Arrow of time problem in gravitational collapse

We investigate the arrow of time problem in the context of gravitational collapse of radiating stars in higher dimensions for both neutral and charged matter. The interior spacetime is described by a shear-free spherically symmetric metric filled with a dissipative fluid. The exterior spacetime of the radiating star is taken as the higher dimensional Vaidya metric. We establish that the arrow of time associated with gravitational entropy is opposite to the thermodynamic arrow of time for all dimensions. The physical consequences of our results are considered. Our result conforms with previous studies on shear-free spherical collapse, which suggests, avoidance of the naked singularity as the end state results in a wrong arrow of time, indicating a fundamental problem with the local application of the Weyl curvature hypothesis.

💡 Research Summary

The paper addresses the long‑standing “arrow of time” problem in the context of gravitational collapse, extending previous four‑dimensional analyses to an arbitrary number of spacetime dimensions (N ≥ 4) and to both neutral and charged matter. The authors model a radiating star as a two‑region system: an interior filled with a shear‑free, spherically symmetric dissipative fluid, and an exterior described by the higher‑dimensional Vaidya metric, which carries outward null radiation. The interior metric is taken in the general form

 ds² = –A(r,t)² dt² + B(r,t)² dr² + Y(r,t)² dΩ_{N‑2}²,

with two concrete subclasses. In the first subclass the lapse function A depends only on the radial coordinate (static seed solution) while the overall scale factor Y = r B is multiplied by a positive time‑dependent function f(t). In the second subclass A itself acquires a time dependence. In both cases the fluid four‑velocity is comoving, the shear vanishes, and the heat flux vector points radially outward. Matching the interior to the exterior across the stellar surface Σ yields the standard junction condition

 p|_Σ = (q B)|_Σ,

which guarantees that the pressure at the surface is non‑zero and directly related to the outward heat flow. Because the Vaidya exterior carries a mass function m(u,R) that decreases with retarded time u, the radiation always moves outward, providing an unambiguous thermodynamic arrow of time.

To quantify the gravitational contribution to entropy, the authors adopt the Weyl‑curvature‑hypothesis (WCH) framework. Two “epoch functions” are employed:

 P = W/R = (C_{abcd}C^{abcd})/(R_{ab}R^{ab}),

 P₁ = W/K = (C_{abcd}C^{abcd})/(R_{abcd}R^{abcd}),

where W is the Weyl scalar, R the Ricci‑tensor square, and K the Kretschmann scalar. P is well‑defined only in non‑vacuum regions (the Ricci tensor must be non‑zero), whereas P₁ remains finite even in vacuum (e.g., outside the star). Both functions are dimension‑sensitive, and the authors explicitly write their dependence on N.

The core of the analysis consists of inserting the metric ansätze into the Einstein field equations (with a higher‑dimensional coupling constant κ_N) and solving for the energy density µ, isotropic pressure p, and heat flux q. The resulting expressions contain the time‑derivative of f(t) and various radial derivatives of the static seed functions A₀(r) and B₀(r). Importantly, the heat flux is proportional to the time derivative of f(t) and is always positive, confirming the outward radiation.

The authors then evaluate P and P₁ for the interior solution as functions of f(t) and its derivatives. In all examined dimensions (N = 4, 5, 6, … up to at least 10) and for both neutral and charged configurations (the latter introduces additional terms in the Ricci tensor but does not alter the dominant Weyl contribution), they find that both epoch functions decrease monotonically as the collapse proceeds (i.e., as f(t) shrinks). Consequently, the gravitational entropy, as defined by the WCH, decreases with time, in stark contrast to the thermodynamic entropy of the emitted radiation, which increases. This reversal holds irrespective of the number of dimensions or the presence of electric charge.

The paper interprets this result as a failure of the local application of the Weyl curvature hypothesis. While the WCH successfully accounts for the global increase of gravitational entropy during cosmological structure formation (where the Weyl tensor grows relative to the Ricci tensor), it predicts the opposite behavior in a collapsing star, where the free‑gravitational field (Weyl curvature) actually diminishes relative to the matter content as the star radiates away energy. The authors argue that this “arrow of time problem” is intrinsic to the definition of gravitational entropy based solely on curvature invariants and suggests that a more refined or entirely different notion of gravitational entropy is required for local, dynamical processes.

In the concluding discussion, the authors emphasize several points:

1. Dimension Independence: The opposite orientation of the gravitational arrow persists for all N ≥ 4, indicating that the issue is not an artifact of four‑dimensional geometry but a fundamental feature of the curvature‑based entropy measures.

2. Charge Independence: Adding electric charge to the interior fluid modifies the Ricci tensor but leaves the qualitative behavior of P and P₁ unchanged; the gravitational arrow remains opposite to the thermodynamic arrow.

3. Implications for Cosmic Censorship: The authors note that earlier work linked avoidance of naked singularities to a “correct” arrow of time. Their findings suggest that insisting on a consistent arrow may inadvertently favor scenarios that violate cosmic censorship, highlighting a deeper tension between entropy concepts and singularity theorems.

4. Future Directions: The paper calls for alternative gravitational entropy candidates (e.g., those based on quasi‑local mass, holographic principles, or thermodynamic potentials derived from the Noether charge) and for investigations of how such measures behave in dynamical collapse across dimensions.

Overall, the study provides a thorough, mathematically detailed examination of the arrow‑of‑time problem in higher‑dimensional radiative collapse, demonstrates the robustness of the paradox across dimensions and charges, and underscores the need for a revised understanding of gravitational entropy in local, non‑equilibrium settings.