Pith Number

pith:3OETCSVB

pith:2026:3OETCSVBGGAMZI7HYGO5ZMLGFH

not attested not anchored not stored refs resolved

Thermodynamics and optical aspects of ModMax black holes in higher order curvature gravity with quintessence dark energy

Abdul Jawad, Ahmad Al-Badawi, Kazuharu Bamba, Usman Zafar

Exact solutions show quintessence and higher curvature corrections enlarge ModMax black hole shadows while charge reduces them.

arxiv:2605.13908 v1 · 2026-05-13 · gr-qc · hep-th

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{3OETCSVBGGAMZI7HYGO5ZMLGFH}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

Exact analytical expressions for the photon-sphere radius are derived, revealing that higher-order curvature corrections and quintessence significantly enhance the shadow size, whereas the electric charge has the opposite effect. Notably, quintessence is found to have a more pronounced impact on the shadow than the charge.

C2weakest assumption

The derivation assumes a specific form of the higher-order curvature action and a linear coupling to the quintessence fluid that permits an exact closed-form solution; if this coupling choice is not realized in a fundamental theory, the metric and all derived quantities cease to apply.

C3one line summary

Exact ModMax black hole solution in higher-order gravity with quintessence yields analytical photon-sphere radius and shows dark energy and curvature corrections enlarge shadow size more than charge.

References

170 extracted · 170 resolved · 1 Pith anchors

[1] 4668, 89. 286, and 17502 . 8. Similarly, in the context of higher-order curvature gravity ( F (R)-gravity), the ModMax BH (black curve) has one point of heat capacity at 0 . 4309, while in the case of

[2] 6, R 0 = − 0

[3] ×10-7 S ϵωt ω=-1 ω=-2/3 ω=-1/3 0 5 10 15 20 -5. × 10-8 0

[4] 11: Plot of εtΩ in the form of the S is illustrated by substituting q = 0

[5] 9, c = 0 . 1, γ = 0 . 6, R 0 = − 0. 01 and w = − 2/ 3. For various models of ModMax BH, we get diﬀerent trajec- tories, such as red, black, blue, and green curves for ModMax, F (R)− ModMax BH, ModMax

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-17T23:39:18.858345Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

db89314aa13180cca3e7c19ddcb16629d7a89e41a995d0d24aee356e7d4c2d4a

Aliases

arxiv: 2605.13908 · arxiv_version: 2605.13908v1 · doi: 10.48550/arxiv.2605.13908 · pith_short_12: 3OETCSVBGGAM · pith_short_16: 3OETCSVBGGAMZI7H · pith_short_8: 3OETCSVB

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/3OETCSVBGGAMZI7HYGO5ZMLGFH \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: db89314aa13180cca3e7c19ddcb16629d7a89e41a995d0d24aee356e7d4c2d4a

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "99e1f6191846ffa48af0fd89a04fff3e76990cd8a2797f569f3c68333435babb",
    "cross_cats_sorted": [
      "hep-th"
    ],
    "license": "http://creativecommons.org/publicdomain/zero/1.0/",
    "primary_cat": "gr-qc",
    "submitted_at": "2026-05-13T04:35:16Z",
    "title_canon_sha256": "223b85b923a3e15c2636ef8adc6c7c22bbc399757c9365a368ef97ddccad1d30"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13908",
    "kind": "arxiv",
    "version": 1
  }
}