Schwarzschild Spacetime in STEGR Gravity

• Schwarzschild spacetime in STEGR is defined by a flat, torsionless connection and nonmetricity that fully encapsulates gravitational dynamics, paralleling GR field equations.
• The coincident gauge eliminates the affine connection, simplifying computations and clarifying the gauge dependence of energy-momentum complexes.
• Nonmetricity scalar analysis and energy-momentum prescriptions in STEGR provide alternative insights into gravitational localization and conserved charges.

The Schwarzschild solution in the context of Symmetric Teleparallel Equivalent of General Relativity (STEGR) constitutes a paradigmatic testbed for dissecting gravitational energy-momentum, localization, and gauge properties in the framework of non-metric gravity. STEGR, formulated in terms of a flat, torsionless affine connection and nonmetricity, is dynamically indistinguishable from general relativity (GR) at the level of field equations, yet it ascribes gravitational phenomena strictly to the nonmetricity tensor rather than curvature or torsion. This alternative geometrization offers distinctive prescriptions for energy-momentum complexes, conserved quantities, and their gauge-dependent and invariant features, especially when computed in the coincident gauge, where the affine connection vanishes globally.

1. Foundations of STEGR and the Role of the Coincident Gauge

STEGR is defined through two fundamental variables: the metric $g_{\mu\nu}$ and a flat ($R^\alpha{}_{\beta\mu\nu}(\Gamma)=0$), torsionless ($T^\alpha{}_{\mu\nu}=0$) connection $\Gamma^\alpha{}_{\mu\nu}$. The nonmetricity tensor,

$$Q_{\alpha\mu\nu} \equiv \nabla_\alpha g_{\mu\nu} = \partial_\alpha g_{\mu\nu} - \Gamma^\beta{}_{\alpha\mu} g_{\beta\nu} - \Gamma^\beta{}_{\alpha\nu} g_{\beta\mu},$$

encodes the entirety of gravitational dynamics in STEGR. The action is

$$S_{\text{STEGR}} = \frac{1}{2\kappa}\int d^4x\,\sqrt{-g} \left[ g^{\mu\nu}(L^\alpha{}_{\beta\mu}L^\beta{}_{\nu\alpha} - L^\alpha{}_{\beta\alpha}L^\beta{}_{\mu\nu}) \right],$$

with $L^\alpha{}_{\mu\nu}$ (disformation) built from nonmetricity and $\kappa=8\pi G/c^4$ (Capozziello et al., 17 Jan 2026, Emtsova et al., 2024, Emtsova et al., 2022).

The coincident gauge, defined by $\Gamma^\alpha{}_{\mu\nu}=0$, eliminates the affine connection entirely from calculations, rendering the covariant derivative simply a partial derivative, and driving all gravitational effects into the nonmetricity structure. This gauge is not unique, but it provides maximal calculational simplification and clarifies the gauge (coordinate) dependence of constructed gravitational energy densities.

2. Schwarzschild Solution in STEGR

The Schwarzschild metric in standard coordinates $(t, r, \theta, \varphi)$ reads

$$ds^2 = \left(1 - \frac{r_s}{r}\right)c^2 dt^2 - \left(1 - \frac{r_s}{r}\right)^{-1} dr^2 - r^2(d\theta^2 + \sin^2\theta\, d\varphi^2),$$

with $r_s=2GM/c^2$ the Schwarzschild radius (Capozziello et al., 17 Jan 2026, D'Ambrosio et al., 2021, Emtsova et al., 2024, Emtsova et al., 2022). In STEGR, the connection is fixed to be either pure gauge or, for calculational clarity, identically zero in the coincident gauge. All curvature invariants coincide with GR, and the vacuum field equations reduce identically to $G_{\mu\nu}=0$, guaranteeing that the Schwarzschild metric solves both GR and STEGR field equations. The nonmetricity tensor reduces to $Q_{\alpha\mu\nu} = \partial_\alpha g_{\mu\nu}$ in the coincident gauge, and the nonmetricity scalar—key for dynamics and energy definitions—evaluates as $Q(r) = -2/r^2$ for Schwarzschild (Capozziello et al., 17 Jan 2026).

3. Nonmetricity Scalar and Energy-Momentum Complex

The nonmetricity scalar $Q$ in STEGR is given by

$$Q = -\tfrac{1}{4} Q_{\alpha\mu\nu} Q^{\alpha\mu\nu} + \tfrac{1}{2} Q_{\alpha\mu\nu} Q^{\mu\alpha\nu} + \tfrac{1}{4} Q_\alpha Q^\alpha - \tfrac{1}{2} \tilde{Q}_\alpha Q^\alpha,$$

where $Q_\alpha = Q_{\alpha\lambda}{}^\lambda$ and $\tilde{Q}_\alpha = Q^\lambda{}_{\lambda\alpha}$. In Schwarzschild, one may alternatively express $Q$ as a quadratic function of the Levi-Civita connection (Capozziello et al., 17 Jan 2026, D'Ambrosio et al., 2021).

STEGR defines the gravitational energy-momentum pseudo-tensor $\tau^\alpha{}_\lambda$ (in the coincident gauge) as

$$\tau^\alpha{}_\lambda = \frac{1}{2\kappa^2} [ P^\alpha{}_{\mu\nu} \partial_\lambda g^{\mu\nu} + Q\, \delta^\alpha_\lambda ],$$

with $P^\alpha{}_{\mu\nu}$ the nonmetricity conjugate ("superpotential"). For a static, spherically symmetric Schwarzschild solution, $\partial_0 g_{\mu\nu}=0$ so the first term vanishes for $\alpha=0, \lambda=0$, yielding

$$\tau^0{}_0(r) = \frac{Q(r)}{2\kappa^2} = -\frac{c^4}{8\pi G}\frac{1}{r^2},$$

or, in terms of $r_s$, $$\tau^0{}_0(r) = -\frac{Mc^2}{4\pi r_s}\frac{1}{r^2}$$, interpreting $\tau^0{}_0$ as the local gravitational energy density (Capozziello et al., 17 Jan 2026).

4. Conserved Energy, Noether Charges, and Their Gauge Dependence

Global conserved quantities in STEGR are defined using Lorentz-invariant Noether superpotentials. The total superpotential for a vector field $\xi^\mu$ is

$$\mathcal{U}^{\alpha\beta}[\xi] = \mathcal{J}_{\mathrm{GR}}^{\alpha\beta}[\xi] + \mathcal{J}_{\mathrm{div}}^{\alpha\beta}[\xi],$$

with

$$\mathcal{J}_{\mathrm{GR}}^{\alpha\beta} = \frac{\sqrt{-g}}{\kappa} \nabla^{[\alpha}\xi^{\beta]}, \quad \mathcal{J}_{\mathrm{div}}^{\alpha\beta} = \frac{\sqrt{-g}}{\kappa} \delta_\sigma^{[\alpha}\left(Q^{\beta]}-\hat Q^{\beta]}\right)\xi^\sigma,$$

where $\hat Q_\alpha = g^{\mu\nu}Q_{\mu\alpha\nu}$ (Emtsova et al., 2024, Emtsova et al., 2022).

Selecting the time translation Killing vector $\xi^\mu=(1,0,0,0)$, the dominant asymptotic component is

$$\mathcal{U}^{0 1} = -\frac{M\sin\theta}{16\pi} + O(r^{-2}).$$

The total energy (mass) is obtained by a surface integral at infinity,

$$E = \oint_{S^2_{\infty}}\mathcal{U}^{0 1}\,d\theta\,d\varphi = M.$$

This reproduces the standard ADM/Komar mass as in GR and is invariant under asymptotically flat coordinate transformations (Emtsova et al., 2024, Emtsova et al., 2022).

The local energy density, however, is gauge-dependent and becomes intricately connected to the chosen coincident gauge. The energy measured by an observer with four-velocity $\xi^{\mu}_{\text{fall}}$ does not in general vanish for geodesic (freely falling) observers when using static coordinate-based coincident gauges (Emtsova et al., 2022). Only in specifically constructed gauges—such as the Painlevé–Gullstrand (PG)–based "Gauge SII**"—can one simultaneously achieve vanishing local current for free-fall ($\mathcal{J}^\alpha(\xi_{\text{fall}})=0$) and correct ADM mass, thereby manifesting the Einstein Equivalence Principle locally (Emtsova et al., 2024).

5. Ambiguity in Connection Choice and Physical Implications

The selection of the flat, torsionless connection ("inertial connection") is not unique in STEGR. Different "gauges" (coordinate systems plus connection prescriptions), even when obtained via the "turning off gravity" principle (setting mass and rotation to zero in the Levi-Civita connection), yield inequivalent local energy densities and can violate the expected vanishing of gravitational energy for a free-falling observer (Emtsova et al., 2024, Emtsova et al., 2022). This gauge freedom is markedly broader than in TEGR, where freeness is limited by Lorentz frames.

For Schwarzschild, the static coordinates yield a connection where local energy density for free-fall is nonzero, while the PG-based Gauge SII** eliminates it, restoring the direct equivalence-principle interpretation. These ambiguities suggest the need for additional physical criteria beyond coordinate/gauge choices to select a preferred inertial connection uniquely, such as demanding both correct global charge (ADM mass) and vanishing local energy density for certain geodesic frames (Emtsova et al., 2022).

6. Comparison to GR and the Physical Status of Gravitational Energy in STEGR

In STEGR, total conserved charges (ADM mass, Komar mass) evaluated via surface integrals of the Noether superpotential coincide with those of GR for asymptotically flat spacetimes. However, the localization of gravitational energy and the distribution of energy-momentum density differ: STEGR attributes them to nonmetricity, in contrast to curvature-based definitions in GR. The gauge-dependent nature of local densities and the boundary-term ambiguities echo the non-localizability of gravitational energy in pseudotensor approaches within GR, but are sharpened in STEGR by the freedom in inertial connection selection (Capozziello et al., 17 Jan 2026, Emtsova et al., 2022).

A further distinction is that in the coincident gauge, all geometric data apart from the metric are encoded in nonmetricity, making the construction and interpretation of gravitational energy-momentum especially transparent but highlighting its coordinate (gauge) dependence.

7. Summary Table: Key Features of the Schwarzschild Spacetime in STEGR

Feature	Formulation in STEGR	Comments
Metric	$$ds^2 = (1 - r_s/r)c^2dt^2 - (1-r_s/r)^{-1} dr^2 - r^2d\Omega^2$$	Identical to GR (Capozziello et al., 17 Jan 2026)
Flat, torsionless connection	$\Gamma^\alpha{}_{\mu\nu} = 0$ (coincident gauge) or via turning-off gravity	Pure gauge, not unique
Nonmetricity scalar $Q$	$Q(r) = -2/r^2$ (coincident gauge)	Encodes gravitation entirely
Gravitational energy density	$\tau^0{}_0(r) = -\frac{c^4}{8\pi G}\frac{1}{r^2}$	Gauge-dependent
Total mass/conserved charge	$M$ via superpotential surface integral	Matches ADM/Komar mass of GR
Local energy (free fall, PG gauge)	$\mathcal{J}^\alpha(\xi_{\text{fall}}) = 0$	Direct EEP realization, unique gauge

The construction and analysis of the Schwarzschild solution in STEGR demonstrate equivalence with GR at the level of field equations and global conserved charges, while elucidating the role of nonmetricity and gauge freedom in defining local gravitational energy and clarifying the underlying ambiguities inherent to teleparallel and non-metric gravity formulations (Capozziello et al., 17 Jan 2026, D'Ambrosio et al., 2021, Emtsova et al., 2024, Emtsova et al., 2022).