Deriving the cosmological constant from the Euler–Lagrange equation of second-order differentiable gravitational field Lagrangian

Abstract

The principles underlying the variational approach prove to be invaluable tools in articulating physical phenomena, particularly when dealing with conserved quantities. The derivation of gravitational field equations from the Euler–Lagrange equation, a result of the first-order derivative of Lagrange field density, is a well-established concept. However, within the context of this paper, the field equations stem from the second-order differentiable field Lagrangian in the Euler–Lagrange equation. The Euler–Lagrange equation of the first-order differentiable field Lagrangian yields the gravitational field equations incorporating matter fields and the cosmological constant. On the other hand, the Euler–Lagrange equation of the second-order differentiable field Lagrangian gives rise to the equation governing the cosmological constant. The procedure of determining the cosmological constant through the Euler–Lagrange equation adheres to fundamental mathematical theory encompassing higher-order Euler–Lagrange equations. This theory implies that utilizing higher-order Euler–Lagrange equations may result in instability due to negative energy. The cosmological constant is postulated to be intertwined with negative vacuum energy density. The value assigned to the gravitational constant, as ascertained through the employment of the general relativistic equation, is not only significantly small and nonzero but also demonstrates remarkable compatibility with the value inferred from observational deductions.