Bimetric interactions based on metric congruences

Mikica Kocic

arxiv: 1906.11841 · v1 · pith:MGQ6AEFNnew · submitted 2019-06-27 · ✦ hep-th · gr-qc

Bimetric interactions based on metric congruences

Mikica Kocic This is my paper

Pith reviewed 2026-05-25 14:22 UTC · model grok-4.3

classification ✦ hep-th gr-qc

keywords bimetric gravitymassive gravitycongruence matrixsquare root matrixvielbein formulationspin-2 interactionsN+1 decomposition

0 comments

The pith

Bimetric gravity interactions can be built from a congruence matrix between two metrics, where the square root is the only power-series solution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs spin-2 interactions in massive gravity and bigravity by replacing the usual square-root matrix with a congruence matrix that relates the two metrics. It proves that the primary square root function is the sole power series satisfying the equations of motion for this congruence. The same equations in N+1 form recover the shift-vector redefinition used in ghost-free proofs. The bimetric congruence formulation is shown to be algebraically identical to the unconstrained vielbein formulation. A reader would care because this supplies an alternative starting point that makes the algebraic structure of consistent interactions more transparent.

Core claim

In massive gravity and bigravity, spin-2 interactions are defined in terms of a square root matrix that involves two metrics. In this work, the interactions are constructed using a congruence matrix between the metrics. It is established that the primary square root matrix function is the only power series solution to the equations of motion for the congruence. Moreover, the shift vector redefinition that is used in the bimetric ghost-free proofs follows from the N+1 form of the equations of motion. The analysis also gives an insight into the vielbein formulation of spin-2 interactions since the bimetric formulation in terms of a congruence is algebraically equivalent to the unconstrained 4D

What carries the argument

The congruence matrix relating the two metrics, whose equations of motion admit a power-series expansion whose only solution is the primary square root.

If this is right

The primary square root is the unique power-series solution for the congruence equations.
The shift-vector redefinition used in ghost-free proofs follows directly from the N+1 decomposition.
The congruence formulation is algebraically equivalent to the unconstrained vielbein formulation.
Any consistent interaction built from the congruence must reduce to the known square-root form at the level of the equations of motion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The congruence approach may allow new parametrizations of the interaction potential that are not obvious in the square-root language.
Because the equivalence to the vielbein is algebraic, any constraint or gauge choice derived in one formulation transfers immediately to the other.
The uniqueness result suggests that attempts to deform the interaction beyond the square root would require abandoning the power-series assumption entirely.

Load-bearing premise

The spin-2 interactions must admit a power-series representation in the congruence matrix and the N+1 decomposition of the equations must hold without further constraints on the metrics.

What would settle it

An explicit second power-series solution to the congruence equations of motion that differs from the primary square root, or a metric pair where the congruence formulation fails to match the unconstrained vielbein equations.

read the original abstract

In massive gravity and bigravity, spin-2 interactions are defined in terms of a square root matrix that involves two metrics. In this work, the interactions are constructed using a congruence matrix between the metrics. It is established that the primary square root matrix function is the only power series solution to the equations of motion for the congruence. Moreover, the shift vector redefinition that is used in the bimetric ghost-free proofs follows from the $N+1$ form of the equations of motion. The analysis also gives an insight into the vielbein formulation of spin-2 interactions since the bimetric formulation in terms of a congruence is algebraically equivalent to the unconstrained vielbein formulation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

read the letter

The paper recasts bimetric interactions via a congruence matrix and shows the square root is the unique power-series solution while recovering the shift redefinition from the N+1 split and claiming algebraic equivalence to unconstrained vielbeins. This is the core new piece: a different matrix object whose equations of motion admit only the standard square root inside the power-series class, plus a direct derivation of the shift vector used in ghost-free arguments. The equivalence to the vielbein side is presented as algebraic rather than through additional constraints, which is a clean technical step relative to the usual square-root starting point. For readers already working inside massive gravity and bigravity, this framing can make certain manipulations more transparent without changing the final physics. The N+1 decomposition is handled in a way that appears to follow from the equations rather than being imposed by hand. That part of the argument looks internally consistent on the terms given. The main limitation is that uniqueness is proved only within power series. No discussion appears of the radius of convergence or spectral conditions on the eigenvalues, so it remains open whether other solutions exist outside the analytic class for metrics that arise in solutions. The equivalence claim to the unconstrained vielbein formulation is stated directly, but the explicit map or check that no hidden constraints are introduced is not visible in the abstract-level description. If that map turns out to be fully general, the result strengthens; if it tacitly restricts the space, the equivalence weakens. The work stays inside the standard literature on ghost-free bimetric theories and does not claim broader reach. It is aimed at specialists who already care about alternative matrix constructions or the technical details of the N+1 proofs. A reader outside that subfield will find little to take away. The paper deserves peer review because the congruence construction and the uniqueness statement inside the series class are distinct enough from prior definitions to warrant checking the derivations, even if the convergence question needs to be addressed in revision.

Referee Report

2 major / 1 minor

Summary. The manuscript constructs spin-2 interactions in massive gravity and bigravity using a congruence matrix between two metrics rather than the conventional square-root matrix. It claims that the primary square-root function is the unique power-series solution to the equations of motion satisfied by the congruence, that the shift-vector redefinition employed in ghost-free proofs follows directly from the N+1 decomposition of those equations, and that the resulting bimetric formulation is algebraically equivalent to the unconstrained vielbein formulation.

Significance. If the uniqueness and equivalence results are rigorously established, the work supplies a useful justification for the square-root choice and clarifies the relation between metric-congruence and vielbein formulations of bimetric theories. The derivation of the shift redefinition from the N+1 equations is a concrete technical contribution that could strengthen existing ghost-free analyses. The significance is limited by the absence of explicit spectral conditions or convergence arguments for the power-series ansatz, which are load-bearing for the uniqueness statement.

major comments (2)

[derivation of uniqueness result (abstract and main text)] The uniqueness result for the primary square root as the only power-series solution to the congruence equations of motion (stated in the abstract) rests on the assumption that the interactions admit a power-series representation in the congruence matrix. No explicit radius-of-convergence estimate or spectral conditions on the eigenvalues of the matrix are provided; if eigenvalues lie outside the disk of convergence, other solutions could satisfy the original bimetric equations but be missed by the ansatz. This assumption is central to the primary claim.
[N+1 decomposition and equivalence to vielbein formulation] The claim that the shift-vector redefinition follows from the N+1 form of the equations of motion, and that this yields algebraic equivalence to the unconstrained vielbein formulation, requires verification that the N+1 split introduces no hidden constraints on the metrics. Any such constraint would undermine the asserted equivalence; the manuscript should exhibit the explicit N+1 equations and the redefinition step to confirm independence from prior assumptions.

minor comments (1)

[abstract] The abstract would benefit from a brief statement of the precise class of interactions considered and the domain of the power series.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We respond point-by-point to the major comments below.

read point-by-point responses

Referee: The uniqueness result for the primary square root as the only power-series solution to the congruence equations of motion (stated in the abstract) rests on the assumption that the interactions admit a power-series representation in the congruence matrix. No explicit radius-of-convergence estimate or spectral conditions on the eigenvalues of the matrix are provided; if eigenvalues lie outside the disk of convergence, other solutions could satisfy the original bimetric equations but be missed by the ansatz. This assumption is central to the primary claim.

Authors: The manuscript establishes uniqueness strictly within the class of power-series solutions to the congruence equations of motion; it does not claim uniqueness for non-analytic solutions. We will revise the abstract and introduction to state this scope explicitly. A general radius-of-convergence estimate for arbitrary metrics lies outside the paper's focus, but we can add a brief remark referencing standard results on matrix power series convergence when eigenvalues satisfy the appropriate spectral condition. revision: partial
Referee: The claim that the shift-vector redefinition follows from the N+1 form of the equations of motion, and that this yields algebraic equivalence to the unconstrained vielbein formulation, requires verification that the N+1 split introduces no hidden constraints on the metrics. Any such constraint would undermine the asserted equivalence; the manuscript should exhibit the explicit N+1 equations and the redefinition step to confirm independence from prior assumptions.

Authors: We will add an appendix containing the explicit N+1 decomposition of the equations of motion together with the algebraic steps deriving the shift redefinition. This will demonstrate that the decomposition introduces no additional constraints beyond those already present in the original metric equations and that the equivalence to the unconstrained vielbein formulation remains purely algebraic. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained within stated assumptions.

full rationale

The paper's central results—the uniqueness of the primary square root as the only power-series solution to the congruence EOM, the derivation of the shift-vector redefinition from the N+1 decomposition, and the algebraic equivalence to the unconstrained vielbein formulation—are presented as direct consequences of solving the equations of motion under an explicit power-series ansatz and performing the N+1 split. No load-bearing step reduces by construction to a fitted parameter, a self-citation chain, or a renaming of inputs; the power-series assumption is stated upfront rather than smuggled, and the uniqueness claim is scoped precisely to that class of solutions. The analysis therefore remains independent of external fitted data or prior self-referential theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, invented entities, or non-standard axioms are stated. The work rests on the standard domain assumptions of bimetric gravity (two metrics, diffeomorphism invariance, and the requirement of ghost-free interactions).

axioms (1)

domain assumption Bimetric gravity theories are defined by two metrics whose interaction must preserve diffeomorphism invariance and eliminate ghosts.
The abstract opens by placing the new construction inside the existing massive-gravity and bigravity framework.

pith-pipeline@v0.9.0 · 5629 in / 1227 out tokens · 27845 ms · 2026-05-25T14:22:35.133026+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · 22 internal anchors

[1]

Resummation of Massive Gravity

C. de Rham, G. Gabadadze and A. J. Tolley, Resummation of Massive Gravity , Phys. Rev. Lett. 106 (2011) 231101, [ 1011.1232]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[2]

S. F. Hassan and R. A. Rosen, Resolving the Ghost Problem in non-Linear Massive Gravity , Phys. Rev. Lett. 108 (2012) 041101, [ 1106.3344]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[3]

S. F. Hassan, R. A. Rosen and A. Schmidt-May, Ghost-free Massive Gravity with a General Reference Metric, JHEP 02 (2012) 026, [ 1109.3230]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[4]

S. F. Hassan and R. A. Rosen, Bimetric Gravity from Ghost-free Massive Gravity , JHEP 02 (2012) 126, [ 1109.3515]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[5]

S. F. Hassan and R. A. Rosen, Conﬁrmation of the Secondary Constraint and Absence of Ghost in Massive Gravity and Bimetric Gravity , JHEP 04 (2012) 123, [ 1111.2070]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[6]

S. F. Hassan and A. Lundkvist, Analysis of constraints and their algebra in bimetric theory , JHEP 08 (2018) 182, [ 1802.07267]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[7]

Massive Gravity

C. de Rham, Massive Gravity, Living Rev. Rel. 17 (2014) 7, [ 1401.4173]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[8]

Theoretical Aspects of Massive Gravity

K. Hinterbichler, Theoretical Aspects of Massive Gravity, Rev. Mod. Phys. 84 (2012) 671–710, [1105.3735]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[9]

Recent developments in bimetric theory

A. Schmidt-May and M. von Strauss, Recent developments in bimetric theory , J. Phys. A49 (2016) 183001, [ 1512.00021]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[10]

D. G. Boulware and S. Deser, Can gravitation have a ﬁnite range? , Phys. Rev. D6 (1972) 3368–3382

work page 1972
[11]

I. G. Macdonald, Symmetric functions and orthogonal polynomials . AMS, 1998

work page 1998
[12]

Interacting Spin-2 Fields

K. Hinterbichler and R. A. Rosen, Interacting Spin-2 Fields , JHEP 07 (2012) 047, [1203.5783]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[13]

Zumino, Eﬀective Lagrangians and Broken Symmetries , in Lectures on Elementary Particles and Quantum Field Theory vol.2, Cambridge, Mass.: Brandeis Univ., pp

B. Zumino, Eﬀective Lagrangians and Broken Symmetries , in Lectures on Elementary Particles and Quantum Field Theory vol.2, Cambridge, Mass.: Brandeis Univ., pp. 437–500 , 1970

work page 1970
[14]

A note on "symmetric" vielbeins in bimetric, massive, perturbative and non perturbative gravities

C. Deﬀayet, J. Mourad and G. Zahariade, A note on ‘symmetric’ vielbeins in bimetric, massive, perturbative and non perturbative gravities , JHEP 03 (2013) 086, [ 1208.4493]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[15]

S. F. Hassan, A. Schmidt-May and M. von Strauss, Metric Formulation of Ghost-Free Multivielbein Theory, 1204.5202

work page internal anchor Pith review Pith/arXiv arXiv
[16]

S. F. Hassan and M. Kocic, On the local structure of spacetime in ghost-free bimetric theory and massive gravity , JHEP 05 (2018) 099, [ 1706.07806]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[17]

Scharlau, Quadratic and hermitian forms

W. Scharlau, Quadratic and hermitian forms . Grundlehren der mathematischen Wissenschaften. Springer, 1985

work page 1985
[18]

Lang, Algebra

S. Lang, Algebra. Graduate Texts in Mathematics. Springer New York, 2002

work page 2002
[19]

Meinrenken, Cliﬀord algebras and Lie theory

E. Meinrenken, Cliﬀord algebras and Lie theory . Springer, 2013, 10.1007/978-3-642-36216-3

work page doi:10.1007/978-3-642-36216-3 2013
[20]

Hawking and G

S. Hawking and G. Ellis, The Large Scale Structure of Space-Time . Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1973

work page 1973
[21]

G. H. Golub and C. F. Van Loan, Matrix Computations (3rd Ed.) . Johns Hopkins University Press, Baltimore, MD, USA, 1996. – 18 –

work page 1996
[22]

Obsieger, Numerical Methods II - Roots and Equation Systems

B. Obsieger, Numerical Methods II - Roots and Equation Systems . Numerical Methods. Boris Obsieger, 2015

work page 2015
[23]

Horn and C

R. Horn and C. Johnson, Topics in Matrix Analysis . Cambridge University Press, 1994

work page 1994
[24]

Higham, Functions of Matrices: Theory and Computation

N. Higham, Functions of Matrices: Theory and Computation . SIAM, 2008

work page 2008
[25]

Konstantinov, D

M. Konstantinov, D. Gu, V. Mehrmann and P. Petkov, Perturbation Theory for Matrix Equations. Studies in Computational Mathematics. Elsevier Science, 2003

work page 2003
[26]

Gourgoulhon, 3+1 Formalism in General Relativity: Bases of Numerical Relativity

´E. Gourgoulhon, 3+1 Formalism in General Relativity: Bases of Numerical Relativity . Lecture Notes in Physics. Springer Berlin Heidelberg, 2012

work page 2012
[27]

R. L. Arnowitt, S. Deser and C. W. Misner, The dynamics of general relativity , Gen. Rel. Grav. 40 (2008) 1997–2027, [ gr-qc/0405109]

work page internal anchor Pith review Pith/arXiv arXiv 2008
[28]

J. W. York, Jr., Kinematics and Dynamics of General Relativity , pp. 83–126

work page
[29]

Kocic, Geometric mean of bimetric spacetimes , 1803.09752

M. Kocic, Geometric mean of bimetric spacetimes , 1803.09752

work page arXiv
[30]

S. F. Hassan, M. Kocic and A. Schmidt-May, Absence of ghost in a new bimetric-matter coupling, 1409.1909

work page internal anchor Pith review Pith/arXiv arXiv 1909
[31]

S. F. Hassan and A. Schmidt-May, Interactions of multiple spin-2 ﬁelds beyond pairwise couplings, 1804.09723

work page internal anchor Pith review Pith/arXiv arXiv
[32]

Geometry of physical dispersion relations

D. Raetzel, S. Rivera and F. P. Schuller, Geometry of physical dispersion relations , Phys. Rev. D83 (2011) 044047, [ 1010.1369]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[33]

F. P. Schuller and C. Witte, How quantizable matter gravitates: A practitioner’s guide , Phys. Rev. D89 (2014) 104061, [ 1402.6548]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[34]

F. P. Schuller, N. Stritzelberger, F. Wolz and M. D¨ ull, Gravitational closure of matter ﬁeld equations, 1611.08878

work page internal anchor Pith review Pith/arXiv arXiv
[35]

K. B. Petersen and M. S. Pedersen, The matrix cookbook, tech. rep., nov, 2012

work page 2012
[36]

No consistent cross-interactions for a collection of massless spin-2 fields

N. Boulanger, T. Damour, L. Gualtieri and M. Henneaux, No consistent cross interactions for a collection of massless spin-2 ﬁelds , in Quantization, gauge theory, and strings. Proceedings, International Conference dedicated to the memory of Professor Eﬁm Fradkin, Moscow, Russia, June 5-10, 2000. Vol. 1+2 , pp. 150–161, 2000, hep-th/0009109

work page internal anchor Pith review Pith/arXiv arXiv 2000
[37]

Inconsistency of interacting, multi-graviton theories

N. Boulanger, T. Damour, L. Gualtieri and M. Henneaux, Inconsistency of interacting, multigraviton theories, Nucl. Phys. B597 (2001) 127–171, [ hep-th/0007220]

work page internal anchor Pith review Pith/arXiv arXiv 2001
[38]

Effective Lagrangians and Universality Classes of Nonlinear Bigravity

T. Damour and I. I. Kogan, Eﬀective Lagrangians and universality classes of nonlinear bigravity, Phys. Rev. D66 (2002) 104024, [ hep-th/0206042]. – 19 –

work page internal anchor Pith review Pith/arXiv arXiv 2002

[1] [1]

Resummation of Massive Gravity

C. de Rham, G. Gabadadze and A. J. Tolley, Resummation of Massive Gravity , Phys. Rev. Lett. 106 (2011) 231101, [ 1011.1232]

work page internal anchor Pith review Pith/arXiv arXiv 2011

[2] [2]

S. F. Hassan and R. A. Rosen, Resolving the Ghost Problem in non-Linear Massive Gravity , Phys. Rev. Lett. 108 (2012) 041101, [ 1106.3344]

work page internal anchor Pith review Pith/arXiv arXiv 2012

[3] [3]

S. F. Hassan, R. A. Rosen and A. Schmidt-May, Ghost-free Massive Gravity with a General Reference Metric, JHEP 02 (2012) 026, [ 1109.3230]

work page internal anchor Pith review Pith/arXiv arXiv 2012

[4] [4]

S. F. Hassan and R. A. Rosen, Bimetric Gravity from Ghost-free Massive Gravity , JHEP 02 (2012) 126, [ 1109.3515]

work page internal anchor Pith review Pith/arXiv arXiv 2012

[5] [5]

S. F. Hassan and R. A. Rosen, Conﬁrmation of the Secondary Constraint and Absence of Ghost in Massive Gravity and Bimetric Gravity , JHEP 04 (2012) 123, [ 1111.2070]

work page internal anchor Pith review Pith/arXiv arXiv 2012

[6] [6]

S. F. Hassan and A. Lundkvist, Analysis of constraints and their algebra in bimetric theory , JHEP 08 (2018) 182, [ 1802.07267]

work page internal anchor Pith review Pith/arXiv arXiv 2018

[7] [7]

Massive Gravity

C. de Rham, Massive Gravity, Living Rev. Rel. 17 (2014) 7, [ 1401.4173]

work page internal anchor Pith review Pith/arXiv arXiv 2014

[8] [8]

Theoretical Aspects of Massive Gravity

K. Hinterbichler, Theoretical Aspects of Massive Gravity, Rev. Mod. Phys. 84 (2012) 671–710, [1105.3735]

work page internal anchor Pith review Pith/arXiv arXiv 2012

[9] [9]

Recent developments in bimetric theory

A. Schmidt-May and M. von Strauss, Recent developments in bimetric theory , J. Phys. A49 (2016) 183001, [ 1512.00021]

work page internal anchor Pith review Pith/arXiv arXiv 2016

[10] [10]

D. G. Boulware and S. Deser, Can gravitation have a ﬁnite range? , Phys. Rev. D6 (1972) 3368–3382

work page 1972

[11] [11]

I. G. Macdonald, Symmetric functions and orthogonal polynomials . AMS, 1998

work page 1998

[12] [12]

Interacting Spin-2 Fields

K. Hinterbichler and R. A. Rosen, Interacting Spin-2 Fields , JHEP 07 (2012) 047, [1203.5783]

work page internal anchor Pith review Pith/arXiv arXiv 2012

[13] [13]

Zumino, Eﬀective Lagrangians and Broken Symmetries , in Lectures on Elementary Particles and Quantum Field Theory vol.2, Cambridge, Mass.: Brandeis Univ., pp

B. Zumino, Eﬀective Lagrangians and Broken Symmetries , in Lectures on Elementary Particles and Quantum Field Theory vol.2, Cambridge, Mass.: Brandeis Univ., pp. 437–500 , 1970

work page 1970

[14] [14]

A note on "symmetric" vielbeins in bimetric, massive, perturbative and non perturbative gravities

C. Deﬀayet, J. Mourad and G. Zahariade, A note on ‘symmetric’ vielbeins in bimetric, massive, perturbative and non perturbative gravities , JHEP 03 (2013) 086, [ 1208.4493]

work page internal anchor Pith review Pith/arXiv arXiv 2013

[15] [15]

S. F. Hassan, A. Schmidt-May and M. von Strauss, Metric Formulation of Ghost-Free Multivielbein Theory, 1204.5202

work page internal anchor Pith review Pith/arXiv arXiv

[16] [16]

S. F. Hassan and M. Kocic, On the local structure of spacetime in ghost-free bimetric theory and massive gravity , JHEP 05 (2018) 099, [ 1706.07806]

work page internal anchor Pith review Pith/arXiv arXiv 2018

[17] [17]

Scharlau, Quadratic and hermitian forms

W. Scharlau, Quadratic and hermitian forms . Grundlehren der mathematischen Wissenschaften. Springer, 1985

work page 1985

[18] [18]

Lang, Algebra

S. Lang, Algebra. Graduate Texts in Mathematics. Springer New York, 2002

work page 2002

[19] [19]

Meinrenken, Cliﬀord algebras and Lie theory

E. Meinrenken, Cliﬀord algebras and Lie theory . Springer, 2013, 10.1007/978-3-642-36216-3

work page doi:10.1007/978-3-642-36216-3 2013

[20] [20]

Hawking and G

S. Hawking and G. Ellis, The Large Scale Structure of Space-Time . Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1973

work page 1973

[21] [21]

G. H. Golub and C. F. Van Loan, Matrix Computations (3rd Ed.) . Johns Hopkins University Press, Baltimore, MD, USA, 1996. – 18 –

work page 1996

[22] [22]

Obsieger, Numerical Methods II - Roots and Equation Systems

B. Obsieger, Numerical Methods II - Roots and Equation Systems . Numerical Methods. Boris Obsieger, 2015

work page 2015

[23] [23]

Horn and C

R. Horn and C. Johnson, Topics in Matrix Analysis . Cambridge University Press, 1994

work page 1994

[24] [24]

Higham, Functions of Matrices: Theory and Computation

N. Higham, Functions of Matrices: Theory and Computation . SIAM, 2008

work page 2008

[25] [25]

Konstantinov, D

M. Konstantinov, D. Gu, V. Mehrmann and P. Petkov, Perturbation Theory for Matrix Equations. Studies in Computational Mathematics. Elsevier Science, 2003

work page 2003

[26] [26]

Gourgoulhon, 3+1 Formalism in General Relativity: Bases of Numerical Relativity

´E. Gourgoulhon, 3+1 Formalism in General Relativity: Bases of Numerical Relativity . Lecture Notes in Physics. Springer Berlin Heidelberg, 2012

work page 2012

[27] [27]

R. L. Arnowitt, S. Deser and C. W. Misner, The dynamics of general relativity , Gen. Rel. Grav. 40 (2008) 1997–2027, [ gr-qc/0405109]

work page internal anchor Pith review Pith/arXiv arXiv 2008

[28] [28]

J. W. York, Jr., Kinematics and Dynamics of General Relativity , pp. 83–126

work page

[29] [29]

Kocic, Geometric mean of bimetric spacetimes , 1803.09752

M. Kocic, Geometric mean of bimetric spacetimes , 1803.09752

work page arXiv

[30] [30]

S. F. Hassan, M. Kocic and A. Schmidt-May, Absence of ghost in a new bimetric-matter coupling, 1409.1909

work page internal anchor Pith review Pith/arXiv arXiv 1909

[31] [31]

S. F. Hassan and A. Schmidt-May, Interactions of multiple spin-2 ﬁelds beyond pairwise couplings, 1804.09723

work page internal anchor Pith review Pith/arXiv arXiv

[32] [32]

Geometry of physical dispersion relations

D. Raetzel, S. Rivera and F. P. Schuller, Geometry of physical dispersion relations , Phys. Rev. D83 (2011) 044047, [ 1010.1369]

work page internal anchor Pith review Pith/arXiv arXiv 2011

[33] [33]

F. P. Schuller and C. Witte, How quantizable matter gravitates: A practitioner’s guide , Phys. Rev. D89 (2014) 104061, [ 1402.6548]

work page internal anchor Pith review Pith/arXiv arXiv 2014

[34] [34]

F. P. Schuller, N. Stritzelberger, F. Wolz and M. D¨ ull, Gravitational closure of matter ﬁeld equations, 1611.08878

work page internal anchor Pith review Pith/arXiv arXiv

[35] [35]

K. B. Petersen and M. S. Pedersen, The matrix cookbook, tech. rep., nov, 2012

work page 2012

[36] [36]

No consistent cross-interactions for a collection of massless spin-2 fields

N. Boulanger, T. Damour, L. Gualtieri and M. Henneaux, No consistent cross interactions for a collection of massless spin-2 ﬁelds , in Quantization, gauge theory, and strings. Proceedings, International Conference dedicated to the memory of Professor Eﬁm Fradkin, Moscow, Russia, June 5-10, 2000. Vol. 1+2 , pp. 150–161, 2000, hep-th/0009109

work page internal anchor Pith review Pith/arXiv arXiv 2000

[37] [37]

Inconsistency of interacting, multi-graviton theories

N. Boulanger, T. Damour, L. Gualtieri and M. Henneaux, Inconsistency of interacting, multigraviton theories, Nucl. Phys. B597 (2001) 127–171, [ hep-th/0007220]

work page internal anchor Pith review Pith/arXiv arXiv 2001

[38] [38]

Effective Lagrangians and Universality Classes of Nonlinear Bigravity

T. Damour and I. I. Kogan, Eﬀective Lagrangians and universality classes of nonlinear bigravity, Phys. Rev. D66 (2002) 104024, [ hep-th/0206042]. – 19 –

work page internal anchor Pith review Pith/arXiv arXiv 2002