The strongest conventional four-dimensional description

The strongest conventional four-dimensional description

Homogeneous sector

Define $$T^{\mathrm{eff}}_{\mu\nu} =T_{\mu\nu} +\frac{\kappa_5^4}{\kappa_4^2}\Pi_{\mu\nu} -\frac{1}{\kappa_4^2}E_{\mu\nu}.\tag{90}$$ The projected equation (38) becomes ordinary four-dimensional Einstein gravity sourced by $T^{\mathrm{eff}}_{\mu\nu}$. In the homogeneous sector, $$\rho_{\mathrm{eff}} =\rho+\frac{\rho^2}{2\lambda} +\frac{3\mu}{\kappa_4^2a^4}, \qquad p_{\mathrm{eff}} =p+\frac{\rho(\rho+2p)}{2\lambda} +\frac{\mu}{\kappa_4^2a^4}.\tag{91}$$ This four-dimensional fluid description reproduces the background expansion exactly, and may perfectly well be regarded as fundamental rather than induced. Background cosmological data alone cannot decide between the two readings.

Linear sector, and the CFT realization of the $1/r^3$ residue

The quadratic action of Section 6 is already a four-dimensional nonlocal action once $\mathcal G_R$ is taken as its primitive kernel. That such nonlocal braneworld actions with explicit KK form factors can be written down constructively is shown, in the two-brane setting, by Barvinsky et al. [11]; we invoke that work only as a constructive control on the form of the eliminated action, not as an identity for RS2, whose single-brane spectrum is a continuum rather than a discrete tower. It contains the same massless pole, the same continuum branch cut, the same causal response, and the same weak-field coefficients as the five-dimensional description.

The four-dimensional alternative is not hypothetical. The stress-tensor vacuum-polarization loop of an appropriate four-dimensional conformal sector corrects the graviton propagator with the same $1/r^3$ coefficient as the classical RS2 bulk calculation [9, 10, 18]. Thus even the most characteristic weak-field residue of the model has a conventional four-dimensional realization. This is the concrete form taken, inside RS2, by the general obstruction proved in Section 11.

The discrimination supplied by the model is therefore $$\text{minimal RS2} \quad\text{versus}\quad \text{4D GR or another \emph{independently restricted} model class},\tag{92}$$ and not $$\text{source projection} \quad\text{versus}\quad \text{every possible brane-only theory}.\tag{93}$$