacetime in the appropriate limit' reduces to the input step function. A further internal-consistency problem is that with the sigmoid (31), Θ(0)=1/2, so p(R) is not strictly zero on the star side of the nominal surface; this is a correctness caveat, while the definitional encoding of the RBH limit is the circular step.

full rationale

The model is an openly phenomenological ansatz, and the paper does not pretend to derive its EoS from a fundamental Lagrangian. However, the central structural claims are definitional: Eq. (28) is written so that p(R)=0, p(0)=-ϵ, and p'(0)=0 are identities of the chosen interpolation factors, and Eq. (30) inserts the de Sitter limit through the step function Θ. Consequently, the headline correspondence between the anisotropic gravastar and the regular black hole is an input of the construction, not a derived prediction. The shadow-image section is different: the geodesic integration, GLM emission profiles, redshift factors, and inclined ray tracing are all forward computations from the metric, and no image feature is fitted to a target. The QHCO-like inner light-ring structure is a qualitative consequence of the designed near-zero eν, not a fitted output. The paper's transparency about the ansatz reduces the severity, but the definitional character of the central claim remains. I therefore set the circularity score at 6: partial circularity in the model-construction claim, with genuine independent content in the image predictions.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The model depends on several hand-chosen functions and parameters: the radius definition, the pressure ansatz, the cut-off kappa, the exponent n, and the profile g(r). None are fitted to observational data, but they are also not derived from microphysics. The exterior Schwarzschild matching and the free-streaming photon assumption are additional imposed inputs. No new physical entities are introduced.

free parameters (4)
  • kappa (transition smoothness) = 0.01
    Chosen by hand after inspecting Fig. 3; controls the width of the transition zone and the value of x_t in Fig. 10.
  • n (cut-off exponent in pressure profile) = 2
    Set 'for simplicity' in Sec. III A; ensures the cut-off dominates in the limit, but the finite-x behavior of the pressure depends on this choice.
  • omega (Model 2 profile constant) = omega = (alpha_c^3 - 2)/(alpha_c^3 + 1)
    Derived by imposing p(R)=p'(R)=0 in Eq. (38); this is a design condition, not a measurement.
  • g(r) (dimensionless profile) = 1 or [1+omega(1-r/R)]^{-1}
    Arbitrary function satisfying Eq. (27); the region boundaries and transition region width depend on it.
assumptions (5)
  • domain assumption Static and spherically symmetric anisotropic-fluid spacetime described by Eqs. (1)-(3).
    This is the standard background for the TOV construction; it excludes rotation and time dependence.
  • domain assumption Hayward density and mass function (Eq. 15) are the regular-black-hole seed for the star configuration.
    The paper builds the gravastar by truncating the Hayward density, rather than deriving this density from a fundamental theory.
  • ad hoc to paper Exterior is exactly Schwarzschild with mass m(R) for r >= R (Eq. 32).
    The Hayward density is forced to vanish at R, creating a surface with continuous radial pressure but no physical mechanism for the cutoff.
  • ad hoc to paper The pressure ansatz Eq. (28) is a valid equation of state.
    No microphysical derivation is offered; the ansatz is designed to satisfy the gravastar boundary conditions.
  • domain assumption Photons traverse the interior freely without interacting.
    Stated in Sec. IV C; the GLM2 images and central dark patch depend on this premise.

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Pith. "Pith review of Anisotropic gravastar as horizonless regular black hole spacetime and its images illuminated by thin accretion disk." pith.science (2026). https://pith.science/paper/MLPWG6HO

@misc{pith2026241112358,
  author       = {Pith},
  title        = {Pith review of: Anisotropic gravastar as horizonless regular black hole spacetime and its images illuminated by thin accretion disk},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MLPWG6HO}},
  note         = {Machine review of arXiv:2411.12358}
}
abstract

A connection between regular black holes and horizonless ultracompact objects was proposed in~\cite{Carballo-Rubio:2022nuj}. In this paper, we construct a model of a horizonless compact object, specifically an anisotropic gravastar with continuous pressure, that corresponds to regular black hole spacetime in the appropriate limit. The construction begins by modeling an equation of state that satisfies the anisotropic gravastar conditions and transitions to the de Sitter ($p=-\epsilon$) upon horizon formation. The spacetime structure is similar to the {\it Quantum Horizonless Compact Object} (QHCO) described in~\cite{Chen:2024ibc}. Within this model, we also generate images of the corresponding objects surrounded by a thin accretion disk. The resulting images reveal that assuming that the emitting matter exists only outside the object, the inner light ring structure closely resembles that of the horizonless configuration of a regular black hole and the QHCO, yet it exhibits a distinct light ring structure compared to the thin-shell gravastar model. However, the opposite occurs when emitting matter is taken into account inside the object.

Figures

Figures reproduced from arXiv: 2411.12358 by the authors.

Figure 1
Figure 1. FIG. 1: Hayward RBH metrics behavior. For [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Qualitative sketch of anisotropic gravastar’s pressure profile proposed by [32]. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The total mass [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Radial (top left) and transverse (bottom right) pressure profile, and energy conditions [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Inner structure of the corresponding proposed gravastar model with [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Same description as Fig. 4 with [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: The e ν component value near the core region is slightly higher than what we obtain on the previous model, partially confirming our argument on the contribution of mass on the positive pressure region. 2 This condition comes from the fact that there should be only one …
Figure 8
Figure 8. Figure 8: FIG. 8: Same description as Fig. 5 with [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Similar plot as Fig. 6 with [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Impact of [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: The [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Photon effective potential for our gravastar model (solid lines) and Hayward spacetime [PITH_FULL_IMAGE:figures/full_fig_p021_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Emission profile for GLM1 (blue) and GLM2 (red) in arbitrary units. In this graph, we [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: The axial observation helps study the shadow and light ring structure around the [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15: Photon trajectory around our corresponding gravastar model with [PITH_FULL_IMAGE:figures/full_fig_p026_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16: Number of photon orbits [PITH_FULL_IMAGE:figures/full_fig_p027_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17: The observed intensity of GLM1 (ISCO) accretion disk model in axial observation for [PITH_FULL_IMAGE:figures/full_fig_p028_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18: Shadow images of GLM1 (ISCO) accretion disk model in axial observation for our gravas [PITH_FULL_IMAGE:figures/full_fig_p029_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19: The observed intensity of GLM2 (Center) accretion disk model in axial observation for [PITH_FULL_IMAGE:figures/full_fig_p030_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20: Shadow images of GLM2 (Center) accretion disk model in axial observation for our [PITH_FULL_IMAGE:figures/full_fig_p031_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21: Shadow images of inclined observation with [PITH_FULL_IMAGE:figures/full_fig_p032_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22: Observed intensities from the center-horizontal cross section of the filtered inclined ob [PITH_FULL_IMAGE:figures/full_fig_p032_22.png]
Figure 23
Figure 23. Figure 23: FIG. 23: Shadow images of inclined observation with [PITH_FULL_IMAGE:figures/full_fig_p033_23.png]
Figure 24
Figure 24. Figure 24: FIG. 24: Observed intensities from the center-horizontal cross section of the filtered inclined ob [PITH_FULL_IMAGE:figures/full_fig_p033_24.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

80 extracted references · 32 canonical work pages

  1. [1]

    the de Sitter core requires p(r = 0) = pt(r = 0) = −ϵ(r = 0),

    work page

  2. [2]

    there must be two radial points where the p vanishes: at r = r0 and r = R; r0 is the transition point between the negative pressure of de Sitter vacuum and positive pressure of the atmosphere, and R is the surface of the star,

    work page

  3. [3]

    Some restrictions were also given regarding the energy conditions

    p′(r = 0) = 0 to maintain the regularity. Some restrictions were also given regarding the energy conditions. The null energy condition (NEC) given by ϵ + p ≥ 0 and ϵ + pt ≥ 0 together with the weak energy condition (WEC) ϵ ≥ 0 are always satisfied throughout the entire region of the star. The strong energy condition (SEC) given by ϵ + p + 2pt ≥ 0 is indee...

    work page

  4. [4]

    the core, located from the star center r = 0 to a point where the SEC satisfied,

    work page

  5. [5]

    the crust, located from the outer side of the core to the point of maximum positive radial pressure r = rmax where dp/dr = 0,

    work page

  6. [6]

    7 Rr0 rmax p r Core Crust Atmosphere ε+p+2p =0t FIG

    the atmosphere, located outside the crust to the star surface R. 7 Rr0 rmax p r Core Crust Atmosphere ε+p+2p =0t FIG. 2: Qualitative sketch of anisotropic gravastar’s pressure profile proposed by [32]. DeBenedictis, et al. [33] realized Cattoen’s idea by introducing an ansatz of anisotropy which is related to the local compactness; ∆(r) = pt(r) − p(r) ϵ(r...

    work page

  7. [7]

    This makes it a constant and still fulfills the requirement of Eq

    Model 1 For the first model, we choose a simple form of the dimensionless function, specifically g(r) = 1. This makes it a constant and still fulfills the requirement of Eq. (27). The pressure profile is explicitly written as p(r) = −ϵ(r) 1 − R r m(r) m(R) Θ(x − 1)2 . (35) The resulting pressure profile and its energy conditions for this model are present...

    work page

  8. [8]

    We see that this function satisfies the requirement of Eq

    Model 2 We consider a simple form of g(r); g(r) = h 1 + ω 1 − r R i−1 , (36) with ω is a constant that can be adjusted or constrained. We see that this function satisfies the requirement of Eq. (27). The pressure profile can then be expressed as p(r) = −ϵ(r) 1 − 1 + ω 1 − r R −1 R r m(r) m(R) Θ(x − 1)2 . (37) The free constant ω generalize the constant g(...

    work page

Show all 80 references
  1. [9]

    5 and 8, the interior eν component gives rise to strong gravitational redshifts, similar to those observed in the QHCO model of Ref

    Strong redshifts inside the star As shown in Figs. 5 and 8, the interior eν component gives rise to strong gravitational redshifts, similar to those observed in the QHCO model of Ref. [2]. It can be seen, partic- ularly in Fig. 8, that higher redshifts are produced at larger v...

  2. [10]

    (44) 20 For photon trajectories, substituting Eq

    null (43) With the static and spherically symmetric ansatz (41), and by restricting the geodesic motion to the equatorial plane θ = π/2, ˙θ = 0, we obtain two conserved quantities: A(r) ˙t = E, ˙ϕr2 = L. (44) 20 For photon trajectories, substituting Eq. (44) into Eq. (43) for ...

  3. [11]

    (47) We plot the photon effective potential in Fig. 12. 0 0.5 1 1.5 2 r=R 0 0.05 0.1 0.15 0.2 V x=xps x=0:9 x=0:98 FIG. 12: Photon effective potential for our gravastar model (solid lines) and Hayward spacetime (dashed lines) with several value of x. To solve (45), we perform ...

  4. [12]

    For simplicity, we consider an accretion disk emission profile Ie(r), which is related to the matter density and the temperature of the accretion disk

    Emission profile It is usually expected that the accumulation of matter on the accretion disk leads to electromagnetic emission of the accretion disk. For simplicity, we consider an accretion disk emission profile Ie(r), which is related to the matter density and the temperatu...

  5. [13]

    This type of accretion disk assumes that the emission peaks and stops near the ISCO radius

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Reviewed August 12, 2026 · model on record in the stance chip above.