Effective Scalar–Gravitational Coupling with Spectral Backreaction
A covariant effective-field-theory model is introduced in which a scalar field couples to spacetime geometry via spectrally constrained backreaction terms. The framework preserves diffeomorphism invariance and yields testable dispersion effects without invoking full quantum gravity.
1. Field equation
We consider a scalar field \(\Psi\) propagating on a Lorentzian manifold \((\mathcal M, g_{\mu\nu})\), governed by the generalized Klein–Gordon equation
\[ \boxed{ \mathcal K \, \Box \Psi + m_{\mathrm{eff}}^2 \, \Psi + \lambda \, \mathcal F_{\mathrm{tot}}(\nu) \, \Psi + \nabla_\mu \!\left( \Gamma^\mu \nabla^\mu \Psi \right) = 0 } \]
where \(\Box = g^{\mu\nu}\nabla_\mu\nabla_\nu\). The function \(\mathcal F_{\mathrm{tot}}(\nu)\) represents an externally constrained spectral distribution, while \(\Gamma^\mu\) encodes higher-order effective corrections (e.g. dissipative or non-local contributions).
2. Effective action and symmetries
The dynamics follow from the local effective action
\[ S_\Psi = \int d^4x \, \sqrt{-g} \left[ \frac{\mathcal K}{2} \nabla_\mu \Psi \nabla^\mu \Psi - \frac{1}{2} \left( m_{\mathrm{eff}}^2 + \lambda \, \mathcal F_{\mathrm{tot}}(\nu) \right) \Psi^2 - \frac{1}{2} \Gamma^\mu \Psi \nabla_\mu \Psi \right]. \]
This action is:
- diffeomorphism invariant,
- locally Lorentz covariant,
- valid within the regime of effective field theory.
3. Generalized conservation law
The scalar field defines a conserved current
\[ J^\mu = i \left( \Psi^\ast \nabla^\mu \Psi - \Psi \nabla^\mu \Psi^\ast \right), \]
while the total energy–momentum balance obeys
\[ \boxed{ \nabla_\mu T^{\mu\nu} + \partial_\mu^{(\nu)} J^\mu = 0 } \]
indicating a controlled exchange between the scalar sector and unresolved degrees of freedom. Conservation is recovered in the decoupling limit.
4. Coupling to geometry
Backreaction on spacetime is encoded through a modified Einstein equation
\[ \boxed{ G_{\mu\nu} = \frac{8\pi G}{c^4} \left( T^{\Psi}_{\mu\nu} + \Theta^{(\Psi)}_{\mu\nu} \right) } \]
where \(\Theta^{(\Psi)}_{\mu\nu}\) collects higher-order or non-minimal effective contributions.
5. Spectral energy density
An effective scalar energy density is defined by
\[ \boxed{ \rho_{\mathrm{eff}} = \int d\nu \; \mathcal F_{\mathrm{tot}}(\nu) \, |\Psi|^2 } \]
which acts as a spectral weight rather than a fundamental mass parameter.
6. Physical status and testability
- The framework is compatible with relativistic quantum fields and general relativity at the EFT level.
- It predicts potential dispersion and amplitude modulations in wave propagation.
- All additional couplings are experimentally falsifiable via spectral or propagation constraints.
No violation of causality or assumption of microscopic spacetime quantization is introduced.
Keywords: effective field theory, scalar fields, general relativity, spectral backreaction, wave dispersion