The five-dimensional source model

The five-dimensional source model

Action and conventions

Let $(\mathcal M_5,g_{AB})$ be a five-dimensional Lorentzian manifold containing a timelike hypersurface $\Sigma$ (the brane). Capital Latin indices $A,B,\dots$ run over the five dimensions; Greek indices $\mu,\nu,\dots$ are tangent to $\Sigma$; Latin $i,j,\dots$ are spatial on a brane Cauchy slice. The signature is $(-++++)$, and $c=\hbar=1$. The classical action is $$\begin{aligned} S_5[g,X,\psi] ={}&\frac{1}{2\kappa_5^2}\int_{\mathcal M_5}d^5x\,\sqrt{-g_5}\, \bigl(R_5-2\Lambda_5\bigr)\\ &+\frac{1}{\kappa_5^2}\sum_{\pm}\int_{\Sigma}d^4x\,\sqrt{-q}\,K_\pm +\int_{\Sigma}d^4x\,\sqrt{-q}\, \bigl[-\lambda+\mathcal L_m(q,\psi)\bigr]. \end{aligned}\tag{3}$$ Gibbons–Hawking–York terms appear on both sides of the thin brane, with the corresponding outward normals. The embedding is $X^A(x^\mu)$, the tangent basis $e^A{}_\mu=\partial_\mu X^A$, the unit spacelike normal $n^A$, and $$q_{\mu\nu}=g_{AB}e^A{}_\mu e^B{}_\nu, \qquad K_{\mu\nu}=e^A{}_\mu e^B{}_\nu\nabla_A n_B .\tag{4}$$ The sign convention for $\Lambda_5$ is fixed by requiring the bulk field equation away from the brane to read $${}^{(5)}G_{AB}+\Lambda_5g_{AB}=0 ,\tag{5}$$ so $[\Lambda_5]=L^{-2}$. The brane stress tensor is $$T_{\mu\nu}=-\frac{2}{\sqrt{-q}} \frac{\delta(\sqrt{-q}\,\mathcal L_m)}{\delta q^{\mu\nu}} .\tag{6}$$ Dimensionally, $[\kappa_5^2]=L^3$, $[\kappa_4^2]=L^2$, $[T_{\mu\nu}]=L^{-4}$, and $[E_{\mu\nu}]=L^{-2}$. The brane tension appears in (3) integrated over $d^4x$, exactly like $\mathcal L_m$; it is therefore a four-dimensional energy density, $$[\lambda]=L^{-4}, \qquad [\kappa_5^2\lambda]=L^{3}\cdot L^{-4}=L^{-1}.\tag{7}$$ This is consistent throughout: $K_{\mu\nu}\sim\kappa_5^2\lambda$ has dimension $L^{-1}$, as an extrinsic curvature must; and $\kappa_4^2=\kappa_5^4\lambda/6$ gives $[\kappa_4^2]=L^{6}\cdot L^{-4}=L^{2}$, as required.

Varying (3) with respect to $g_{AB}$ gives (5) in the bulk together with the Israel junction condition at $\Sigma$; varying with respect to $X^A$ gives the brane equation of motion, which for a $Z_2$-symmetric configuration is implied by the junction condition and the Codazzi identity. The response tensor $\mathcal C^{\mathrm{RS}}_{\mu\nu}$ below is obtained from this variation; it is not introduced by hand.

Physical equivalence and the reduced source space

This quotient does not remove bulk transverse-traceless gravitational waves, Kaluza–Klein continuum amplitudes, or the AdS–Schwarzschild mass parameter. Each changes gauge-invariant five-dimensional curvature invariants, energy flux, horizon structure, or the later induced brane metric, and hence is not pure gauge. This point is what makes the witness of Section 3 a genuine physical distinction rather than a coordinate artifact.

The instantaneous brane readout

Choose a five-dimensional Cauchy surface $\mathcal C_5$ intersecting the brane worldvolume in a three-dimensional surface $\mathcal C_3$.

Two features of this definition matter. First, $\Theta_{ij}$ records the gauge-independent first normal derivative of the induced metric along the brane without presupposing that the brane possesses a closed intrinsic Einstein Hamiltonian , which is exactly what is at issue. Second, the quotient by spatial diffeomorphisms and the imposition of the constraints means that no lapse-, shift-, or time-coordinate-dependent velocity enters, so the readout is not made noninjective by a trivial gauge artifact.

The readout is physically motivated inside the model: standard-model matter is confined to $\Sigma$, and local rods, clocks, detectors, and gravitational-wave interferometers respond to induced brane fields. Bulk data can be inferred only through their subsequent influence on those fields; they are not components of the present brane state.