Warped Braneworld Gravity

A noncompact extra dimension can still yield four-dimensional gravity if warping makes the graviton zero mode normalizable. The Randall–Sundrum construction demonstrates the mechanism, but localization is only half the calculation: the brane tension must satisfy the junction condition, and the effective brane equations retain bulk Weyl data that no four-dimensional initial-value problem determines by itself.

Required background. Gravitational Actions and Boundary Terms supplies the hypersurface term and fixed-boundary-data variation used at the brane; Gauge Theory and String-Theory Preparation supplies the brane, orbifold, and Kaluza–Klein mode language.

Helpful background. Einstein Equations supplies the bulk and induced equations, Formulations of General Relativity clarifies normal evolution and constraint data, and Exact Spacetimes supplies the AdS background and branch checks.

Warping rather than a small circle

Use signature $(-,+,+,+,+)$ in five dimensions and $\kappa_5^2=8\pi G_5$. For a timelike boundary with outward spacelike unit normal, define $K_{AB}=q_A{}^Cq_B{}^D\nabla_Cn_D$ and $K=q^{AB}K_{AB}$, where $q_{AB}=g_{AB}-n_An_B$. For a bulk region with a timelike brane boundary, a convention-complete action is

$$S=\frac{1}{2\kappa_5^2}\int_{\mathcal M}d^5x\sqrt{-g_5} \left(\mathcal R_5-2\Lambda_5\right) +\frac{1}{\kappa_5^2}\int_{\partial\mathcal M}d^4x\sqrt{-q}\,K -\int_{\rm brane}d^4x\sqrt{-q}\,\lambda+S_m[q,\psi].$$

For an internal thin brane the Gibbons–Hawking–York contribution is understood from both sides. The induced metric is $q_{AB}=g_{AB}-n_An_B$. The one-brane Randall–Sundrum background is

$$ds_5^2=e^{-2|y|/\ell}\eta_{\mu\nu}dx^\mu dx^\nu+dy^2, \qquad \Lambda_5=-\frac{6}{\ell^2}.$$

Unlike ordinary compactification, $y$ ranges over an infinite line. Its unwarped proper length is infinite, but the integral relevant to the graviton kinetic term is finite. Maartens and Koyama derive the AdS curvature, the one- and two-brane branches, and the zero-mode equations in Maartens and Koyama 2010, §2 and §2.1, PDF.

Junction conditions fix the tension

Let $S_{\mu\nu}=-\lambda q_{\mu\nu}+T_{\mu\nu}$ be the stress tensor localized on the brane and define $[X]=X_+-X_-$. In this jump formula, use one continuous normal $n=\partial_y$ pointing from the minus region to the plus region on both sides. It differs from the two outward normals used for the boundary terms of the separate regions. Integrating the field equations across the delta-function source gives the Israel condition

$$\bigl[K_{\mu\nu}-Kq_{\mu\nu}\bigr]=-\kappa_5^2S_{\mu\nu},$$

or equivalently $[K_{\mu\nu}]=-\kappa_5^2(S_{\mu\nu}-Sq_{\mu\nu}/3)$. This form, including the factor $1/3$ appropriate to a four-dimensional hypersurface in five dimensions, follows in Maartens and Koyama 2010, §3.1, Eqs. (63)–(68), PDF.

For the vacuum warped metric and this single normal, $K_{\mu\nu}=\partial_y q_{\mu\nu}/2$, so

$$K^+_{\mu\nu}=-\frac1\ell q_{\mu\nu}, \qquad K^-_{\mu\nu}=+\frac1\ell q_{\mu\nu}.$$

Consequently $[K_{\mu\nu}-Kq_{\mu\nu}]=6q_{\mu\nu}/\ell$. Since $S_{\mu\nu}=-\lambda q_{\mu\nu}$, the positive-tension branch requires

$$\lambda=\frac{6}{\kappa_5^2\ell}.$$

Reversing the normal reverses both the definition of $K_{\mu\nu}$ and the jump ordering; changing only one is a common source of an erroneous negative tension. The original localization proposal and its $Z_2$ matching are given in Randall and Sundrum 1999, pp. 2–5, PDF.

The normalizable graviton and its continuum

Write the reduced Planck conventions as $M_5^3=\kappa_5^{-2}$ and $M_4^2=\kappa_4^{-2}$. Write the tensor perturbation as $g_{\mu\nu}=e^{-2|y|/\ell}(\eta_{\mu\nu}+h_{\mu\nu})$. The transverse-traceless zero mode of $h_{\mu\nu}$ is constant in $y$, while its norm includes the warp factor:

$$M_4^2=M_5^3\int_{-\infty}^{\infty}dy\,e^{-2|y|/\ell} =M_5^3\ell.$$

This equality is an independent normalization check: using the tension above in $\kappa_4^2=\kappa_5^4\lambda/6$ gives $\kappa_4^2=\kappa_5^2/\ell$, the same result. The massive modes form a continuum. For a static source on the brane they correct the potential at $r\gg\ell$ as

$$V(r)=-\frac{G_Nm}{r}\left[1+\frac{2\ell^2}{3r^2}+o\!\left(\frac{\ell^2}{r^2}\right)\right],$$

whereas $r\ll\ell$ probes the five-dimensional $1/r^2$ behavior. These two limits and the Neumann condition inherited from the junction equation are shown in Maartens and Koyama 2010, §2.1, Eqs. (29)–(41), PDF. The expansion is not a universal prediction of every warped model: it assumes the RS2 background, a source on the brane, linear response, and regular bulk boundary behavior.

The compact RS1 regime is different. Two branes at finite separation give a discrete tower and a radion measuring their separation; four-dimensional GR requires that radion to be stabilized. Sending the second brane to infinite proper distance gives RS2, whose warped volume is finite despite its noncompact fifth coordinate. “Compact,” “large,” and “warped infinite” therefore name physically distinct spectra and boundary problems.

Effective brane equations are not closed

Projecting the five-dimensional equations with Gauss–Codazzi and applying the junction condition yields

$$G_{\mu\nu}=-\Lambda_4q_{\mu\nu}+\kappa_4^2T_{\mu\nu} +\kappa_5^4\Pi_{\mu\nu}-E_{\mu\nu},$$

with

$$\Lambda_4=\frac12\left(\Lambda_5+\frac{\kappa_5^4\lambda^2}{6}\right), \qquad \kappa_4^2=\frac{\kappa_5^4\lambda}{6},$$

and

$$\Pi_{\mu\nu}=-\frac14T_{\mu\alpha}T_\nu{}^\alpha +\frac1{12}TT_{\mu\nu} +\frac1{24}q_{\mu\nu}\left(3T_{\alpha\beta}T^{\alpha\beta}-T^2\right).$$

$E_{\mu\nu}=C^{(5)}{}_{ACBD}n^Cn^Dq^A{}_\mu q^B{}_\nu$ is the electric part of the bulk Weyl tensor. Substituting $\Lambda_5=-6/\ell^2$ and the tuned tension gives $\Lambda_4=0$, another sign and factor check. The projected equation and its precise hypotheses originate with Shiromizu, Maeda, and Sasaki 2000, pp. 2–5, Eqs. (8)–(17), PDF.

The four-dimensional Bianchi identity constrains the divergence of $E_{\mu\nu}$ but not all of its components. Bulk initial data, regularity at the AdS horizon, incoming radiation, or a second-brane condition is still required. Maartens and Koyama make this nonclosure explicit in Maartens and Koyama 2010, §§3.1–3.2, PDF. Thus a metric that solves a guessed four-dimensional equation need not possess a regular five-dimensional completion.

Empirical limits on $ℓ$ and the KK continuum belong in Tests of Gravity; ultraviolet realizations of the brane and AdS region belong in Quantum Gravity. The next page contrasts this ultraviolet crossover with DGP’s infrared leakage.

Common pitfalls

Using the warp volume instead of the graviton norm. The measure in the Einstein kinetic term is $e^{-2|y|/\ell}$, not the five-dimensional volume measure $e^{-4|y|/\ell}$.

Dropping the Weyl projection. $E_{\mu\nu}=0$ is a bulk boundary choice, not a consequence of the brane stress tensor or four-dimensional symmetry in general.

Mixing RS1 and RS2 spectra. A finite two-brane interval has a radion and discrete massive modes; the one-brane infinite geometry has a normalizable zero mode plus a continuum.

Exercises

Verify both fine-tuning relations for the vacuum RS2 branch: use the jump in extrinsic curvature to find $λ$, and then show that the effective cosmological constant vanishes and the Planck normalization agrees with the zero-mode integral.

Solution

For $K^+_{\mu\nu}=-q_{\mu\nu}/\ell$ and $K^-_{\mu\nu}=+q_{\mu\nu}/\ell$, the jump of $K_{\mu\nu}-Kq_{\mu\nu}$ is $6q_{\mu\nu}/\ell$. Equating it to $-\kappa_5^2S_{\mu\nu}=\kappa_5^2\lambda q_{\mu\nu}$ gives $\lambda=6/(\kappa_5^2\ell)$. Then $\kappa_5^4\lambda^2/6=6/\ell^2$, which cancels $\Lambda_5=-6/\ell^2$ inside $\Lambda_4$. Finally $\kappa_4^2=\kappa_5^4\lambda/6=\kappa_5^2/\ell$, hence $M_4^2=\ell M_5^3$, identical to integrating $e^{-2|y|/\ell}$. Agreement between a local junction calculation and a global mode normalization is the independent check; either result alone could conceal a normal-orientation or factor-of-two error.

References

• Maartens, R., and K. Koyama, “Brane-World Gravity,” Living Reviews in Relativity 13 (2010) 5, DOI, Open PDF.
• Randall, L., and R. Sundrum, “An Alternative to Compactification,” Physical Review Letters 83 (1999) 4690–4693, DOI, Open PDF.
• Shiromizu, T., K. Maeda, and M. Sasaki, “The Einstein Equations on the 3-Brane World,” Physical Review D 62 (2000) 024012, DOI, Open PDF.