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arxiv: 2606.01980 · v1 · pith:3KGLKEMAnew · submitted 2026-06-01 · 🌀 gr-qc

Stress-energy tensor of quantized scalar fields in thermal states on a zero-tidal wormhole

Shun Jiang , Xiangdong Zhang This is my paper

Pith reviewed 2026-06-28 13:37 UTC · model grok-4.3

classification 🌀 gr-qc
keywords stress-energy tensorquantum scalar fieldthermal stateswormholeMorris-Thorne conditionstraversable wormholezero-tidal forceexotic matter
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The pith

The stress-energy tensor of a massive scalar field in thermal states on a zero-tidal wormhole satisfies the Morris-Thorne conditions only for masses in a bounded interval and below a critical temperature.

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

The paper calculates the stress-energy tensor for a quantized massive scalar field in thermal states on the throat of a zero-tidal-force wormhole. This tensor is examined to determine whether it can serve as the exotic matter required for a static traversable wormhole. Parameter scans over dimensionless temperature and mass show that the Morris-Thorne conditions hold only inside a restricted mass window; inside that window a mass-dependent critical temperature exists such that the conditions are met only below the threshold.

Core claim

We present the first calculation of the stress-energy tensor for a quantum massive scalar field in thermal states localized on the throat of a zero-tidal-force wormhole. By varying the dimensionless temperature and dimensionless mass of the scalar field, the Morris-Thorne conditions can only be satisfied when the scalar field mass falls within a specific bounded interval. For any scalar field mass within this interval there always exists a mass-dependent dimensionless critical temperature such that the conditions are fulfilled only if the temperature remains below this threshold.

What carries the argument

The renormalized stress-energy tensor of the quantized massive scalar field in a thermal state, whose components at the wormhole throat are compared against the Morris-Thorne requirements.

If this is right

  • Wormhole throats can be supported by the scalar-field stress-energy tensor only when the field mass lies inside the identified interval.
  • For any mass inside the interval the temperature must stay below the mass-dependent critical value.
  • Outside the mass interval or above the critical temperature the tensor violates the Morris-Thorne conditions at every temperature.
  • The critical temperature is a continuous function of the mass inside the allowed interval.

Where Pith is reading between the lines

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

  • The temperature dependence implies that any practical construction using such fields would require active cooling to remain traversable.
  • The same regularization procedure could be applied to other static wormhole metrics to map out their allowable mass-temperature windows.
  • Analog laboratory systems that simulate curved-space thermal states might be used to check the predicted critical temperatures.

Load-bearing premise

The stress-energy tensor has been correctly computed, regularized, and renormalized for the quantized massive scalar field in a thermal state on the given zero-tidal-force wormhole metric.

What would settle it

A direct computation of the renormalized stress-energy tensor for a scalar-field mass lying outside the reported bounded interval that nevertheless satisfies the Morris-Thorne conditions at some temperature would falsify the claimed restriction to that interval.

Figures

Figures reproduced from arXiv: 2606.01980 by Shun Jiang, Xiangdong Zhang.

Figure 1
Figure 1. Figure 1: The curve in this figure displays the value of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The curve in this figure displays the value of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The curve in this figure displays the value of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The curve in this figure displays the value of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The two curves in this figure display the value of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: This figure shows the dependence of the Morris-Thorne conditions on the dimensionless temperature [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

The construction of a static traversable wormhole requires exotic matter that satisfies the Morris-Thorne conditions. Quantum energy-momentum tensors have long been considered the most promising candidate for such exotic matter. In this paper, we present the first calculation of the stress-energy tensor for a quantum massive scalar field in thermal states localized on the throat of a zero-tidal-force wormhole. By varying the dimensionless temperature and dimensionless mass of the scalar field, we find that the Morris-Thorne conditions can only be satisfied when the scalar field mass falls within a specific bounded interval. Furthermore, for any scalar field mass within this interval, there always exists a mass-dependent dimensionless critical temperature: the Morris-Thorne conditions are fulfilled only if the temperature remains below this critical threshold.

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 paper computes the renormalized stress-energy tensor of a massive quantized scalar field in thermal states at the throat of a zero-tidal-force wormhole (ds² = -dt² + dl² + r(l)² dΩ²). It reports that the Morris-Thorne conditions hold only for scalar-field masses in a bounded interval and, within that interval, only below a mass-dependent critical dimensionless temperature.

Significance. If the regularization and mode-sum procedure are correct, the result supplies concrete, falsifiable bounds on mass and temperature for which a thermal quantum scalar field can violate the null energy condition at the throat. This would be a notable advance over generic statements that quantum fields can support wormholes, as it identifies restricted parameter windows rather than open sets.

major comments (2)
  1. [Computation of <T_{\mu\nu}> (likely §3 or §4)] The central claim (bounded mass interval and critical temperature) rests entirely on the sign of the renormalized ρ + p_r at the throat. The manuscript must therefore supply the explicit regularization procedure (Hadamard parametrix for the massive Klein-Gordon operator, subtraction of the divergent terms, and retention of the finite thermal and mass-dependent pieces) together with the mode expansion or Green's function used on the given metric. Without these steps, the reported intervals cannot be verified and the result is not reproducible.
  2. [Thermal-state construction and Matsubara sum] The thermal state is defined via a Matsubara sum with respect to the static Killing vector. The manuscript should state the precise frequency spacing, the value of the curvature coupling ξ, and how the zero-mode or infrared contributions are handled for the massive field; any inconsistency here directly alters the temperature dependence of the critical threshold.
minor comments (2)
  1. [Abstract] The abstract states the headline intervals but supplies no numerical values or functional form for the mass bounds or critical temperature; these should appear in the main text or a table for immediate reference.
  2. [Introduction and parameter definitions] Notation for the dimensionless temperature and dimensionless mass should be defined once at first use and used consistently thereafter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed reading and for identifying the need for greater explicitness in the computational sections. We address each major comment below and will revise the manuscript accordingly to improve reproducibility.

read point-by-point responses

  1. Referee: [Computation of <T_{\mu\nu}> (likely §3 or §4)] The central claim (bounded mass interval and critical temperature) rests entirely on the sign of the renormalized ρ + p_r at the throat. The manuscript must therefore supply the explicit regularization procedure (Hadamard parametrix for the massive Klein-Gordon operator, subtraction of the divergent terms, and retention of the finite thermal and mass-dependent pieces) together with the mode expansion or Green's function used on the given metric. Without these steps, the reported intervals cannot be verified and the result is not reproducible.

    Authors: We agree that the current presentation does not supply the full regularization details at the level required for independent verification. In the revised manuscript we will expand the relevant section to include the explicit Hadamard parametrix for the massive scalar, the precise subtraction of the divergent geometric terms, and the mode-sum expression for the Green's function adapted to the zero-tidal wormhole metric. This addition will make the sign of ρ + p_r at the throat directly traceable to the stated procedure. revision: yes

  2. Referee: [Thermal-state construction and Matsubara sum] The thermal state is defined via a Matsubara sum with respect to the static Killing vector. The manuscript should state the precise frequency spacing, the value of the curvature coupling ξ, and how the zero-mode or infrared contributions are handled for the massive field; any inconsistency here directly alters the temperature dependence of the critical threshold.

    Authors: We will add an explicit statement of the Matsubara frequencies (ω_n = 2π n T), the value of the non-minimal coupling ξ employed in the calculation, and the regularization of the zero-mode contribution for the massive field. These clarifications will be inserted in the section describing the thermal state so that the mass-dependent critical temperature can be reproduced without ambiguity. revision: yes

Circularity Check

0 steps flagged

Direct numerical evaluation of renormalized stress-energy tensor yields mass and temperature bounds

full rationale

The paper's central result is obtained by explicit computation of the mode-sum or Green's function for the massive Klein-Gordon field in a thermal state on the given static metric, followed by regularization, renormalization, and direct evaluation of the NEC-violating combinations at the throat. No equation is defined in terms of its own output, no fitted parameter is relabeled as a prediction, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The bounded mass interval and critical temperature are outputs of the calculation rather than inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The work rests on the standard framework of quantum field theory in curved spacetime and the Morris-Thorne wormhole metric.

axioms (1)
  • domain assumption Validity of renormalization procedures for the stress-energy tensor of a massive scalar field in a static curved spacetime.
    Required to obtain a finite <T_{\mu\nu}> from the mode sum.

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Reference graph

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