Four-Scalar Field Formulation

• Four-scalar field formulation is a method that employs four independent scalar fields mapped to spacetime coordinates to reconstruct arbitrary geometries in modified gravity.
• It utilizes Lagrange multipliers to enforce constraints, freezing scalar dynamics and preventing ghost or unwanted propagating modes.
• The framework generalizes mimetic and two-scalar models, enabling the construction of diverse astrophysical and cosmological solutions in f(T) and f(Q) theories.

A four-scalar field formulation refers to a class of field-theoretic constructions in which four independent scalar fields—typically denoted $\phi^{(\rho)}$ for $\rho=0,1,2,3$—are introduced in a gravitational or modified-gravity context. Such models play a key role in geometric engineering of spacetime solutions, generalizing mimetic and two-scalar approaches to scenarios where reconstruction of arbitrary (including non-symmetric) spacetime geometries, particularly in modified gravity, is required. The formulation's power derives from the additional degrees of scalar freedom in correspondence with spacetime coordinates, enabling the realization of diverse metrics while controlling unwanted propagating modes via Lagrange multipliers. Four-scalar field models have found significant application in teleparallel $f(\mathcal{T})$ gravity, metric-affine $f(Q)$ gravity, and as generalizations of non-linear sigma models, expanding the landscape of physically viable solutions beyond what is accessible with minimal or two-field constructions.

1. Motivations and Context in Modified Gravity

Four-scalar field frameworks arise primarily to overcome limitations present in conventional mimetic gravity or its two-scalar extensions, which are generally insufficient for reconstructing arbitrary spherically symmetric or non-symmetric spacetimes in modified gravity theories, especially in non-linear $f(\mathcal{T})$ or $f(Q)$ models (Nashed et al., 20 Feb 2024, Nashed et al., 6 Aug 2025). In $f(\mathcal{T})$ gravity, for instance, standard mimetic and two-scalar approaches can only reproduce solutions if the torsion scalar is constant or if $f(\mathcal{T})$ is linear, thus restricting the theory to the teleparallel equivalent of General Relativity (TEGR). The four-scalar field extension generalizes the coordinatization of spacetime, enabling arbitrary metrics (including those with arbitrary dependence on all four coordinates) to be identified as solutions. This is achieved by associating each scalar with a spacetime coordinate direction, thus capturing the full local structure of the manifold.

In addition, these models have utility beyond teleparallelism, notably in $f(Q)$ gravity where the scalar $Q$ is constructed from non-metricity tensors and traditional reconstructive methods fail to accommodate general spacetimes in non-linear $f(Q)$ theories (Nashed et al., 20 Feb 2024). The flexibility of the four-scalar approach is instrumental in the construction of broad classes of solutions including static, time-dependent, and non-symmetric geometries.

2. Core Formalism and Action Structure

The canonical action for a four-scalar model is constructed by supplementing a gravitational action $S_\text{gravity}$ with a scalar sector and a constraint term: $$S = S_\text{gravity} + \int d^4 x\, \sqrt{-g} \left[\frac{1}{2} A_{(\rho\sigma)}(\phi)\, g^{\mu\nu} \partial_\mu \phi^{(\rho)} \partial_\nu \phi^{(\sigma)} - V_\phi(\phi) \right] + S_\lambda,$$ where $A_{(\rho\sigma)}(\phi)$ plays the role of a kinetic metric on the target space and $V_\phi(\phi)$ is a scalar potential (Nashed et al., 20 Feb 2024, Nashed et al., 6 Aug 2025). The crucial ingredient is the inclusion of Lagrange multipliers enforcing constraints,

$$S_\lambda = \int d^4 x\, \sqrt{-g}\; \lambda^{(\rho)} [g^{\mu\nu}\partial_\mu\phi^{(\rho)}\partial_\nu\phi^{(\rho)} - 1],$$

ensuring that each scalar is effectively aligned with a spacetime coordinate, i.e., $\phi^{(\rho)}=x^\rho$ in a suitable gauge. This identification encodes a coordinate gauge-fixing, with the target-space metric $A_{(\rho\sigma)}$ tailored so that the field equations reduce to the Einstein equations (or their modified gravity analogs) for any desired metric $g_{\mu\nu}$.

The constraints enforced by $\lambda^{(\rho)}$ are essential for the non-propagation of the scalars, freezing their fluctuations and eliminating sound-like modes or ghost degrees of freedom. The removal of propagating degrees is strictly guaranteed when the Lagrange multipliers fully fix the four scalars, allowing only the background solution (Nashed et al., 6 Aug 2025).

3. Overcoming the Limitations of Two-Scalar Models

Two-scalar field approaches, commonly found in mimetic gravity and its extensions, are intrinsically limited in reconstructing only spherically symmetric solutions or spacetime geometries subject to specific symmetry criteria. In $f(\mathcal{T})$ and $f(Q)$ gravities, the field equations in the presence of two field degrees of freedom impose that either the torsion/non-metricity scalar must be constant or that the action reduces to the standard (linear) form of the gravity theory, thereby excluding genuinely new solutions (Nashed et al., 20 Feb 2024, Nashed et al., 6 Aug 2025). The four-field generalization, by contrast, provides the requisite algebraic freedom to sidestep these constraints, making it possible to embed any given spacetime as a solution—regardless of symmetry—when the scalar fields are locked to the spacetime coordinates and the relevant target metric $A_{(\rho\sigma)}$ is appropriately chosen.

This capability is especially important for the construction of nontrivial spherically symmetric solutions in non-linear $f(\mathcal{T})$ or $f(Q)$ gravity, for which two-scalar models fail. In the case of the quadratic teleparallel action $$f(\mathcal{T}) = \mathcal{T} + \frac\alpha2 \mathcal{T}^2$$, the four-scalar formulation enables explicit reconstruction of black hole and astrophysically relevant solutions not available to prior approaches.

4. Constraints, Non-Propagation, and Extensions of Mimetic Theory

A defining property of the four-scalar framework is the imposition of constraints,

$$g^{\mu\nu}(x=\phi) \partial_\mu \phi^{(\rho)} \partial_\nu \phi^{(\rho)} = 1,$$

for each scalar $\rho$. These constraints "freeze" all dynamical fluctuations of the fields, leaving as admissible solutions only those corresponding to coordinate maps, e.g., $\phi^{(\rho)} = x^\rho$ (Nashed et al., 20 Feb 2024, Nashed et al., 6 Aug 2025). The constraints serve a dual purpose: they prevent the scalar fields from introducing new propagating degrees of freedom (thus avoiding ghost or sound excitations), and they ensure that the energy-momentum content induced by the scalars mimics that of a perfect, pressureless fluid (the so-called "frozen" fluid), in direct analogy with non-propagating dark matter found in mimetic gravity.

This motivates the interpretation of the four-scalar construction as a generalization of the mimetic paradigm, now capable of describing not just cold dark matter, but a wide array of non-dynamical fluid-like sources, including those with more exotic equations of state or effective behaviors (phantom fluids, non-collapsing fluids, etc.) (Nashed et al., 20 Feb 2024). In this sense, the four-scalar field construction is a "general extension of the mimetic theory" suitable for encoding arbitrary non-dynamical matter.

5. Mathematical and Model-Theoretic Features

In addition to its coordinate identification role, the four-scalar formulation can be recast as a general non-linear $\sigma$-model by letting the potential $V_\phi$ vanish. The kinetic term then defines a metric $A_{(\rho\sigma)}(\phi)$ on the target space of scalars, yielding

$$\int d^4 x\, \sqrt{-g}\, \frac12\, A_{(\rho\sigma)}(\phi)\, g^{\mu\nu} \,\partial_\mu \phi^{(\rho)}\, \partial_\nu \phi^{(\sigma)},$$

and via the identification $\phi^{(\rho)}=x^\rho$ one obtains a mapping from the manifold's geometry to the geometry of the target space, establishing a structural bridge to non-linear $\sigma$-model theory (Nashed et al., 20 Feb 2024). The physical significance of this mapping, for example within quantum gravity or holographic dualities, remains an open area for investigation.

Key formulas appearing in the construction include:

• The full action:

$$S = S_\text{gravity} + \int d^4x \sqrt{-g}\, \tfrac12 A_{(\rho\sigma)}(\phi) g^{\mu\nu} \partial_\mu\phi^{(\rho)}\partial_\nu\phi^{(\sigma)} - V_\phi(\phi) + S_\lambda,$$

• The constraint action:

$$S_\lambda = \int d^4x\, \sqrt{-g}\, \lambda^{(\rho)} [g^{\mu\nu}\partial_\mu\phi^{(\rho)}\partial_\nu\phi^{(\rho)} - 1],$$

and the identification $\phi^{(\rho)} = x^\rho$.

6. Applications, Model Reconstruction, and Extensions

The four-scalar model is directly applicable to:

• Realization and reconstruction of arbitrary spacetime geometries as explicit solutions in $f(\mathcal{T})$ gravity, including metrics of the form:

$$ds^2 = -e^{2\mu(r)}dt^2 + e^{2\nu(r)}dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2),$$

with $\phi^{(\rho)}= x^\rho$ and a specified target-space metric $\Phi_{(\rho\sigma)}$ determined algebraically from the gravitational field equations and the source energy-momentum structure (Nashed et al., 6 Aug 2025).

• Overcoming the constraint that in two-scalar theories, a quadratic $$f(\mathcal{T}) = \mathcal{T} + \frac\alpha2 \mathcal{T}^2$$ requires constant torsion or reduction to TEGR; the four-scalar model allows reconstruction without this restriction (Nashed et al., 6 Aug 2025).
• Constructing astrophysically relevant interior (stellar) solutions, quantum-corrected black hole spacetimes, and potentially the paper of black hole thermodynamics (entropy, Hawking temperature, heat capacity) in modified gravity (Nashed et al., 6 Aug 2025).

The enforced constraints ensure that all reconstructed solutions are free from ghost instabilities, as demonstrated by positivity and absence of growing modes in explicit models.

Typical model-building workflow:

Step	Description	Dependent Variables
Specify metric ansatz	Choose $ds^2$ (e.g., spherically symmetric)	$\mu(r), \nu(r)$
Identify scalars	Set $\phi^{(\rho)} = x^\rho$	Coordinates $t, r, \theta, \phi$
Compute target metric	Solve for $\Phi_{(\rho\sigma)}$ algebraically from field equations	$\Phi_{(\rho\sigma)}(\phi)$
Impose constraints	Enforce $$g^{\mu\nu}\partial_\mu\phi^{(\rho)}\partial_\nu\phi^{(\rho)}=1$$	$\lambda^{(\rho)}$
Check ghost-freedom	Analyze spectrum in constrained linearization	Model parameters

7. Physical and Model-Theoretic Implications

The four-scalar formulation's distinguishing feature is its universal applicability: arbitrary spacetime geometries, not restricted by symmetry or field content, can be embedded as solutions, including non-linear $f(Q)$ and $f(\mathcal{T})$ cases. By aligning the four scalars with the full coordinate set of the manifold and "freezing" their fluctuations, these models both generalize mimetic gravity and provide a systematic route toward the modeling of any kind of non-dynamical fluid. When the scalar sector reduces to a non-linear $\sigma$-model, there is a direct mapping between spacetime and target geometry, possibly suggestive of novel geometric or quantum implications.

In modified gravity and cosmology, this transforms the landscape of viable solutions and allows for the systematic treatment of dark matter, non-collapsing fluids, and extensions to non-metric theories. The approach preserves ghost-freedom via constraint enforcement and grants model-builders a versatile protocol for engineering spacetimes to specification without propagating extraneous degrees of freedom.

References: (Nashed et al., 20 Feb 2024, Nashed et al., 6 Aug 2025)