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Paper Citation Record · LEDGER

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes

As of 16 August 2026, this Paper Citation Record lists 52 of 52 outbound references and 0 inbound Pith citation observations for arXiv:2608.12423.

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pith.paper-citation-record.v1
2608.12423 v1

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measured 52 of 52 reference resolution

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Outbound references

Observation 01dce4cf-3193-4f76-b0ee-d50eeb162dec · outbound

This paper cites Einstein, The foundation of the general theory of relativity., Annalen Phys.49, 769 (1916).

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Einstein, The foundation of the general theory of relativity., Annalen Phys.49, 769 (1916)

Reference 1

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Observation 78cda0c3-78bf-41d2-9095-663cc51084ed · outbound

This paper cites an unresolved cited work.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Unresolved cited work

Reference 2

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Observation 172cfc4c-fd20-4ab5-99eb-6fd8ee8dc7db · outbound

This paper cites −dy dq q=ϵsin 3 Θ −1 # dΘ.(4.28) The symmetry sin(π−Θ) = sin Θ(4.29) implies that the integrand is symmetric about closest approach. We may therefore write α= 2 Z π/2 0.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes −dy dq q=ϵsin 3 Θ −1 # dΘ.(4.28) The symmetry sin(π−Θ) = sin Θ(4.29) implies that the integrand is symmetric about closest approach. We may therefore write α= 2 Z π/2 0

Reference 3

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Observation ce7e1d86-cd19-4818-9317-92f5abd3dc8f · outbound

This paper cites (4.12) givesq c = 1/(3 √ 3).

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes (4.12) givesq c = 1/(3 √ 3)

Reference 4

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Observation 9b334dca-172b-4685-b77e-a9ae83f35ad0 · outbound

This paper cites an unresolved cited work.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Unresolved cited work

Reference 5

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Observation d5500fcf-e2bd-4a60-8ea7-c7ea8e348b8a · outbound

This paper cites Einstein, Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field, Science84, 506 (1936).

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Einstein, Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field, Science84, 506 (1936)

Reference 6

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Observation fb32bbf4-dec0-4406-9bc0-9bd5a92fbdac · outbound

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Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Unresolved cited work

Reference 7

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Observation bb12405c-58d4-45ae-a2be-334c37a0642c · outbound

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Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Unresolved cited work

Reference 8

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Observation a4f702a8-a087-4173-ae89-e3ffc2d060a9 · outbound

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Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Unresolved cited work

Reference 9

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Observation 1efe6196-75db-4dc9-addc-6b5a572f6b97 · outbound

This paper cites An Exact Universal Gravitational Lensing Equation.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes An Exact Universal Gravitational Lensing Equation

Reference 10

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Observation 4b85b87c-454f-454b-90fc-feb773084725 · outbound

This paper cites Spacetime perspective of Schwarzschild lensing.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Spacetime perspective of Schwarzschild lensing

Reference 11

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Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Unresolved cited work

Reference 12

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Observation 9689cf4c-5549-4d7f-80c5-15185a95dbe9 · outbound

This paper cites On the exact gravitational lens equation in spherically symmetric and static spacetimes.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes On the exact gravitational lens equation in spherically symmetric and static spacetimes

Reference 13

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Observation a66b4c0e-3866-48c2-81ee-2a484196b5f5 · outbound

This paper cites Perlick, Gravitational lensing from a spacetime perspective, Living Rev.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Perlick, Gravitational lensing from a spacetime perspective, Living Rev

Reference 14

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Observation 38c07550-6e3c-4232-a003-fca8232a3b76 · outbound

This paper cites Epstein and I.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Epstein and I

Reference 15

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Observation 9d583117-13c5-4d2e-94cb-6a6a067a024e · outbound

This paper cites Fischbach and B.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Fischbach and B

Reference 16

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Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Unresolved cited work

Reference 17

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Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Unresolved cited work

Reference 18

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Observation 888714d9-7daa-4aad-bd22-9d8fb0593abe · outbound

This paper cites Bodenner and C.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Bodenner and C

Reference 19

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Observation ea0b4c8e-67b0-40b1-8ba6-db10b6a92f1f · outbound

This paper cites Gravitational lensing in metric theories of gravity.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Gravitational lensing in metric theories of gravity

Reference 20

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Observation d3fcc88f-0b02-4870-898d-127f4f3bdf6d · outbound

This paper cites Formalism for Testing Theories of Gravity Using Lensing by Compact Objects. I: Static, Spherically Symmetric Case.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Formalism for Testing Theories of Gravity Using Lensing by Compact Objects. I: Static, Spherically Symmetric Case

Reference 21

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Observation 3ecf8a6e-945b-4144-8539-9af9e794092f · outbound

This paper cites Formalism for testing theories of gravity using lensing by compact objects. II: Probing Post-Post-Newtonian metrics.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Formalism for testing theories of gravity using lensing by compact objects. II: Probing Post-Post-Newtonian metrics

Reference 22

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Observation 2bd8df5b-0f36-4000-be14-51105ff85aba · outbound

This paper cites Light's Bending Angle due to Black Holes: From the Photon Sphere to Infinity.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Light's Bending Angle due to Black Holes: From the Photon Sphere to Infinity

Reference 23

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Observation 1c576e21-d1ee-403a-8c1d-f20c3fd957ed · outbound

This paper cites A comparison of approximate gravitational lens equations and a proposal for an improved new one.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes A comparison of approximate gravitational lens equations and a proposal for an improved new one

Reference 24

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Observation 9dbe35ec-a86c-4525-8e86-e7f15474c183 · outbound

This paper cites A generalized lens equation for light deflection in weak gravitational fields.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes A generalized lens equation for light deflection in weak gravitational fields

Reference 25

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Observation 5c29058d-a9c5-4109-b9f6-946496d9af99 · outbound

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Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Analytical formulas for gravitational lensing

Reference 26

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Observation 7b384564-09c0-4fe9-95c5-65daabda44ff · outbound

This paper cites Analytical formulas for gravitational lensing: higher order calculation.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Analytical formulas for gravitational lensing: higher order calculation

Reference 27

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Observation 92b64dc5-f43d-4713-a3e3-e2d97adda015 · outbound

This paper cites Computation of the general relativistic perihelion precession and of light deflection via the Laplace-Adomian Decomposition Method.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Computation of the general relativistic perihelion precession and of light deflection via the Laplace-Adomian Decomposition Method

Reference 28

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This paper cites Higher-order corrections for the deflection of light around a massive object.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Higher-order corrections for the deflection of light around a massive object

Reference 29

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Observation 1a8fc997-4eb1-4644-a304-cc94e00194c5 · outbound

This paper cites The perturbative approach for the weak deflection angle.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes The perturbative approach for the weak deflection angle

Reference 30

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This paper cites Bending of Light and Inhomogeneous Picard-Fuchs Equation.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Bending of Light and Inhomogeneous Picard-Fuchs Equation

Reference 31

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This paper cites Applications of the Gauss-Bonnet theorem to gravitational lensing.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Applications of the Gauss-Bonnet theorem to gravitational lensing

Reference 32

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This paper cites Universal properties of the near-horizon optical geometry.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Universal properties of the near-horizon optical geometry

Reference 33

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This paper cites Finite-distance corrections to the gravitational bending angle of light in the strong deflection limit.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Finite-distance corrections to the gravitational bending angle of light in the strong deflection limit

Reference 34

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Observation 9256a10c-9f4d-4b5f-a574-78f4ee5659cd · outbound

This paper cites Gravitational bending angle of light for finite distance and the Gauss-Bonnet theorem.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Gravitational bending angle of light for finite distance and the Gauss-Bonnet theorem

Reference 35

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Observation bd3bfe79-8281-4622-8d35-6f919cb49190 · outbound

This paper cites Weak lensing in a plasma medium and gravitational deflection of massive particles using the Gauss-Bonnet theorem. A unified treatment.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Weak lensing in a plasma medium and gravitational deflection of massive particles using the Gauss-Bonnet theorem. A unified treatment

Reference 36

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source=pdf_text observed=2026-08-16T00:33:41.159485Z digest=sha256:23b749bdc47b772762f5ac81026b35b0aa629c1e900277ab44b73c58825d5ad2

Observation 0dabbaf5-239b-45fa-9abb-4b5b16bc62bf · outbound

This paper cites Gravitational lensing in dispersive media and deflection angle of charged massive particles in terms of curvature scalars and energy-momentum tensor.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Gravitational lensing in dispersive media and deflection angle of charged massive particles in terms of curvature scalars and energy-momentum tensor

Reference 37

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Observation c1a0bb27-d733-4d29-8337-87b3416a3b01 · outbound

This paper cites The effects of finite distance on the gravitational deflection angle of light.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes The effects of finite distance on the gravitational deflection angle of light

Reference 38

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source=pdf_text observed=2026-08-16T00:33:41.168773Z digest=sha256:394bf5813f2569d61542ff729d436ca792e3d353b39fa981fd8976c30c0e4e50

Observation 190633fe-c6ec-4ff8-8fa2-287e8c5f99a8 · outbound

This paper cites Equivalence of Gibbons-Werner method to geodesics method in the study of gravitational lensing.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Equivalence of Gibbons-Werner method to geodesics method in the study of gravitational lensing

Reference 39

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source=pdf_text observed=2026-08-16T00:33:41.174888Z digest=sha256:d1c12d8ba8a775db9d3e5e575338964ab11097153f2f5a45e2729658b9b68741

Observation 7a3a1739-e089-4100-8f2b-c35768cf9f75 · outbound

This paper cites Reference-renormalized curvature-primitive Gauss-Bonnet formalism for finite-distance weak gravitational lensing in static spherical spacetimes.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Reference-renormalized curvature-primitive Gauss-Bonnet formalism for finite-distance weak gravitational lensing in static spherical spacetimes

Reference 40

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source=pdf_text observed=2026-08-16T00:33:41.179999Z digest=sha256:70df13ee3eb60a17da8505983636703e06d317019ec6471c02ef2c52c5cb391e

Observation e2558c1a-27ce-4cb2-946c-d0ed3f4f5de0 · outbound

This paper cites The geometry of photon surfaces.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes The geometry of photon surfaces

Reference 41

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source=pdf_text observed=2026-08-16T00:33:41.247846Z digest=sha256:d0b769c736cb87ab4800ebd388b21ef34e6a6161961c9fc2e10bed2dd139e397

Observation feac103d-261d-4713-b092-726ba8b664ac · outbound

This paper cites Strong field limit of black hole gravitational lensing.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Strong field limit of black hole gravitational lensing

Reference 42

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source=pdf_text observed=2026-08-16T00:33:41.338246Z digest=sha256:361eb0ee11dd83c1fa384724f87ee3cc608cbfab4869c8cf56e58d586753c5c1

Observation 17c2878c-23a0-42c2-bd32-de42a5b0b5c0 · outbound

This paper cites Gravitational lensing in the strong field limit.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Gravitational lensing in the strong field limit

Reference 43

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no resolver link, observed 2026-08-16T00:33:41.342597Z

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source=pdf_text observed=2026-08-16T00:33:41.342597Z digest=sha256:cb09785036a94c113accda5bec942bbeaa8b7c0c1b6aa7108d28f53d25d45529

Observation d060df74-6fa7-4514-92c7-c086771e3ac8 · outbound

This paper cites Reissner-Nordstrom black hole lensing.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Reissner-Nordstrom black hole lensing

Reference 44

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no resolver link, observed 2026-08-16T00:33:41.347532Z

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source=pdf_text observed=2026-08-16T00:33:41.347532Z digest=sha256:8ce86336ee6363dbfd180125aada9874dabc21f4728776c27ccd32c14d400866

Observation 854974c6-3dd7-44f4-9e61-3465d97a42b9 · outbound

This paper cites Strong deflection limit of black hole gravitational lensing with arbitrary source distances.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Strong deflection limit of black hole gravitational lensing with arbitrary source distances

Reference 45

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no resolver link, observed 2026-08-16T00:33:41.353251Z

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source=pdf_text observed=2026-08-16T00:33:41.353251Z digest=sha256:67e1bf1a4e355d2e5e797e30a09bcc2fc9aed354c06ad77874bf9fb82d1ce2d8

Observation 320922dc-dfba-4b90-a595-3be7022b62f1 · outbound

This paper cites Gravitational Lensing by Black Holes.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Gravitational Lensing by Black Holes

Reference 46

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no resolver link, observed 2026-08-16T00:33:41.358418Z

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source=pdf_text observed=2026-08-16T00:33:41.358418Z digest=sha256:5ab2dc73501a08466f1db340e6e7909b7f33e5804c3fcfd05dc6e58d230bbcf7

Observation 66111674-d9b8-472e-86cd-52cfbe887a98 · outbound

This paper cites Deflection angle in the strong deflection limit in a general asymptotically flat, static, spherically symmetric spacetime.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Deflection angle in the strong deflection limit in a general asymptotically flat, static, spherically symmetric spacetime

Reference 47

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no resolver link, observed 2026-08-16T00:33:41.428681Z

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source=pdf_text observed=2026-08-16T00:33:41.428681Z digest=sha256:d28b03042e196fd016e9adaf3c48f9f99f0167d445ed3dcca6f1d40033a37911

Observation 8d4a0d64-a956-4aec-a04e-45f422a213ee · outbound

This paper cites Perturbative deflection angle, gravitational lensing in the strong field limit and the black hole shadow.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Perturbative deflection angle, gravitational lensing in the strong field limit and the black hole shadow

Reference 48

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local_arxiv, observed 2026-08-16T00:33:41.659458Z

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No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

source=pdf_text observed=2026-08-16T00:33:41.505549Z digest=sha256:a09e4469233a3856ac511a8aa27aa4f0335501a669961ac90275076a85990368

Observation 6565b845-8336-4799-8f1f-d7aa8f203ec9 · outbound

This paper cites Gravitational lensing inside and outside of a marginally unstable photon sphere in a general, static, spherically symmetric, and asymptotically-flat spacetime in strong deflection limits.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Gravitational lensing inside and outside of a marginally unstable photon sphere in a general, static, spherically symmetric, and asymptotically-flat spacetime in strong deflection limits

Reference 49

Resolution
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no resolver link, observed 2026-08-16T00:33:41.513366Z

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source=pdf_text observed=2026-08-16T00:33:41.513366Z digest=sha256:b5afcf3eac0429a7d4c4d04ce3d5b655befa1b8e72bbeebef5b6e2f4c1a623d1

Observation 453db8fe-b87b-456a-9f97-66d515219ec5 · outbound

This paper cites Universal time delay in static spherically symmetric spacetimes for null and timelike signals.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Universal time delay in static spherically symmetric spacetimes for null and timelike signals

Reference 50

Resolution
verified exact
local_arxiv, observed 2026-08-16T00:33:41.618056Z

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No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

source=pdf_text observed=2026-08-16T00:33:41.518560Z digest=sha256:20ac7b99e7c8d4b8ace754730a5aace715151cec6063176c8321f6ce530034e2

Observation 501f5aad-1e58-4598-b3aa-01f3098e6e46 · outbound

This paper cites Mayol, R.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Mayol, R

Reference 51

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verified fuzzy
raw_fallback, observed 2026-08-16T00:33:43.176022Z

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No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

source=pdf_text observed=2026-08-16T00:33:41.523339Z digest=sha256:162deaeb05d52536a52e2fe1a22f894fca5de7bc9a09cc7e16b365f264db1bbd

Observation b1b5cce0-fb62-480b-b3ce-b1059c06b10d · outbound

This paper cites an unresolved cited work.

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes Unresolved cited work

Reference 52

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unresolved
raw_fallback, observed 2026-08-16T00:33:43.117438Z

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No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

source=pdf_text observed=2026-08-16T00:33:41.527785Z digest=sha256:76c5346f7bf2a5734772c8edfb1e1eded86384b22cb596ed402027ca3790a7c4

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