Gravitational lensing and shadows in the toron solution of Einstein's equations using ray tracing methods

D. Korotkin; Eddy B. de Leon. C. Klein

REVIEW 3 major objections 4 minor 33 references

Gravitational lensing and shadows in the toron solution of Einstein's equations using ray tracing methods

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The toron spacetime, with zero ordinary mass but a gravimagnetic parameter, produces NUT-like lensing and shadows, not Kerr-like.

desk verdict First ray tracing of the toron spacetime: the NUT-like shadow and blind-region claims rest on an unjustified absorbing boundary at the ergosphere; deserves a referee but needs a fix. read the letter →

arxiv 2501.04509 v2 pith:NWJM5ZEI submitted 2025-01-08 gr-qc nlin.SI

Eddy B. de Leon. C. Klein , D. Korotkin This is my paper · ORCID

classification gr-qcnlin.SI MSC 83C1083C5783C15

keywords toronspacetimeErnstequationraytracinggravitationallensingblackholeshadowNUTnullgeodesicsergosphere

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

This paper aims to establish that the toron, a vacuum solution of the stationary axisymmetric Einstein equations built from elliptic functions, behaves as a gravimagnetic lens. It has zero ordinary mass and an asymptotic Ernst expansion whose leading correction is purely imaginary, $E = 1 - i\sigma\sinh(2\pi\alpha)/z + O(1/z^2)$, so its large-distance behaviour matches the NUT solution with vanishing mass. Using backwards ray tracing, the authors compare null geodesics, light rings, shadows, and images of the celestial sphere in Schwarzschild, Kerr, NUT, and toron spacetimes. Their central finding is that the primary and secondary images in NUT and toron spacetimes look qualitatively similar to each other and quite different from Kerr: both spacetimes hide a region of the celestial sphere and twist the apparent image, while the toron additionally shifts and deforms its shadow because the same parameter $\alpha$ also produces angular momentum. The significance is that a massless exact solution, not just a black hole, can produce NUT-style lensing signatures that could be sought in images of compact objects.

What carries the argument

The central object is the toron Ernst potential, $$E(\xi,\bar\xi)=\frac{\vartheta\left(\int_{\infty_+}^{\xi}\omega+i\$\alpha$,\tau\right)}{\vartheta\left(\int_{\infty_-}^{\xi}\omega+i\$\alpha$,\tau\right)},$$ with $p=0$, $q=i\alpha$, where $\omega$ is the normalized holomorphic differential on the elliptic curve $\mu^2=(\lambda-i\sigma)(\lambda+i\sigma)(\lambda-\xi)(\lambda-\bar\xi)$ and the $\theta$ functions encode the dependence on the Weyl coordinates. This object carries the argument because the single real parameter $\alpha$ controls both the gravimagnetic mass $i\sigma\sinh(2\pi\alpha)$ and the angular momentum of the disk-like source. The metric functions $f$, $A$, and $k$ are obtained from it by quadratures, and the fact that $A$ cannot be made to vanish on the whole symmetry axis produces the twisting of images and the out-of-plane deflection. The ray tracing is driven by backwards integration of the null geodesic equations from a virtual camera, with a termination rule that paints a pixel black once $f=\Re E<\epsilon$ at the approach to the ergosphere.

What would settle it

Integrate backward null geodesics in the toron spacetime using coordinates that do not stop at the surface where $f=0$ and check whether any ray that reaches the observer passes through the first ergosphere to a luminous source; if any do, the black-pixel termination criterion misclassifies them and the shadow and blind region are not physical.

Watch

Extended reading notes

Core claim

The paper's claim is that the toron solution of [20], written as a ratio of $\theta$ functions on an elliptic curve, is asymptotically identical to the massless NUT spacetime: on the symmetry axis the Ernst potential is $E(z,0)=(z\pm\sqrt{z^2+\sigma^2}-i\sigma e^{-2\pi\alpha})/(z\pm\sqrt{z^2+\sigma^2}-i\sigma e^{2\pi\alpha})$, which gives the imaginary mass $i\sigma\sinh(2\pi\alpha)$. Ray tracing then shows the geodesic flow is NUT-like near the object: initially equatorial photons leave the plane, light rings lift off the equatorial plane, the Einstein ring disappears, and a quasi-circular blind region appears on the celestial sphere behind the lens. In the toron the blind region is shifted sideways and slightly deformed, and the shadow is non-circular, because the disk also carries angular momentum. The paper further argues that the toron has an infinite family of toroidal ergospheres that accumulate on a ring at $\rho=1$, $z=0$, and that the first ergosphere, approached by any ray with $f<\epsilon$, cannot be crossed since Weyl coordinate time would go to infinity.

Load-bearing premise

The load-bearing premise is that no light ray can cross the surface where the metric function $f$ vanishes, so the simulation stops and colors the pixel black there; if photons can cross that surface, as they do in Kerr, the shadow and blind region would be enlarged or misplaced.

Editorial extensions

If this is right

A massless vacuum compact object with a gravimagnetic parameter would show a NUT-type blind region on the celestial sphere and no Einstein ring for an equatorial observer, unlike Kerr.
Larger $\alpha$ means stronger out-of-plane deflection, stronger twisting of the apparent image, and a larger shadow that shifts further sideways.
The toroidal ergosphere topology leaves a signature: the shadow boundary is set by the outermost ergosphere rather than by a spherical photon sphere.
Comparing primary and secondary images in the same frame gives a qualitative test: NUT-like images keep the upper and lower celestial hemispheres on the same side, while Kerr-like images flip the secondary copy.

Reading between the lines

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

If the ergosphere impenetrability assumption is wrong, the reported shadow and blind region could be artifacts of the $f<\epsilon$ cut-off; integrating rays in coordinates that continue past $f=0$ would test this directly.
The authors' closing question suggests a testable programme: higher-genus toron-type solutions without a NUT parameter should produce lensing closer to Kerr; if so, the NUT-likeness identified here is tied to the imaginary mass term, not to the elliptic construction itself.
The shadow offset and blind-region shift could be inverted to estimate $\alpha$ from a single image of an exotic compact object, since these are the only parameters shaping the lensing signature.
Because the branch disk is treated as totally absorbing, the shadow interior conflates true photon capture with absorption at the disk; a disk with finite emissivity would brighten part of the reported shadow.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

You should know this paper applies an established ray tracing code to the toron, a vacuum solution with zero mass and NUT parameter, and claims qualitatively NUT-like lensing: a shifted asymmetric shadow and a blind region on the celestial sphere. The benchmarks are real: the code reproduces known images for Schwarzschild, Kerr, and NUT, and the analytic axis limits and asymptotic expansion match [20]. If you only read one thing, read the termination criterion in §4.3.

What is actually new: the toron solution and the ray tracing method both predate this paper, but this is the first application of the method to the toron. The deflection patterns in §6.1 give a concrete picture of gravimagnetic dragging, and the parameter study in α is systematic. The analytical part is consistent with prior work and the code passes sanity checks, which is genuine support.

The soft spot is load-bearing. The code stops every backwards ray at the first ergosphere (f < ε) and colors the pixel black, with the argument that coordinate time goes to infinity and therefore the ergosphere cannot be crossed in Weyl coordinates. That argument does not hold up. In Weyl coordinates for Kerr, the same coordinate singularity occurs at the ergosurface, and photons do cross it in finite affine parameter; the ergosphere is not a boundary of the physical spacetime. For NUT, f = 0 is the horizon rod, so the same criterion is really a horizon stop, which is fine. For the toron, E0 is a toroidal surface where g_tt = 0 but no horizon condition is computed. If null geodesics cross E0 and re-emerge, the shifted shadow in Fig. 27 and the blind region in Fig. 28 are at least partly numerical artifacts. Since the qualitative 'NUT-like' claim rests on those figures, this is a serious weakness, not a cosmetic one.

Two smaller issues: there are no convergence or resolution checks, and the interpretation that α controls angular momentum is read off from the images rather than computed from an independent integral (say, a Komar charge). Neither is fatal, but both add to the sense that the numerical claims need hardening.

Who should read it: anyone working on exact solutions and shadow phenomenology. The analytical parts are solid and the ray tracing setup is reusable. But I would not cite the shadow or blind-region results as they stand. My recommendation: send it to peer review—a good referee can push for a proper treatment of the ergosphere, either by integrating in a regular coordinate system or by proving that E0 is truly impenetrable. That is fixable, but it is real work.

Referee Report

3 major / 4 minor

Summary. The paper studies the 'toron' solution of the stationary axisymmetric Einstein equations in vacuum, an elliptic-function solution whose asymptotic Ernst potential contains a purely imaginary mass parameter iσ sinh(2πα). The authors derive the axis limit and the ergosphere structure, and then use the ray-tracing code of [8] to compare null geodesics and synthetic images in Schwarzschild, Kerr, NUT, and toron spacetimes. The main observational claim is that toron lensing is qualitatively NUT-like: the primary and secondary images are twisted, there is a quasi-circular blind region on the celestial sphere, and the shadow is non-circular and shifted. The numerical code is benchmarked against known Schwarzschild, Kerr, and NUT images, and the analytical asymptotics are consistent with the earlier toron literature.

Significance. If the main claim holds, the paper provides a nontrivial example of a vacuum spacetime with vanishing real mass and pure NUT-type gravimagnetic mass, whose lensing differs qualitatively from Kerr and resembles NUT despite having no event horizon. The analytical part is valuable: the axis limit and the imaginary-mass asymptotics are derived explicitly, and the paper reproduces several known ray-tracing results as checks. The main fragility is that the shadow and blind-region statements rely on a numerical termination rule whose physical justification is not established; this is a load-bearing issue for the central conclusions.

major comments (3)

[Sec. 4.3 (termination criterion 2)] The termination rule that stops every backward integration once f < ε and colors the pixel black is load-bearing for the shadow and blind-region claims, but its physical justification is not established. The sentence 'the coordinate time will go to infinity; therefore, the ergosphere cannot be crossed by any light ray in Weyl coordinates' is an assertion, not a derivation. In Kerr spacetimes, the ergosurface is a regular, crossable surface and null geodesics pass through the ergoregion even though f vanishes there in Weyl-Lewis-Papapetrou coordinates; the divergence of f^{-1} is a coordinate effect, not a spacetime boundary. For the toron, E0 is an ergosurface, not an event horizon or the branch disk, and the paper gives no proof that null geodesics cannot cross it. If such crossing is possible, the black pixels in Figs. 27 and 28 can include rays that actually reach the celestial sphere, so the shifted shadow and the hidden region would be at least partly numerical artifacts. The authors should either prove, by a coordinate-independent argument, that E0 is impenetrable to null geodesics in the toron, or change the termination condition and recompute the images.
[Sec. 7 (conclusion)] The central conclusion that 'the primary images in NUT and toron spacetimes look qualitatively similar' inherits the same assumption, because the comparison uses the same f < ε stopping rule for both spacetimes. For NUT the rule coincides with the event horizon, which is genuinely absorbing, but for toron it does not. The paper should state explicitly which reported features—shadow boundary, blind region, image twist—persist if geodesics are allowed to cross E0, or restrict the claims to the exterior of E0 with a clear caveat.
[Secs. 3.1 and 7] Equation (3.8) determines the imaginary mass parameter σ sinh(2πα), but the claim that α also controls angular momentum is not derived from an invariant quantity. The numerical deflection of rays in Figs. 21, 22, and 25 is suggestive, but it does not by itself establish the Komar angular momentum or the relevant multipole moments. Either compute an invariant angular-momentum measure or soften the statement that α controls both gravimagnetic mass and angular momentum.

minor comments (4)

[Sec. 4.3] The threshold ε in termination criterion 2 is never specified; without a value, the effective stopping surface is not well defined and the images could depend on this free numerical parameter.
[Captions of Figs. 5 and 6] The captions refer to 'Af' without making clear whether the plotted quantity is the product Af or the function A; this should be clarified to avoid confusion.
[Table 2 and Fig. 25] The apparent disk images for α = 0.2 and α = 0.3 are computed with different focal lengths, so the visual comparison mixes the physical effect of α with the changed camera setting; the text should state this explicitly when discussing apparent sizes.
[Appendix B] The paper states that the null condition L = 0 is used to control numerical accuracy, but no quantitative error estimate or convergence test is reported; a short statement of the achieved conservation would strengthen confidence in the images.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the toron ray-tracing results are computed from the fixed toron metric with independent Schwarzschild/Kerr/NUT benchmarks; self-citations are source attributions rather than load-bearing reductions.

full rationale

The paper's central claims are numerical solutions of the null geodesic equations (B.4)-(B.5) for the toron metric (3.1)-(3.5). The metric parameters, such as alpha and sigma=1, are fixed inputs; no observable is fitted and no target quantity is used to define the model. The comparisons against Schwarzschild, Kerr, and NUT spacetimes are independent external benchmarks run with the same integrator, so the central NUT-toron similarity claim is not manufactured by construction. The toron solution is imported from Korotkin's earlier work [20], and the ray-tracing approach from the authors' prior paper [8], but these are source attributions and numerical tools, not premises that already contain the lensing and shadow conclusions. The interpretation that alpha controls angular momentum (§6.3 and Conclusion) is read off from the same shadow images used to motivate it, so it is an interpretation rather than a derivation; it does not reduce any equation to an input. The paper's termination criterion 2 in §4.3 stops integration at f < epsilon and colors pixels black, asserting that 'the ergosphere cannot be crossed by any light ray in Weyl coordinates'; this is an unproved physical assumption for the toron and is a genuine correctness risk for the shadow and blind-region claims, but it is not circular in the enumerated sense: the stopping rule is not fitted from the output and the claimed result is not equivalent to the rule by definition. Therefore the derivation chain is self-contained apart from this physical-assumption risk, and the appropriate circularity score is low.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The lensing results depend on the toron metric, taken from [16,20] as theta-function expressions, and on the numerical integration of the geodesic ODEs (B.4). The free parameters are alpha and sigma (the latter normalized to 1). The paper introduces two ad hoc modeling choices: the branch cut disk is treated as totally absorbing and the first ergosphere as an impenetrable boundary. No new physical entities are postulated.

free parameters (2)

alpha (toron parameter) = scanned: 0.2 and 0.3 in simulations; tabulated from 0 to 0.5
Sets q = i*alpha in the Ernst potential (3.1); controls the imaginary mass sigma*sinh(2*pi*alpha) and apparent angular momentum. Chosen by hand to explore the solution family, not derived from first principles.
sigma (fixed branch point) = 1 (normalization)
Appendix A: 'Without loss of generality, we will assume sigma = 1', a rescaling of xi; the dimensionless combination sigma*sinh(2*pi*alpha) sets the physical scale of the gravimagnetic mass.

assumptions (4)

domain assumption The Ernst equation and its elliptic solutions given by theta functions (3.1)-(3.5) correctly describe a vacuum spacetime.
Taken from [16,20]; the paper does not re-derive the Einstein equations, only uses the metric.
ad hoc to paper The branch cut z=0, 0<rho<1 is a totally absorbing disk; the spacetime is considered on the first sheet only.
Section 3: the Ernst potential is discontinuous for 0<rho<1; the authors choose to treat the disk as light absorbing and ignore the infinite sheeted structure.
ad hoc to paper Light rays reaching the first ergosphere (f < epsilon) are absorbed and cannot contribute to the image.
Section 4.3, termination criterion 2; this is asserted without proof and is questionable since null geodesics can cross ergospheres.
domain assumption The Runge-Kutta scheme (B.5) with polynomial interpolation on a grid is sufficiently accurate for the qualitative claims.
Appendix B; no convergence study is provided.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Gravitational lensing and shadows in the toron solution of Einstein's equations using ray tracing methods." pith.science (2026). https://pith.science/paper/NWJM5ZEI

@misc{pith2026250104509,
  author       = {Pith},
  title        = {Pith review of: Gravitational lensing and shadows in the toron solution of Einstein's equations using ray tracing methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NWJM5ZEI}},
  note         = {Machine review of arXiv:2501.04509}
}

read the original abstract

We present a numerical and analytical study of the so-called `toron' solution of the stationary axisymmetric Einstein equations in vacuum expressed in terms of elliptic functions. The asymptotic behavior of this solution coincides with the one of the NUT solution, i.e., it has a `gravimagnetic' mass known as the NUT parameter while the ordinary mass vanishes. The physical properties of this spacetime are studied via ray tracing. The results are compared to known geodesic flows in Schwarzschild, Kerr and NUT spacetimes to discuss similarities and differences, with a particular emphasis on the comparison of NUT and toron spacetimes.

Figures

Figures reproduced from arXiv: 2501.04509 by the authors.

**Figure 1.** Ernst potential and metric functions for the Kerr solution for m = 1 and φ = 1. In Weyl coordinates the horizon of the Kerr black hole is located on the axis for z ∈ [−m cos φ, m cos φ]. The ergoregion, where gtt ≥ 0, i.e. f ≤ 0, is bounded by the ergosphere defined by the equation ℜE = 0, where an observer stationary at infinity will note an infinite redshift. The ergosphere of the Kerr solution touches the horizon… view at source ↗

**Figure 2.** Metric functions for the NUT solutions (2.10) for m = ℓ = 1, on the left f, on the right e 2k . Similarly to the Kerr solution, there is an event horizon of spherical topology occupying the interval of the symmetry axis z ∈ [− √ m2 + ℓ 2, √ m2 + ℓ 2]. The metric functions f and e 2k plotted in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** However, A can be alternatively chosen to vanish on the positive (respectively negative) regular part of the axis by replacing A by A−2ℓ (respectively A+ 2ℓ), which would correspond to the choice of a rotating coordinate system. It is not possible in NUT spacetimes to choose A to vanish on the whole regular part of the z-axis [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (28 more)

**Figure 3.** Figure 3: Metric function Af for the NUT solutions (2.10) for m = ℓ = 1, on the left for A odd, on the right such that it vanishes for z > √ m2 + ℓ 2. Since the metric function A is at the origin of so-called gravimagnetic effects as frame dragging, there is a considerable diffe…

**Figure 4.** Figure 4: Ernst potential (A.9) for α = 0.3, p = 0. On the left the real part, on the right the imaginary part. The metric functions corresponding to the Ernst potential (3.1) can be given in closed form, see [16]: (3.2) f = Q ϑ(iα)ϑ(iα + 1/2) ϑ( R ∞− ξ ω + iα)ϑ( R ∞− ξ¯ ω + iα)…

**Figure 5.** Figure 5: Metric function Af on the left and e 2k on the right for the Ernst potential shown in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: shows the three-dimensional representation of the ergospheres in [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: Diagram of the virtual camera (not to scale for representation purposes) with respect to the black hole and an enlarged image of the screen with some incoming light rays. The following are the chosen criteria to terminate the time integration of the geodesics, namely, …

**Figure 8.** Figure 8: Vertical section of the celestial and observer’s spheres, showing the relation between the angular aperture δ c V of the virtual camera and the effective angle of view of the celestial sphere δV . The same relation applies to the angles δ c H and δH. bending of light, …

**Figure 9.** Figure 9: All-sky survey by the eROSITA X-ray telescope. The celestial sphere is projected onto an ellipse with the Aitoff projection. Credit: J. Sanders, H. Brunner, A. Merloni and the eSASS Team (MPE); E. Churazov, M. Gilfanov, R. Sunyaev (IKI) [PITH_FULL_IMAGE:figures/full_f…

**Figure 10.** Figure 10: Aitoff projection of the celestial sphere with constant coordinates ϕ and θ = arctan(z/ρ). The boundary of the recorded image in Minkowski spacetime with angles of view δH = δV = 90◦ is indicated by the blue contour. 4.4. Shadows. One important consequence of strongly…

**Figure 11.** Figure 11: Initially-parallel light rays in the equatorial plane for Schwarzschild (φ = 0) and various Kerr spacetimes. For each example, two pairs of light rays corresponding to the same initial conditions except for the y-component (we chose y0 > 0 for one of the rays and y ′ …

**Figure 12.** Figure 12: Initially-parallel light rays in the equatorial plane for various NUT spacetimes and their projection on the xy-plane. For each example, two pairs of light rays corresponding to the same initial conditions except for the y-component (we chose y0 > 0 for one of the ray…

**Figure 13.** Figure 13: Initially-parallel light rays on the xz-plane for various NUT spacetimes and their projection on the xy-plane. For each example, two pairs of light rays corresponding to the same initial conditions except for the z-component (we chose z0 > 0 for one of the rays and z…

**Figure 14.** Figure 14: Light rings corresponding to spacetimes with a fixed m and various ℓ and their projection onto the xy-plane. Both the radius and the elevation of each ring increase with ℓ. The light rays represented in Fig. (14) correspond to different spacetimes with a fixed m = 1 …

**Figure 15.** Figure 15: Light rings on the photon spheres in a Schwarzschild and a NUT spacetime. 5.3. Shadows and apparent image of the background. It has been shown in [14] that the shadows of NUT black holes are circular. Given a fixed m, the size of the shadow increases in dependence of …

**Figure 16.** Figure 16: Boundary of the primary image on the celestial sphere in Minkowski, Schwarzschild, Kerr and NUT spacetimes. In all cases, the focal length of the virtual camera is chosen such that the angles of view in Minkowski spacetime are δH = δV = 90◦ . We continue the discussi…

**Figure 17.** Figure 17: Primary and secondary images in Schwarzschild and NUT spacetimes (with m = 0 and ℓ = 1). The dashed lines correspond to the border between the primary and secondary images, while the solid ones show the parts of the celestial sphere with constant z = 0, ϕ = 0 and ϕ =…

**Figure 18.** Figure 18: Shadows and apparent images of the background space in Schwarzschild, Kerr and NUT spacetimes. To explain why a region of the celestial sphere is hidden when ℓ ̸= 0, we show the backwards tracing of some light rays reaching the observer in both Schwarzschild and NUT s…

**Figure 19.** Figure 19: Multiple light rays traced back in time from the position of an observer on the equatorial plane. We show one of these lines in green to highlight the twisting about the x-axis. It can be observed in [PITH_FULL_IMAGE:figures/full_fig_p022_19.png]

**Figure 20.** Figure 20: Origin on the celestial sphere of every pixel in Kerr and NUT spacetimes. 6. Geodesics in the toron spacetime In this section we present simulations analogous to those presented in the preceding section, both for individual geodesics and for the apparent images of ext…

**Figure 21.** Figure 21: Initially-parallel light rays on the equatorial plane in toron spacetimes for various values of α and their projection onto the xy-plane. In each example, two pairs of light rays corresponding to the same initial conditions except for the y-component (we chose y0 > 0…

**Figure 22.** Figure 22: Initially-parallel light rays on the xz-plane in a toron spacetimes with various values of α and their projection on the xy-plane. In each example, two pairs of light rays corresponding to the same initial conditions except for the z-component (we chose z0 > 0 for one…

**Figure 23.** Figure 23: Light rays approaching photon spheres in a toron spacetime with α = 0.3. The lower figures show how these light rays either approach the ergosphere or escape to infinity. 6.3. Apparent image of a disk. We simulate the apparent image of the disk described by the toron …

**Figure 24.** Figure 24: Diagram of the virtual camera, the ergosphere and the disk with its artificial coloring pattern. Since we only consider the exterior of the first ergosphere for the numerical computation of the geodesics, we represent the ergosphere as a shadow. If the line of sight o…

**Figure 25.** Figure 25: Apparent images of disks in two different spacetimes. The images at the top correspond to an observer (virtual camera) with an inclination angle ψ = 10◦ with respect to the symmetry axis and those at the bottom correspond to ψ = 90◦ . In order to visualize further eff…

**Figure 26.** Figure 26: Boundary of the primary image on the celestial sphere in various toron spacetimes. In all cases, the focal length of the virtual camera is chosen such that the angles of view in Minkowski spacetime are δH = δV = 90◦ . We continue the discussion by showing simulations …

**Figure 27.** Figure 27: Apparent image of the celestial sphere in toron spacetimes. To finish the discussion, let us look at the blind region on the celestial sphere in a toron spacetime. Analogous to [PITH_FULL_IMAGE:figures/full_fig_p030_27.png]

**Figure 28.** Figure 28: Origin of each light ray that reached the observer in the toron spacetime with α = 0.2. The quasi-circular region corresponds to the hidden region. 7. Conclusion In this paper we have presented a discussion of physical properties of toron solutions to the stationary a…

**Figure 29.** Figure 29: Degeneration of the curve Lξ into the rational curve in the limit ρ → 0 for z < 0. Therefore, ω → − 1 2πi p (z − iσ)(z + iσ) (λ − z) p (λ − iσ)(λ + iσ) dλ and Z ∞+ iσ ω = 1 2πi log z + √ z 2 + σ 2 −iσ . Moreover, Z ∞+ ξ ω = Z ∞+ iσ ω − τ 2 . As the period of the curve…

**Figure 30.** Figure 30: Degeneration of the curve Lξ into the rational curve in the limit ρ → 0 for z > 0. Thus, as ρ → 0, ω → 1 2πi p (z − iσ)(z + iσ) (λ − z) p (λ − iσ)(λ + iσ) dλ and Z ∞+ iσ ω = − 1 2πi log z + √ z 2 + σ 2 −iσ . Therefore, ϑ Z ∞+ ξ ω − iα ! = ϑ Z ∞+ iσ w − τ 2 − iα ! → 1 …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 31 canonical work pages

[20]

Korotkin, Solutions of the vacuum Einstein equation having toroidal infinite red-shift surface , Class

D. Korotkin, Solutions of the vacuum Einstein equation having toroidal infinite red-shift surface , Class. Quantum Grav. 8 L219 - L222 (1991)

work page 1991
[8]

de Leon, J

E.B. de Leon, J. Frauendiener and C. Klein. Visualisation of counter-rotating dust disks using ray tracing methods. Class. Quantum Grav. 41 155005 (2024)

work page 2024
[1]

Abdujabbarov, F

A. Abdujabbarov, F. Atamurotov, Y. Kucukakca, B. Ahmedov and U. Camci. Shadow of Kerr-Taub-NUT black hole . Astrophys. Space Sci. 344, 429 (2012)

work page 2012
[2]

J. M. Bardeen, in Black Holes (Les Astres Occlus), edited by C. DeWitt and B. S. DeWitt (Gordon and Breach, New York, 1973) p. 215

work page 1973
[3]

Belinskii, V.E

V.A. Belinskii, V.E. Zakharov, Integration of the Einstein equations by the methods of inverse scattering theory and construction of explicit multisoliton solutions , Soviet Phys. JETP 48 (1978) 985–994

work page 1978
[4]

Belokolos, A.I

E.D. Belokolos, A.I. Bobenko, V.Z. Enol’skii, A.R. Its and V.B. Matveev. Algebro-geometric approach to nonlinear integrable equations. Springer, Berlin (1994)

work page 1994
[5]

W.B. Bonnor. A new interpretation of the NUT metric in general relativity . Math. Proc. Cambridge Philos. Soc. 66, 145 (1969)

work page 1969
[6]

Chandrasekharan

K. Chandrasekharan. Elliptic Functions . Volume 281, Grundlehren der mathematischen Wissenschaften (1985)

work page 1985

Show all 33 references

[7]

https://doi.org/10.1007/s10714-018-2361-9

P.V.P.Cunha, C.A.R.Herdeiro, Shadows and strong gravitational lensing: a brief review , Gen Relativ Gravit 50, 42 (2018). https://doi.org/10.1007/s10714-018-2361-9

2018 doi
[9]

E.B. de Leon. On a class of algebro-geometric solutions to the Ernst equation , arXiv:2310.19095v1 (2023)

2023 arXiv
[10]

Einstein, Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field, Science 84, 506 (1936)

A. Einstein, Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field, Science 84, 506 (1936)

1936
[11]

Ernst, New formulation of the axially symmetric gravitational field problem I , Phys

F.J. Ernst, New formulation of the axially symmetric gravitational field problem I , Phys. Rev. D 167, 1175
[12]

Ernst, New formulation of the axially symmetric gravitational field problem II , Phys

F.J. Ernst, New formulation of the axially symmetric gravitational field problem II , Phys. Rev. D 168, 1415
[13]

Frauendiener and C

J. Frauendiener and C. Klein, Computational approach to hyperelliptic Riemann surfaces , Lett. Math. Phys. 105(3), 379-400
[14]

Grenzebach, V

A. Grenzebach, V. Perlick and C. L¨ ammerzahl.Photon regions and shadows of Kerr-Newman-NUT black holes with a cosmological constant . Phys. Rev. D 89, 124004 (2014)

2014
[15]

Hackmann and C

E. Hackmann and C. L¨ ammerzahl.Observables for bound orbital motion in axially symmetric space-times . Phys. Rev. D 85, 044049 (2012)

2012
[16]

Klein, D

C. Klein, D. Korotkin and V. Shramchenko. Ernst equation, Fay identities and variational formulas on hyperelliptic curves. Mathematical Research Letters (2004) 27–45

2004
[17]

Klein, O

C. Klein, O. Richter, Exact relativistic gravitational treatment of a stationary counter- rotating dust disk , Phys. Rev. Lett. 83 (1999) 2884

1999
[18]

Klein and O

C. Klein and O. Richter. Ernst Equation and Riemann Surfaces: Analytical and Numerical Methods . Lecture Notes in Physics, Vol. 685. Springer (2005)

2005
[19]

Korotkin

D. Korotkin. Finite-gap solutions of the stationary axisymmetric Einstein equation in vacuum . Theor. Math. Phys. 77, 1018 (1988)

1988
[21]

Korotkin, Algebraic Geometric Solutions of Einstein ’s Equations: Some Physical Properties , Commun

D. Korotkin, Algebraic Geometric Solutions of Einstein ’s Equations: Some Physical Properties , Commun. Math. Phys. 37, 383-398 (1991)

1991
[22]

Maison, Are the stationary axially symmetric Einstein equations completely integrable? Phys

D. Maison, Are the stationary axially symmetric Einstein equations completely integrable? Phys. Rev. Lett. 41 (1978) 521–524

1978
[23]

Manko and E

V.S. Manko and E. Ruiz. Physical interpretation of NUT solution. Class. Quantum Grav. 22 3555-60 (2005)

2005
[24]

Moreira, C.A

Z.S. Moreira, C.A. Herdeiro and L.C. Crispino. Twisting shadows: Light rings, lensing, and shadows of black holes in swirling universes . Phys. Rev. D 109, 104020 (2024)

2024
[25]

Neugebauer, Backlund transformations of axially symmetric stationary gravitational fields J

G. Neugebauer, Backlund transformations of axially symmetric stationary gravitational fields J. Phys. A 12, L67 (1979)

1979
[26]

Neugebauer, R.Meinel, General relativistic gravitational field of a rigidly rotating disk of dust: Solution in terms of ultraelliptic functions , Phys

G. Neugebauer, R.Meinel, General relativistic gravitational field of a rigidly rotating disk of dust: Solution in terms of ultraelliptic functions , Phys. Rev. Lett. 75 3046 (1995)

1995
[27]

Perlick, O.Y

V. Perlick, O.Y. Tsupko, Calculating black hole shadows: Review of analytical studies , Physics Reports, 2022, arxiv 2105.07101

2022 arXiv
[28]

Silverman

J.H. Silverman. The arithmetic of elliptic curves . Graduate texts in mathematics (1986)

1986
[29]

Stephani, D

H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt. Exact Solutions of Einstein ’s Field Equations. 2nd ed. Cambridge University Press (2003)

2003
[30]

J. L. Synge, The Escape of Photons from Gravitationally Intense Stars , Mon. Not. R. Astron. Soc. 131, 463 (1966)

1966
[31]

Teo, Spherical Photon Orbits Around a Kerr Black Hole ,General Relativity and Gravitation

E. Teo, Spherical Photon Orbits Around a Kerr Black Hole ,General Relativity and Gravitation. 35 (11): 1909–1926 (2003)

2003
[32]

F. H. Vincent, M. Wielgus, M. A. Abramowicz,, E. Gourgoulhon, J.-P. Lasota, T. Paumard and G. Perrin, Geometric modeling of M87* as a Kerr black hole or a non-Kerr compact object, A& A, 646, A37 (2021) 38 E. DE LEON, C. KLEIN, AND D. KOROTKIN

2021
[33]

Wilkins, D. C. Bound geodesics in the Kerr metric , Phys. Rev. D 5, 814–822 (1972) Email address : eddybrandon11@hotmail.com Institut de Math ´ematiques de Bourgogne, Universit ´e de Bourgogne-Europe, 9 avenue Alain Savary, 21078 Dijon Cedex, France Institut de Math´ematiques ...

1972

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [20]

Korotkin, Solutions of the vacuum Einstein equation having toroidal infinite red-shift surface , Class

D. Korotkin, Solutions of the vacuum Einstein equation having toroidal infinite red-shift surface , Class. Quantum Grav. 8 L219 - L222 (1991)

work page 1991

[2] [8]

de Leon, J

E.B. de Leon, J. Frauendiener and C. Klein. Visualisation of counter-rotating dust disks using ray tracing methods. Class. Quantum Grav. 41 155005 (2024)

work page 2024

[3] [1]

Abdujabbarov, F

A. Abdujabbarov, F. Atamurotov, Y. Kucukakca, B. Ahmedov and U. Camci. Shadow of Kerr-Taub-NUT black hole . Astrophys. Space Sci. 344, 429 (2012)

work page 2012

[4] [2]

J. M. Bardeen, in Black Holes (Les Astres Occlus), edited by C. DeWitt and B. S. DeWitt (Gordon and Breach, New York, 1973) p. 215

work page 1973

[5] [3]

Belinskii, V.E

V.A. Belinskii, V.E. Zakharov, Integration of the Einstein equations by the methods of inverse scattering theory and construction of explicit multisoliton solutions , Soviet Phys. JETP 48 (1978) 985–994

work page 1978

[6] [4]

Belokolos, A.I

E.D. Belokolos, A.I. Bobenko, V.Z. Enol’skii, A.R. Its and V.B. Matveev. Algebro-geometric approach to nonlinear integrable equations. Springer, Berlin (1994)

work page 1994

[7] [5]

W.B. Bonnor. A new interpretation of the NUT metric in general relativity . Math. Proc. Cambridge Philos. Soc. 66, 145 (1969)

work page 1969

[8] [6]

Chandrasekharan

K. Chandrasekharan. Elliptic Functions . Volume 281, Grundlehren der mathematischen Wissenschaften (1985)

work page 1985