Eternal Trumpet Spacetime: Definitions & Applications

• Eternal Trumpet Spacetime is a geometric framework describing stationary, asymptotically cylindrical end-states in numerical relativity and analytic black-hole slicings.
• It unites diverse formulations including 2D gravity, holographic BTZ interiors, and topological-cosmological models under a common trumpet motif.
• The construct offers practical insights into gauge dynamics, regularization techniques, and the causal structure of spacetimes across multiple dimensions.

Searching arXiv for the cited literature on “Eternal Trumpet Spacetime” and closely related trumpet geometries. “Eternal Trumpet Spacetime” is a polysemous term in the literature. In numerical relativity it most commonly denotes the stationary, asymptotically cylindrical end-state reached by black-hole evolutions in moving-puncture gauges, where the spatial slice ends at a finite nonzero areal radius and the exterior geometry becomes time-independent to numerical accuracy (Thierfelder et al., 2010). In other settings the same expression is used for analytically constructed trumpet foliations of black-hole spacetimes, for two-dimensional wormhole geometries built by gluing trumpet amplitudes, for BTZ interior constructions associated with temporal quantum correlations, and for a nonreflective cosmological spacetime that is the universal cover of the “Itty-Bitty Blender” (Dennison et al., 2014, Blommaert et al., 28 Jan 2025, Racorean, 2023, Bray et al., 15 Sep 2025). This suggests a family resemblance rather than a single invariant definition.

1. Terminological scope

The principal uses of the term fall into a small number of technically distinct categories.

Context	Meaning of “eternal trumpet spacetime”	Representative papers
Numerical relativity	Stationary trumpet end-state of moving-puncture black-hole evolutions	(Thierfelder et al., 2010, Miller et al., 2016)
Analytic black-hole slicings	Time-independent trumpet slices of Schwarzschild, Kerr, higher-dimensional, or SdS spacetimes	(Dennison et al., 2014, Dennison et al., 2014, Dennison et al., 2010, Dennison et al., 2017)
2D gravity and matrix models	Trumpet or double-trumpet geometries in JT-like or sine dilaton gravity	(Blommaert et al., 28 Jan 2025, Freivogel et al., 2019)
AdS$_3$/BTZ holography	BTZ interior or non-orientable eternal constructions described in trumpet language	(Racorean, 2023, Racorean, 20 Aug 2025)
Topological cosmology	Universal cover of the Itty-Bitty Blender spacetime, with a trumpet-like causal structure	(Bray et al., 15 Sep 2025)

A common misconception is to treat all of these usages as referring to a single Schwarzschild trumpet geometry. The papers do not support that identification. The shared element is the trumpet motif itself: a finite inner end or geodesic boundary, a flaring exterior or two-boundary gluing, and an associated stationary or global causal structure.

2. Moving-puncture black holes and the standard numerical-relativity meaning

In moving-puncture evolutions, an eternal trumpet spacetime is the stationary, asymptotically cylindrical end-state that the coordinates settle into at late times (Thierfelder et al., 2010). For spherical collapse, the exterior geometry outside the matter evolves under puncture gauges to the same stationary trumpet slice obtained by evolving a single Schwarzschild wormhole puncture. “Eternal” means that once the exterior has reached this stationary slice, it persists for arbitrarily long evolution times under the same gauge; stationarity was demonstrated through long 1D and 3D runs and gauge-invariant diagnostics (Thierfelder et al., 2010).

In spherical symmetry the line element is

$$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$

with

$$\gamma_{rr}=\psi^4,\qquad \gamma_{\theta\theta}=\psi^4 r^2,\qquad R=r\psi^2.$$

The defining geometric feature is a nonzero minimal areal radius $R_0$ at the puncture: as $r\to 0$, the slice asymptotes to a cylindrical end at $R\to R_0>0$. For the Schwarzschild trumpet obtained with puncture gauges, $R_0/M \approx 1.3$ (Thierfelder et al., 2010).

Near the puncture, the late-time collapse end-state reproduces the trumpet scalings. The lapse collapses as

$\alpha(r)\simeq 0.54\,(r/M)^{1.09},$

the BSSN conformal variable behaves as

$\chi(r)\simeq 1.22\,(r/M)^2,$

implying $\psi(r)\sim \text{const}\times r^{-1/2}$, and the extrinsic-curvature trace satisfies

$$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$0

in agreement with the analytic trumpet expansion

$$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$1

for $$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$2 (Thierfelder et al., 2010).

The gauge conditions are the Bona-Massó $$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$3 lapse with $$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$4,

$$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$5

and the Gamma-driver shift

$$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$6

with $$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$7 or $$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$8 and $$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$9 (Thierfelder et al., 2010). With $$\gamma_{rr}=\psi^4,\qquad \gamma_{\theta\theta}=\psi^4 r^2,\qquad R=r\psi^2.$$0, the exterior becomes vacuum and matches the single-puncture trumpet exceptionally well, with $$\gamma_{rr}=\psi^4,\qquad \gamma_{\theta\theta}=\psi^4 r^2,\qquad R=r\psi^2.$$1 at $$\gamma_{rr}=\psi^4,\qquad \gamma_{\theta\theta}=\psi^4 r^2,\qquad R=r\psi^2.$$2. With $$\gamma_{rr}=\psi^4,\qquad \gamma_{\theta\theta}=\psi^4 r^2,\qquad R=r\psi^2.$$3, or with $$\gamma_{rr}=\psi^4,\qquad \gamma_{\theta\theta}=\psi^4 r^2,\qquad R=r\psi^2.$$4, the matter remains resolved on the grid and the match degrades substantially; for $$\gamma_{rr}=\psi^4,\qquad \gamma_{\theta\theta}=\psi^4 r^2,\qquad R=r\psi^2.$$5 the paper reports $$\gamma_{rr}=\psi^4,\qquad \gamma_{\theta\theta}=\psi^4 r^2,\qquad R=r\psi^2.$$6 for $$\gamma_{rr}=\psi^4,\qquad \gamma_{\theta\theta}=\psi^4 r^2,\qquad R=r\psi^2.$$7 (Thierfelder et al., 2010).

The mechanism is gauge-dynamical rather than topological. The Gamma-driver shift with $$\gamma_{rr}=\psi^4,\qquad \gamma_{\theta\theta}=\psi^4 r^2,\qquad R=r\psi^2.$$8 stretches spatial coordinates strongly near the center, advects grid points outward, and effectively removes matter from the resolved domain; simultaneously, $$\gamma_{rr}=\psi^4,\qquad \gamma_{\theta\theta}=\psi^4 r^2,\qquad R=r\psi^2.$$9 slicing collapses the lapse near the center and the shift arrests radial coordinate motion outside the matter. The exterior then evolves exactly as vacuum Schwarzschild under the same puncture gauges and relaxes to the known $R_0$0 trumpet slice (Thierfelder et al., 2010). In this sense the puncture gauges act as a natural excision.

3. Analytic trumpet slicings and geometric generalizations

The moving-puncture end-state has several analytic relatives. In the conformal thin-sandwich formalism, maximally sliced trumpet-puncture data can be constructed in quasiequilibrium, unlike wormhole data. The essential distinction is that for trumpet data the Killing lapse remains strictly positive for all $R_0$1 and vanishes only at $R_0$2, where the puncture method absorbs the singular behavior. For the maximal Schwarzschild trumpet, the slice asymptotes to $R_0$3, the conformal factor scales as $R_0$4, and the slice connects spatial infinity in one universe to future timelike infinity $R_0$5 of the other (Baumgarte, 2011).

A particularly simple analytic family of Schwarzschild trumpets is obtained from

$R_0$6

with

$R_0$7

where

$R_0$8

Constant-$R_0$9 slices are spatially isotropic, horizon-penetrating, asymptotically flat, and asymptotically cylindrical inside the horizon at the prescribed areal radius $r\to 0$0. In the limit $r\to 0$1, the family reduces to Painlevé-Gullstrand coordinates and loses its trumpet geometry (Dennison et al., 2014).

The trumpet construction generalizes beyond four-dimensional Schwarzschild. For Schwarzschild-Tangherlini spacetimes in $r\to 0$2, time-independent maximal and generalized $r\to 0$3 foliations admit special members for which the lapse vanishes at a finite $r\to 0$4 and the proper distance to that surface diverges, defining a trumpet slice. In $r\to 0$5 with advective $r\to 0$6 and $r\to 0$7, the paper gives $r\to 0$8, $r\to 0$9, and $R\to R_0>0$0, and numerical BSSN evolutions settle to the analytic trumpet solution (Dennison et al., 2010).

Rotating black holes also admit stationary trumpet slices. In a new family of analytical Kerr coordinates, the trumpet surface is the coordinate sphere $R\to R_0>0$1, with lapse

$R\to R_0>0$2

where

$R\to R_0>0$3

The trumpet surface has finite area

$R\to R_0>0$4

finite $R\to R_0>0$5, and infinite proper distance because the radial lapse $R\to R_0>0$6 diverges like $R\to R_0>0$7 (Dennison et al., 2014). These slices are stationary and horizon-penetrating, although they are neither maximal nor standard $R\to R_0>0$8 slices.

In Schwarzschild-de Sitter, static constant-mean-curvature trumpet slicings with $R\to R_0>0$9 generalize the maximal Schwarzschild trumpet. The inner trumpet radius is

$R_0/M \approx 1.3$0

with $R_0/M \approx 1.3$1 determined by $R_0/M \approx 1.3$2 and $R_0/M \approx 1.3$3. After a transformation to comoving isotropic coordinates, the metric asymptotes at large distances to flat FLRW with $R_0/M \approx 1.3$4, while as the isotropic radius goes to zero it approaches a horizon-penetrating trumpet geometry rather than a second asymptotically de Sitter end (Dennison et al., 2017).

Trumpet coordinates are also useful for matter flows. In Bondi accretion, transforming the stationary Schwarzschild flow into maximal or analytical trumpet coordinates yields regular expressions for $R_0/M \approx 1.3$5, $R_0/M \approx 1.3$6, $R_0/M \approx 1.3$7, and $R_0/M \approx 1.3$8 across the horizon and into the black-hole interior up to the puncture at $R_0/M \approx 1.3$9. When the initial data are given directly in trumpet coordinates consistent with the chosen gauge, the evolved solution remains time-independent, which is precisely the sense in which the trumpet spacetime is “eternal” in that context (Miller et al., 2016).

4. Trumpets and double trumpets in two-dimensional gravity

In two-dimensional gravity, the term shifts from a foliation of a four-dimensional black hole to a building block of the path integral. In sine dilaton gravity, the trumpet is a two-dimensional geometry with one geodesic boundary and one asymptotic boundary, studied through minisuperspace Wheeler-DeWitt quantization (Blommaert et al., 28 Jan 2025). The gauge-invariant closed geodesic length in the effective AdS metric is discrete,

$\alpha(r)\simeq 0.54\,(r/M)^{1.09},$0

so physical trumpet states are labeled by an integer $\alpha(r)\simeq 0.54\,(r/M)^{1.09},$1 (Blommaert et al., 28 Jan 2025).

The exact minisuperspace Wheeler-DeWitt operator is

$\alpha(r)\simeq 0.54\,(r/M)^{1.09},$2

and the asymptotic trumpet amplitude is

$\alpha(r)\simeq 0.54\,(r/M)^{1.09},$3

with $\alpha(r)\simeq 0.54\,(r/M)^{1.09},$4 a modified Bessel function of the first kind. Gluing two trumpets with the same geodesic length gives the double trumpet, or two-boundary wormhole,

$\alpha(r)\simeq 0.54\,(r/M)^{1.09},$5

which the paper interprets as the hallmark of “eternal” two-sided geometries (Blommaert et al., 28 Jan 2025). The same construction matches the universal spectral correlation of a finite-cut matrix integral, strongly suggesting an identification with a $\alpha(r)\simeq 0.54\,(r/M)^{1.09},$6-deformed JT gravity matrix model (Blommaert et al., 28 Jan 2025).

The comparison with JT gravity is precise. In JT, geodesic sizes are continuous and the physical resolution of the identity is

$\alpha(r)\simeq 0.54\,(r/M)^{1.09},$7

whereas in sine dilaton gravity the sizes are discrete and

$\alpha(r)\simeq 0.54\,(r/M)^{1.09},$8

The trumpet amplitudes are correspondingly $\alpha(r)\simeq 0.54\,(r/M)^{1.09},$9-Bessel in JT and $\chi(r)\simeq 1.22\,(r/M)^2,$0 in sine dilaton gravity (Blommaert et al., 28 Jan 2025).

This two-dimensional setting is also the place where trumpet geometries are explicitly contrasted with higher-dimensional no-go results. Under boundary Poincaré invariance, static inter-boundary couplings, semiclassical gravity, and the NEC for added conventional matter, static traversable wormholes in $\chi(r)\simeq 1.22\,(r/M)^2,$1 cannot be made semiclassical; the core obstruction is $\chi(r)\simeq 1.22\,(r/M)^2,$2 with perturbative $\chi(r)\simeq 1.22\,(r/M)^2,$3 (Freivogel et al., 2019). By contrast, in AdS$\chi(r)\simeq 1.22\,(r/M)^2,$4/JT, Euclidean “trumpet” and “double trumpet” geometries are standard and their Lorentzian continuations describe eternal wormholes (Freivogel et al., 2019).

5. BTZ interior constructions, temporal correlations, and non-orientable eternal spacetimes

A different use of the term appears in AdS$\chi(r)\simeq 1.22\,(r/M)^2,$5/BTZ discussions of temporal quantum correlations. In one construction, the Eternal Trumpet Spacetime is the BTZ interior geometry that holographically encodes high-temperature temporal entanglement of a single CFT measured at two times (Racorean, 2023). The key quantum-information statement is that the temporal-correlations state equals the partially transposed density matrix of a maximally entangled pair,

$\chi(r)\simeq 1.22\,(r/M)^2,$6

and in the $\chi(r)\simeq 1.22\,(r/M)^2,$7 limit the thermofield-double state approaches the maximally entangled state, so spatial TFD correlations become equivalent to temporal correlations (Racorean, 2023).

On the gravity side, the proposed interior metric is

$\chi(r)\simeq 1.22\,(r/M)^2,$8

valid for $\chi(r)\simeq 1.22\,(r/M)^2,$9. The horizon lies at $\psi(r)\sim \text{const}\times r^{-1/2}$0, and the spatial circle grows with $\psi(r)\sim \text{const}\times r^{-1/2}$1, producing the trumpet-like interior profile discussed in the paper (Racorean, 2023). The bridge is “temporal” because it connects the same boundary at different times rather than two distinct boundaries at equal time.

A related BTZ paper derives the interior metric by explicitly interchanging the characteristics of space and time coordinates. The maximally extended geometry has the same Kruskal form as the exterior counterpart, but the full bulk partition function obtained by including both exterior and interior thermofield-double constructions is non-orientable and is identified with a Klein bottle amplitude (Racorean, 20 Aug 2025). In that setting there are two independent thermofield-double states, one dual to the exterior solution and one dual to the interior solution, and the resulting eternal BTZ spacetime is described as non-orientable (Racorean, 20 Aug 2025).

These AdS$\psi(r)\sim \text{const}\times r^{-1/2}$2 usages differ sharply from the puncture-gauge meaning in numerical relativity. The common word “trumpet” is doing geometric work, but the underlying objects are a BTZ interior or an AdS path-integral building block rather than a $\psi(r)\sim \text{const}\times r^{-1/2}$3 Schwarzschild end-state.

6. The topological-cosmology Eternal Trumpet

In “Topological Big Bangs: Reflection, Itty-Bitty Blenders, and Eternal Trumpets,” the Eternal Trumpet is neither a black-hole slice nor a two-dimensional wormhole amplitude. It is the universal cover of the “Itty-Bitty Blender” spacetime, obtained by unwinding the two $\psi(r)\sim \text{const}\times r^{-1/2}$4 factors of $\psi(r)\sim \text{const}\times r^{-1/2}$5. The resulting manifold is

$\psi(r)\sim \text{const}\times r^{-1/2}$6

with coordinates $\psi(r)\sim \text{const}\times r^{-1/2}$7 and metric

$\psi(r)\sim \text{const}\times r^{-1/2}$8

(Bray et al., 15 Sep 2025).

The spacetime is time-oriented by the global unit timelike vector field

$\psi(r)\sim \text{const}\times r^{-1/2}$9

and the future-directed null directions are

$$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$00

The smooth temporal function

$$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$01

has timelike gradient everywhere, and its level sets are Cauchy surfaces; the spacetime is therefore globally hyperbolic (Bray et al., 15 Sep 2025).

Far in the cosmological region $$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$02, the metric asymptotes to

$$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$03

With

$$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$04

the exact rewrite becomes

$$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$05

which for $$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$06 approaches flat FLRW with

$$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$07

Thus the Eternal Trumpet asymptotes to a radiation-dominated flat FLRW universe while remaining smooth at $$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$08 (Bray et al., 15 Sep 2025).

Its physical role is explicitly topological. The model is nonreflective, not of the reflective type $$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$09, and the “earliest” structure is the tubular core rather than a one-sided reflection surface (Bray et al., 15 Sep 2025). The paper states that the spacetime is geodesically complete and globally hyperbolic. It also states that the standard energy conditions fail in the core $$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$10, which is how the construction avoids the hypotheses of the Hawking and Penrose singularity theorems in its present form (Bray et al., 15 Sep 2025). Because every point in the large-$$ds^2 = -\alpha^2 dt^2 + \gamma_{rr}(dr+\beta^r dt)^2 + \gamma_{\theta\theta} d\Omega^2,$$11 cosmological region shares a causal past in the tubular core, the paper argues that there is no horizon problem, and it does so without invoking inflation (Bray et al., 15 Sep 2025).

Here again, “Eternal Trumpet” has a distinct technical meaning. It is a smooth, nonreflective, topological big-bang model with FLRW asymptotics, not a puncture black hole and not a double-trumpet amplitude. What remains common across the literature is the geometric image of a core or boundary from which the spacetime opens outward while preserving a regular global structure.