Energy transfer between the sources in gravitational decoupling

TL;DR

The paper addresses how energy transfer can occur between multiple gravitational sources inside self‑gravitating, relativistic stars by extending Gravitational Decoupling (GD) with a polytropic component. It presents an analytic framework in static, spherically symmetric spacetimes that decomposes the total energy–momentum into a seed source and an auxiliary sector, linked by a controlled metric deformation and a nonperturbative energy‑exchange term. The key contributions are the explicit GD scheme for two interacting sources, the energy‑exchange relation $\Delta E = \frac{g'}{2}(\rho + p_r)$, and analytic expressions for the effective density and pressures under several admissible $\Delta E(r)$ profiles, ensuring smooth Israel–Darmois matching to Schwarzschild. The findings demonstrate that polytropic energy transfer can modify global stellar properties, including mass, radius, and anisotropy, offering a tractable path toward realistic multi‑component models in GR and beyond, with several extensions proposed for future work.

Abstract

A straightforward and fully analytic approach is introduced to examine how polytropic fluids influence arbitrary gravitational sources in static, spherically symmetric spacetimes. As a concrete application, we explore the internal mechanism of energy transfer between gravitational sources embedded within a self-gravitating system.

Figures (3)

• Figure 1: The spatial dependence of the energy–exchange term $\Delta E(r)$ for the eight suggested profiles. The panels are ordered from left to right in the first row and then in the second row. The top row displays, respectively, the power-law growth profile $\Delta E_1$, the inverted power-law profile $\Delta E_2$, the exponential decay profile $\Delta E_3$, and the peaked profile $\Delta E_4$. The bottom row shows, in order, the logarithmic profile $\Delta E_5$, the smooth-decay profile $\Delta E_6$, the oscillatory profile $\Delta E_7$, and the saturating profile $\Delta E_8$.
• Figure 2: Radial pressure $p_r(r)+\mathcal{P}_{r}(r)$ (left column), tangential pressure $p_t(r)+\mathcal{P}_{t}(r)$ (middle column), and energy density $\rho(r)+E(r)$ (right column) for different choices of the energy–transfer profile $\Delta E(r)$. From top to bottom, the rows correspond to the power–law growth profile $\Delta E_1$, the peaked profile $\Delta E_4$, the oscillatory profile $\Delta E_7$, and the saturating profile $\Delta E_8$.
• Figure 3: Radial pressure (top row) and energy density (bottom row) of the highest curve of each selected energy transfer function. From left to right, each column corresponds to the power–law growth profile $\Delta E_1$, the peaked profile $\Delta E_4$, the oscillatory profile $\Delta E_7$, and the saturating (sigmoidal) profile $\Delta E_8$.