Static Spherical Vacuum Solution

• Static spherical vacuum solutions are spacetime metrics that, in the absence of matter, utilize torsion to extend the classical Schwarzschild framework.
• The introduction of a contorsion tensor via a scalar torsion potential redefines the field equations, yielding explicit modifications to metric functions and observable phenomena.
• Parameter α influences horizon structure and black hole thermodynamics, offering testable deviations from general relativity through effects on light bending and perihelion advance.

A static spherical vacuum solution is a spacetime metric that is both static and spherically symmetric, and which solves the field equations of a given gravity theory in the absence of matter. In classical General Relativity (GR), the Schwarzschild metric uniquely fills this role. However, when generalizing gravity to include torsion, higher-order curvature, non-Riemannian geometry, or alternative conservation laws, the structure, uniqueness, and physical properties of static spherical vacuum solutions become substantially more varied. In non-Riemannian extensions, such as those including torsion, additional degrees of freedom lead to novel static vacuum geometries that exhibit signatures and observables deviating from the standard Schwarzschild predictions, with consequences for both fundamental theory and observational tests.

1. Geometric and Field-Theoretic Framework with Torsion

In non-Riemannian gravity, the fundamental connection $\Gamma^\alpha{}_{\beta\gamma}$ includes both the metric-compatible Christoffel symbols $\{\}^{\alpha}_{\beta\gamma}$ and an independent contorsion tensor $K^\alpha{}_{\beta\gamma}$: $\Gamma^\alpha{}_{\beta\gamma} = \left\{\!^{\alpha}_{\beta\gamma}\!\right\} + K^\alpha{}_{\beta\gamma}, \qquad T^\alpha{}_{\beta\gamma} = \Gamma^\alpha{}_{\beta\gamma} - \Gamma^\alpha{}_{\gamma\beta}.$ For static, spherically symmetric vacuum solutions, all relevant tensors are formed from the metric $g_{\mu\nu}$ and a contorsion ansatz respecting symmetry constraints. A particularly effective choice is

$$K_{\alpha\beta\gamma} = g_{\gamma\beta} \, \partial_\alpha \phi(r) - g_{\alpha\beta} \, \partial_\gamma \phi(r),$$

reducing the problem to determining a scalar "torsion potential" $\phi(r)$ alongside the metric functions, and rendering the system formally equivalent to a scalar-tensor theory.

2. Explicit Metric Structure and Integration of Field Equations

The general static, spherically symmetric line element is

$$ds^2 = F(r) \, dt^2 - \frac{1}{G(r)} \, dr^2 - r^2 (d\theta^2 + \sin^2\theta\, d\varphi^2),$$

with $F(r)$ and $G(r)$ determined by the coupled system derived from the modified Einstein equations. For the above contorsion, the field equations integrate to

$$F(r) = \exp\left[2\phi(r)\right] \left(1 - \frac{2GM}{r} \, e^{\phi(r)} \right),\qquad G(r) = \frac{1}{\left[1 - r\phi'(r)\right]^2} \left(1 - \frac{2GM}{r} \, e^{\phi(r)} \right).$$

To obtain an asymptotically flat metric and explicit torsion corrections, the scalar $\phi(r)$ is chosen as

$\phi(r) = \ln\left(1 + \frac{\alpha}{r}\right),$

with $\alpha$ serving as a torsion parameter. Explicit metric coefficients then become

$$F(r) = \frac{(\alpha + r)^2}{r^2} \left[1 - \frac{2GM}{r}\left(1 + \frac{\alpha}{r}\right)\right], \quad G(r) = \frac{(\alpha + r)^2 r^2}{(2\alpha + r)^2 \left[r^2 - 2GM(r+\alpha)\right]}.$$

This form includes the Schwarzschild metric as the $\alpha\to 0$ limit.

3. Causal Structure: Horizons and Singularities

The location and number of horizons are governed by the real roots of the polynomial

$P(r) = (\alpha + r)^2 [ r^2 - 2GM(r + \alpha) ],$

yielding roots at $R_{\rm in} = -\alpha$, $R_- = GM - \sqrt{GM(GM+2\alpha)}$, and $R_+ = GM + \sqrt{GM(GM+2\alpha)}$. The structure is parameter-dependent:

• For $\alpha > 0$: there is a single (event) horizon $R_+ > 0$ outside $r=0$.
• For $-\frac{GM}{2} < \alpha < 0$: three real roots $R_{\rm in} < R_- < R_+$ exist, giving one exterior and two interior horizons surrounding the singularity.
• For small torsion ($|\alpha|\ll GM$), the outer horizon is located at $R_+ = 2GM + \alpha + O(\alpha^2)$.

In the Schwarzschild limit ($\alpha = 0$), $R_+$ reduces to $2GM$ and the multiple-horizon structure collapses accordingly.

4. Observational Effects and Corrections to Classical Tests

Torsion modifications propagate into classical observables relevant for experimental tests of gravity:

• Perihelion advance per revolution for a planet of semi-major axis $a$ and eccentricity $e$:

$\Delta\varpi = \frac{6\pi (GM+\alpha)}{a(1-e^2)}.$

• Light-bending angle for a light ray passing at impact parameter $R$:

$\Delta\varphi = \frac{4(GM+\alpha)}{R}.$

Both formulas reproduce the standard general relativistic results for $\alpha=0$, while nonzero $\alpha$ induces perturbative corrections that could, in principle, be constrained by precision observation of solar-system phenomena.

5. Black Hole Thermodynamics and Extended First Law

Introducing torsion influences semiclassical black hole thermodynamics, modifying both the Hawking temperature and the first law:

• Hawking temperature:

$$T_H = \frac{1}{2\pi\delta}, \qquad \delta = 2R_+ = 2\left(GM + \sqrt{GM(GM+2\alpha)}\right),$$

which for small $\alpha$ yields

$$T_H = \frac{1}{4\pi GM} \left(1 - \frac{\alpha}{2GM} + \cdots \right).$$

• First law extension: Let $A_{eh} = 4\pi R_+^2$ (event horizon area), then

$$\frac{1}{4G} dA_{eh} = \frac{1}{T_H} \left(dM + \frac{1}{2G} d\alpha\right),$$

so

$$dM = T_H dS_{\text{geom}} - \Phi_\alpha d\alpha, \quad \Phi_\alpha = -\frac{1}{2G}.$$

This additional work term $\Phi_\alpha d\alpha$ identifies $\alpha$ as a work variable conjugate to a "torsion potential." Such an extension parallels previous generalizations of black hole mechanics by including new physical degrees of freedom.

6. Relation to General Relativity and Limiting Behavior

In the limit $\alpha \to 0$, all features of the solution continuously deform to those of Schwarzschild geometry: $$\phi \to 0, \quad F(r) \to 1 - \frac{2GM}{r}, \quad G(r) \to 1 - \frac{2GM}{r},$$ reinstating the standard uniqueness of the static, spherically symmetric vacuum solution. Conversely, nonzero $\alpha$ explicitly parametrizes departures from GR predictions, allowing tests of the presence of spacetime torsion with astrophysical and gravitational-wave observations.

7. Broader Context and Theoretical Implications

Static spherical vacuum solutions with torsion exemplify the structure of exact solutions in non-Riemannian theories, offering insight into:

• The interplay between additional geometric degrees of freedom (here, torsion) and observable quantities.
• How classical and semiclassical black hole thermodynamics generalize when new fields or connection components enter the theory.
• The way non-Schwarzschild horizon structure (including multiple or shifted horizons) arises generically in extensions beyond pure Riemannian geometry.

These solutions also highlight how even small, theoretically plausible extensions to the connection structure of gravity can systematically modify predictions for strong- and weak-field tests, and how first-law thermodynamics must be adapted to include novel physical work terms. The non-Riemannian framework thus provides both a mathematically precise and observationally testable generalization of the classical vacuum paradigm.