First Law of Black Hole Interior Dynamics

Abstract: We obtain a local first law of dynamics within the interior geometries that are governed by cosmological solutions connecting the event horizon and the Kasner-like singularity. Evaluating the Iyer-Wald identity between the horizon and near-singularity geometries, we define a Kasner potential and transported response coefficients. We test the first law by both exact and numerical solutions. Our formalism provides a self-consistent approach to study the interior dynamics without having to appeal to the outside geometry and the asymptotic charges. It may provide a new approach to study the phase structures of the interior dynamics.

• The paper establishes a local variational law, using the Iyer–Wald covariant phase-space formalism, that relates the Kasner potential to horizon entropy, electric charge, and normalized scalar charge without using asymptotic infinity.
• The framework defines transported temperature and electric potential that remain invariant under interior time rescaling, and verifies the three-term law analytically for Schwarzschild, charged EMD, and scalar-hairy solutions.
• Numerical Einstein-scalar families confirm the response relations for independent interior parameters, while a modified Smarr relation exposes how Kasner scalar hair and potential-energy bulk terms alter standard black-hole thermodynamics.

Overview

Xiong and Lü establish a local first law of dynamics for the interior of black holes in Einstein-Maxwell-Scalar (EMS) theories, formulated entirely within the cosmological region connecting the event horizon to a Kasner-like spacelike singularity (2608.18583). The central result is the variational identity

$$\delta M_{\rm K} = \mathcal{T}\,\delta S + \Phi_e\,\delta Q_e + \phi_{\rm K}\,\delta\Sigma_{\rm K},$$

where $M_{\rm K}$ is a newly defined "Kasner potential" extracted from the near-singularity geometry, $S$ is the horizon entropy, $Q_e$ the electric charge, $\Sigma_{\rm K}$ a normalized scalar charge at the singularity, and $\mathcal{T} = T/\Xi$ and $\Phi_e = (\Phi_H - \Phi_K)/\Xi$ are "transported response coefficients." The derivation uses the covariant phase-space (Iyer-Wald) formalism evaluated on a hypersurface spanning from the horizon to a regulated Kasner surface, and is verified against both exact analytic solutions and numerical horizon-to-Kasner geometries.

The significance of this construction lies in its locality: it requires no reference to asymptotic infinity or to asymptotic charges. This addresses a genuine conceptual obstacle, since inside the horizon the coordinate $t$ becomes spacelike and admits no canonical normalization; the authors show that all quantities entering the first law — including $M_{\rm K}$ itself — are invariant under constant rescalings $t \to \lambda t$.

Interior parameters: horizon and Kasner data

The framework is developed for EMS theories with action containing generic couplings $M_{\rm K}$0 between the scalar and Maxwell field and a potential $M_{\rm K}$1, using the static spherically symmetric ansatz with metric functions $M_{\rm K}$2, $M_{\rm K}$3. For $M_{\rm K}$4 this is the general static ansatz in areal gauge; for $M_{\rm K}$5 the solution is cosmological, with $M_{\rm K}$6 playing the role of time. Horizon quantities are standard: temperature $M_{\rm K}$7, entropy $M_{\rm K}$8, and electric potential $M_{\rm K}$9. Local charges on a sphere of radius $S$0 include the radially conserved Maxwell charge $S$1 and a non-conserved scalar momentum $S$2 conjugate to $S$3.

For the scalar-hairy configurations considered, no inner (Cauchy) horizon forms, consistent with earlier no-inner-horizon results (Cai et al., 2020), and the interior terminates at a Kasner singularity. Near $S$4,

$S$5

with the two Kasner constraints imposing $S$6, directly tying the metric power law to the scalar hair. The combination $S$7 is finite by the first Kasner constraint. The normalized scalar charge is $S$8, and the Kasner potential is defined as

$S$9

The authors emphasize that $Q_e$0 is not an energy or ADM mass; it is fixed uniquely by requiring that the Iyer-Wald Hamiltonian variation close into an exact 1-form.

Iyer-Wald derivation

For any Killing vector $Q_e$1, Stokes' theorem applied to the Iyer-Wald 2-form $Q_e$2 yields $Q_e$3 between the horizon and a regulated surface $Q_e$4. This holds even though the interior is cosmological and $Q_e$5 is timelike. On the horizon, $Q_e$6. At the regulated surface, the potentially logarithmically divergent term $Q_e$7 vanishes identically as a consequence of the Kasner constraint — a nontrivial consistency check of the whole construction. Equating the finite parts gives

$Q_e$8

which, upon defining $Q_e$9, $\Sigma_{\rm K}$0, and $\Sigma_{\rm K}$1, is precisely the interior first law. An important structural point: for genuine black holes obeying the no-scalar-hair theorem, $\Sigma_{\rm K}$2 is not independent but a function of $\Sigma_{\rm K}$3, so the third term is trivially constrained. The first law becomes fully nontrivial only for more general "black interiors" — cosmological regions bounded by a horizon but not smoothly connected to asymptotic Minkowski space — where $\Sigma_{\rm K}$4 are all independent.

Analytic and numerical verification

The first law is checked exactly in three settings:

• Schwarzschild: $\Sigma_{\rm K}$5, $\Sigma_{\rm K}$6, and $\Sigma_{\rm K}$7 holds identically.
• Charged EMD black holes ($\Sigma_{\rm K}$8): with closed-form expressions such as $\Sigma_{\rm K}$9 and $\mathcal{T} = T/\Xi$0, the full three-term first law is satisfied.
• Neutral scalar hairy black holes with potentials admitting exact solutions (Feng et al., 2013): again verified analytically, though here $\mathcal{T} = T/\Xi$1 is fixed by coupling constants, so $\mathcal{T} = T/\Xi$2.

To exercise the scalar-charge term nontrivially, the authors construct numerical two-parameter families of horizon-to-Kasner solutions in pure Einstein-scalar gravity with potentials $\mathcal{T} = T/\Xi$3 and $\mathcal{T} = T/\Xi$4, scanning $\mathcal{T} = T/\Xi$5 and $\mathcal{T} = T/\Xi$6. None of these horizons integrate smoothly to asymptotic flatness except along fine-tuned subfamilies. Data fitting extracts $\mathcal{T} = T/\Xi$7, from which theoretical response functions $\mathcal{T} = T/\Xi$8 and $\mathcal{T} = T/\Xi$9 are computed and compared to direct numerical values; the agreement is reported as exact within numerical accuracy, including along the black-hole subfamily. This confirms that the first law governs the entire two-parameter class of black interiors, not merely black holes proper.

Smarr relations and scaling subtleties

Because $\Phi_e = (\Phi_H - \Phi_K)/\Xi$0 is not a mass, the usual Smarr relation fails. Under the scaling induced by $\Phi_e = (\Phi_H - \Phi_K)/\Xi$1 (with $\Phi_e = (\Phi_H - \Phi_K)/\Xi$2), the charges transform homogeneously but $\Phi_e = (\Phi_H - \Phi_K)/\Xi$3 acquires a logarithmic shift,

$\Phi_e = (\Phi_H - \Phi_K)/\Xi$4

yielding via Euler's theorem the modified Smarr-like relation

$\Phi_e = (\Phi_H - \Phi_K)/\Xi$5

This is corroborated independently by a generalized Komar charge $\Phi_e = (\Phi_H - \Phi_K)/\Xi$6, whose radial conservation for $\Phi_e = (\Phi_H - \Phi_K)/\Xi$7 reproduces the same relation via Stokes' theorem. For Schwarzschild and EMD black holes it reduces to the standard Smarr relation. When $\Phi_e = (\Phi_H - \Phi_K)/\Xi$8, an additional bulk integral appears:

$\Phi_e = (\Phi_H - \Phi_K)/\Xi$9

Notably, whenever the solution happens to be the interior of a genuine black hole, this relation reproduces the familiar exterior Smarr relation — for instance, the Schwarzschild-AdS relation $t$0 follows with $t$1, $t$2. The interior and exterior Smarr structures are thus equivalent where they overlap, while the interior version remains well-defined without asymptotic normalization.

Limitations and open questions

Several restrictions should be noted. The derivation assumes spherical symmetry, staticity outside the horizon, and a single scalar field; rotating interiors, higher dimensions beyond the ansatz used, and multi-scalar systems are not covered. The Kasner regime relies on scalar kinetic dominance so that the two Kasner constraints hold; whether the formalism extends to BKL-type oscillatory singularities is left open. The cancellation of the logarithmic divergence in $t$3 depends explicitly on the Kasner constraint, so the closure of the 1-form is guaranteed only within this kinetic-dominated class. The authors also note that no exact solutions with independent scalar hair were available for analytic testing of the full three-term law — the confirmation there rests on numerics alone. Finally, they conjecture without proof that the formalism extends to spacetimes with unavoidable naked singularities (e.g., free massless scalar configurations admitting no horizon at all), which remains to be demonstrated.

Conclusion

This paper supplies a self-consistent thermodynamic-style variational principle for black hole interiors, anchored in the Iyer-Wald formalism and insensitive to the time-rescaling ambiguity inherent to the cosmological region. The introduction of the Kasner potential $t$4, the transported coefficients $t$5 and $t$6, and the modified Smarr relation provides a concrete bridge between horizon data and near-singularity dynamics, validated by both exact and numerical solutions. The framework opens a systematic route to characterizing phase structure of interior dynamics, with the principal open questions being its extension beyond spherical symmetry, beyond kinetic-dominated Kasner regimes, and to naked-singularity geometries.