Bulk-to-Brane Projection, Dynamical Nonclosure, and Observable Residues in Randall–Sundrum Gravity

The parameter-eliminated cross-regime prediction

Jeremy RodgersSection 8 of 18Original PDF ↗Publication DOI ↗

The parameter-eliminated cross-regime prediction

Under the RS tuning $\Lambda_5=-\frac{6}{\ell^2}, \qquad \lambda=\frac{6}{\kappa_5^2\ell}, \qquad \kappa_4^2=\frac{\kappa_5^2}{\ell},\tag{81}$ one also has $\lambda=6/(\kappa_4^2\ell^2)$ , and (39) gives $\Lambda_4=\tfrac12\bigl(-6/\ell^2+\tfrac16\kappa_5^4\lambda^2\bigr)=0$ exactly, since $\kappa_5^4\lambda^2=\kappa_5^4\cdot36/(\kappa_5^4\ell^2) =36/\ell^2$ .

Remark 7.1 (Status of $\Lambda_4$ below). The symbol $\Lambda_4$ retained in (82) is a phenomenological background term that may be fitted or supplied by a much smaller late-time brane energy component. It is not an independent parameter of the exactly tuned flat-brane theorem. Applying the coefficient relation on a weakly curved cosmological background requires $H\ell\ll1$ and negligible detuning corrections in the compared regime. See Section 15, item 9.

Write the two independently measurable departures from four-dimensional GR, without using RS2 in the fits, as $H^2+\frac{k}{a^2} =\frac{\Lambda_4}{3}+\frac{\kappa_4^2}{3}\rho +B\rho^2+\frac{\mu}{a^4},\tag{82}$ $\Phi(r)=-\frac{G_4M}{r}\Bigl(1+\frac{A_\Phi}{r^2}+\cdots\Bigr), \qquad \Psi(r)=-\frac{G_4M}{r}\Bigl(1+\frac{A_\Psi}{r^2}+\cdots\Bigr).\tag{83}$

Theorem 7.2 (Cross-regime coefficient relation).

In minimal, exactly tuned, positive-tension RS2 with the regular retarded bulk state, the source calculation gives $B=\frac{\kappa_5^4}{36}=\frac{\kappa_4^4\ell^2}{36}, \qquad A_\Phi=\frac{2\ell^2}{3}, \qquad A_\Psi=\frac{\ell^2}{3},\tag{84}$ and eliminating $\ell$ yields $\boxed{\; A_\Psi=\tfrac12 A_\Phi, \qquad B=\frac{\kappa_4^4}{24}A_\Phi =\frac{\kappa_4^4}{12}A_\Psi . \;}\tag{85}$

Proof. $B=\kappa_5^4/36$ is read off from (48). Using $\kappa_5^2=\kappa_4^2\ell$ from (79) gives $\kappa_5^4=\kappa_4^4\ell^2$ , hence $B=\kappa_4^4\ell^2/36$ . The values $A_\Phi=2\ell^2/3$ , $A_\Psi=\ell^2/3$ are read off from (75). Eliminating, $\ell^2=3A_\Phi/2=3A_\Psi$ , so $A_\Psi=A_\Phi/2$ , and $B=\tfrac{\kappa_4^4}{36}\cdot\tfrac{3A_\Phi}{2} =\tfrac{\kappa_4^4}{24}A_\Phi =\tfrac{\kappa_4^4}{12}A_\Psi$ . ◻

Dimensional consistency.

$[\kappa_4^2]=L^2$ , $[A_\Phi]=L^2$ , hence $[B]=L^4\cdot L^2=L^6$ ; and with $[\rho]=L^{-4}$ , $[B\rho^2]=L^{6-8}=L^{-2}=[H^2]$ , as required.

Scope, and what it does and does not do.

The relation (85) contains no unknown brane parameter once $G_4$ and the phenomenological coefficients are measured. It is independent of the dark-radiation constant $\mu$ , whose $a^{-4}$ scaling is separable from $\rho^2$ when the matter equation of state and energy density are independently constrained. Its content is exact within its intersecting regimes:

$B$ comes from the exact homogeneous thin-brane equation (48);
$A_\Phi,A_\Psi$ are the leading terms of the controlled $r\gg\ell$ weak-field expansion (75);
the same $\ell$ and the same tuned source action govern both sectors;
no fit of $\ell$ remains after the coefficients are compared.

A violation of either equality falsifies minimal classical RS2 under the stated assumptions. A statistically significant nonzero common signal satisfying both equalities is a parameter-saving, out-of-sample success for RS2 and favours it over pure four-dimensional GR in model comparison. It does not falsify a four-dimensional theory that independently parameterizes all three coefficients: such a theory fits either agreement or violation. Only a restricted alternative predicting an incompatible relation is excluded. Note also the all-zero limit: $A_\Phi=A_\Psi=B=0$ satisfies both equalities trivially and merely bounds $\ell$ from above. And even nonzero agreement does not establish that the five-dimensional variables are ontologically fundamental, as Section 11 proves.