[Paper Review] Einstein-Maxwell and Einstein-Proca theory from a modified gravitational action

Dan N. Vollick|ArXiv.org|Jan 4, 2006

Relativity and Gravitational Theory1 references3 citations

TL;DR

This paper derives both Einstein-Proca and Einstein-Maxwell equations from a modified gravitational action by introducing a term proportional to $ F^{ ho au}F_{ ho au} $, where $ F_{ ho au} = abla_ ho ar{ abla}_ au - abla_ au ar{ abla}_ ho $, and $ \Gamma_\mu = \Gamma^\alpha_{\mu\alpha} $ transforms like a $ U(1) $ gauge field. The key result is that with a specific choice of coupling constant $ \beta = -3/(4\kappa) $, the theory reproduces the Einstein-Maxwell equations from a geometric scalar density $ g^{-ie/2} $, unifying massive and massless vector fields within a Palatini formalism.

ABSTRACT

A modified gravitational action is considered which involves the quantity $F_{μν}=\partial_μΓ_ν-\partial_νΓ_μ$, where $Γ_μ=Γ^α_{μα}$. Since $Γ_μ$ transforms like a U(1) gauge field under coordinate transformations terms such as $F^{μν}F_{μν}$ are invariant under coordinate transformations. If such a term is added to the usual gravitational action the resulting field equations, obtained from a Palatini variation, are the Einstein-Proca equations. The vector field can be coupled to point charges or to a complex scalar density of weight $ie$, where $e$ is the charge of the field. If this scalar density is taken to be $g^{-ie/2}$ and the overall factor of the scalar density Lagrangian takes on a particular value the resulting field equations are the Einstein-Maxwell equations.

Motivation & Objective

• To unify massive and massless vector fields within a geometric framework of general relativity using a modified gravitational action.
• To show that the contraction $ \Gamma_\mu = \Gamma^\alpha_{\mu\alpha} $ behaves like a $ U(1) $ gauge field under coordinate transformations, enabling gauge-invariant field strength $ F_{\mu\nu} $.
• To derive the Einstein-Proca equations via Palatini variation of a gravitational action including $ F^{\mu\nu}F_{\mu\nu} $.
• To demonstrate that coupling to a complex scalar density of weight $ ie $, particularly $ g^{-ie/2} $, yields the Einstein-Maxwell equations when $ \beta = -3/(4\kappa) $.

Proposed method

• Introduce a modified gravitational action $ S = \int \left[ -\frac{1}{2\kappa}R - \frac{\alpha}{4}F^{\mu\nu}F_{\mu\nu} + L_M \right] \sqrt{g} \, d^4x $, with $ F_{\mu\nu} = \partial_\mu \Gamma_\nu - \partial_\nu \Gamma_\mu $.
• Use Palatini variation treating $ g_{\mu\nu} $ and $ \Gamma^\alpha_{\mu\nu} $ as independent variables to derive field equations.
• Show that $ \Gamma_\mu $ transforms like a $ U(1) $ gauge field under coordinate transformations, making $ F_{\mu\nu} $ a tensor and enabling gauge-invariant terms.
• Couple the vector field $ V_\mu $ to point charges via a current $ J^\mu $, and to a complex scalar density $ \phi $ of weight $ ie $, with $ \phi = g^{-ie/2} $.
• Derive the Proca equation for $ V_\mu $ from the connection variation, and show that with $ \beta = -3/(4\kappa) $, the scalar density Lagrangian yields the Einstein-Maxwell equations.
• Verify current conservation $ \tilde{\nabla}_\mu J^\mu = 0 $ and the traceless nature of the connection variation, ensuring consistency with the Palatini formalism.

Experimental results

Research questions

• RQ1 Can the Einstein-Proca equations be derived from a modified gravitational action involving $ F^{\mu\nu}F_{\mu\nu} $ with $ \Gamma_\mu = \Gamma^\alpha_{\mu\alpha} $?
• RQ2 Does the field $ \Gamma_\mu $ transform like a $ U(1) $ gauge field under coordinate transformations, and is $ F_{\mu\nu} $ a tensor under such transformations?
• RQ3 Can a complex scalar density of weight $ ie $, specifically $ g^{-ie/2} $, reproduce the Einstein-Maxwell equations when coupled to the vector field?
• RQ4 What value of the coupling constant $ \beta $ in the scalar density Lagrangian leads to the massless limit and the Einstein-Maxwell equations?
• RQ5 How does the inclusion of point charge sources affect the resulting field equations in this geometric framework?

Key findings

• The field equations derived from the Palatini variation of the modified action are the Einstein-Proca equations, with $ V_\mu $ satisfying the Proca equation $ \tilde{\nabla}_\beta F^{\beta\mu} = \frac{3}{2\alpha\kappa} V^\mu + J^\mu $.
• The vector field $ V_\mu $ is sourced by point charges through $ J^\mu $, and current conservation $ \tilde{\nabla}_\mu J^\mu = 0 $ holds.
• When coupled to a complex scalar density $ \phi $ of weight $ ie $, the field equations reduce to Einstein gravity minimally coupled to a massive vector field $ V_\mu $ and a complex scalar field $ \phi $.
• For the geometric scalar density $ g^{-ie/2} $, the Lagrangian $ L = -\frac{1}{4}\beta \sqrt{g} g^{-2} \nabla_\mu g \nabla^\mu g $ yields the Einstein-Maxwell equations when $ \beta = -3/(4\kappa) $.
• The vector field equation becomes $ \tilde{\nabla}_\beta F^{\beta\mu} = \left[ \frac{3}{2\alpha\kappa} + \frac{2\beta}{\alpha} \right] V^\mu $, and setting $ \beta = -3/(4\kappa) $ makes the mass term vanish, yielding the Maxwell field equation.
• The trace of the connection variation vanishes, indicating four arbitrary functions in the solution, consistent with the underdetermined system arising from the Palatini formalism.