Chern-Simons Modification of General Relativity

R. Jackiw; S.-Y. Pi

REVIEW

Chern-Simons Modification of General Relativity

R. Jackiw, S.-Y. Pi

General Relativity and Quantum Cosmology · Submitted 21 Aug 2003 · 7 references

Pith's verdict: The founding paper of dynamical Chern-Simons gravity: careful derivations, an honest but under-explored *RR = 0 constraint, and a 'hidden symmetry breaking' claim that is really a projection onto Pontryagin-vanishing metrics.

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arXiv IDgr-qc/0308071 (version 2)

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CitationR. Jackiw, S.-Y. Pi. "Chern-Simons Modification of General Relativity." arXiv:gr-qc/0308071 (2003). https://pith.science/paper/PJDV3UUK

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Editor's note

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This is the paper that became dynamical Chern-Simons gravity, and it earns that status on the derivations. Jackiw and Pi promote the three-dimensional gravitational Chern-Simons term to four dimensions by coupling the Pontryagin density to an external scalar with constant gradient, then work the consequences through cleanly. Appendix A gets the four-dimensional Cotton tensor by varying the action; Appendix B gets the divergence identity D_mu C^{mu nu} = v^nu *RR/(8 sqrt(-g)); Appendix C shows the stationary Cotton tensor vanishes on spherically symmetric metrics, so Schwarzschild survives; Appendix D shows *RR vanishes at quadratic order for plane waves. The linearized analysis gives two polarizations at speed c and the polarization-dependent intensity factors 1/(1 ∓ k mu)^2 in Eq. 4.22. That last result is the one with observational teeth, and it is derived, not asserted. No fitted parameters, no data, no circularity. The added note disclosing overlap with Lue et al. on gravity waves is handled the right way, and the Stueckelberg analogy in Section 5 is genuinely illuminating about what kind of symmetry breaking this is. One correction to the record: the classifier found no stated limitations, which is wrong. The paper says plainly that Kerr is excluded because *RR is nonzero, that the linear theory has a pole at k = mu, and that the improved Noether pseudotensor is left for later. It is more candid about its limits than a quick read suggests. The stress-test is right that the sharper issue is not the choice of Eq. 2.8 among possible extensions, which the authors flag themselves, but the *RR = 0 constraint. That constraint is what makes Schwarzschild pass and waves pass, and it also projects out generic rotating backgrounds. Calling the breaking 'hidden' is a little generous: the dynamics is not insensitive to the breaking, the solution space is being truncated. Whether what survives beyond Schwarzschild and linearized waves is physically rich is left open, and that is the question that mattered historically. The k = mu zero-mode discussion is honest but thin. None of this is fatal; it is the natural edge of a first paper. Send it to referees, and cite it if you work on parity violation in gravity.

Referee report

Significance

The paper is a careful and self-contained construction, and its strengths are worth naming. The action is stated explicitly, the only new quantity is the coupling μ, and every downstream claim is derived from that action rather than fitted: the Cotton tensor, the divergence identity, the Schwarzschild check, and the linearized spectrum all follow in closed form, with the algebra relegated to Appendices A–D so that each step can be checked. The result is a sharp, falsifiable statement about the one place the deformation is observable, namely the polarization-dependent intensity (4.22), which is a genuine prediction rather than a consistency remark. The authors are also candid about what does not work: they state in Section 3 that Kerr is excluded, call the deformed Kerr solution an outstanding open question, and flag the k = μ singularity in Section 4. The Added Note crediting Lue, Wang and Kamionkowski for the overlapping wave analysis is appropriate scholarly practice.

Major objections (3)

[Section 3, Eq. (2.12)] The divergence identity (2.12) is the hinge of the whole construction: with v^ν fixed and non-vanishing, every solution of (2.11) must satisfy *RR = 0. The authors state this plainly and acknowledge that Kerr is therefore excluded, calling the replacement 'an outstanding open question.' That acknowledgment is appropriate, but the paper does not characterize how large the surviving solution space is. The central claim of the abstract (s3), that the symmetry breaking is 'hidden,' rests on that space being physically non-trivial, and the paper verifies only two members of it: Schwarzschild and linearized plane waves. A short paragraph in Section 3 stating what is known about the class of Pontryagin-vanishing metrics, or at least bounding how restrictive the condition is, would materially strengthen the claim and would let a reader judge whether the deformation is a small perturbation of GR on generic backgrounds or a projection that removes them. As written, the reader is asked to accept 'hidden' on the basis of two special cases.
[Section 4 and Appendix D] The constraint *RR = 0 is exact, but Appendix D verifies it only to quadratic order in h_μν, and the paper does not address whether the linearized plane waves can be promoted to exact solutions, i.e. whether *RR remains zero at third and higher orders. This matters for how the Section 4 result should be read: 'gravitational waves possess two polarizations, each traveling at the velocity of light' is, on the present analysis, a statement about the linearized truncation. A sentence in Section 4 or Appendix D making explicit whether the wave solutions are exact or only approximate solutions of (2.11), and if only approximate, at what order the constraint first bites, would remove the ambiguity without requiring new calculation.
[Section 4, Eqs. (4.20)–(4.22)] The modified source (4.20) carries the denominator (1 - k²/μ²), and the radiated power (4.22) inherits it, so the map from T^{ijTT} to the physical intensity is singular at k = μ. The authors note the singularity and the associated zero modes and say they 'find no further information about this in the linear theory.' Since (4.22) is the paper's one observable prediction, the domain of validity should be stated explicitly: is the enhancement/suppression factor (1 ∓ k/μ)^{-2} to be trusted only for k ≪ μ, and does the singularity at k = μ signal a breakdown of the linear treatment or a genuine feature of the extended theory? A short clarifying sentence would keep the prediction from being read outside its regime.

Minor objections (10)

[Abstract] 'enchanced' should read 'enhanced.'
[Eqs. (4.20)–(4.22)] The combination written 'kµ' is easily misread as a four-vector contraction; writing it as k/μ (or |k|/μ) throughout, and stating that k ≡ |k|, would remove the ambiguity.
[Eq. (3.7)] '∗Roνiµ' should read '∗R_{0νiμ}'.
[Eq. (3.8)] The index 'J' in '³G_{nJ}' and '³G_{mJ}' should be lower case 'j' to match the surrounding notation.
[Eq. (4.11b)] The factor 'ε_{jmnγ}' carries a stray index γ that does not belong; it should presumably be 'ε_{jmn}'.
[Eq. (D.1)] 'h_{rα}' should read 'h_{τα}'.
[Appendix D] The statement that (D.4) 'vanishes for a variety of reasons' would be clearer if the reasons were named, e.g. the nullness of k_γ and the antisymmetry of ε contracted with symmetric combinations.
[Section 2] The claim that the theory with external θ is 'identical' to the one with dynamical θ holds at the level of equations of motion, not of the off-shell actions; a clause making that explicit would prevent misreading.
[Appendix A, Appendix B, Section 5] Typographical: 'intergrand' should be 'integrand' (Appendix A), 'explicity' should be 'explicitly' (Appendix B), and 'possiblility' should be 'possibility' (Section 5).
[Added Note] The Added Note is good practice; consider also citing Ref. [7] at the point in Section 4 where the wave analysis is presented, rather than only in the note, so that the overlap is visible to a reader who stops before the end.

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The paper

The paper claims

Adding a parity-violating Chern-Simons term to Einstein's gravity leaves Schwarzschild intact and keeps gravitational waves at light speed, but makes the two polarizations radiate with unequal intensities.

The reading

The paper asks what happens if you take the gravitational Chern-Simons term — a three-dimensional object that gives rise to parity-violating ``topologically massive gravity'' — and promote it to four dimensions inside Einstein's theory. The promotion requires an extra ingredient, an ``embedding coordinate'' $v_$ whose derivative is taken to be a fixed, non-zero constant in its time component, so the action is no longer diffeomorphism invariant. The authors show that the breaking is largely hidden by the dynamics: consistency of the field equations forces the Pontryagin density ${{}^{*}}RR$ to vanish, the Schwarzschild metric still solves the equations, gravitational waves still have exactly two polarizations travelling at $c$, and a conserved symmetric energy-momentum pseudotensor can still be built. What does survive is a parity-violating signature in radiation: the two circular polarizations of a gravitational wave are emitted with intensities scaled by $1/(1-k)^2$ and $1/(1+k)^2$, where $k$ is the wave number and $$ the extension parameter. For a general reader the interest is that a theory which looks like it violates a sacred symmetry turns out to violate it only in a place — the relative brightness of left- and right-handed gravity waves — that no experiment has yet reached.

What carries the argument

The load-bearing object is the four-dimensional Cotton tensor $C^{}$ obtained by varying $4} d^4x {{}^{*}}RR$ with respect to the metric: a traceless, symmetric tensor built from the embedding-coordinate derivative $v_$ and the dual Riemann tensor. Its non-zero divergence, $D_C^{} = v^{} {{}^{*}}RR/(8{-g})$, converts the Bianchi identity into a constraint that the gravitational Pontryagin density must vanish, and this one identity does all the work — it supplies the dynamical suppression of symmetry breaking, shows the external-$ and dynamical-$ theories coincide, and guarantees that the Schwarzschild metric (whose Cotton tensor vanishes) survives.

Extended reading notes

Core claim

The authors extend general relativity by adding $I_{CS} = 4} d^4x {{}^{*}}RR$ to the Hilbert-Einstein action, with $ = x^ v_$ an externally prescribed quantity and $v_ = (1/,0)$. Varying the metric produces a traceless, symmetric four-dimensional Cotton tensor $C^{}$, and the modified field equation is $G^{} + C^{} = -8GT^{}$. The central technical result is that the covariant divergence of this Cotton tensor does not vanish but equals $v^{} {{}^{*}}RR/(8{-g})$, so any solution of the modified equations must live on a background with ${{}^{*}}RR = 0$. That consistency condition is what hides the symmetry breaking: it is also exactly the equation one would get by varying $ as a dynamical field, so the external-$ theory and the fully diffeomorphism-invariant dynamical-$ theory have identical equations of motion. Evaluating $$C^{{mn}}$$ on a stationary, spherically symmetric metric shows it vanishes identically, so Schwarzschild remains a solution and the classical solar-system tests are unaffected; the Kerr metric, for which ${{}^{*}}RR 0$, cannot be a solution. In the linearized theory the quadratic action still propagates only the transverse-traceless $$h^{{TT}}$_{ij}$, two polarizations at $ = |k|$, while the extension operators can be moved onto the source, replacing $$T^{{ij}}$_{TT}$ by a filtered $^{ij}_{TT}$ and yielding the unequal radiation intensities of Eq. (4.22).

Load-bearing premise

The results depend on one specific way of promoting the three-dimensional Chern-Simons term to four dimensions — the action $4} d^4x {{}^{*}}RR$ with an external, linear $ — and the paper itself notes that other extensions of Einstein's theory by a Chern-Simons term are possible.

Editorial extensions

If this is right

Schwarzschild remains an exact solution, so the modified theory passes the classical solar-system tests of general relativity without adjustment.
Gravitational waves keep exactly two polarization states, both propagating at the speed of light, because the extension only adds spatial derivative operators that can be moved from the field side to the source side.
The observable signature is a parity asymmetry in emission: the two circular polarizations of a wave from a monochromatic source carry power scaled by $1/(1-k)^2$ and $1/(1+k)^2$, a correction that grows with energy.
A symmetric, divergenceless energy-momentum pseudotensor exists for the modified theory, constructed by moving the non-linear part of $G^{}+C^{}$ to the source side.
The Kerr metric is excluded as an exact solution, since ${{}^{*}}RR 0$ there, leaving open which deformed rotating solution replaces it.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the correction to radiated power scales as $k/$, precision measurements of gravitational-wave polarization from spinning sources would bound $$ — a channel the paper notes is not yet feasible but does not quantify.
The authors report a singularity at $k = $ where the spatial operator acting on $h^{TT}_{ij}$ has zero modes; this looks like a strong-coupling scale of the effective theory rather than a physical pole, and reading it that way would suggest the linear analysis breaks down before the parity asymmetry becomes large.
The equivalence with a dynamical $$ suggests reading the fixed embedding coordinate as a Stueckelberg field: the symmetry breaking is a gauge choice, and the physical content sits entirely in the ${{}^{*}}RR = 0$ constraint.
Since the exclusion of Kerr is exact rather than approximate, black-hole ringdown spectroscopy matching the Kerr spectrum to high precision converts directly into a bound on the Chern-Simons coupling.

Original abstract

General relativity is extended by promoting the three-dimensional gravitational Chern-Simons term to four dimensions. This entails choosing an embedding coordinate v_\mu -- an external quantity, which we fix to be a non-vanishing constant in its time component. The theory is identical to one in which the embedding coordinate is itself a dynamical variable, rather than a fixed, external quantity. Consequently diffeomorphism symmetry breaking is hidden in the modified theory: the Schwarzschild metric is a solution; gravitational waves possess two polarizations, each traveling at the velocity of light; a conserved energy-momentum (pseudo-) tensor can be constructed. The modification is visible in the intensity of gravitational radiation: the two polarizations of a gravity wave carry intensities that are suppressed/enchanced by the extension.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,322 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.24 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.27 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

General relativity is extended by promoting the three-dimensional gravitational Chern-Simons term to four dimensions. This entails choosing an embedding coordinate v_\mu -- an external quantity, which we fix to be a non-vanishing constant in its time component. The theory is identical to one in which the embedding coordinate is itself a dynamical variable, rather than a fixed, external quantity. Consequently diffeomorphism symmetry breaking is hidden in the modified theory: the Schwarzschild metric is a solution; gravitational waves possess two polarizations, each traveling at the velocity of light; a conserved energy-momentum (pseudo-) tensor can be constructed. The modification is visible in the intensity of gravitational radiation: the two polarizations of a gravity wave carry intensities that are suppressed/enchanced by the extension.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	General relativity is extended by promoting the three-dimensional gravitational Chern-Simons term to four dimensions.	contribution 0.97	0.73	0.11	asserted 1.00
s1	This entails choosing an embedding coordinate v_\mu -- an external quantity, which we fix to be a non-vanishing constant in its time component.	method 0.90	0.75	0.05	asserted 1.00
s2	The theory is identical to one in which the embedding coordinate is itself a dynamical variable, rather than a fixed, external quantity.	interpretation 0.46	0.75	0.33	asserted 1.00
s3	Consequently diffeomorphism symmetry breaking is hidden in the modified theory: the Schwarzschild metric is a solution; gravitational waves possess two polarizations, each traveling at the velocity of light; a conserved energy-momentum (pseudo-) tensor can be constructed.	result 0.68	0.80	0.47	asserted 1.00
s4	The modification is visible in the intensity of gravitational radiation: the two polarizations of a gravity wave carry intensities that are suppressed/enchanced by the extension.	result 0.98	0.78	0.41	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (7)

[1]

S. Deser, R. Jackiw and S. Templeton, Ann. Phys. (NY) 140, 372 (1982); (E) 185, 406 (1988); Phys. Rev. Lett. 48, 975 (1982); (C) 59, 1981 (1987)

View on Pith 1982
[2]

S. Carroll, G. Field and R. Jackiw, Phys. Rev. D 41, 123 (1990)

View on Pith 1990
[3]

S. Carroll and G. Field, Phys. Rev. Lett. 79, 2394 (1997)

View on Pith 1997
[4]

S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972)

View on Pith 1972
[5]

A. Garcia, F. Hehl, C. Heinicke and A. Macias, gr-qc/0309008

View on Pith Pith review arXiv
[6]

D. Bak, D. Cangemi and R. Jackiw, Phys. Rev. D 49, 5173 (1994)

View on Pith 1994
[7]

A. Lue, L. Wang and M. Kamionkowski, Phys. Rev. Lett. 83, 1506 (1999)

View on Pith 1999

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A Hamiltonian count shows that among degenerate higher-derivative metric theories in four dimensions only f(R) is ghost-free in covariant form, while Chern-Simons gravity and new parity-violating scalar-tensor theorie...
Black hole spectroscopy with coherent mode stacking
gr-qc 2017-01 conditional novelty 6.0 of 10

Coherently stacking ringdown signals from many binary black hole mergers, with phases predicted from each event's inspiral, boosts the collective signal-to-noise of the subdominant l=m=3 quasinormal mode enough for Ad...
Binary Love Relations
gr-qc 2015-12 conditional novelty 6.0 of 10

Approximately universal relations between combinations of the two neutron stars' tidal deformabilities break the degeneracy that otherwise prevents gravitational wave detectors from measuring each star's deformability...
Third order equations of motion and the Ostrogradsky instability
physics.class-ph 2014-11 accept novelty 6.0 of 10

Any Lagrangian in point-particle mechanics that yields third-order (or higher odd-order) equations of motion has an unbounded Hamiltonian, so the Ostrogradsky instability is unavoidable.
The Initial Value Formulation of Dynamical Chern-Simons Gravity
gr-qc 2014-07 conditional novelty 6.0 of 10

The full field equations of dynamical Chern-Simons gravity do not admit a first-order-in-time reduction, so they are not hyperbolic in the standard sense, though an order-reduced effective version is.
Isolated and Binary Neutron Stars in Dynamical Chern-Simons Gravity
gr-qc 2013-02 conditional novelty 6.0 of 10

Dynamical Chern-Simons corrections to neutron stars and binary pulsars are too small, and too degenerate with the neutron-star equation of state, for current pulsar timing to constrain the theory.
Gravitational Waves from Quasi-Circular Black Hole Binaries in Dynamical Chern-Simons Gravity
gr-qc 2012-08 conditional novelty 6.0 of 10

A first self-consistent waveform for inspiraling black hole binaries in dynamical Chern-Simons gravity yields a 2PN phase correction that Advanced LIGO could use to tighten constraints on the theory by six orders of m...
Slowly Rotating Black Holes in Dynamical Chern-Simons Gravity: Deformation Quadratic in the Spin
gr-qc 2012-06 accept novelty 6.0 of 10

A stationary, axisymmetric black hole metric in dynamical Chern-Simons gravity is derived to quadratic order in spin, revealing a Petrov type I classification and no exact Carter-like constant.
Gravitational waves from extreme mass-ratio inspirals in Dynamical Chern-Simons gravity
gr-qc 2011-04 conditional novelty 6.0 of 10

In dynamical Chern-Simons gravity, a stellar-mass object inspiraling into a Schwarzschild black hole radiates slightly more total energy than in general relativity, reducing the number of LISA-band gravitational-wave ...
Gravitational signature of Schwarzschild black holes in dynamical Chern-Simons gravity
gr-qc 2010-04 accept novelty 6.0 of 10

Perturbations of a Schwarzschild black hole in dynamical Chern-Simons gravity decay stably and ring with two distinct frequency branches, one matching General Relativity's gravitational modes and one its scalar modes.
Perturbations of Schwarzschild black holes in Dynamical Chern-Simons modified gravity
gr-qc 2009-07 conditional novelty 6.0 of 10

In dynamical Chern-Simons gravity with no background scalar field, axial gravitational perturbations of a Schwarzschild black hole are unstable for coupling βM^4 below roughly 2π, implying β ≳ 10^-2 km^-4.
Circular Polarization of Primordial Gravitational Waves in String-inspired Inflationary Cosmology
astro-ph 2007-06 conditional novelty 6.0 of 10

A Gauss-Bonnet-driven super-inflationary phase plus Chern-Simons parity violation produces nearly 100% circularly polarized primordial gravitational waves, and a two-field model can place that signal in the BBO/DECIGO band.
Quantum Gravity, Torsion, Parity Violation and all that
hep-th 2005-07 conditional novelty 6.0 of 10

Parity-violating gravitational interactions require a non-minimal fermion-torsion coupling, so real values of the Immirzi parameter remain unconstrained by parity experiments.
First-order corrections to equatorial periodic orbits around rapidly rotating black holes in dynamical Chern--Simons gravity
gr-qc 2026-10 conditional novelty 5.0 of 10

First-order corrections to equatorial periodic orbits around rapidly rotating black holes are derived in dynamical Chern-Simons gravity and shown to be reproducible by a Kerr mass rescaling.
New constraints on modified gravity with dimension-six operators from gravitational waves
gr-qc 2026-08 conditional novelty 5.0 of 10

A dimension-six Lorentz-violating gravity operator is linearized to obtain gravitational-wave dispersion relations, and time-of-flight data from GW170817 and GW150914 bound the coefficients to 10^-5 to 10^-4 m^2 (nonb...
Inflation with the Gauss-Bonnet term in the Palatini formulation
astro-ph.CO 2026-03 accept novelty 5.0 of 10

In Palatini gravity the inflaton–Gauss–Bonnet coupling yields a negative Chern–Simons-like kinetic correction and similar GW modifications, with metric-like behaviour unless the kinetic term nearly flips sign.
Sensitivity forecasts for gravitational-wave detectors to dark matter decaying into gravitons
hep-ph 2025-10 unverdicted novelty 5.0 of 10

Model-independent forecasts for the stochastic gravitational-wave background from ultralight dark matter decaying into gravitons and the sensitivity of current and future detectors to this signal.
Neural Post-Einsteinian Test of General Relativity with the Third Gravitational-Wave Transient Catalog
gr-qc 2025-10 unverdicted novelty 5.0 of 10

Neural post-Einsteinian analysis of GWTC-3 finds no GR violation and sets constraints covering both post-Newtonian and beyond-post-Newtonian deviations in a single theory-agnostic setup.
Axial quasi-normal modes of slowly rotating black holes in dynamical Chern-Simons gravity to second-order in spin and coupling
gr-qc 2025-06 conditional novelty 5.0 of 10

For the n=0, l=m=2 axial ringdown mode, dynamical Chern-Simons gravity lowers the frequency and shortens the damping time, and the paper gives a polynomial fit for these shifts.

See all 95 Pith citations

How later papers used it

339 later papers cite it

Evidence: substantial. 337 of the citing papers share no author with it, and 18 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 596 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

5 later papers by other authors are listed below: 1 used its method or tool, 3 compared their own results with it and 1 cited it as support.

Uses its method or tool 1 of 2 papers with this label

Followed this paper to decompose the perturbations into irreducible pieces to write the field equations in terms of gauge invariants.

Polarizations of Gravitational Waves in the Bumblebee Gravity Model (2022) · 1.00 · paraphrased: the licence does not allow quoting

Compares with it 3 of 11 papers with this label

Noted that its own one-loop result is very similar to the known four-dimensional gravitational Chern-Simons term of this paper, with the constant vector b_μ replaced by the axial-vector torsion S_μ.

Induced gravitational topological term and the Einstein-Cartan modified theory (2021) · 1.00 · paraphrased: the licence does not allow quoting
Observed that the general structure of its P reproduces that of the parity anomaly of this paper, but the relative coefficients in front of the Chern-Simons and extrinsic curvature terms are different.

Gravitational parity anomaly with and without boundaries (2018) · 0.99 · paraphrased: the licence does not allow quoting
Said that its action is different from the action considered in this paper in that here θ is a dynamical scalar field with a canonical kinetic term, so the ℓ parameter is required to make the action dimensionless.

The effects of Chern-Simons gravity on bodies orbiting the Earth (2007) · 0.98 · paraphrased: the licence does not allow quoting

Cites it as support 1 of 4 papers with this label

Used this paper as evidence that moving derivatives off the perturbed quantities must be possible, since the Pontryagin density can be written as the divergence of a 4-current.

Effective Gravitational Wave Stress-energy Tensor in Alternative Theories of Gravity (2010) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

279 arXiv papers cite this in 493 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 267 sentences without a confident label

Contests it 1 paper

The tensor $C_ab$ that appears in Eq. [ref] is a $4$-dimensional generalization of the $3$-dimensional Cotton-York tensor, which in order to distinguish it from the latter we shall call the C-tensor[In the original work of [ref] , the C-tensor was incorrectly called “Cotton tensor”, but the concept of a higher-dimensional Cotton-York tensor already exists [ref] and differs from the definition of Eq. [ref]].

Chern-Simons Modified General Relativity · Formulation · contest 0.74

Builds on it 1 paper

Following the analysis of [ref] for the Nieh-Yan case, we can consider an analogous scenario in metric Chern-Simons gravity [ref] , looking again at the propagation of gravitational waves on a Minkowski background.

Landau damping for gravitational waves in parity-violating theories · Metric Chern-Simons gravity · 0.93

Uses its method or tool 2 papers

Such solutions can, however, be constructed numerically, following [ref] , in the corresponding non-linear model: [equation] This model is a variation of the usual dynamical Chern-Simons modified gravity, wherein the coupling of the scalar field to the Pontryagin density is of the form $ϕ ^∗R^μ_ν^αβ R^ν_μαβ$ [ref] .

Isolated black holes without $\mathbb{Z}_2$ isometry · A gravitational source · 0.81
To write the field equations in terms of gauge invariants, we first follow Refs. [ref] to decompose the perturbations into irreducible pieces: [equation] together with the constraints [equation] The transformation under infinitesimal particle diffeomorphisms is given by [ref] : [equation] It is easy to show that $h_ij^TT$ and the following combinations are gauge invariants [ref] : [equation]

Polarizations of Gravitational Waves in the Bumblebee Gravity Model · Linearized Regime · 1.00

Compares with it 11 papers

Apart from that, the divergence of Eq. ([ref]) imposes another constraint or consistency condition to hold the diffeomorphism invariance of the modified theory [equation] which is somewhat similar to the Pontryagin constraint in CSMGR [ref] .

Induced gravitational topological term and the Einstein-Cartan modified theory · EINSTEIN-CARTAN MODIFIED THEORY WITH ADDITIVE TOPOLOGICAL TERM · 0.96
This needs to be contrasted with the gravitational anomaly terms in [ref], whose variation with respect to the metric field $g_μν$ yields non-trivial results [ref] : [equation] where [equation] is the (tracelss) Cotton tensor [ref] , [equation] with $u_σ ≡∂_σb = b_;σ, u_στ ≡u_τ; σ = b_;τ;σ$.

Lorentz Symmetry Violation in String-Inspired Effective Modified Gravity Theories · A string-inspired gravitational theory with torsion and anomalies · 0.95
This action is different from the action considered in Ref. [ref] in that here $θ$ is a dynamical scalar field with a canonical kinetic term, so the $ℓ$ parameter is required to make the action dimensionless.

The effects of Chern-Simons gravity on bodies orbiting the Earth · CHERN-SIMONS GRAVITY · 0.98

Cites it as support 2 papers

In fact, we know that this must be possible from [ref] : the Pontryagin density can be written as the divergence of a 4-current, so at least one derivative can be moved off of $^μν$.

Effective Gravitational Wave Stress-energy Tensor in Alternative Theories of Gravity · Variation with respect to the Background · 0.98
Nevertheless, the Chern-Simons gravity [ref] and the recently studied ghostfree quadratic gravity [ref] show the possibility of the existence of ghostfree high curvature gravity, which also indicates the existence of ghostfree scalar-tensor theories with derivatives beyond the second order.

Higher derivative scalar-tensor theory and spatially covariant gravity: the correspondence · Introduction · 0.98

Cites it as background 162 papers

In dSCG, a dynamical scalar field is introduced to couple to the gravitational Chern-Simons invariant non-minimally [ref] .

Magnetic-induced Spontaneous Scalarization in Dynamcial Chern-Simons Gravity · Introduction · 0.97
The CS gravitational theory [ref] introduces a linear coupling of the (pseudo)scalar field, $b$, with the gravitational CS (gCS) term, $ℒ_int∼b R_CS$, where $R_CS$ is given in [ref], and is a total derivative.[We should stress at this point that the gCS term, unlike the gauge CS term, which the axion also couples to, yields a non trivial variation with the metric tensor, namely the Cotton tensor (cf. [ref], below) [ref] .

Condensate-Induced Inflation from Primordial Gravitational Waves in String-Inspired Chern-Simons Gravity · Producing The Linear Potential Through Gravitational Condensate · 0.87
Consider an axion matter field coupled to the gravitational CS topological term known as the Pontryagin density term [ref] .

Scalarization of Chern-Simons Kerr Black Hole Solutions and Wormholes · Chern-Simons Gravitational Theory · 0.99

Discussion

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