Pith Number

pith:PDX5Y545

pith:2009:PDX5Y545XSZOSPQDUVUM6PJIUT

not attested not anchored not stored refs pending

Gravitational Lensing by Black Holes

V. Bozza

arxiv:0911.2187 v2 · 2009-11-11 · gr-qc · astro-ph.GA

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{PDX5Y545XSZOSPQDUVUM6PJIUT}

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.

Cited by

5 papers in Pith

Boundary-only weak deflection angles from isothermal optical geometry

Strong-deflection expansion of the deflection angle near a degenerate photon sphere

Effective null geodesics and black hole images in Kruglov nonlinear electrodynamics

Relative Magnification Factor of Point Sources on Accretion Disks

Effective null geodesics and black hole images in Kruglov nonlinear electrodynamics

Receipt and verification

First computed	2026-05-18T02:34:11.217647Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

78efdc779dbcb2e93e03a568cf3d28a4ee38bbf49146ccb619f5826acdd617a4

Aliases

arxiv: 0911.2187 · arxiv_version: 0911.2187v2 · doi: 10.48550/arxiv.0911.2187 · pith_short_12: PDX5Y545XSZO · pith_short_16: PDX5Y545XSZOSPQD · pith_short_8: PDX5Y545

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/PDX5Y545XSZOSPQDUVUM6PJIUT \
  | 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: 78efdc779dbcb2e93e03a568cf3d28a4ee38bbf49146ccb619f5826acdd617a4

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "85a57e24895a6dd0aff838ea16cf1542a5fdb757341b38ad4158a0790a0ee953",
    "cross_cats_sorted": [
      "astro-ph.GA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "gr-qc",
    "submitted_at": "2009-11-11T17:19:37Z",
    "title_canon_sha256": "f6ef2eac8d5fd2684b62f1a97556417d879c9f8d60965675c84762c57655e5eb"
  },
  "schema_version": "1.0",
  "source": {
    "id": "0911.2187",
    "kind": "arxiv",
    "version": 2
  }
}