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arxiv: 2605.21823 · v1 · pith:XEKZRWJRnew · submitted 2026-05-20 · 🌀 gr-qc · hep-th

Maximal extension of Schwarzschild-like spacetimes in Lorentz gauge theory

Mohsen Fathi This is my paper

Pith reviewed 2026-05-22 08:19 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords black holemaximal analytic extensionKruskal-Szekeres coordinatesLorentz gauge theorySchwarzschild spacetimewhite holeCarter-Penrose diagramcausal topology
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The pith

Schwarzschild-like solutions in Lorentz gauge theory extend maximally to include a white-hole region while preserving the same causal topology.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper adapts the Kruskal-Szekeres construction to a black-hole metric whose lapse function carries an extra constant factor A0. This produces an extended manifold containing two exterior regions, a black-hole interior, and a white-hole interior. A reader would care because the construction shows that a modified theory can still recover the familiar global causal arrangement of the Schwarzschild solution even though lengths, surface gravity, and radial scales are rescaled by powers of A0. The result therefore separates topological features from geometric details that depend on the theory parameter.

Core claim

Starting from the lapse f(r) = A0^{-2} - 2m/r with horizon at r+ = 2m A0^2, the Kruskal-Szekeres chart is built so that the maximal analytic extension consists of two asymptotically flat exterior regions connected through a black-hole region and a white-hole region. The conformal skeleton of the Carter-Penrose diagram remains identical to the Schwarzschild case, yet the physical horizon radius, surface gravity kappa = 1/(4m A0^4), and constant-radius curves continue to depend on A0. Hence the spacetime shares the causal topology of Schwarzschild while remaining geometrically inequivalent whenever A0 differs from one.

What carries the argument

The Kruskal-Szekeres coordinates adapted to the given lapse function, which remove the coordinate singularity at the horizon and connect the interior and exterior regions through null geodesics.

If this is right

  • The extended manifold necessarily includes a white-hole region in addition to the black-hole interior.
  • Surface gravity at the horizon is fixed at 1 over 4m A0 to the fourth.
  • The Carter-Penrose compactification retains the same conformal skeleton as Schwarzschild but with A0-dependent physical scales.
  • Radial null curves in both the Schwarzschild-Droste and ingoing Eddington-Finkelstein charts remain regular across the horizon once expressed in the new coordinates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar coordinate extensions could be attempted for other static solutions that arise in the same gauge theory.
  • If A0 differs from one, light-cone tilting and proper-time intervals near the horizon would differ from their Schwarzschild values.
  • The separation between causal topology and metric scales suggests that observational tests sensitive to surface gravity could distinguish the two geometries.

Load-bearing premise

The given functional form of the lapse is taken to be the exact solution of the Lorentz gauge theory field equations on which all coordinate extensions are built.

What would settle it

An explicit check that the proposed Kruskal-Szekeres metric fails to make all radial null geodesics complete or produces a curvature singularity inside the claimed extended regions would refute the maximal extension.

Figures

Figures reproduced from arXiv: 2605.21823 by Mohsen Fathi.

Figure 1
Figure 1. Figure 1: U-constant hypersurfaces in the IEF chart (t˜,r, θ, ϕ) for three representative values of A0. The curves are obtained from U = (1 − r/r+) exp[κ(r − t˜)]. The green curves correspond to U > 0 inside the horizon, while the blue curves correspond to U < 0 outside the horizon. Different line styles show |U| = 0.25, 1, 4. The red dashed vertical line denotes r = r+. Therefore the coordinate speed goes to zero a… view at source ↗
Figure 2
Figure 2. Figure 2: Outgoing and ingoing radial null geodesics in the SD coordinates for three representative values of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Outgoing and ingoing radial null geodesics in the IEF coordinates for three representative values of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Domain of the IEF coordinates, MIEF, represented in the KS (X, T) plane for three representative values of A0. The blue solid curves correspond to t˜ = constant and are spaced by ∆t˜ = r+/2. The green dotted curves are representative constant-radius curves chosen at r/r+ = 0.20, 0.40, 0.60, 0.80, 1.20, 1.50, 2.00, 2.60. The red dashed line denotes the future horizon H + , the grey dashed line denotes V = 0… view at source ↗
Figure 5
Figure 5. Figure 5: KS diagrams of the LGT Schwarzschild-like black hole for three representative values of [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Standard CP compactification of the LGT Schwarzschild-like black hole for three representative values of [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Regular compact CP diagrams of the LGT Schwarzschild-like black hole for three representative values of [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
read the original abstract

We study the maximal analytic extension of the Schwarzschild-like black hole solution in Lorentz gauge theory. The lapse function is $f(r)=A_0^{-2}-2\m/r$, so the horizon is located at $r_+=2\m A_0^2$ and the non-affinity coefficient of the horizon generator is $\kappa=1/(4\m A_0^4)$. We first analyze the radial null curves in the Schwarzschild-Droste (SD) and ingoing Eddington-Finkelstein (IEF) charts, and then construct the Kruskal-Szekeres (KS) chart adapted to the LGT geometry. The KS extension contains two exterior regions, a black-hole region and a white-hole region. We also present the standard and regular Carter-Penrose (CP) compactifications. The conformal skeleton is Schwarzschild-like, but the physical scale of the horizon, the surface gravity and the constant-radius curves remain controlled by $A_0$. Hence the solution has the same causal topology as Schwarzschild, while it is geometrically inequivalent to it when $A_0\neq1$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to construct the maximal analytic extension of a Schwarzschild-like black hole solution arising in Lorentz gauge theory. The lapse function is given explicitly as f(r) = A_0^{-2} - 2m/r, yielding a horizon at r_+ = 2m A_0^2 and surface gravity κ = 1/(4m A_0^4). The authors analyze radial null geodesics in the Schwarzschild-Droste and ingoing Eddington-Finkelstein charts, build an adapted Kruskal-Szekeres coordinate system, and present both standard and regular Carter-Penrose compactifications. The central result is that the extended manifold possesses the same causal topology as the Schwarzschild spacetime (two exterior regions together with black-hole and white-hole interiors) while remaining geometrically inequivalent for A_0 ≠ 1.

Significance. If the coordinate constructions are shown to be regular and geodesically complete across the horizons, the work would establish that Lorentz gauge theory admits black-hole solutions whose causal structure is identical to that of general relativity, with the free parameter A_0 rescaling the horizon radius, surface gravity, and proper distances while leaving the Penrose-diagram skeleton unchanged. This supplies a concrete, tunable example of a modified-gravity black hole whose observational signatures could differ from Schwarzschild while sharing the same global causal topology, and the explicit chart constructions provide a reproducible template for similar extensions in other gauge-theoretic models.

major comments (2)
  1. The adaptation of the Kruskal-Szekeres chart to the given lapse function is load-bearing for the claim of a regular maximal extension; the manuscript should explicitly verify that the transformed metric remains regular across the horizons and that null geodesics satisfy the geodesic equation throughout the extended manifold, rather than assuming the standard Schwarzschild rescaling carries over unchanged.
  2. Abstract and the section on the Kruskal-Szekeres construction: the stated surface gravity κ = 1/(4m A_0^4) follows from the standard formula once f'(r_+) is computed, but the paper must display the intermediate steps (including the explicit tortoise-coordinate integral) to confirm that the exponential rescaling factor in the KS coordinates is correctly normalized for this particular f(r).
minor comments (2)
  1. The physical interpretation and dimensions of the free parameters A_0 and m should be stated explicitly at the first appearance of the lapse function.
  2. Figure captions for the Carter-Penrose diagrams should indicate the coordinate ranges and mark the locations of the horizons and the central singularity for both A_0 = 1 and A_0 ≠ 1 cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its potential significance in providing a concrete example of a modified-gravity black hole with Schwarzschild-like causal topology. We address each major comment below and have revised the manuscript accordingly to strengthen the explicit verifications and derivations.

read point-by-point responses

  1. Referee: The adaptation of the Kruskal-Szekeres chart to the given lapse function is load-bearing for the claim of a regular maximal extension; the manuscript should explicitly verify that the transformed metric remains regular across the horizons and that null geodesics satisfy the geodesic equation throughout the extended manifold, rather than assuming the standard Schwarzschild rescaling carries over unchanged.

    Authors: We agree that explicit verification is necessary to support the regularity claim. In the revised manuscript we have added a dedicated subsection that performs the coordinate transformation explicitly for f(r) = A_0^{-2} - 2m/r. We compute the metric components in the new (U,V) coordinates and confirm they remain finite and smooth at the horizons. We further substitute the null geodesic solutions from the Eddington-Finkelstein chart into the geodesic equation for the extended metric and verify that they continue to satisfy it, thereby demonstrating that the extension is regular without relying on an unexamined carry-over from the Schwarzschild case. revision: yes

  2. Referee: Abstract and the section on the Kruskal-Szekeres construction: the stated surface gravity κ = 1/(4m A_0^4) follows from the standard formula once f'(r_+) is computed, but the paper must display the intermediate steps (including the explicit tortoise-coordinate integral) to confirm that the exponential rescaling factor in the KS coordinates is correctly normalized for this particular f(r).

    Authors: We thank the referee for this suggestion. The revised manuscript now includes the full intermediate steps: we first differentiate f(r) to obtain f'(r) = 2m/r^2, evaluate at r_+ = 2m A_0^2 to find f'(r_+) = 1/(2m A_0^4), and recover κ = f'(r_+)/2 = 1/(4m A_0^4). We then display the explicit tortoise-coordinate integral r^* = ∫ dr/f(r) for our specific lapse function, showing that the resulting exponential factor in the Kruskal-Szekeres coordinates is correctly normalized and yields the stated surface gravity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard coordinate extension of given metric

full rationale

The paper assumes the lapse function f(r) = A0^{-2} - 2m/r as the exact solution from Lorentz gauge theory and applies standard Kruskal-Szekeres and Carter-Penrose constructions. The radial null geodesics, tortoise coordinate, and exponential rescaling proceed identically to the Schwarzschild case, with A0 only rescaling surface gravity and proper distances while preserving causal ordering. No derivation step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the topology claim follows directly from the metric form without internal forcing. The result is self-contained against the supplied metric.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the given lapse function being a valid solution of Lorentz gauge theory and on the standard differential-geometric properties of null geodesics and conformal compactifications.

free parameters (2)
  • A0
    Constant appearing in the lapse function that rescales the horizon and surface gravity; treated as a free parameter of the solution.
  • m
    Mass parameter in the lapse function, standard in Schwarzschild-like metrics.
axioms (2)
  • domain assumption The metric with the stated lapse function solves the field equations of Lorentz gauge theory.
    Invoked at the outset to justify building the coordinate extensions on this particular f(r).
  • standard math Standard properties of radial null geodesics and conformal compactifications hold in the extended manifold.
    Used to construct the KS and CP charts without additional proof in the abstract.

pith-pipeline@v0.9.0 · 5719 in / 1676 out tokens · 29773 ms · 2026-05-22T08:19:20.446827+00:00 · methodology

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Reference graph

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