Abbassi Coordinates in SdS Spacetime

• Abbassi coordinates are a class of coordinate systems for Schwarzschild–de Sitter spacetime that combine local black hole properties with global cosmological expansion using a FRW asymptotic form.
• They are derived from a spherically symmetric, spatially flat metric ansatz, ensuring regularity across black hole and cosmological horizons when the cosmological constant is positive.
• By retaining an explicit cosmological time coordinate, Abbassi coordinates provide a practical alternative to Kruskal–Szekeres and Israel coordinates for studying black holes in an expanding universe.

Abbassi coordinates are a class of coordinate systems for the Schwarzschild–de Sitter (SdS) spacetime, constructed to be regular (i.e., nonsingular) across both the black hole and cosmological horizons, while also incorporating the cosmological expansion manifest in Friedmann–Robertson–Walker (FRW) form at large distances. They provide a framework that explicitly links the local gravitational field of a black hole (via mass parameter $m$) with the global effects of cosmological expansion (through the cosmological constant $\Lambda$ and scale factor $a(\eta)$), and are particularly well suited for analyzing the interplay between local black hole physics and the evolving universe background (Lima et al., 1 Aug 2025).

1. Derivation and Mathematical Structure

The derivation begins by demanding a spherically symmetric, spatially flat metric ansatz that tends to the FRW form at large scales: $$ds^2 = a^2(\eta)\left[ -e^{\alpha(\eta, r)}\, d\eta^2 + e^{\beta(\eta, r)}\, dr^2 + r^2 d\Omega^2 \right]$$ where $a(\eta)$ is the scale factor and $\eta$ is conformal time. Defining $\ell \equiv a(\eta) r$, and using the constraint from the Einstein equations that $\alpha + \beta$ is constant (absorbed by rescaling coordinates), the metric coefficients reduce to: $$e^{\alpha} = A(\ell), \qquad e^{\beta} = A^{-1}(\ell)$$ with $A(\ell)$ determined by: $$A(\ell) = \frac12 \Big[ f(\ell) \pm \sqrt{f^2(\ell) + 4\ell^2 (\Lambda/3)}\,\Big] \,, \qquad f(\ell) = 1 - \frac{2m}{\ell} - \frac{\Lambda}{3}\ell^2$$ Two branches arise, denoted $A_+$ and $A_-$. The $A_+$ solution, first constructed by Abbassi in 1999, is the standard form: $$A_+(\ell) = \frac12 \left[ f(\ell) + \sqrt{f^2(\ell) + 4\ell^2 (\Lambda/3)}\,\right]$$ and the line element becomes

$$ds^2_{(+)} = a^2(\eta) \left[ -A_+(\ell)\, d\eta^2 + \frac{dr^2}{A_+(\ell)} + r^2 d\Omega^2 \right] \,, \qquad \ell = a(\eta) r$$

The $A_-$ branch is mathematically equivalent, differing by an inversion of temporal/spatial coordinate roles, but the $A_+$ (Abbassi) branch remains the physically accepted form for cosmological matching (Lima et al., 1 Aug 2025).

2. Regularity and the Role of $\Lambda > 0$

Regularity of the Abbassi coordinates across both the black hole and cosmological horizons is ensured for strictly positive cosmological constant, $\Lambda > 0$. The discriminant $\sqrt{f^2(\ell) + 4\ell^2 (\Lambda/3)}$ remains nonvanishing and real in this case, so the metric is regular (except at the physical singularity $\ell = 0$):

• For $\Lambda=0$, the expression reduces and degeneracies appear: $A_+$ vanishes for $\ell < 2m$, while $A_-$ vanishes for $\ell > 2m$; neither covers the whole manifold individually.
• Hence, the Abbassi coordinates are not a mere extension of Kruskal–Szekeres or Israel coordinates to $\Lambda>0$; rather, the additional $\Lambda$ term ensures regularity through the horizons, encoding the cosmological background smoothly.

This makes the coordinates intrinsically cosmological: $\eta$ is conformal time, and the local and global metric structure is consistently linked by $a(\eta)$. The choices in constructing these coordinates align the time coordinate explicitly with cosmological evolution.

3. Comparison with Kruskal–Szekeres and Israel Coordinates

Kruskal–Szekeres and Israel coordinates for (A)dS–Schwarzschild spacetimes are constructed to remove coordinate singularities via null coordinates, which necessitate signature flips (temporal and spatial roles invert across the horizon):

• Abbassi coordinates, however, retain an explicit cosmological time, producing diagonal metrics that preserve the character of the time and radial coordinates across the horizons.
• No patchwork covering or signature inversion is required; time remains an explicit, globally defined coordinate.
• The construction demands FRW form in the asymptotic limit, a property not present in Kruskal–Szekeres or Israel-type extensions.

These properties make Abbassi coordinates effective for studying black holes embedded in expanding universes, where faithful representation of cosmological time is crucial.

4. Explicit Coordinate Transformations

Abbassi coordinates are diffeomorphic to standard forms via explicit coordinate transformations. Two important examples are to Kottler coordinates (the static SdS chart) and to the maximally extended Lake–Israel coordinates.

(a) Transformation to Kottler Coordinates

The static SdS metric (Kottler): $$ds^2 = -f(\ell)\, d\tilde{t}^2 + \frac{d\ell^2}{f(\ell)} + \ell^2 d\Omega^2$$ is related to the Abbassi chart by:

• $\ell = a(\eta) r$
• Physical time $t$ such that $dt = a\, d\eta$

The nontrivial transformation between time coordinates involves removing $d\ell\, d\eta$ cross terms, yielding: $$\pm d\tilde{t} = a\, d\eta + \sqrt{\frac{\Lambda}{3}\frac{\ell}{f(\ell) A_+(\ell)}}\, d\ell$$ This explicit mapping demonstrates the regularity and physical equivalence of the Abbassi and Kottler forms.

(b) Transformation to Lake–Israel Coordinates

The maximally extended Lake–Israel coordinates write the metric as: $$ds^2 = g(u,w)\, du^2 + 2 du\, dw + v^2(u,w)\, d\Omega^2,\qquad v(u,w) = a(\eta) r = \ell$$ The mapping involves expressing $dt$ in terms of $(u, w)$, $v$, and $f(v)$. The full expression is complicated but confirms that both coordinate systems are regular at the horizons; their relation is globally well-defined except at isolated transformation singularities. This mapping underscores that the Abbassi chart, though only covering a particular patch, provides a regular description of the SdS manifold matching the Lake–Israel extension.

5. Physical Interpretation and Applications to Black Holes in Cosmology

Abbassi coordinates encode both local black hole effects ($m$) and global cosmological expansion ($\Lambda$, $a(\eta)$) via explicit metric dependence on $\ell = a(\eta) r$. Consequences include:

• The metric outside the black hole interpolates continuously to a de Sitter (cosmologically expanding) regime at large $\ell$.
• The proper distance from the center is dynamically linked to universal expansion, modeling physical scenarios (e.g., potential “cosmological coupling of black hole masses”) relevant to current research.
• Only outgoing null geodesics cross both horizons for an expanding universe, as demonstrated by causal structure analysis.
• These features make the coordinates pertinent for examining black hole mass evolution, cosmological coupling hypotheses, and the influence of dark energy on compact objects.

A plausible implication is that studies of black holes in non-static, cosmologically realistic backgrounds benefit from adopting Abbassi coordinates for both analytic and numerical analyses.

6. Summary and Distinctive Features

Abbassi coordinates are derived by requiring simultaneous regularity at all SdS horizons and asymptotic FRW behavior, yielding a diagonal metric in cosmological time for $\Lambda > 0$. Their salient points are summarized in the following table:

Coordinate System	Time Coordinate	Regular at Horizons	$\Lambda > 0$ Required	Covers Full SdS Manifold
Abbassi (A₊ branch)	Cosmological (FRW)	Yes	Yes	FRW Patch
Kruskal–Szekeres	Null-like	Yes	No	Maximal
Kottler (static SdS)	Static	No	No	Static Patch

• Abbassi coordinates cover those regions of SdS matched to expanding (or contracting) FRW universes; they are not suitable for global maximal extension but are optimal for cosmological studies.
• The time and space roles are preserved globally across all horizons due to the asymptotic cosmological construction.
• Explicit transformation formulas relate Abbassi coordinates to Kottler and Lake–Israel charts, confirming their full geometric content.

The coordinates provide a powerful tool for investigating the interplay between local dynamics of black holes and global cosmological evolution, an area of increasing interest in gravitational physics (Lima et al., 1 Aug 2025).