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

Exact evaluation in the homogeneous sector

Jeremy RodgersSection 6 of 18Original PDF ↗Publication DOI ↗

Exact evaluation in the homogeneous sector

Take an FRW brane and a perfect fluid, $ds_4^2=-dt^2+a(t)^2\,d\Sigma_k^2, \qquad T_{\mu\nu}=\rho\,u_\mu u_\nu+p\,h_{\mu\nu}, \qquad h_{\mu\nu}=q_{\mu\nu}+u_\mu u_\nu,\tag{43}$ with $u^\mu u_\mu=-1$ . Then $T=-\rho+3p$ and $T_{\alpha\beta}T^{\alpha\beta}=\rho^2+3p^2$ .

Lemma 5.1 (Perfect-fluid $\Pi_{\mu\nu}$ ).

For (43), (37) evaluates to $\boxed{\; \Pi_{\mu\nu} =\frac{\rho}{12}\Bigl[\rho\,u_\mu u_\nu+(\rho+2p)\,h_{\mu\nu}\Bigr]. \;}\tag{44}$

Proof. Direct substitution. In the rest frame, $T_{00}=\rho$ , $T_{ij}=p\,q_{ij}$ , and $\Pi_{00} =\frac{\rho^2}{4} +\frac{(-\rho+3p)\rho}{12} -\frac{\rho^2+3p^2}{8} +\frac{(-\rho+3p)^2}{24} =\frac{\rho^2}{12},$ where the first term carries a $+$ sign because $-\tfrac14T_{0\beta}T_0{}^\beta =-\tfrac14 T_{00}T_0{}^0=-\tfrac14\rho\cdot(-\rho)=+\tfrac14\rho^2$ , and the third and fourth carry the signs induced by $q_{00}=-1$ . Assembling the terms: $\tfrac{\rho^2}{4} -\tfrac{\rho^2}{12}+\tfrac{p\rho}{4} -\tfrac{\rho^2}{8}-\tfrac{3p^2}{8} +\tfrac{\rho^2}{24}-\tfrac{p\rho}{4}+\tfrac{3p^2}{8} =\rho^2\bigl(\tfrac14-\tfrac1{12}-\tfrac18+\tfrac1{24}\bigr) =\tfrac{\rho^2}{12}$ . Similarly $\Pi_{ij}=\tfrac{1}{12}\rho(\rho+2p)\,q_{ij}$ . These are the $u_\mu u_\nu$ and $h_{\mu\nu}$ components of (44). ◻

Homogeneity, isotropy, tracelessness, and (42) (whose right-hand side vanishes in this sector because $\Pi_{\mu\nu}$ is a conserved perfect-fluid combination) force $E_{\mu\nu}=-\mathcal U(t)\Bigl(u_\mu u_\nu+\tfrac13h_{\mu\nu}\Bigr), \qquad \dot{\mathcal U}+4H\mathcal U=0 \quad\Longrightarrow\quad \mathcal U(t)=\frac{3\mu}{a^4(t)} ,\tag{46}$ where $\mu$ is a constant fixed by the bulk geometry , in an AdS–Schwarzschild realization, proportional to the bulk black-hole mass parameter [5, 4]. It is not determined by the brane matter state.

The response is now fully evaluated: $\boxed{\; \mathcal C^{\mathrm{RS}}_{00} =\frac{\kappa_5^4}{12}\rho^2+\frac{3\mu}{a^4}, \qquad \mathcal C^{\mathrm{RS}}_{ij} =\Bigl[\frac{\kappa_5^4}{12}\rho(\rho+2p)+\frac{\mu}{a^4}\Bigr]q_{ij}. \;}\tag{47}$

Proposition 5.2 (Exact brane Friedmann equations).

For the model of Section 2 with an FRW brane, conserved brane matter, and no bulk matter flux, $\boxed{\; H^2+\frac{k}{a^2} =\frac{\Lambda_4}{3} +\frac{\kappa_4^2}{3}\rho +\frac{\kappa_5^4}{36}\rho^2 +\frac{\mu}{a^4}. \;}\tag{48}$ Using $\kappa_4^2=\kappa_5^4\lambda/6$ this is equivalently $H^2+\frac{k}{a^2} =\frac{\Lambda_4}{3} +\frac{\kappa_4^2}{3}\rho\Bigl(1+\frac{\rho}{2\lambda}\Bigr) +\frac{\mu}{a^4},\tag{49}$ and, with $\dot\rho+3H(\rho+p)=0$ , $\dot H-\frac{k}{a^2} =-\frac{\kappa_4^2}{2}(\rho+p)\Bigl(1+\frac{\rho}{\lambda}\Bigr) -\frac{2\mu}{a^4}.\tag{50}$

Proof. Take the $00$ component of (38) with (44)–(46). The Einstein tensor gives $G_{00}=3(H^2+k/a^2)$ ; the correction contributes $\kappa_5^4\Pi_{00}-E_{00} =\tfrac{\kappa_5^4}{12}\rho^2+3\mu/a^4$ . Dividing by $3$ gives (48). Substituting $\kappa_5^4=6\kappa_4^2/\lambda$ gives $\kappa_5^4\rho^2/36=\kappa_4^2\rho^2/(6\lambda)$ , i.e. the $\rho^2/(2\lambda)$ term in (49). Differentiating (49) and eliminating $\dot\rho$ with conservation gives (50). ◻

The signs and scalings are fixed and physically transparent. Positive brane tension gives a positive $\rho^2$ correction, enhancing the expansion rate at high density. The Weyl contribution scales as radiation, but its magnitude is set by a bulk-state parameter, not by brane matter. For $\rho/\lambda\ll1$ and $\mu/a^4\ll H^2$ , standard four-dimensional cosmology is recovered.

The homogeneous sector already exhibits the exact division between what the projection fixes and what it leaves open:

the form and coefficient of the $\rho^2$ term are fixed once $\kappa_5$ and $\lambda$ are specified;
the exponent $a^{-4}$ is fixed by symmetry and conservation alone;
the constant $\mu$ is bulk initial/boundary data, undetermined by the brane matter state. It is not a prediction of the model without a bulk-state condition.