Shiromizu–Maeda–Sasaki Formalism in Braneworld Gravity

• The Shiromizu–Maeda–Sasaki formalism is a braneworld projection method that embeds a 4D brane in a 5D bulk to derive effective gravitational dynamics with novel correction terms.
• It introduces local quadratic stress-energy corrections and nonlocal Weyl tensor contributions that modify the standard Einstein equations.
• Generalizations to f(R) and Einstein–Gauss–Bonnet frameworks extend its applications to exact black hole, wormhole, and cosmological solutions with observable signatures.

Searching arXiv for the original SMS formalism and closely related braneworld generalizations. The Shiromizu–Maeda–Sasaki formalism is a braneworld projection framework in which a 4D brane is embedded as a hypersurface in a 5D bulk, and the effective gravitational dynamics on the brane are obtained by projecting the higher-dimensional field equations. In the formulations discussed across the literature considered here, the induced 4D equations are not identical to ordinary Einstein equations: besides the effective cosmological term and the brane matter contribution, they contain a local quadratic stress-energy correction and a nonlocal term given by the projected 5D Weyl tensor. In later extensions, the same projection logic is generalized to $f(R)$ bulks and to Einstein–Gauss–Bonnet settings, where additional curvature-induced structures appear on the brane (Kuerten, 2017, Rocha et al., 2014, Ramirez, 2017). Within this framework, the bulk can act as an effective source for exact vacuum brane geometries, including continuous families interpolating between black holes and traversable wormholes, with perturbative signatures such as black-hole-like ringdown followed by echoes (Bronnikov et al., 2019).

1. Geometric setup and projected field equations

In the standard SMS setting, the bulk coordinates are $x^a$ and the brane coordinates are $x^\mu$, with the bulk metric decomposed as

$g_{ab}=q_{ab}+n_a n_b,$

where $q_{\mu\nu}$ is the induced metric on the brane and $n^a$ is the unit normal. In Gaussian normal coordinates,

$$ds^2=g_{ab}dx^a dx^b=q_{\mu\nu}(x^\alpha,y)\,dx^\mu dx^\nu+dy^2.$$

This identifies the brane as a codimension-one hypersurface, commonly taken at fixed transverse coordinate, and provides the geometric basis for the projection procedure (Kuerten, 2017).

The derivation uses the Gauss equation together with Codazzi relations and Israel junction conditions. In the notation summarized for the formalism,

$$R_{abcd}=q_a{}^e q_b{}^f q_c{}^g q_d{}^h\,{}^{(5)}\!R_{efgh}+2K_{a[c}K_{d]b},$$

while, assuming $Z_2$-symmetry, the extrinsic curvature is related to the brane stress tensor and brane tension by

$$K_{\mu\nu}=-\frac{\kappa_5^2}{2} \left(T_{\mu\nu}+\frac{1}{3}(\lambda-T)\,q_{\mu\nu}\right).$$

Combining these ingredients yields the effective 4D equation

$x^a$0

or, in the mixed-index form used for exact vacuum brane solutions,

$x^a$1

The effective cosmological constant is written as

$x^a$2

in one convention, and as

$x^a$3

up to the conventions used in another presentation. This suggests that normalization differences across papers are not intrinsic to the projection idea but to the chosen coupling conventions (Bronnikov et al., 2019, Kuerten, 2017).

2. Correction terms and their physical content

The SMS equation contains two corrections beyond ordinary 4D Einstein gravity. The first is the quadratic stress-energy term. In one convention it is written as

$x^a$4

while in the mixed-index vacuum-brane presentation it appears as

$x^a$5

In both forms, it is the local high-energy correction quadratic in the brane stress tensor (Kuerten, 2017, Bronnikov et al., 2019).

The second is the projected Weyl term,

$x^a$6

or equivalently

$x^a$7

This is the “electric” part of the 5D Weyl tensor. It is traceless, encodes nonlocal bulk gravitational influence on the brane, and in cosmological language behaves like dark radiation (Bronnikov et al., 2019, Kuerten, 2017).

A central feature of the formalism is therefore the split between local and nonlocal corrections. The local term $x^a$8 depends only on brane matter variables, whereas $x^a$9 carries information not determined by 4D brane data alone. A plausible implication is that exact solution-building in SMS typically requires either bulk input, symmetry assumptions, or an ansatz for the projected Weyl contribution.

3. Vacuum branes, exact geometries, and black hole–wormhole interpolation

A particularly important SMS sector is the vacuum brane, for which

$x^\mu$0

Then the effective equation reduces to a geometry sourced entirely by the tidal Weyl term $x^\mu$1. In the exact solutions considered for static spherical symmetry, the line element is taken as

$x^\mu$2

and the bulk Weyl curvature acts as an effective source even in the absence of ordinary 4D matter (Bronnikov et al., 2019).

Within this setting, exact vacuum brane solutions can form continuous families containing black holes, an extremal black hole at threshold, and traversable wormholes. The paper on echoes examines three representative families. The Casadio–Fabbri–Mazzacurati metric uses an interpolating parameter $x^\mu$3, with the special cases $x^\mu$4 giving Schwarzschild, $x^\mu$5 giving a double-horizon extremal black hole, $x^\mu$6 giving black holes, and $x^\mu$7 giving a symmetric traversable wormhole. The Bronnikov–Kim-1 family has the extremal black hole only at

$x^\mu$8

while other parameter values yield either wormholes or a naked singularity. The Bronnikov–Kim-2 family uses a threshold parameter $x^\mu$9, with $g_{ab}=q_{ab}+n_a n_b,$0 at the double horizon, $g_{ab}=q_{ab}+n_a n_b,$1 producing a horizon at $g_{ab}=q_{ab}+n_a n_b,$2, and $g_{ab}=q_{ab}+n_a n_b,$3 producing a wormhole throat at $g_{ab}=q_{ab}+n_a n_b,$4, where

$g_{ab}=q_{ab}+n_a n_b,$5

These families show that, in exact SMS vacuum solutions, changing a continuous parameter can change the induced geometry from horizon-bearing to throat-bearing while preserving exact solvability on the brane (Bronnikov et al., 2019).

The formal reason is that the Weyl term can generate effective geometries not sourced by ordinary 4D matter. The threshold extremal black hole is structurally important because nearby wormholes inherit its near-horizon geometry outside a small region near the throat. This explains why such wormholes can behave as black-hole mimickers in perturbation theory (Bronnikov et al., 2019).

4. Perturbations, quasinormal ringing, and echo formation

For massless scalar and electromagnetic test fields, perturbations reduce to a Schrödinger-like wave equation in the tortoise coordinate $g_{ab}=q_{ab}+n_a n_b,$6,

$g_{ab}=q_{ab}+n_a n_b,$7

The effective potentials are

$g_{ab}=q_{ab}+n_a n_b,$8

for the scalar field and

$g_{ab}=q_{ab}+n_a n_b,$9

for the electromagnetic field. These potentials determine whether the signal exhibits ordinary black-hole ringdown or partial trapping between multiple barriers (Bronnikov et al., 2019).

The perturbation problem is treated by a higher-order WKB method for black holes with a single barrier and two turning points, and by time-domain integration for both black holes and wormholes, particularly when the potential develops multiple barriers. The WKB expression employed is

$q_{\mu\nu}$0

with the improved Padé version used with $q_{\mu\nu}$1. The time-domain integration scheme is

$q_{\mu\nu}$2

Near the black-hole/wormhole threshold, the effective potential develops a second symmetric peak far to the left in tortoise coordinate. The waveform then shows an initial stage essentially indistinguishable from threshold-black-hole ringdown, followed at later times by echoes generated by repeated reflections between the two barriers (Bronnikov et al., 2019).

In the Casadio–Fabbri–Mazzacurati family, the near-threshold wormhole signal is almost indistinguishable from the threshold black hole at early times, while later echoes appear. As the parameter moves farther into the wormhole side, the echoes fade into the wormhole’s own characteristic ringing. In that regime, the WKB method ceases to be applicable because the potential has two barriers and four turning points. The Bronnikov–Kim-1 and Bronnikov–Kim-2 families display the same general pattern, with a longer initial black-hole-like stage in the former and a threshold at $q_{\mu\nu}$3 in the latter (Bronnikov et al., 2019).

5. Generalizations to $q_{\mu\nu}$4 bulk gravity and curvature dynamics

The SMS projection has been generalized to bulks whose gravitational dynamics are already modified. In the $q_{\mu\nu}$5-bulk version, the 5D Einstein equations with bulk cosmological constant are written as

$q_{\mu\nu}$6

where the effective bulk stress tensor is that of 5D $q_{\mu\nu}$7 gravity,

$q_{\mu\nu}$8

The brane is assumed at $q_{\mu\nu}$9, with $n^a$0, $n^a$1, a warped 5D geometry, $n^a$2 symmetry, and localized brane matter

$n^a$3

The fine-tuning relations are given as

$n^a$4

Projecting onto the brane produces a generalized SMS equation with an extra term $n^a$5,

$n^a$6

with $n^a$7 encoding $n^a$8 bulk effects (Rocha et al., 2014).

In the explicit expression quoted for this generalization,

$n^a$9

and, for a conformally flat bulk, it is conserved (Rocha et al., 2014).

A further conceptual extension reinterprets the BSSY $$ds^2=g_{ab}dx^a dx^b=q_{\mu\nu}(x^\alpha,y)\,dx^\mu dx^\nu+dy^2.$$0 term as a generator of effective $$ds^2=g_{ab}dx^a dx^b=q_{\mu\nu}(x^\alpha,y)\,dx^\mu dx^\nu+dy^2.$$1 gravity on the brane. In that view, instead of treating $$ds^2=g_{ab}dx^a dx^b=q_{\mu\nu}(x^\alpha,y)\,dx^\mu dx^\nu+dy^2.$$2 only as an additional stress correction, it is moved to the geometric side so that the induced brane dynamics becomes $$ds^2=g_{ab}dx^a dx^b=q_{\mu\nu}(x^\alpha,y)\,dx^\mu dx^\nu+dy^2.$$3-type. This is accompanied by the introduction of a curvature dynamical constraint, or CDC, which supplements the trace of the Gauss relation. The trace of the Gauss relation is called the curvature geometrical constraint,

$$ds^2=g_{ab}dx^a dx^b=q_{\mu\nu}(x^\alpha,y)\,dx^\mu dx^\nu+dy^2.$$4

while the CDC links the bulk and brane scalaron dynamics through

$$ds^2=g_{ab}dx^a dx^b=q_{\mu\nu}(x^\alpha,y)\,dx^\mu dx^\nu+dy^2.$$5

with

$$ds^2=g_{ab}dx^a dx^b=q_{\mu\nu}(x^\alpha,y)\,dx^\mu dx^\nu+dy^2.$$6

This construction is used to map bulk $$ds^2=g_{ab}dx^a dx^b=q_{\mu\nu}(x^\alpha,y)\,dx^\mu dx^\nu+dy^2.$$7 theories to induced brane $$ds^2=g_{ab}dx^a dx^b=q_{\mu\nu}(x^\alpha,y)\,dx^\mu dx^\nu+dy^2.$$8 theories, and in the special case $$ds^2=g_{ab}dx^a dx^b=q_{\mu\nu}(x^\alpha,y)\,dx^\mu dx^\nu+dy^2.$$9 it leads to a braneworld version of $$R_{abcd}=q_a{}^e q_b{}^f q_c{}^g q_d{}^h\,{}^{(5)}\!R_{efgh}+2K_{a[c}K_{d]b},$$0-unimodular gravity formally identical to that of Nojiri et al. (Kuerten, 2017).

6. Einstein–Gauss–Bonnet extensions, junction conditions, and cosmology

The SMS projection philosophy also extends to 5D Einstein–Gauss–Bonnet gravity, where the brane remains a timelike hypersurface but the matching conditions are modified by Gauss–Bonnet terms. In the isotropic setting considered, the bulk metric on each side is

$$R_{abcd}=q_a{}^e q_b{}^f q_c{}^g q_d{}^h\,{}^{(5)}\!R_{efgh}+2K_{a[c}K_{d]b},$$1

with

$$R_{abcd}=q_a{}^e q_b{}^f q_c{}^g q_d{}^h\,{}^{(5)}\!R_{efgh}+2K_{a[c}K_{d]b},$$2

where $$R_{abcd}=q_a{}^e q_b{}^f q_c{}^g q_d{}^h\,{}^{(5)}\!R_{efgh}+2K_{a[c}K_{d]b},$$3 is the Gauss–Bonnet coupling, $$R_{abcd}=q_a{}^e q_b{}^f q_c{}^g q_d{}^h\,{}^{(5)}\!R_{efgh}+2K_{a[c}K_{d]b},$$4 the bulk cosmological constant, $$R_{abcd}=q_a{}^e q_b{}^f q_c{}^g q_d{}^h\,{}^{(5)}\!R_{efgh}+2K_{a[c}K_{d]b},$$5 the mass parameter, and $$R_{abcd}=q_a{}^e q_b{}^f q_c{}^g q_d{}^h\,{}^{(5)}\!R_{efgh}+2K_{a[c}K_{d]b},$$6 labels the GR and stringy branches. The paper assumes

$$R_{abcd}=q_a{}^e q_b{}^f q_c{}^g q_d{}^h\,{}^{(5)}\!R_{efgh}+2K_{a[c}K_{d]b},$$7

For $$R_{abcd}=q_a{}^e q_b{}^f q_c{}^g q_d{}^h\,{}^{(5)}\!R_{efgh}+2K_{a[c}K_{d]b},$$8, the asymptotic effective cosmological constant is

$$R_{abcd}=q_a{}^e q_b{}^f q_c{}^g q_d{}^h\,{}^{(5)}\!R_{efgh}+2K_{a[c}K_{d]b},$$9

The central brane has metric

$Z_2$0

with perfect-fluid surface stress-energy

$Z_2$1

All standard matter is confined to the brane and the bulk is vacuum (Ramirez, 2017).

The generalized matching condition is

$Z_2$2

with

$Z_2$3

where $Z_2$4 is cubic in $Z_2$5 and $Z_2$6 is built from the intrinsic curvature of the brane. In this sense, the SMS idea survives, but the extrinsic-curvature sector becomes nonlinear already at the junction condition level (Ramirez, 2017).

For the central brane, the $Z_2$7 component yields a Friedmann-like equation,

$Z_2$8

This is recast as

$Z_2$9

with the effective potential $$K_{\mu\nu}=-\frac{\kappa_5^2}{2} \left(T_{\mu\nu}+\frac{1}{3}(\lambda-T)\,q_{\mu\nu}\right).$$0 specified in terms of

$$K_{\mu\nu}=-\frac{\kappa_5^2}{2} \left(T_{\mu\nu}+\frac{1}{3}(\lambda-T)\,q_{\mu\nu}\right).$$1

and an auxiliary function $$K_{\mu\nu}=-\frac{\kappa_5^2}{2} \left(T_{\mu\nu}+\frac{1}{3}(\lambda-T)\,q_{\mu\nu}\right).$$2. The continuity equation remains

$$K_{\mu\nu}=-\frac{\kappa_5^2}{2} \left(T_{\mu\nu}+\frac{1}{3}(\lambda-T)\,q_{\mu\nu}\right).$$3

At large $$K_{\mu\nu}=-\frac{\kappa_5^2}{2} \left(T_{\mu\nu}+\frac{1}{3}(\lambda-T)\,q_{\mu\nu}\right).$$4, the effective Friedmann equation becomes

$$K_{\mu\nu}=-\frac{\kappa_5^2}{2} \left(T_{\mu\nu}+\frac{1}{3}(\lambda-T)\,q_{\mu\nu}\right).$$5

with explicit identifications for $$K_{\mu\nu}=-\frac{\kappa_5^2}{2} \left(T_{\mu\nu}+\frac{1}{3}(\lambda-T)\,q_{\mu\nu}\right).$$6 and $$K_{\mu\nu}=-\frac{\kappa_5^2}{2} \left(T_{\mu\nu}+\frac{1}{3}(\lambda-T)\,q_{\mu\nu}\right).$$7. This is the SMS low-energy limit corrected by Gauss–Bonnet terms (Ramirez, 2017).

An additional feature of the EGB extension is the existence of vacuum thin shells satisfying

$$K_{\mu\nu}=-\frac{\kappa_5^2}{2} \left(T_{\mu\nu}+\frac{1}{3}(\lambda-T)\,q_{\mu\nu}\right).$$8

but with nontrivial discontinuity of the extrinsic curvature. Such shells do not exist in GR but are allowed by the nonlinear EGB junction conditions. The same framework is used to construct a pair of $$K_{\mu\nu}=-\frac{\kappa_5^2}{2} \left(T_{\mu\nu}+\frac{1}{3}(\lambda-T)\,q_{\mu\nu}\right).$$9-symmetric vacuum thin shells splitting from the central brane, with existence controlled by an inequality comparing shell and brane accelerations immediately after splitting. A necessary condition is

$x^a$00

and no solution exists if $x^a$01. The resulting dynamics permits either shell collapse with a lasting stringy bulk or expansion followed by recoil and recombination (Ramirez, 2017).

7. Observational and conceptual scope

The $x^a$02-bulk generalization shows how the SMS procedure enters phenomenology through effective brane black hole metrics. For a static spherically symmetric vacuum brane with constant 5D Ricci scalar, the line element is taken as

$x^a$03

and the solution used for Solar System tests is

$x^a$04

Here $x^a$05 is the effective mass, $x^a$06 is the tidal charge parameter, and

$x^a$07

If $x^a$08, the solution reduces to the DMPR black hole; if $x^a$09 the tidal charge vanishes; and if both $x^a$10 and $x^a$11, Schwarzschild is recovered (Rocha et al., 2014).

The same paper derives brane-world corrections to perihelion precession, light deflection, and radar echo delay. The strongest quoted constraint is

$x^a$12

while radar echo delay yields

$x^a$13

implying an upper limit around $x^a$14 for $x^a$15, hence also for the trace $x^a$16. This places the SMS-derived parameters into direct contact with classical tests of general relativity (Rocha et al., 2014).

Across these applications, several common misconceptions can be clarified. The SMS formalism is not merely a re-expression of 4D general relativity in unusual coordinates; its defining content is the appearance of genuinely new effective terms, especially $x^a$17 and $x^a$18. Nor is it restricted to perturbative or cosmological settings: the same projection framework supports exact black hole and wormhole solutions, quasinormal-mode analyses, Solar System constraints, and Gauss–Bonnet branch-changing shell dynamics (Bronnikov et al., 2019, Rocha et al., 2014, Ramirez, 2017). A plausible broader implication is that SMS functions less as a single model than as a geometric projection scheme whose phenomenology depends sensitively on the assumed bulk theory and on how the nonlocal sector is closed.