Grumiller Metric Overview

• Grumiller Metric is a static, spherically symmetric spacetime defined by a lapse function that includes a linear term in the areal radius, yielding a constant Rindler acceleration.
• It is applied in modified gravity and nonlinear electrodynamics models to extend Schwarzschild solutions, providing insights into dark-matter phenomenology and rotation curve behavior.
• The metric serves as a testbed for black hole thermodynamics, quasinormal-mode calculations, and perturbative analyses, offering a controlled benchmark for large-distance modifications of gravity.

Searching arXiv for recent and foundational papers on the Grumiller metric and closely related constructions. The Grumiller metric is a static, spherically symmetric spacetime whose defining feature is a lapse function containing a term linear in the areal radius. In its simplest and most commonly cited four-dimensional form, it is written as

$$ds^2=-A(r)\,dt^2+A(r)^{-1}\,dr^2+r^2\,d\Omega^2,\qquad A(r)=1-\frac{2m}{r}+2ar,$$

where $m$ is the Schwarzschild mass and $a>0$ is a constant termed the Rindler acceleration. In the literature summarized here, the same geometry is also described as Mannheim’s metric, rediscovered by Grumiller, and as the Rindler-modified Schwarzschild black hole. It is studied as an effective large-distance modification of general relativity, as a solution of reduced dilaton or scalar–tensor models, as a background supported by anisotropic fluids or nonlinear electrodynamics, and as a testbed for black-hole thermodynamics, Hawking radiation, and exact quasinormal-mode calculations (Mazharimousavi et al., 2013, Mirekhtiary et al., 2014, Perivolaropoulos et al., 2019).

1. Definition, nomenclature, and standard forms

The minimal Grumiller ansatz adopts the static, spherically symmetric line element

$ds^2=-A(r)\,dt^2+A(r)^{-1}\,dr^2+r^2\,d\Omega^2,$

with

$A(r)=1-\frac{2m}{r}+2ar.$

In the convention used by Mazharimousavi, Kerachian, and Halilsoy, the linear term enters with a positive sign, $+2ar$, and $a>0$ is interpreted as an inward Rindler acceleration. The same work explicitly notes that some literature adopts $A(r)=1-\frac{2m}{r}-2ar$; translation between the conventions is achieved by $a\rightarrow -a$ (Mazharimousavi et al., 2013).

Several extensions appear in the literature. In the nonlinear-electrodynamics construction of Halilsoy, Mazharimousavi, and Amirabi, the metric function becomes

$f(r)=1+2k-\frac{2M}{r}+2ar-\frac{\Lambda}{3}r^2,$

where $m$0 is a dimensionless constant interpreted as a global monopole parameter and $m$1 is the cosmological constant. In that model, both $m$2 and $m$3 are source-dependent quantities induced by the nonlinear electromagnetic sector rather than universal constants (Halilsoy et al., 2012).

A Schwarzschild–Rindler–AdS form also occurs in curved-space soliton studies,

$m$4

where the linear term is again the Rindler contribution, but metastable spherical walls require the opposite sign, $m$5 (Alestas et al., 2019).

The reduced-gravity reconstruction program provides another standard embedding. In the two-dimensional dilaton action studied by Nashed and collaborators, Grumiller’s potential

$m$6

yields the vacuum metric

$m$7

with $m$8 recovering the familiar Rindler term (Perivolaropoulos et al., 2019).

2. Geometric structure and Newtonian interpretation

In the weak-field limit, the temporal component satisfies $m$9, so the effective Newtonian potential of the basic metric is

$a>0$0

The first term is the standard Newtonian contribution, while the linear term produces a constant attractive acceleration,

$a>0$1

This is the core physical meaning of the Grumiller deformation: a uniform inward acceleration superposed on the Schwarzschild field (Mazharimousavi et al., 2013).

For circular motion, the relativistic geodesic condition gives

$a>0$2

and the weak-field circular speed follows as

$a>0$3

Hence, at large radii, $a>0$4, so the velocity grows as $a>0$5. The literature characterizes this as capturing part of the phenomenology of flat or slowly rising rotation curves, rather than yielding an exactly constant asymptotic speed (Mazharimousavi et al., 2013).

The event horizon of the simplest $a>0$6 metric is the positive root of $a>0$7:

$a>0$8

For $a>0$9 and $ds^2=-A(r)\,dt^2+A(r)^{-1}\,dr^2+r^2\,d\Omega^2,$0, there is a single positive horizon. In the black-hole treatment of Mirekhtiary and Sakalli, the second root is negative and therefore not a physical horizon (Mirekhtiary et al., 2014).

The Ricci scalar for the basic Rindler-augmented Schwarzschild metric is

$ds^2=-A(r)\,dt^2+A(r)^{-1}\,dr^2+r^2\,d\Omega^2,$1

which is negative throughout the exterior region when $ds^2=-A(r)\,dt^2+A(r)^{-1}\,dr^2+r^2\,d\Omega^2,$2. In the $ds^2=-A(r)\,dt^2+A(r)^{-1}\,dr^2+r^2\,d\Omega^2,$3-extended form used in exact quasinormal-mode calculations, the Ricci scalar becomes

$ds^2=-A(r)\,dt^2+A(r)^{-1}\,dr^2+r^2\,d\Omega^2,$4

and the spacetime remains algebraically special of type D with Weyl scalar $ds^2=-A(r)\,dt^2+A(r)^{-1}\,dr^2+r^2\,d\Omega^2,$5 (Mazharimousavi et al., 2013, Mi et al., 21 Sep 2025).

3. Realizations in modified gravity and matter models

A major line of research asks which covariant theories admit the Grumiller metric as an exact solution. In $ds^2=-A(r)\,dt^2+A(r)^{-1}\,dr^2+r^2\,d\Omega^2,$6 gravity, the relevant action is

$ds^2=-A(r)\,dt^2+A(r)^{-1}\,dr^2+r^2\,d\Omega^2,$7

with anisotropic matter source

$ds^2=-A(r)\,dt^2+A(r)^{-1}\,dr^2+r^2\,d\Omega^2,$8

Using $ds^2=-A(r)\,dt^2+A(r)^{-1}\,dr^2+r^2\,d\Omega^2,$9, the field equations are written in effective-fluid form as

$A(r)=1-\frac{2m}{r}+2ar.$0

For the metric $A(r)=1-\frac{2m}{r}+2ar.$1, Mazharimousavi, Kerachian, and Halilsoy evaluate the weak energy conditions

$A(r)=1-\frac{2m}{r}+2ar.$2

model by model, together with the viability conditions

$A(r)=1-\frac{2m}{r}+2ar.$3

Their scan finds that several $A(r)=1-\frac{2m}{r}+2ar.$4 classes can support the metric in substantial radial domains, but ghost-freedom and scalaron stability strongly restrict parameter space because the exterior curvature is negative, $A(r)=1-\frac{2m}{r}+2ar.$5. Explicitly discussed viable or partially viable choices include $A(r)=1-\frac{2m}{r}+2ar.$6, $A(r)=1-\frac{2m}{r}+2ar.$7, tuned log-cosh plus $A(r)=1-\frac{2m}{r}+2ar.$8 models, and $A(r)=1-\frac{2m}{r}+2ar.$9 with $+2ar$0; other classes fail the WEC or violate $+2ar$1 (Mazharimousavi et al., 2013).

Nonlinear electrodynamics furnishes a different microphysical realization. For a purely electric NED model with action

$+2ar$2

Halilsoy, Mazharimousavi, and Amirabi derive

$+2ar$3

with

$+2ar$4

Here $+2ar$5 is the NED charge, $+2ar$6 the NED coupling, and $+2ar$7 a uniform background electric field. In this construction the background field is essential: if $+2ar$8, the energy conditions are violated. For the electric model the stress tensor is diagonal and the WEC and SEC hold provided

$+2ar$9

The same paper also presents a purely magnetic NED model producing a logarithmic metric term,

$a>0$0

whose circular speed approaches a constant at large radius; that model is adjacent to, rather than identical with, the original Grumiller geometry (Halilsoy et al., 2012).

Reduced-gravity constructions identify the metric as a vacuum solution of effective infrared theories. In the two-dimensional dilaton framework, choosing $a>0$1 and $a>0$2 with the potential $a>0$3 yields the vacuum line element

$a>0$4

The reconstructed NFW-motivated potential

$a>0$5

generates a modified metric term that reduces to the Grumiller form in the limit $a>0$6. This suggests that the original Rindler term may be viewed as the leading member of a broader family of large-distance geometric potentials (Perivolaropoulos et al., 2019).

A complementary statement appears in the curved-space domain-wall analysis: the Schwarzschild–Rindler–AdS metric emerges generically as a spherically symmetric vacuum solution in a class of scalar–tensor theories, in Weyl conformal gravity, and in GR in the presence of a cosmological constant and a suitable spherically symmetric anisotropic perfect fluid (Alestas et al., 2019).

4. Horizons, Hawking radiation, and thermodynamic behavior

For the uncharged Grumiller black hole,

$a>0$7

the surface gravity is

$a>0$8

and the Hawking temperature is

$a>0$9

where $A(r)=1-\frac{2m}{r}-2ar$0 is the negative root of $A(r)=1-\frac{2m}{r}-2ar$1. In the limit $A(r)=1-\frac{2m}{r}-2ar$2, one recovers the Schwarzschild values $A(r)=1-\frac{2m}{r}-2ar$3 and $A(r)=1-\frac{2m}{r}-2ar$4 (Mirekhtiary et al., 2014).

Mirekhtiary and Sakalli compute Hawking radiation by the Hamilton–Jacobi tunneling method in four coordinate systems: the naïve Schwarzschild-like chart, Painlevé–Gullstrand, ingoing Eddington–Finkelstein, and Kruskal–Szekeres coordinates. In all four descriptions the tunneling probability reproduces the same conventional Hawking temperature, and the analysis explicitly avoids the factor-2 problem by imposing the classical absorption condition $A(r)=1-\frac{2m}{r}-2ar$5 and using the appropriate contour prescription at the horizon (Mirekhtiary et al., 2014).

The same work then applies Parikh–Wilczek tunneling in PG coordinates, with backreaction included through $A(r)=1-\frac{2m}{r}-2ar$6, and incorporates a logarithmic entropy correction

$A(r)=1-\frac{2m}{r}-2ar$7

The resulting corrected temperature is

$A(r)=1-\frac{2m}{r}-2ar$8

so the radiation spectrum is slightly nonthermal once self-gravitation and the entropy correction are included (Mirekhtiary et al., 2014).

Thermodynamic stability has also been examined in the $A(r)=1-\frac{2m}{r}-2ar$9 realizations. For the model

$a\rightarrow -a$0

the entropy at the horizon is

$a\rightarrow -a$1

and the computed heat capacity is positive for $a\rightarrow -a$2, indicating thermodynamic stability and no phase transition in that example. More generally, the thermodynamic outcome is model-dependent even when the same metric ansatz is held fixed (Mazharimousavi et al., 2013).

5. Perturbations, lensing, and galaxy-scale phenomenology

At the level of timelike geodesics, the original metric predicts

$a\rightarrow -a$3

so the linear term supplies a constant inward acceleration and modifies orbital frequencies at large radius. The literature explicitly motivates this as a large-distance effect potentially relevant to dark-matter phenomenology, but it also notes that the asymptotic behavior is slowly rising rather than strictly flat (Mazharimousavi et al., 2013).

Null geodesics are likewise modified. In the electric NED realization with monopole term $a\rightarrow -a$4, the equatorial bending equation becomes

$a\rightarrow -a$5

and the leading bending angle is

$a\rightarrow -a$6

The same paper derives perihelion-precession bounds on $a\rightarrow -a$7 and $a\rightarrow -a$8 and reports Solar System constraints with $a\rightarrow -a$9 typically around $f(r)=1+2k-\frac{2M}{r}+2ar-\frac{\Lambda}{3}r^2,$0–$f(r)=1+2k-\frac{2M}{r}+2ar-\frac{\Lambda}{3}r^2,$1 and $f(r)=1+2k-\frac{2M}{r}+2ar-\frac{\Lambda}{3}r^2,$2 at the level of $f(r)=1+2k-\frac{2M}{r}+2ar-\frac{\Lambda}{3}r^2,$3–$f(r)=1+2k-\frac{2M}{r}+2ar-\frac{\Lambda}{3}r^2,$4 for planetary data (Halilsoy et al., 2012).

Galaxy-scale fitting motivates refinements of the simple Rindler term. The NFW-reconstructed metric

$f(r)=1+2k-\frac{2M}{r}+2ar-\frac{\Lambda}{3}r^2,$5

reduces to the Grumiller form when $f(r)=1+2k-\frac{2M}{r}+2ar-\frac{\Lambda}{3}r^2,$6, but its effective correction softens and decays as $f(r)=1+2k-\frac{2M}{r}+2ar-\frac{\Lambda}{3}r^2,$7. In the rotation-curve analysis summarized in the source material, this reconstructed term yields markedly better fits than the constant Rindler term, especially for flat or slowly declining outer profiles (Perivolaropoulos et al., 2019).

Wave propagation has recently been analyzed exactly in the de Sitter-like extension

$f(r)=1+2k-\frac{2M}{r}+2ar-\frac{\Lambda}{3}r^2,$8

Using a unified spin-$f(r)=1+2k-\frac{2M}{r}+2ar-\frac{\Lambda}{3}r^2,$9 equation in the type-D background and a Heun-polynomial truncation, the quasinormal-mode spectrum is found to be

$m$00

In this construction, bosonic fields with the same integer $m$01 are isospectral, fermionic fields with the same half-integer $m$02 are isospectral, bosonic and fermionic spectra never coincide, and for fixed spin state the modes exhibit an $m$03-fold degeneracy (Mi et al., 21 Sep 2025).

A separate lensing diagnostic arises in the Balasin–Grumiller galaxy model, which is stationary and axisymmetric rather than spherically symmetric. Exact equatorial lensing calculations yield bending angles compatible with typical disc-galaxy observations for Milky Way–like parameter choices, but the time delay between prograde and retrograde images comes out at the level of $m$04–$m$05, far above the observed months-scale delays. The paper attributes this discrepancy to the model’s rigid rotation and excessive frame dragging, concluding that the construction is too crude for a reliable relativistic description of disc galaxies (Galoppo et al., 2022).

6. Related usages, limitations, and open questions

A recurrent limitation of the four-dimensional Grumiller program is that satisfying one set of desiderata does not guarantee satisfying the others. In the $m$06 scan, WEC satisfaction is often confined to restricted radial intervals; ghost-free and scalaron-stable regimes are not automatic; and extensive parameter tuning is common. The negative exterior curvature, $m$07, is a persistent obstruction because otherwise attractive models can develop $m$08 in the physical region (Mazharimousavi et al., 2013).

The domain-wall literature adds a distinct geometric application. In the Schwarzschild–Rindler–AdS metric with $m$09 and $m$10,

$m$11

the quantity

$m$12

can develop a local minimum with $m$13. This permits a static, finite-energy, spherically symmetric scalar configuration that evades Derrick’s theorem in curved space. The numerically obtained wall is well approximated by

$m$14

and backreaction shrinks but does not eliminate the metastable region for sufficiently small gravitational coupling (Alestas et al., 2019).

The name “Grumiller metric” also has a distinct three-dimensional usage. In chiral higher-spin gravity, the term refers to the most general asymptotically AdS$m$15 metric in the sense of Grumiller–Riegler boundary conditions, realized through generalized Fefferman–Graham falloffs and depending, in the spin-3 case discussed there, on 19 functions. This is not the four-dimensional Rindler-modified Schwarzschild geometry, but a separate boundary-condition framework within $m$16 Chern–Simons theory (Krishnan et al., 2017).

Taken together, these results suggest a precise but limited status for the four-dimensional Grumiller metric. It is a compact parametrization of a linear-in-$m$17 deformation of Schwarzschild geometry; it can be realized in several modified-gravity or matter-coupled settings; it has well-defined horizon thermodynamics and analytically tractable perturbation theory in some extensions; and it remains phenomenologically interesting at large distances. A plausible implication is that its lasting value lies less in serving as a final galaxy model than in functioning as a controlled infrared benchmark against which more elaborate large-distance modifications of gravity can be compared.