The discussion of vacuum energy is currently a subject of great theoretical importance, specially concerning the cosmological constant problem in General Relativity. From Quantum Field Theory, it is stated that vacuum states subject to boundary conditions may generate tensions on these boundaries related to a measurable non-zero renormalized vacuum energy: the Casimir Effect. As such, investigating how these vacuum states and energy behave in curved backgrounds is just natural and might provide important results in the near future. In this paper we revisit a model of the Casimir Effect in weak gravitational field background, which has been proposed and further generalized in the literature. A trick originally used to simplify calculations is shown to lead to a wrong value for the energy shift, and by performing explicit mode expansion we arrive at an unexpected result: null gravitational correction even at order (M/R) 2 , in opposition to earlier results. 1

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We show through Schwinger's approach that in a static weak gravitational background, the Casimir Energy for a real massless scalar field obeying Dirichlet boundary conditions on rectangular plates is unaltered from it's flat space-time value. The result is obtained considering an rather general class of backgrounds, adding further generality and consistency test for the previous work. The proposed result has direct consequences on earlier works in the literature that found gravity related corrections for similar setups.

Neste trabalho estudamos um sistema clássico de dois osciladores harmônicos acoplados com massas (mi), constantes de mola (ki) e parâmetro de acoplamento (κ) dependentes do tempo. Para encontrar as soluções das equações de movimento de cada oscilador, usamos uma transformação canônica para reescrever a hamiltoniana do sistema acoplado como a soma das hamiltonianas de dois osciladores harmônicos desacoplados com frequências modificadas e massas unitárias. Analisamos o comportamento de xi, vi =ẋi e do diagrama de fase xi vs. vi para o sistema m1 = m2 = moe γt e k1 = k2 = κ = koe γt . Palavras-chave: osciladores acoplados, transformação canônica.In this work we study a coupled system of two classical oscillators with time-dependent masses (mi), spring constants (ki) and coupling parameter (κ). To obtain the solution of the equation of motion for each oscillator, we use a canonical transformation to rewrite the Hamiltonian of the coupled system as the sum of the hamiltonians of two uncoupled harmonic oscillators with modified frequencies and unitary masses. We analyze the behavior of xi, vi =ẋi and the phase diagram xi vs. vi for the system m1 = m2 = moe γt and k1 = k2 = κ = koe γt