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A free real scalar field with zero mass has the displayed wave equation in Minkowski spacetime. In curved spacetime its dynamics also depend on curvature coupling: a minimally coupled massless scalar field and a conformally coupled scalar field need not have the same cosmological modes.

The displayed action functional contains no mass term and no $R{\varphi }^2$ curvature coupling. Its Euler-Lagrange field equation is ${\partial }_{\mu }(\sqrt{-g}{g}^{\mu \nu }{\partial }_{\nu }\varphi )=0$. In a flat FLRW metric, a Fourier mode obeys ${\varphi }_k^{\prime \prime }+2(a^{\prime }/a){\varphi }_k^{\prime }+k^2{\varphi }_k=0$ in four dimensions. The expansion can therefore freeze these modes; the curvature term of a conformally coupled scalar field changes that conclusion.

For the FLRW metric in conformal time, $\sqrt{-g}=a^4$, $g^{00}=-a^{-2}$ and $g^{ij}=a^{-2}{\delta }^{ij}$. The equation for a minimally coupled massless scalar field is consequentlyFor a Fourier mode of comoving wavevector $\mathbf{k}$, the Laplacian contributes $-k^2$, where $k=|\mathbf{k}|$. ThusSince $a^{\prime }/a=-1/\tau$, this yieldsThe absence of a curvature-coupling term is material: a conformally coupled scalar field obeys a different mode equation.

For a scalar operator of dimension $\Delta =2$ in a boundary theory of dimension $d=3$, the AdS scalar mass--dimension relation givesin unit-radius ${\mathrm{AdS}}_4$. In Poincare coordinates,so the free Klein-Gordon equation isThis mass is exactly that of a conformally coupled scalar field in four-dimensional anti-de Sitter spacetime. The Weyl transformationtherefore maps the equation to the flat half-space wave equation

The Euler-Lagrange equation isAfter a spatial Fourier transform and the change to conformal time, ${\partial }_t=a^{-1}{\partial }_{\tau }$, this becomesFor the exact de Sitter spacetime scale factor $a=-1/(H\tau )$, one has $a^{\prime \prime }/a=2a^2{H}^2$. Setting $u_{\mathbf{k}}=a{\phi }_{\mathbf{k}}$ therefore cancels both the friction term and the effective mass term, leavingThis is the special cancellation for a conformally coupled scalar field.

Varying the scalar action and Fourier transforming the spatial coordinates gives the Klein-Gordon equationIn de Sitter spacetime, $a=-1/(H\eta )$ and $m^2=2H^2$, soDirect substitution shows that $\eta e^{-ik\eta }$ and $\eta e^{ik\eta }$ are independent solutions. HenceThe mass value is the one that gives a conformally coupled scalar field in four-dimensional de Sitter spacetime.