Relativistic Keplerian Disk Model

• The relativistic Keplerian accretion disk model is a thin, optically thick framework where gas moves in nearly circular, geodesic orbits around compact objects in a relativistic spacetime.
• It underpins observational techniques like continuum fitting and X‐ray reflection spectroscopy by linking inner disk dynamics such as ISCO location and Doppler effects to black hole spin and disk structure.
• Modifications including quadrupolar deformations, extra stresses, and disk thickness variations reveal the model’s sensitivity to non-Keplerian effects and the conditions under which its assumptions hold.

Searching arXiv for the cited disk-model papers and closely related thin-disk / Keplerian accretion work to ground the article in the current arXiv record. Searching arXiv for the cited disk-model papers and closely related thin-disk / Keplerian accretion work to ground the article in the current arXiv record. Searching arXiv for "Testing the Keplerian disk hypothesis using X-ray reflection spectroscopy" and related relativistic thin-disk models. A relativistic Keplerian accretion disk model is a description of accreting gas around a compact object in which the disk is treated as geometrically thin, optically thick, equatorial, and axisymmetric, while the gas motion is approximated by circular Keplerian or nearly geodesic circular orbits in a relativistic spacetime. In the arXiv literature, this model appears most commonly in the Novikov–Thorne thin-disk framework and in its observational implementations for continuum fitting, relativistic reflection, and broadened-line calculations, while a parallel literature examines how the same framework is modified by quadrupolar deformations, external distortions, thickness effects, fallback feeding, or explicitly non-Keplerian motion (Tripathi et al., 2020, Yilmaz et al., 2023, Faraji et al., 2020, Faraji, 23 May 2025, Mageshwaran et al., 2020).

1. Standard relativistic thin-disk framework

The standard reference point is the Novikov-Thorne model, described as the standard framework for a geometrically thin and optically thick accretion disk around a black hole. In this framework, the spacetime is stationary, axisymmetric, and asymptotically flat; the disk lies on the equatorial plane; radial heat transport is negligible compared with surface radiation; and matter moves on nearly geodesic circular orbits (Tripathi et al., 2020).

For a stationary, axisymmetric metric,

$$ds^2 = g_{tt} dt^2 + 2 g_{t\phi} dt\, d\phi + g_{rr} dr^2 + g_{\theta\theta} d\theta^2 + g_{\phi\phi} d\phi^2 \, ,$$

the Keplerian angular velocity for circular equatorial geodesics is obtained from the metric derivatives. In the form quoted for the general relativistic thin-disk machinery,

$$\Omega_{\rm K} = \frac{-\partial_r g_{t\phi} \pm \sqrt{\left( \partial_r g_{t\phi} \right)^2 - \left( \partial_r g_{tt} \right)\left( \partial_r g_{\phi\phi} \right)}}{\partial_r g_{\phi\phi}} \, ,$$

and the gas 4-velocity is

$u^\mu = \dot{t}(1,0,0,\Omega)\, ,$

with

$$\dot{t} = \frac{1}{\sqrt{-g_{tt} - 2\Omega g_{t\phi} - \Omega^2 g_{\phi\phi}}} \, .$$

In the Novikov-Thorne picture, the gas motion is therefore fully characterized by $\Omega=\Omega_{\rm K}$ (Tripathi et al., 2020).

This Keplerian prescription is also the basis of the relativistic continuum models KERRBB and KYNBB. Both assume a Novikov–Thorne-type disk: geometrically thin, optically thick, and Keplerian. The local emission is built from blackbody annuli with a radial temperature profile, and the observed spectrum is obtained by ray tracing through the Kerr spacetime, including gravitational redshift, Doppler boosting, frame dragging, light bending, and returning radiation / self-irradiation (Yilmaz et al., 2023). A closely related formulation is used in broadened-line morphology studies, where the “Standard / Novikov–Thorne disk model” means a geometrically thin, optically thick, equatorial, and axisymmetric disk whose emitting particles move on stable circular orbits in the equatorial plane (Gates et al., 2024).

The inner edge is commonly taken to be the innermost stable circular orbit. For Kerr spacetime the ISCO radius is written as

$$R_{\rm ISCO}=R_{g}\left\{3+A_{2} \pm\left[\left(3-A_{1}\right)\left(3+A_{1}+2 A_{2}\right)\right]^{1 / 2}\right\},$$

with

$$A_{1}=1+\left(1-a_{*}^{2}\right)^{1 / 3}\left[\left(1+a_{*}\right)^{1 / 3}+\left(1-a_{*}\right)^{1 / 3}\right],$$

$A_{2}= \left(3 a_{*}^{2}+A_{1}^{2}\right)^{1 / 2},$

and

$a_{*}=\frac{|\mathbf{J}| c}{G M^{2}}.$

Higher prograde spin means a smaller ISCO, so the disk can extend deeper into the potential well, producing a hotter, more relativistically shifted thermal spectrum (Yilmaz et al., 2023).

2. Radiative structure, flux, and local observables

The standard thin-disk model is built from conservation laws for rest mass, energy, and angular momentum, combined with vertical hydrostatic equilibrium, viscosity closure, and an opacity law. In the q-metric thin-disk treatment, the fundamental equations are written as

$(\rho u^\mu)_{;\mu}=0,$

$$\Omega_{\rm K} = \frac{-\partial_r g_{t\phi} \pm \sqrt{\left( \partial_r g_{t\phi} \right)^2 - \left( \partial_r g_{tt} \right)\left( \partial_r g_{\phi\phi} \right)}}{\partial_r g_{\phi\phi}} \, ,$$0

with projections

$$\Omega_{\rm K} = \frac{-\partial_r g_{t\phi} \pm \sqrt{\left( \partial_r g_{t\phi} \right)^2 - \left( \partial_r g_{tt} \right)\left( \partial_r g_{\phi\phi} \right)}}{\partial_r g_{\phi\phi}} \, ,$$1

and the surface density

$$\Omega_{\rm K} = \frac{-\partial_r g_{t\phi} \pm \sqrt{\left( \partial_r g_{t\phi} \right)^2 - \left( \partial_r g_{tt} \right)\left( \partial_r g_{\phi\phi} \right)}}{\partial_r g_{\phi\phi}} \, ,$$2

For steady accretion this gives

$$\Omega_{\rm K} = \frac{-\partial_r g_{t\phi} \pm \sqrt{\left( \partial_r g_{t\phi} \right)^2 - \left( \partial_r g_{tt} \right)\left( \partial_r g_{\phi\phi} \right)}}{\partial_r g_{\phi\phi}} \, ,$$3

so that

$$\Omega_{\rm K} = \frac{-\partial_r g_{t\phi} \pm \sqrt{\left( \partial_r g_{t\phi} \right)^2 - \left( \partial_r g_{tt} \right)\left( \partial_r g_{\phi\phi} \right)}}{\partial_r g_{\phi\phi}} \, ,$$4

The viscous stress is written in thin-disk form as

$$\Omega_{\rm K} = \frac{-\partial_r g_{t\phi} \pm \sqrt{\left( \partial_r g_{t\phi} \right)^2 - \left( \partial_r g_{tt} \right)\left( \partial_r g_{\phi\phi} \right)}}{\partial_r g_{\phi\phi}} \, ,$$5

and the vertically integrated stress as

$$\Omega_{\rm K} = \frac{-\partial_r g_{t\phi} \pm \sqrt{\left( \partial_r g_{t\phi} \right)^2 - \left( \partial_r g_{tt} \right)\left( \partial_r g_{\phi\phi} \right)}}{\partial_r g_{\phi\phi}} \, ,$$6

The pressure is taken as the sum of gas and radiation pressure,

$$\Omega_{\rm K} = \frac{-\partial_r g_{t\phi} \pm \sqrt{\left( \partial_r g_{t\phi} \right)^2 - \left( \partial_r g_{tt} \right)\left( \partial_r g_{\phi\phi} \right)}}{\partial_r g_{\phi\phi}} \, ,$$7

and the vertical hydrostatic relation is approximated by

$$\Omega_{\rm K} = \frac{-\partial_r g_{t\phi} \pm \sqrt{\left( \partial_r g_{t\phi} \right)^2 - \left( \partial_r g_{tt} \right)\left( \partial_r g_{\phi\phi} \right)}}{\partial_r g_{\phi\phi}} \, ,$$8

Radiative diffusion is written as

$$\Omega_{\rm K} = \frac{-\partial_r g_{t\phi} \pm \sqrt{\left( \partial_r g_{t\phi} \right)^2 - \left( \partial_r g_{tt} \right)\left( \partial_r g_{\phi\phi} \right)}}{\partial_r g_{\phi\phi}} \, ,$$9

in the distorted-Schwarzschild construction, while the q-metric treatment uses

$u^\mu = \dot{t}(1,0,0,\Omega)\, ,$0

with $u^\mu = \dot{t}(1,0,0,\Omega)\, ,$1 (Faraji et al., 2020, Faraji, 23 May 2025).

For relativistic thin disks, the emitted flux is determined by the Page–Thorne or Novikov–Thorne expression. In the q-metric paper it is given in standard form as

$u^\mu = \dot{t}(1,0,0,\Omega)\, ,$2

The inner edge is taken to be the ISCO, so the flux profile is controlled by the geodesic functions $u^\mu = \dot{t}(1,0,0,\Omega)\, ,$3, $u^\mu = \dot{t}(1,0,0,\Omega)\, ,$4, $u^\mu = \dot{t}(1,0,0,\Omega)\, ,$5, and the metric determinant (Faraji, 23 May 2025).

A related but observationally distinct construction is X-ray reflection spectroscopy. There the reflection spectrum is computed in two stages: the local reflection spectrum in the gas rest frame, determined by atomic physics, and a relativistic convolution over the whole disk, including gravitational redshift, Doppler boosting, light bending, and projection effects. The observed flux is written as

$u^\mu = \dot{t}(1,0,0,\Omega)\, ,$6

where $u^\mu = \dot{t}(1,0,0,\Omega)\, ,$7 is the redshift factor (Tripathi et al., 2020).

Line-emission studies use the same redshift mapping in a more analytic form. The observed specific flux is written as

$u^\mu = \dot{t}(1,0,0,\Omega)\, ,$8

with

$u^\mu = \dot{t}(1,0,0,\Omega)\, ,$9

For a monochromatic line, a rescaled profile is defined by

$$\dot{t} = \frac{1}{\sqrt{-g_{tt} - 2\Omega g_{t\phi} - \Omega^2 g_{\phi\phi}}} \, .$$0

In this formulation, the broadened profile is shaped by gravitational redshift, special-relativistic Doppler shifts / beaming, and the orbital motion of the emitters (Gates et al., 2024).

3. Model realizations in Kerr and other relativistic backgrounds

In Kerr spacetime, KERRBB and KYNBB provide two implementations of relativistic thin-disk continuum fitting. When KYNBB is configured with

$$\dot{t} = \frac{1}{\sqrt{-g_{tt} - 2\Omega g_{t\phi} - \Omega^2 g_{\phi\phi}}} \, .$$1

the two models provide identical results with black hole spin measurements, disk temperature, and disk luminosity when the inner edge of the accretion disk is set at the innermost stable circular orbit for the same accretion rates. The paper reports only about a 4.5% average difference in spin between the two models, indicating that the two implementations of relativistic Keplerian disk emission are effectively equivalent for this use case (Yilmaz et al., 2023).

Relativistic thin-disk constructions have also been developed for non-Kerr or non-isolated backgrounds. One example is the distorted Schwarzschild black hole, a static and axially symmetric spacetime connected to an external distribution of matter. In prolate spheroidal coordinates the metric is written as

$$\dot{t} = \frac{1}{\sqrt{-g_{tt} - 2\Omega g_{t\phi} - \Omega^2 g_{\phi\phi}}} \, .$$2

Keeping only the quadrupole distortion,

$$\dot{t} = \frac{1}{\sqrt{-g_{tt} - 2\Omega g_{t\phi} - \Omega^2 g_{\phi\phi}}} \, .$$3

the equatorial distortion functions are

$$\dot{t} = \frac{1}{\sqrt{-g_{tt} - 2\Omega g_{t\phi} - \Omega^2 g_{\phi\phi}}} \, .$$4

For circular equatorial geodesics, the explicit orbital quantities $$\dot{t} = \frac{1}{\sqrt{-g_{tt} - 2\Omega g_{t\phi} - \Omega^2 g_{\phi\phi}}} \, .$$5, $$\dot{t} = \frac{1}{\sqrt{-g_{tt} - 2\Omega g_{t\phi} - \Omega^2 g_{\phi\phi}}} \, .$$6, and $$\dot{t} = \frac{1}{\sqrt{-g_{tt} - 2\Omega g_{t\phi} - \Omega^2 g_{\phi\phi}}} \, .$$7 reduce to the Schwarzschild expressions when $$\dot{t} = \frac{1}{\sqrt{-g_{tt} - 2\Omega g_{t\phi} - \Omega^2 g_{\phi\phi}}} \, .$$8 (Faraji et al., 2020).

A second example is the q-metric, a static, axisymmetric, asymptotically flat vacuum solution that includes a quadrupole deformation of the central object. In that interpretation, $$\dot{t} = \frac{1}{\sqrt{-g_{tt} - 2\Omega g_{t\phi} - \Omega^2 g_{\phi\phi}}} \, .$$9 gives Schwarzschild, $\Omega=\Omega_{\rm K}$0 corresponds to an oblate source, and $\Omega=\Omega_{\rm K}$1 to a prolate source. The disk model again uses the standard relativistic thin, optically thick, Keplerian disk, with the angular velocity computed from equatorial circular geodesics (Faraji, 23 May 2025).

The ISCO shift is the main structural effect in both deformed backgrounds. In the distorted Schwarzschild construction, stable circular orbits and the ISCO exist only for

$\Omega=\Omega_{\rm K}$2

with

$\Omega=\Omega_{\rm K}$3

For negative $\Omega=\Omega_{\rm K}$4, the ISCO moves closer to the horizon than in Schwarzschild; for positive $\Omega=\Omega_{\rm K}$5, the ISCO moves outward (Faraji et al., 2020). The q-metric paper states the same qualitative trend: Schwarzschild has $\Omega=\Omega_{\rm K}$6, $\Omega=\Omega_{\rm K}$7 moves the ISCO inward, and $\Omega=\Omega_{\rm K}$8 moves it outward (Faraji, 23 May 2025).

A more specialized relativistic realization is the time-dependent Kerr thin disk for tidal disruption events. There the disk is built in the equatorial plane of a rotating black hole, with Keplerian specific angular momentum

$\Omega=\Omega_{\rm K}$9

and a relativistic diffusion-like equation for the surface density,

$$R_{\rm ISCO}=R_{g}\left\{3+A_{2} \pm\left[\left(3-A_{1}\right)\left(3+A_{1}+2 A_{2}\right)\right]^{1 / 2}\right\},$$0

That model couples continuous mass supply from fallback at the outer boundary to accretion onto the black hole, while retaining Kerr corrections in the viscous transport and in the disk-height relation (Mageshwaran et al., 2020).

4. Observational use: continuum fitting, reflection, and broadened lines

Relativistic Keplerian disk models are used operationally in three observational channels: thermal continuum fitting, X-ray reflection spectroscopy, and relativistically broadened line modeling.

In continuum fitting, the central assumption is that the disk is in the standard thin-disk regime and that the inner edge is at the ISCO. The GRO J1655-40 study tested this with RXTE/PCA data from the 2005 outburst. With a fixed black hole spin value at $$R_{\rm ISCO}=R_{g}\left\{3+A_{2} \pm\left[\left(3-A_{1}\right)\left(3+A_{1}+2 A_{2}\right)\right]^{1 / 2}\right\},$$1, both KERRBB and KYNBB gave poor fits for about 89% of the observations, with reduced chi-square values often in the range

$$R_{\rm ISCO}=R_{g}\left\{3+A_{2} \pm\left[\left(3-A_{1}\right)\left(3+A_{1}+2 A_{2}\right)\right]^{1 / 2}\right\},$$2

Allowing the spin parameter to vary improved the fit statistic significantly, with reduced $$R_{\rm ISCO}=R_{g}\left\{3+A_{2} \pm\left[\left(3-A_{1}\right)\left(3+A_{1}+2 A_{2}\right)\right]^{1 / 2}\right\},$$3 values below 2, and both models revealed black hole spin values varying between

$$R_{\rm ISCO}=R_{g}\left\{3+A_{2} \pm\left[\left(3-A_{1}\right)\left(3+A_{1}+2 A_{2}\right)\right]^{1 / 2}\right\},$$4

The paper interprets this not as genuine spin evolution but as a variable inner edge of the disk throughout different accretion states (Yilmaz et al., 2023).

In reflection spectroscopy, the relativistic shaping of fluorescent lines, especially the Fe K$$R_{\rm ISCO}=R_{g}\left\{3+A_{2} \pm\left[\left(3-A_{1}\right)\left(3+A_{1}+2 A_{2}\right)\right]^{1 / 2}\right\},$$5 complex, and of the Compton hump around 20–30 keV provides a direct probe of the velocity field of the emitting gas. The modified RELXILL implementation introduces a phenomenological parameter $$R_{\rm ISCO}=R_{g}\left\{3+A_{2} \pm\left[\left(3-A_{1}\right)\left(3+A_{1}+2 A_{2}\right)\right]^{1 / 2}\right\},$$6 through

$$R_{\rm ISCO}=R_{g}\left\{3+A_{2} \pm\left[\left(3-A_{1}\right)\left(3+A_{1}+2 A_{2}\right)\right]^{1 / 2}\right\},$$7

with $$R_{\rm ISCO}=R_{g}\left\{3+A_{2} \pm\left[\left(3-A_{1}\right)\left(3+A_{1}+2 A_{2}\right)\right]^{1 / 2}\right\},$$8 corresponding to standard Keplerian motion, $$R_{\rm ISCO}=R_{g}\left\{3+A_{2} \pm\left[\left(3-A_{1}\right)\left(3+A_{1}+2 A_{2}\right)\right]^{1 / 2}\right\},$$9 to super-Keplerian motion, and $$A_{1}=1+\left(1-a_{*}^{2}\right)^{1 / 3}\left[\left(1+a_{*}\right)^{1 / 3}+\left(1-a_{*}\right)^{1 / 3}\right],$$0 to sub-Keplerian motion. The modification enters only in the relativistic transfer function/convolution; the atomic reflection spectrum in the gas frame is unchanged (Tripathi et al., 2020).

The Suzaku application to GRS 1915+105 illustrates the diagnostic power and the degeneracy structure of this approach. With inclination free, the fit obtained $$A_{1}=1+\left(1-a_{*}^{2}\right)^{1 / 3}\left[\left(1+a_{*}\right)^{1 / 3}+\left(1-a_{*}\right)^{1 / 3}\right],$$1, $$A_{1}=1+\left(1-a_{*}^{2}\right)^{1 / 3}\left[\left(1+a_{*}\right)^{1 / 3}+\left(1-a_{*}\right)^{1 / 3}\right],$$2, and $$A_{1}=1+\left(1-a_{*}^{2}\right)^{1 / 3}\left[\left(1+a_{*}\right)^{1 / 3}+\left(1-a_{*}\right)^{1 / 3}\right],$$3, which would suggest a sub-Keplerian disk. However, the paper finds that $$A_{1}=1+\left(1-a_{*}^{2}\right)^{1 / 3}\left[\left(1+a_{*}\right)^{1 / 3}+\left(1-a_{*}\right)^{1 / 3}\right],$$4 and the inclination angle $$A_{1}=1+\left(1-a_{*}^{2}\right)^{1 / 3}\left[\left(1+a_{*}\right)^{1 / 3}+\left(1-a_{*}\right)^{1 / 3}\right],$$5 are strongly degenerate because lowering $$A_{1}=1+\left(1-a_{*}^{2}\right)^{1 / 3}\left[\left(1+a_{*}\right)^{1 / 3}+\left(1-a_{*}\right)^{1 / 3}\right],$$6 weakens Doppler boosting while increasing $$A_{1}=1+\left(1-a_{*}^{2}\right)^{1 / 3}\left[\left(1+a_{*}\right)^{1 / 3}+\left(1-a_{*}\right)^{1 / 3}\right],$$7 strengthens it. Fixing $$A_{1}=1+\left(1-a_{*}^{2}\right)^{1 / 3}\left[\left(1+a_{*}\right)^{1 / 3}+\left(1-a_{*}\right)^{1 / 3}\right],$$8 to $$A_{1}=1+\left(1-a_{*}^{2}\right)^{1 / 3}\left[\left(1+a_{*}\right)^{1 / 3}+\left(1-a_{*}\right)^{1 / 3}\right],$$9, $A_{2}= \left(3 a_{*}^{2}+A_{1}^{2}\right)^{1 / 2},$0, and $A_{2}= \left(3 a_{*}^{2}+A_{1}^{2}\right)^{1 / 2},$1 moves the fitted $A_{2}= \left(3 a_{*}^{2}+A_{1}^{2}\right)^{1 / 2},$2 toward larger values; and if one adopts the independent jet-based estimate

$A_{2}= \left(3 a_{*}^{2}+A_{1}^{2}\right)^{1 / 2},$3

then the fitted $A_{2}= \left(3 a_{*}^{2}+A_{1}^{2}\right)^{1 / 2},$4 is consistent with zero, meaning the disk is consistent with Keplerian motion (Tripathi et al., 2020).

Broadened-line morphology studies provide a complementary analytic interpretation. Under the standard thin Keplerian disk model, the observed line profile is controlled by the topology of constant-redshift contours on the observer’s screen. The flux-contributing region is bounded by the projected inner and outer disk radii, and the critical redshift values define the maximum observable redshift and maximum observable blueshift,

$A_{2}= \left(3 a_{*}^{2}+A_{1}^{2}\right)^{1 / 2},$5

The line-shape features—extent, kinks, and fall-off—encode the black hole spin, viewing inclination, and locations of the disk’s inner and outer edges (Gates et al., 2024).

5. Departures from strict Keplerianity and their interpretation

A recurring result in the literature is that the relativistic Keplerian disk model is both powerful and fragile: it provides clean inferences only to the extent that its kinematic assumptions are accurate.

The clearest phenomenological test is the modified RELXILL parameterization,

$A_{2}= \left(3 a_{*}^{2}+A_{1}^{2}\right)^{1 / 2},$6

introduced specifically to test the Keplerian disk hypothesis. The paper emphasizes that $A_{2}= \left(3 a_{*}^{2}+A_{1}^{2}\right)^{1 / 2},$7 mainly changes the Doppler boosting, so its effect is strongest at high inclination. This is why the iron line profile is much more sensitive to $A_{2}= \left(3 a_{*}^{2}+A_{1}^{2}\right)^{1 / 2},$8 when the disk is viewed edge-on than face-on (Tripathi et al., 2020).

Broadened-line morphology studies reach a related conclusion by relaxing the standard orbital assumption parametrically. In the “Cunningham model,” the disk still follows circular orbits outside the ISCO, but inside the ISCO particles plunge geodesically; and in the phenomenological non-Keplerian circular-orbit model, a “Keplerianity” parameter $A_{2}= \left(3 a_{*}^{2}+A_{1}^{2}\right)^{1 / 2},$9 scales the angular momentum away from the Keplerian value, with $a_{*}=\frac{|\mathbf{J}| c}{G M^{2}}.$0 Keplerian, $a_{*}=\frac{|\mathbf{J}| c}{G M^{2}}.$1 sub-Keplerian, and $a_{*}=\frac{|\mathbf{J}| c}{G M^{2}}.$2 super-Keplerian. The paper states that even a few-percent change in $a_{*}=\frac{|\mathbf{J}| c}{G M^{2}}.$3 can noticeably shift the line-profile kinks and MOB/MOR values, especially at high inclination, and that allowing the disk particles to deviate from stable circular orbits rapidly degenerates the characteristic features of the line profile under the Standard disk model (Gates et al., 2024).

This model dependence also appears in continuum fitting. The GRO J1655-40 analysis shows that relativistic Keplerian disk models can successfully reproduce thermal disk spectra and infer spin only when the disk is close to the standard thin-disk regime. A fixed-spin, fixed-ISCO interpretation fails for most observations of the 2005 outburst; the variable fitted spin is therefore best interpreted as evidence for variable disk truncation or variable inner edge, not genuine spin evolution (Yilmaz et al., 2023).

A plausible implication is that “Keplerian” in this context is not merely a mathematical convenience but an empirical hypothesis whose validity depends on source state, geometry, and the presence or absence of extra stresses or non-geodesic forces. That implication is stated directly in the reflection work, where $a_{*}=\frac{|\mathbf{J}| c}{G M^{2}}.$4 would indicate extra forces or stresses, for example magnetic or viscous effects modifying orbital motion (Tripathi et al., 2020).

6. Thickness, local Eddington limit, and the boundary of validity

The standard relativistic Keplerian disk model relies upon geometrical thinness. Whenever this condition is violated, new physical effects become important such as radial energy advection and mass loss from the disc. Near the local Eddington limit, the relevant balance is not a single global luminosity bound but a local vertical flux condition,

$a_{*}=\frac{|\mathbf{J}| c}{G M^{2}}.$5

where $a_{*}=\frac{|\mathbf{J}| c}{G M^{2}}.$6 is the component of gravity normal to the photosphere (Abolmasov et al., 2015).

For a relativistic thin disc, the Novikov–Thorne flux is written as

$a_{*}=\frac{|\mathbf{J}| c}{G M^{2}}.$7

or in dimensionless form,

$a_{*}=\frac{|\mathbf{J}| c}{G M^{2}}.$8

with

$a_{*}=\frac{|\mathbf{J}| c}{G M^{2}}.$9

The paper’s main quantitative result for the thin relativistic disc is that GR corrections increase the local Eddington limit by about a factor of two compared with the simplest Newtonian estimate, and the critical mass accretion rate increases by about a factor of $(\rho u^\mu)_{;\mu}=0,$0 relative to the non-relativistic thin-disc estimate (Abolmasov et al., 2015).

Once thickness is included, the same paper argues that the effective Eddington threshold can be pushed higher still, though advection lowers the radiative efficiency by about a factor of several. Near the critical rate, the radiative efficiency can fall to roughly 1–2%, and for very large accretion rates the luminosity saturates at only a few times $(\rho u^\mu)_{;\mu}=0,$1, with a characteristic range of roughly $(\rho u^\mu)_{;\mu}=0,$2–$(\rho u^\mu)_{;\mu}=0,$3 in the high-$(\rho u^\mu)_{;\mu}=0,$4 regime (Abolmasov et al., 2015).

The time-dependent TDE disk offers another explicit example of the transition away from the simplest steady thin-disk picture. There the disk remains thin and gas-pressure dominated, but its structure evolves because fallback debris forms a seed disc in time $(\rho u^\mu)_{;\mu}=0,$5 and thereafter adds mass at the outer boundary while accretion removes mass at the inner edge. The model derives an explicit relativistic disk height,

$(\rho u^\mu)_{;\mu}=0,$6

and finds that the disk mass and luminosity decay at late times as

$(\rho u^\mu)_{;\mu}=0,$7

for full disruptions, and

$(\rho u^\mu)_{;\mu}=0,$8

for partial disruptions. The luminosity therefore declines faster than the luminosity inferred using $(\rho u^\mu)_{;\mu}=0,$9 (Mageshwaran et al., 2020).

This suggests that the relativistic Keplerian accretion disk model is best understood not as a single immutable solution, but as a controlled regime whose defining assumptions—thinness, local radiation of dissipated heat, equatorial circular motion, and an ISCO inner edge—remain accurate only over part of the accretion-state space. The arXiv literature consistently treats departures from those assumptions not as minor corrections, but as changes that can alter inferred spin, inclination, flux, and even the interpretation of line morphology itself (Abolmasov et al., 2015, Yilmaz et al., 2023, Gates et al., 2024).