Difference Between Scalar Field Solutions in Curved Spacetime

[Tertius]

TL;DR Summary

A free scalar field solution for the situation of 1) static, and 2) homogeneous and isotropic yield either trigonometric or exponential solutions. The first would seem to be a harmonic oscillator, the second seems to be a simple field being dispersed across space. How can these be interpreted?

A general free field Lagrangian in curved spacetime (- + + +), is given by:
L = -1/2 ∇^cΦ ∇_cΦ - V(Φ)
when the derivative index is lowered, we obtain:
L = -1/2 g^dc∇_dΦ ∇_cΦ - V(Φ)
then we can choose to replace V(Φ) with something like 1/2 b² Φ² so:
L = -1/2 g^dc∇_dΦ ∇_cΦ - 1/2 b² Φ²

** I will only give the fields and solutions in terms of t and x for simplicity (ignoring y and z). And because we are acting on a scalar the covariant derivatives can be replaced with partial derivatives.
Case 1: let's assume the field only changes with time (i.e. is homogeneous and isotropic), Φ(t,x) -> Φ(t), which then yields the Lagrangian (a Lagrangian that contains only non-derivative terms should have the field forced to equal 0):
L = -1/2 * -1 * ∂_tΦ ∂_tΦ - 1/2 b² Φ²
which gives the equations of motion:
Φ''(t) = -1/2 b²Φ(t)
which can be easily solved for a general solution:
Φ(t) = c₁cos(b t) + c₂ sin(b t)

Case 2: let's assume the field only changes with space (i.e. is static), Φ(t,x) -> Φ(x), which then yields the Lagrangian:
L = -1/2 * 1 * ∂_xΦ ∂_xΦ - 1/2 b² Φ²
which gives the equations of motion:
Φ''(x) = 1/2 b²Φ(x)
which can be easily solved for a general solution:
Φ(t) = c₁e^bx + c₂e^-bxIf the math I showed above doesn't have any glaring errors, this leads me to a few questions:
1) why would this field yield a trigonometric solution that clearly oscillates over time when the field is homogeneous and isotropic?
2) and why would it only yield an exponential decay/growth in the spatial direction?
3) I realize plane wave solutions derived using Fourier transform are often used as solutions, are they any more correct than these solutions? It seems like the plane wave solutions result in a vectorized wave that typically corresponds to particle behavior. are the solutions I gave valid for particles as well as fields? any insight here is appreciated.

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[Ambitwistor]

1) The trigonometric solution arises because the field is homogeneous and isotropic, meaning it does not vary with position. This leads to a time-dependent solution, as the field can only vary with time. The oscillatory behavior is a result of the sine and cosine functions being periodic in nature.

2) The exponential solution arises because the field is static, meaning it does not vary with time. This leads to a spatially-dependent solution, as the field can only vary with position. The exponential behavior is a result of the exponential function being a solution to the differential equation.

3) The plane wave solutions derived using Fourier transform are often used because they are a convenient way to represent the solutions of the wave equation. They are also valid solutions, but they may not always be the most physically meaningful. The solutions you gave are valid for both particles and fields, but they may not always correspond to observable behavior. It is important to consider the physical interpretation of the solutions in addition to their mathematical validity.