Quantum fields on self-similar spacetimes (2509.10360v1)

Abstract: We study (scalar, not necessarily conformal) quantum fields on self-similar spacetimes. It is shown that in states respecting the self-similarity the expectation value of the stress tensor gives rise to a quantum Lyapunov exponent $\omega_q = 2$, with a leading coefficient which is geometric. Two examples for states respecting self-similarity are discussed. Our findings are in contradiction with results obtained via dimensional reduction.

• The paper derives a scaling law for the renormalized stress tensor, identifying a quantum Lyapunov exponent of 2 for minimally coupled scalar fields.
• It employs the Hadamard point-split renormalization to reveal geometric universality, linking stress tensor corrections to local curvature invariants.
• Explicit self-similar quantum states are constructed for Minkowski and Hayward spacetimes, questioning the validity of dimensional reduction methods.

Quantum Fields on Self-Similar Spacetimes: Stress Tensor Scaling and Quantum Lyapunov Exponents

Introduction

This paper investigates the behavior of quantum scalar fields on self-similar spacetimes, with particular emphasis on the scaling properties of the renormalized stress-energy tensor and the emergence of a quantum Lyapunov exponent. The analysis is motivated by the appearance of self-similar solutions in critical gravitational collapse, where quantum effects are expected to become significant as curvature diverges. The work generalizes previous results obtained under conformal coupling to the more physically relevant case of minimal coupling, and provides explicit examples of quantum states respecting self-similarity.

Renormalized Stress Tensor and Self-Similarity

The central technical result concerns the scaling of the expectation value of the stress tensor $\langle T_{\mu\nu} \rangle$ under self-similarity transformations. For a spacetime metric $g_{\mu\nu}(u, y^i)$ satisfying

$$g_{\mu\nu}(u+\alpha, y^i) = e^{-2\alpha} g_{\mu\nu}(u, y^i),$$

and a quantum state whose Wightman function transforms as

$$w(u+\alpha, y; u'+\alpha, y') = e^{2\alpha} w(u, y; u', y'),$$

the renormalized stress tensor exhibits the scaling

$$\langle T_{\mu\nu} \rangle(u+\alpha, y) = e^{2\alpha} \left( \langle T_{\mu\nu} \rangle(u, y) + \frac{\alpha}{4\pi} V_{\mu\nu}(u, y) \right),$$

where $V_{\mu\nu}$ is a geometric, state-independent tensor constructed from curvature invariants and their derivatives. This scaling law is derived using the Hadamard point-split renormalization procedure, with careful attention to the transformation properties of the Hadamard parametrix under global scale changes.

For continuously self-similar spacetimes, the result simplifies to

$$\langle T_{\mu\nu} \rangle(u, y) = e^{2u} \left( \langle T_{\mu\nu} \rangle(y) + \frac{u}{4\pi} V_{\mu\nu}(y) \right),$$

demonstrating that the quantum corrections to the classical stress tensor grow exponentially with the self-similarity parameter $u$.

Quantum Lyapunov Exponent and Geometric Leading Coefficient

The scaling behavior of the stress tensor implies that quantum modifications to the classical solution of the Einstein equations must also scale as $e^{2u}$ to maintain consistency. This leads to the identification of a quantum Lyapunov exponent $\omega_q = 2$, in agreement with previous results for conformally coupled fields. Notably, the leading coefficient of the exponential growth is determined solely by local geometric data, independent of the quantum state or renormalization scheme. This geometric universality suggests that quantum effects in self-similar collapse can be incorporated without explicit computation of vacuum expectation values, at least in the regime of weak backreaction.

Explicit Construction of Self-Similar States

Two explicit examples of quantum states respecting self-similarity are provided:

1. Minkowski Vacuum on a Self-Similar Patch: By restricting Minkowski space to the region $t < r$ and introducing suitable coordinates, the vacuum two-point function is shown to possess the required self-similarity property. In this case, the geometric tensor $V_{\mu\nu}$ vanishes, as expected for flat spacetime.
2. Vacuum State on the Hayward Spacetime: The Hayward solution, a self-similar solution to the Einstein-scalar field equations with minimal coupling, is analyzed in detail. The mode decomposition of the scalar field is constructed, and the corresponding vacuum state is shown to be Hadamard and to respect self-similarity. The geometric tensor $V_{\mu\nu}$ is computed explicitly for this spacetime, yielding nontrivial, traceless components:

$$V_{\tau\tau} = \frac{1}{80}, \quad V_{uu} = \frac{17}{240}, \quad V_{\theta\theta} = -\frac{7}{480}, \quad V_{\phi\phi} = V_{\theta\theta} \sin^2\theta,$$

with all other components vanishing. The trace of $V_{\mu\nu}$ vanishes, consistent with the structure of renormalization ambiguities in massless theories.

Contradiction with Dimensional Reduction Techniques

A significant claim of the paper is the demonstration that dimensional reduction techniques, commonly used to simplify the analysis of quantum fields in spherically symmetric spacetimes, yield qualitatively and quantitatively incorrect results for the expectation value of the stress tensor. In particular, the dimensional reduction approach fails to produce stationary states on the Hayward spacetime and imposes constraints on the stress tensor that are not satisfied by the explicit construction presented. This calls into question the validity of dimensional reduction for studying quantum effects in critical collapse and related scenarios.

Implications and Future Directions

The results have both practical and theoretical implications. The geometric universality of the leading quantum correction provides a robust tool for analyzing quantum effects in self-similar collapse, especially in the regime where backreaction is weak and the spacetime remains approximately self-similar. The explicit construction of self-similar states in nontrivial spacetimes suggests that such states may exist more generally, potentially serving as dynamically preferred vacua in critical collapse.

The contradiction with dimensional reduction techniques highlights the need for caution when applying lower-dimensional models to capture quantum effects in higher-dimensional spacetimes. Future work should focus on extending the analysis to more general self-similar solutions, exploring the existence and properties of self-similar quantum states, and investigating the breakdown of self-similarity under strong backreaction.

Conclusion

This paper establishes that, for quantum scalar fields on self-similar spacetimes, the expectation value of the stress tensor scales exponentially with the self-similarity parameter, characterized by a quantum Lyapunov exponent $\omega_q = 2$ and a leading coefficient determined by local geometric data. The analysis generalizes previous results to minimally coupled fields and provides explicit examples of self-similar quantum states. The findings challenge the validity of dimensional reduction techniques in this context and suggest new avenues for the paper of quantum effects in critical gravitational collapse.