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Slowly rotating black hole solution to Einstein-Bel-Robinson gravity

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Einstein-Bel-Robinson gravity admits slowly rotating black holes whose spin enters only through $g_{t\phi}$ at leading order.

desk verdict A useful assembly of slow-rotation observables in EBR gravity, but the printed field equations have internal inconsistencies that must be fixed before the results can be trusted. read the letter →

arxiv 2505.01054 v1 pith:T45VPFLF submitted 2025-05-02 gr-qc

Seyed Naseh Sajadi , Supakchai Ponglertsakul , Robert B. Mann This is my paper · ORCID
classification gr-qc
keywords Einstein-Bel-Robinsongravityslowlyrotatingblackholeshigher-curvaturecorrectionsholeshadowinnermoststablecircularorbitsuperradiancemassivescalarfieldcontinuedfractionexpansion
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 claims that Einstein-Bel-Robinson gravity, a four-dimensional higher-curvature extension of general relativity inspired by M-theory, admits slowly rotating, asymptotically flat black holes. At first order in the spin parameter, the only new metric ingredient is a nonzero $g_{t\phi}$ component, and the paper constructs explicit approximate metric functions $h$, $f$, and $P$ to leading order in the coupling $\beta$. It then shows how this affects horizon angular velocity, photon sphere, shadow, ISCO, and superradiant scattering. A sympathetic reader should care because these are concrete, testable departures from the general-relativistic rotating solution: the horizon shrinks for positive $\beta$, the shadow grows at small mass, and superradiant energy extraction is suppressed as $\beta$ increases.

What carries the argument

The machinery is the slow-rotation ansatz $ds^2=-N f\,dt^2+f^{-1}dr^2-2aP r^2\sin^2\theta\,dt\,d\phi+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$, where the spin enters only through $g_{t\phi}$. To solve the field equations the paper combines a large-$r$ power series, a near-horizon expansion, and a continued-fraction interpolation, then extracts a small-$\beta$ expansion valid everywhere outside the horizon. That perturbative solution carries all later computations: geodesic equations, photon sphere and shadow, ISCO, and the massive-scalar superradiance analysis.

What would settle it

Take the proposed metric functions and substitute them directly into the Einstein-Bel-Robinson field equations at order $a$; in particular, check whether the $E_{rr}$ equation respects the reflection symmetry $\theta\to\pi-\theta$ of the ansatz. The printed Eq. (8) contains a term proportional to $aP'\cos\theta$ that is odd under this symmetry, so a direct computation, or a rerun of the near-horizon expansion, would settle whether the solution actually exists.

Watch

Extended reading notes

Core claim

The central discovery is a slowly rotating black hole solution whose metric functions, in the small-$\beta$ expansion, are $h=1-2M/r+128\beta M^3/r^9(8-11M/r)-\dots$, $f=1-2M/r+128\beta M^3/r^9(36-67M/r)-\dots$, and $P=2M/r^3-128\beta M^3/(11 r^{11})(108-121M/r)+\dots$, with $g_{t\phi}=-2aP r^2\sin^2\theta$. The paper claims this solves the Einstein-Bel-Robinson field equations to $\mathcal{O}(a)$ and $\mathcal{O}(\beta^2)$. It then derives that the horizon radius decreases with positive $\beta$, the horizon angular velocity increases above the general-relativistic value, the photon sphere, shadow radius, and ISCO all shift, and a massive scalar wave is superradiantly amplified when $\mu<\varpi\le p_0 m a$, with the energy flux through the horizon decreasing as $\beta$ grows.

Load-bearing premise

The entire construction assumes that at first order in the spin parameter $a$, the only metric component that changes is $g_{t\phi}$; if Einstein-Bel-Robinson corrections source order-$a$ terms in $g_{tt}$, $g_{rr}$, or $g_{\theta\theta}$, all the derived observables would change.

Editorial extensions

If this is right

  • For positive $\beta$, the event horizon is smaller than $2M$, so Einstein-Bel-Robinson black holes are more compact than their general-relativistic counterparts at the same mass.
  • The horizon angular velocity $\omega_+$ exceeds the general-relativistic value $a/4M^2$ when $\beta>0$.
  • The photon ring radius, shadow diameter, and ISCO all receive $\beta$-dependent shifts, and matching the shadow diameter to observational data yields a bound $0<\beta<2.05\,M^6$.
  • Superradiance of a massive scalar occurs for $\mu<\varpi\le p_0 m a$, and the extracted energy flux is suppressed as $\beta$ increases.

Reading between the lines

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

  • This construction presupposes that the slow-rotation ansatz is self-consistent; the printed $E_{rr}$ equation contains a term proportional to $aP'\cos\theta$ that is not invariant under the metric's reflection symmetry, so a direct order-by-order check is needed before these predictions can be trusted.
  • If the ansatz survives that check, the same continued-fraction method could be applied to other higher-curvature theories, making shadow and superradiance data a generic probe of such couplings.
  • Because the photon-ring angular-velocity ratio depends on both spin and $\beta$, independent spin measurements could turn shadow observations into a constraint on the Einstein-Bel-Robinson coupling.
  • The shifts in ISCO and photon sphere grow most strongly at small mass, suggesting that lower-mass black holes are the likeliest observational testbed for these corrections.
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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 / 5 minor

Summary. The paper studies slowly rotating black holes in Einstein-Bel-Robinson (EBR) gravity in four dimensions. Starting from the slow-rotation metric ansatz (1), it displays the field equations, gives large-r and near-horizon expansions, and constructs a continued-fraction interpolation. The authors then focus on the small-beta perturbative metric functions (24)-(26) and use them to compute the horizon angular velocity, photon sphere, photon rings, black hole shadow, Lyapunov exponent, ISCO radius, and an EHT/SgrA-based bound on the coupling constant. In the second half of the paper they study massive scalar superradiance, deriving the superradiant condition, the energy flux through the horizon, and a thermodynamic argument about mass decrease. The central claim is that Eqs. (24)-(26) provide an O(a, beta^2) solution of the EBR field equations and that the subsequent observables follow from them.

Significance. If the central derivation is correct, the paper provides a useful phenomenological template for EBR gravity: explicit closed-form corrections to the Kerr slow-rotation metric, concrete predictions for shadow and ISCO observables, and a first bound on the EBR coupling from EHT data. The authors work from the action, display substantial parts of the field equations, give both large-r and near-horizon series, and provide many explicit coefficients. However, the manuscript contains internal inconsistencies in central displayed equations that prevent the reader from verifying the main solution. These must be resolved before the physical predictions can be accepted.

major comments (3)
  1. [Section 2, Eq. (1) and Eq. (12)] The metric ansatz (1) sets g_{t\phi} = -2 a P(r) r^2 sin^2(theta), which gives omega(r) = -g_{t\phi}/g_{\phi\phi} = 2 a P(r). Equation (12), however, defines omega(r)=aP(r), and every subsequent spin-dependent formula, including the geodesic equations (33), the photon-ring equations (48), and the shadow radius (55), uses the single-aP convention. The claimed large-r Kerr match, P -> 2M/r^3 and omega -> 2Ma/r^3, is only consistent with omega=aP. If Eq. (1) is literal, all linear-in-a observables in Sections 3 and 4 are too large by a factor of 2; if Eq. (12) is the intended convention, the factor 2 in Eq. (1) must be removed and the convention propagated consistently. This is load-bearing because the spin corrections to r_ph, R_s, r_ISCO, and omega_+ all depend on this normalization.
  2. [Section 2, Eq. (8)] The displayed E_rr equation contains a term proportional to a P'(r) sqrt(f) cos(theta). For the metric (1), the only O(a) metric component is g_{t\phi} proportional to sin^2(theta), so the spacetime is invariant under theta -> pi - theta at this order. Consequently every curvature invariant, and in particular the tensor component E_rr, which transforms as a scalar under this reflection because r and r are unchanged, must be invariant under theta -> pi - theta. A term proportional to cos(theta) changes sign under this transformation and cannot appear. Since Eq. (8) is the 'simplest' field equation used to determine h, f, and P, the printed equation is either mis-transcribed or the ansatz is inconsistent at O(a). The derivation of Eqs. (24)-(26) cannot be verified as printed until this term is corrected or explained.
  3. [Section 3, Eqs. (55) and (59)-(62)] The shadow-diameter derivation has a dimensionally inconsistent spin term. In Eq. (55), the spin correction a carries the dimension of length and multiplies a dimensionless bracket. Equation (59) then writes d_sh/M = 6 sqrt(3) - ... - 2a[1+...], which requires a to be dimensionless; the subsequent substitution beta = B M^6 and the bound (61)-(62) treat a as a dimensionless spin. As printed, the factor a/M is missing in Eq. (59). Because the numerical bound (62) is one of the paper's main applications, the authors should state the normalization of a in this section and re-derive the constraint consistently.
minor comments (5)
  1. [Section 2, Eqs. (24)-(26)] The text says the small-beta expansion is given 'up to order O(beta^3)', but Eqs. (24)-(26) display terms only through O(beta^2). Please clarify whether the expansion is truncated at O(beta^2) or whether O(beta^3) terms are omitted for brevity.
  2. [Section 2, Figures 1-2] The continued-fraction plots are not reproducible as presented because the undetermined near-horizon constants h1, f1, p0, and p1, which enter the coefficients in Appendix B, are not listed for the plotted curves.
  3. [Section 3.2.1, Eqs. (45)-(47)] The ISCO expansion states that the positive sign corresponds to prograde orbits, but the sign convention relating J_ISCO to the direction of rotation is not defined before Eq. (44). A short statement of the convention would improve clarity.
  4. [Section 4, heat flux discussion] The abstract says superradiance is studied 'using direct integration', but Section 4 presents only an asymptotic Wronskian analysis and plots; no numerical integration scheme, error tolerance, or convergence test is described. Please either describe the numerical method or reword the claim.
  5. [Section 4, Eq. (80)-(82)] The first-law argument uses psi_beta from Ref. [45] without demonstrating that the same coefficient applies to the slowly rotating solution studied here. Since the second-law conclusion relies on this identification, a brief derivation or an explicit statement of the assumption is needed.

Circularity Check

0 steps flagged · score 2.0 of 10

Derivation is self-contained; central metric and observable results are not circular, with only peripheral self-citations.

full rationale

The central metric construction is self-contained: Eqs. (24)-(26) are obtained by inserting the slow-rotation ansatz (1) into the EBR field equations and solving order by order in the spin and coupling parameters, with only the Kerr-matching coefficients h1, f1, P3 fixed at the β=0 limit. All subsequent predictions—horizon angular velocity (30), photon sphere (40), ISCO (45)-(47), photon rings (51)-(52), shadow radius (55), and superradiance flux (77)—are computed directly from those metric functions rather than fitted to the target observables. The EHT shadow comparison is an application of the derived formula, not a fit. The geodesic equations and shadow relations cited from [35]-[38] are standard textbook-type results, and the only author-overlapping citations ([7] for static-AdS context and constraints, [45] for ψβ = -4π√(f1h1) in the first-law discussion around Eq. (80)) are peripheral to the main derivation of the solution and its observable consequences. Neither supplies the EBR metric corrections nor forces the spin-dependent predictions by construction. The factor-of-two discrepancy between the g_tφ term in Eq. (1) and the frame-dragging definition in Eq. (12) noted by a skeptic is an internal consistency or correctness issue, not a case of a prediction reducing to its inputs.

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

The paper introduces no new particles, forces, or fields; the EBR theory and its coupling β predate this work. The free parameters of the solution (mass M, spin a) are physical, and β is a theory parameter that the paper later constrains rather than fits. The main unproven inputs are the completeness of the slow-rotation ansatz, the convergence of the continued fraction, and the cited ψβ formula, which is a self-citation.

assumptions (5)
  • domain assumption The EBR action (2) with coupling constant β is the correct theory to start from.
    The paper takes Einstein-Bel-Robinson gravity as given; its M-theory motivation is cited but not re-derived.
  • domain assumption The slow-rotation ansatz (1), in which only gtφ changes at first order in a, is complete.
    Standard for slowly rotating metrics but unproven for this higher-derivative theory; the printed Err equation (8) contains a θ-odd term that makes this assumption suspect.
  • ad hoc to paper The continued fraction truncation at order four converges to the true solution.
    The paper asserts convergence based on figures and omits several lengthy coefficients; no error bound is provided.
  • domain assumption The separation constant Λ=l(l+1) and the leading-order effective potential (69) correctly describe massive scalar scattering.
    Standard partial-wave approximation; frame-dragging corrections beyond leading order and the exact eigenvalue of (65) are not computed.
  • domain assumption The first-law coefficient ψβ=-4π√(f1h1) taken from reference [45] is valid.
    Reference [45] is by two of the current authors and is not yet published; used in the thermodynamic-superradiance argument.

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Pith. "Pith review of Slowly rotating black hole solution to Einstein-Bel-Robinson gravity." pith.science (2026). https://pith.science/paper/T45VPFLF

@misc{pith2026250501054,
  author       = {Pith},
  title        = {Pith review of: Slowly rotating black hole solution to Einstein-Bel-Robinson gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T45VPFLF}},
  note         = {Machine review of arXiv:2505.01054}
}
abstract

We study slowly rotating black hole solutions in the Einstein-Bel-Robinson gravity (EBR) in four dimensions. At the leading order in the rotation parameter, the only modification with respect to the static case is the appearance of a non-vanishing $g_{t\phi}$ component. We construct approximate solutions to these equations and study how physical properties of the solutions, such as the angular velocity, photon sphere, black hole shadow, and innermost stable circular orbit, are modified, working to leading order in the coupling constant and the rotation parameter. Finally, we study the superradiance of a massive scalar wave scattering off slowly rotating black holes. Using direct integration, we derive the superradiant conditions and compute the energy flux through the event horizon and amplification factor. We demonstrate how the flux and amplification factor will change as a function of the black hole rotation and frequency of the incident wave.

Figures

Figures reproduced from arXiv: 2505.01054 by the authors.

Figure 1
Figure 1. Left: The behavior of the full continued fraction s [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Angular velocity P(r) as a function of r for p0 = 0.5, p1 = −1.2. The solid line is the full continued fraction solution, the near-horizon approximation is the red dot-dash line, and the large r approximation is the blue dashed line. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. The behavior of ˆω+ in terms of βˆ is shown. 3 Geodesics We now consider geodesic motion about the slowly rotating black hole solution. We shall employ the spacetime metric in the small-β approximation (24)–(26) in our analysis. Using the Hamilton-Jacobi Method, it is easy to obtain the system of equations [35, 36, 37] ˙t = 1 Nf (E − aP Lz), φ˙ = aEP Nf + Lz r 2 sin2 θ , r2 ˙θ = ± s J 2 − L 2 z sin2 θ , r˙ 2 = 1 N … view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The behavior of ISCO quantities in terms of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: The left panel shows the radius of the photon ring, w [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: The ratio of angular velocities for the photon ring [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: The behavior of λ (0), and λ (1)M in terms of M for the photon ring. The blue solid line is the zeroth-order term in the static solution, and black solid line is the leading-order correction due to rotation. The red-dashed curves represent the result from Einstein’s gr…
Figure 8
Figure 8. Figure 8: The behavior of FE/Ω in terms of ̟ for a/Mˆ = 0.1, 0.2, 0.25, 0.3, l = m = 1, µ/Mˆ = 0.01, r+ = 3Mˆ and βˆ = 0 (left), βˆ = 0.1 and a/Mˆ = 0.05, 0.1, 0.15, 0.2 (middle). The behavior of FE/Ω in terms βˆ for ˆω = 0.1, a = 0.2 (right). Here Ω = R dθ sin θ|S(θ)| 2 Conside…

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.