---
title: Grumiller Metric Overview
url: https://www.emergentmind.com/topics/grumiller-metric
type: topic
---

# Grumiller Metric Overview

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 [1309.4768] [1402.5514] [1903.06554].

## 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 $a\rightarrow -a$ [1309.4768].

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 $k$ is a dimensionless constant interpreted as a global monopole parameter and $\Lambda$ is the cosmological constant. In that model, both $a$ and $k$ are source-dependent quantities induced by the nonlinear electromagnetic sector rather than universal constants [1212.2159].

A Schwarzschild–Rindler–AdS form also occurs in curved-space soliton studies,
$$
f(r)=1-\frac{2Gm}{r}+2br-\frac{\Lambda}{3}r^2,\qquad \Lambda<0,\quad b<0,
$$
where the linear term is again the Rindler contribution, but metastable spherical walls require the opposite sign, $b<0$ [1901.06659].

The reduced-gravity reconstruction program provides another standard embedding. In the two-dimensional dilaton action studied by Nashed and collaborators, Grumiller’s potential
$$
V(\Phi)=1+4\alpha\Phi-3\Lambda\Phi^2
$$
yields the vacuum metric
$$
f(r)=1-\frac{2GM}{r}+2\alpha r-\Lambda r^2,
$$
with $a\equiv \alpha$ recovering the familiar Rindler term [1903.06554].

## 2. Geometric structure and Newtonian interpretation

In the weak-field limit, the temporal component satisfies $g_{tt}\approx -(1+2\Phi)$, so the effective Newtonian potential of the basic metric is
$$
\Phi(r)\approx \frac{A(r)-1}{2}=-\frac{m}{r}+ar.
$$
The first term is the standard Newtonian contribution, while the linear term produces a constant attractive acceleration,
$$
-\frac{d\Phi}{dr}= -\left(\frac{m}{r^2}+a\right).
$$
This is the core physical meaning of the Grumiller deformation: a uniform inward acceleration superposed on the Schwarzschild field [1309.4768].

For circular motion, the relativistic geodesic condition gives
$$
\Omega^2(r)=\frac{A'(r)}{2r}=\frac{m}{r^3}+\frac{a}{r},
$$
and the weak-field circular speed follows as
$$
v^2(r)=r\,\frac{d\Phi}{dr}=\frac{m}{r}+ar.
$$
Hence, at large radii, $v(r)\approx \sqrt{ar}$, so the velocity grows as $r^{1/2}$. 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 [1309.4768].

The event horizon of the simplest $a>0$ metric is the positive root of $A(r)=0$:
$$
2ar_h^2+r_h-2m=0,
\qquad
r_h=\frac{-1+\sqrt{1+16am}}{4a}.
$$
For $a>0$ and $m>0$, 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 [1402.5514].

The Ricci scalar for the basic Rindler-augmented Schwarzschild metric is
$$
R(r)=-\frac{12a}{r},
$$
which is negative throughout the exterior region when $a>0$. In the $\Lambda$-extended form used in exact quasinormal-mode calculations, the Ricci scalar becomes
$$
R=12\left(-\frac{a}{r}+\Lambda\right),
$$
and the spacetime remains algebraically special of type D with Weyl scalar $\psi_2=-M/r^3$ [1309.4768] [2509.16991].

## 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 $f(R)$ gravity, the relevant action is
$$
S=\frac{1}{2}\int \sqrt{-g}\,f(R)\,d^4x+S_M,
$$
with anisotropic matter source
$$
T^\nu{}_\mu=\mathrm{diag}[-\rho,\,p,\,q,\,q].
$$
Using $F\equiv f_R=df/dR$, the field equations are written in effective-fluid form as
$$
G^\nu{}_\mu=\frac{1}{F}T^\nu{}_\mu+\check T^\nu{}_\mu.
$$
For the metric $A(r)=1-\frac{2m}{r}+2ar$, Mazharimousavi, Kerachian, and Halilsoy evaluate the weak energy conditions
$$
\rho\ge 0,\qquad \rho+p\ge 0,\qquad \rho+q\ge 0
$$
model by model, together with the viability conditions
$$
f_R>0,\qquad f_{RR}>0.
$$
Their scan finds that several $f(R)$ 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, $R<0$. Explicitly discussed viable or partially viable choices include $f(R)=R+\frac{1}{R}+R^2$, $f(R)=R+\frac{1}{R}-R^3$, tuned log-cosh plus $R^2$ models, and $f(R)=R+b e^{\alpha R}$ with $\alpha=-1$; other classes fail the WEC or violate $f_{RR}>0$ [1309.4768].

Nonlinear electrodynamics furnishes a different microphysical realization. For a purely electric NED model with action
$$
S=\int d^4x\sqrt{-g}\,\big[R-2\Lambda+L(\mathcal F)\big],
$$
Halilsoy, Mazharimousavi, and Amirabi derive
$$
f(r)=1+2k-\frac{2M}{r}+2ar-\frac{\Lambda}{3}r^2,
$$
with
$$
a=2^{-5/4}\sqrt{\alpha Q},\qquad k=-QE_0.
$$
Here $Q$ is the NED charge, $\alpha$ the NED coupling, and $E_0$ a uniform background electric field. In this construction the background field is essential: if $E_0=0$, the energy conditions are violated. For the electric model the stress tensor is diagonal and the WEC and SEC hold provided
$$
2ar\le |k|.
$$
The same paper also presents a purely magnetic NED model producing a logarithmic metric term,
$$
f(r)=1-\frac{2M}{r}+2ar_0\ln\!\left(\frac{r}{r_0}\right),
$$
whose circular speed approaches a constant at large radius; that model is adjacent to, rather than identical with, the original Grumiller geometry [1212.2159].

Reduced-gravity constructions identify the metric as a vacuum solution of effective infrared theories. In the two-dimensional dilaton framework, choosing $F(\Phi)=\Phi^2$ and $Z(\Phi)=-2$ with the potential $V(\Phi)=1+4\alpha\Phi-3\Lambda\Phi^2$ yields the vacuum line element
$$
f(r)=1-\frac{2GM}{r}+2\alpha r-\Lambda r^2.
$$
The reconstructed NFW-motivated potential
$$
V(\Phi)=1+\frac{4\alpha\Phi}{(1+\beta\Phi)^2}-3\Lambda\Phi^2
$$
generates a modified metric term that reduces to the Grumiller form in the limit $\beta\to 0$. This suggests that the original Rindler term may be viewed as the leading member of a broader family of large-distance geometric potentials [1903.06554].

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 [1901.06659].

## 4. Horizons, Hawking radiation, and thermodynamic behavior

For the uncharged Grumiller black hole,
$$
H(r)=1-\frac{2M}{r}+2ar,
$$
the surface gravity is
$$
\kappa=\frac{1}{2}H'(r_h),
$$
and the Hawking temperature is
$$
T_H=\frac{\kappa}{2\pi}
=\frac{a}{2\pi}\frac{r_h-r_o}{r_h}
=\frac{\sqrt{1+16aM}}{4\pi r_h},
$$
where $r_o$ is the negative root of $H(r)=0$. In the limit $a\to 0$, one recovers the Schwarzschild values $r_h=2M$ and $T_H=1/(8\pi M)$ [1402.5514].

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 $P_{\rm in}=1$ and using the appropriate contour prescription at the horizon [1402.5514].

The same work then applies Parikh–Wilczek tunneling in PG coordinates, with backreaction included through $M\to M-\omega$, and incorporates a logarithmic entropy correction
$$
S_{QG}=\frac{A}{4}+\alpha\ln A+O(A^{-1}).
$$
The resulting corrected temperature is
$$
T_{QG}=T_H\left[1+\frac{\alpha}{\pi r_h^2}\right]^{-1},
$$
so the radiation spectrum is slightly nonthermal once self-gravitation and the entropy correction are included [1402.5514].

Thermodynamic stability has also been examined in the $f(R)$ realizations. For the model
$$
f(R)=\sqrt{R^2+b^2},
$$
the entropy at the horizon is
$$
S=\pi r_h^2\,F(R_h),
$$
and the computed heat capacity is positive for $a>0$, 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 [1309.4768].

## 5. Perturbations, lensing, and galaxy-scale phenomenology

At the level of timelike geodesics, the original metric predicts
$$
v^2(r)=\frac{m}{r}+ar,
$$
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 [1309.4768].

Null geodesics are likewise modified. In the electric NED realization with monopole term $k$, the equatorial bending equation becomes
$$
\frac{d^2u}{d\phi^2}+(1-2|k|)u=3Mu^2-a,\qquad u\equiv \frac{1}{r},
$$
and the leading bending angle is
$$
2\psi_0\approx \frac{4M}{R}\,\big[1+2Ma+2|k|(1+6Ma)\big].
$$
The same paper derives perihelion-precession bounds on $|a|$ and $|k|$ and reports Solar System constraints with $|a|$ typically around $10^{-12}$–$10^{-10}\,\mathrm{m\,s^{-2}}$ and $|k|$ at the level of $10^{-12}$–$10^{-8}$ for planetary data [1212.2159].

Galaxy-scale fitting motivates refinements of the simple Rindler term. The NFW-reconstructed metric
$$
f(r)=1-\frac{2GM}{r}-\frac{4\alpha}{\beta^2 r}\left[1-\frac{1}{1+\beta r}-\ln(1+\beta r)\right]-\Lambda r^2
$$
reduces to the Grumiller form when $\beta\to 0$, but its effective correction softens and decays as $\sim (\ln r)/r$. 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 [1903.06554].

Wave propagation has recently been analyzed exactly in the de Sitter-like extension
$$
f(r)=1-\frac{2M}{r}-\Lambda r^2+2ar.
$$
Using a unified spin-$p$ equation in the type-D background and a Heun-polynomial truncation, the quasinormal-mode spectrum is found to be
$$
\omega_{n,p}=-\,i\,\kappa_b\,(n+p+1)\equiv -\,i\,\kappa_b N.
$$
In this construction, bosonic fields with the same integer $N$ are isospectral, fermionic fields with the same half-integer $N$ are isospectral, bosonic and fermionic spectra never coincide, and for fixed spin state the modes exhibit an $(n+1)$-fold degeneracy [2509.16991].

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 $10^9$–$10^{10}\,\mathrm{s}$, 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 [2212.10290].

## 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 $f(R)$ 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, $R<0$, is a persistent obstruction because otherwise attractive models can develop $f_R<0$ in the physical region [1309.4768].

The domain-wall literature adds a distinct geometric application. In the Schwarzschild–Rindler–AdS metric with $b<0$ and $\Lambda<0$,
$$
f(r)=1-\frac{2Gm}{r}+2br-\frac{\Lambda}{3}r^2,
$$
the quantity
$$
r^2f(r)=-2Gmr+r^2+2br^3-\frac{\Lambda}{3}r^4
$$
can develop a local minimum with $f(r_{\min})>0$. 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
$$
\Phi(r)\simeq \tanh\!\big[q(r-r_0)\big],
$$
and backreaction shrinks but does not eliminate the metastable region for sufficiently small gravitational coupling [1901.06659].

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$_3$ 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 $SL(3,\mathbb R)\oplus SL(3,\mathbb R)$ Chern–Simons theory [1703.01769].

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-$r$ 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.

Source: https://www.emergentmind.com/topics/grumiller-metric