Gravitational memory and compact extra dimensions

An Einstein frame observer is insensitive to the overall breathing mode.

“Thus the Einstein news tensor is equivalent to the trace-free Bondi news tensor - an Einstein frame observer is insensitive to the overall breathing mode as we expect [24].”

• cites [24] [24] S. Tahura, D. A. Nichols, A. Saffer, L. C. Stein, and K. Yagi, 'Brans-Dicke theory in Bondi-Sachs form: Asymptotically flat solutions, asymptotic symmetries and gravitational-wave memory effects,' 2020. - sourceDisputed

  The Introduction (p. 2) says test particles in the Einstein frame accelerate in response to scalar-field gradients; it does not describe them as insensitive to the scalar mode. It also says physical predictions, including gravitational-wave memory effects, are conformal-frame invariant. The paper’s description of the breathing mode as an expansion or contraction of freely falling observers (p. 1) further contradicts the claimed physical insensitivity. (LLM, conf=0.82)

  Checked by a reasoning model — may be wrong.

  Experimental — accuracy not yet measured

For a binary black hole merger the contribution of the null memory is roughly 10^3 times larger than the ordinary memory.

“For a binary black hole merger the contribution of the null memory is roughly ∼ 10 3 times larger than the ordinary memory [22].”

• cites [22] [22] M. Favata, 'NONLINEAR GRAVITATIONAL-WAVE MEMORY FROM BINARY BLACK HOLE MERGERS,' The Astrophysical Journal 696 (apr, 2009) L159-L162. - sourceCould not verify

  The Introduction and Motivation distinguishes linear memory, which can occur in recoiling black holes, from the nonlinear memory studied for binary black hole mergers. Neither that section nor the retrieved passages comparing memory models gives a null-to-ordinary memory ratio; searches for a 10^3 comparison and recoil-related comparisons found none. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

From known constructions of Calabi-Yau 4-fold geometries, there are examples with O(10^5) complex structure moduli.

“From known constructions of Calabi-Yau 4-fold geometries, there are examples with of O (10 5 ) such moduli [34, 35].”

• cites [34] [34] W. Taylor and Y.-N. Wang, 'Scanning the skeleton of the 4D F-theory landscape,' JHEP 01 (2018) 111, 1710.11235 . - sourceSupported
• cites [35] [35] J. Halverson, C. Long, and B. Sung, 'Algorithmic universality in F-theory compactifications,' Phys. Rev. D 96 (2017), no. 12, 126006, 1706.02299 . - sourceCould not verify

  The introductory discussion counts approximately 10^755 geometries, not complex-structure moduli. The discussion of multiplicities of vanishing mentions tuning in complex-structure moduli space but gives no moduli count. Searches for 10^5, Hodge numbers, and complex-structure moduli found no example with O(10^5) such moduli in our copy. (LLM, conf=0.73)

  Experimental — accuracy not yet measured

Gravitational radiation for spacetimes with compact dimensions has not yet been studied beyond linearized gravity, or in the special case of a circle compactification.

“Somewhat surprisingly, gravitational radiation for spacetimes with compact dimensions has not yet been studied beyond linearized gravity, or in the special case of a circle compactification [10-12].”

• cites [10] [10] D. Andriot and G. Lucena G´ omez, 'Signatures of extra dimensions in gravitational waves,' JCAP 06 (2017) 048, 1704.07392 . [Erratum: JCAP 05, E01 (2019)]. - sourcePartly supported

  The Background and state of the art (p. 3) says earlier work often used flat extra dimensions or circles, and identifies a study of a single-circle polarization effect. But it also reports a waveform correction for fairly general compact extra dimensions and does not establish that gravitational radiation had not been studied beyond linearized gravity. Section 2 (p. 6) says this paper derives linearized equations; that describes its own approach, not the limits of all prior work. (LLM, conf=0.82)

  Experimental — accuracy not yet measured
• cites [11] [11] H. L¨ u, P. Mao, and J.-B. Wu, 'Asymptotic Structure of Einstein-Maxwell-Dilaton Theory and Its Five Dimensional Origin,' JHEP 11 (2019) 005, 1909.00970 . -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [12] [12] W.-D. Tan, 'Einstein-Maxwell-dilaton theory in Newman-Penrose formalism,' Phys. Rev. D 102 (2020), no. 4, 044054, 2007.11802 . - sourceSupported

The variation of the warp factor could be much larger because the D3-brane tadpole found in F-theory on a Calabi-Yau 4-fold, which determines the maximum amount of background flux, can be as large as O(10^4) in known examples.

“In principle, the variation of the warp factor could be much larger because the D3-brane tadpole found in F-theory on a Calabi-Yau 4-fold [40, 41], which determines the maximum amount of background flux, can be as large as O (10 4 ) in known examples.”

• cites [40] [40] S. Sethi, C. Vafa, and E. Witten, 'Constraints on low dimensional string compactifications,' Nucl. Phys. B 480 (1996) 213-224, hep-th/9606122 . -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [41] [41] K. Becker and M. Becker, 'M theory on eight manifolds,' Nucl. Phys. B 477 (1996) 155-167, hep-th/9605053 . - sourcePartly supported

  Section 2 (p. 12) relates the four-form flux to the Euler number of the eight-manifold and describes the resulting topological constraint; it also gives an equation for the warp factor. The paper does not state that a D3-brane tadpole in known F-theory examples reaches O(10^4), or quantify a correspondingly larger warp-factor variation. (LLM, conf=0.86)

  Experimental — accuracy not yet measured

The ten-dimensional Planck scale takes the range M_D=10p ∼ 10 keV - 10 TeV, assuming an upper bound on the size of any compact dimension of roughly order microns (or eV) from gravitational bounds to approximately 10^-18 m (or a TeV) from collider bounds, and six compact dimensions.

“If we assume an upper bound on the size of any compact dimension of roughly order microns, or equivalently eV, from gravitational bounds [29] to approximately 10 -18 m or a TeV from collider bounds [30], and six compact dimensions then the ten-dimensional Planck scale takes the range M D =10 p ∼ 10 keV -10 TeV.”

• cites [29] [29] D. Kapner, T. Cook, E. Adelberger, J. Gundlach, B. R. Heckel, C. Hoyle, and H. Swanson, 'Tests of the gravitational inverse-square law below the dark-energy length scale,' Phys. Rev. Lett. 98 (2007) 021101, hep-ph/0611184 . - sourceSupported
• cites [30] [30] Particle Data Group Collaboration, P. Zyla et al. , 'Review of Particle Physics,' PTEP 2020 (2020), no. 8, 083C01. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: The size of any compact dimension has an upper bound of roughly order microns (or eV) from gravitational bounds and approximately 10^-18 m (or a TeV) from collider bounds, and there are six compact dimensions.

If a one-form V_m is harmonic then g^mn V_n is also a Killing vector.

“To complete the correspondence we now show that if a one-form V m is harmonic then ˆ g mn V n is also a Killing vector [56].”

• cites [56] [56] S. Bochner, 'Vector fields and Ricci curvature,' Bull. Amer. Math. Soc. 52 (09, 1946) 776-797. - sourcePartly supported

  In §2, “Positive curvature” (page unavailable), Theorem 1 considers fields with zero curl and divergence, but concludes they are parallel only under additional restrictions on the field and Ricci curvature. The paper also identifies a separate condition for a vector field to generate motions. It does not establish that every harmonic one-form, when raised with the metric, is a Killing vector. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

Even in the setting of light degrees of freedom, there are exotic compact objects that can support scalar hair.

“Surprisingly, even in this setting there are exotic compact objects that can support scalar hair, which is our basic signature of extra dimensions.”

• cites [42] [42] S. L. Liebling and C. Palenzuela, 'Dynamical Boson Stars,' Living Rev. Rel. 20 (2017), no. 1, 5, 1202.5809 . - sourceSupported

Solutions with mass M are dynamically stable only for a certain range of the ratio M/R because of the Gregory-Laflamme instability.

“Solutions with mass M are dynamically stable only for a certain range of the ratio M/R because of the Gregory-Laflamme instability [48].”

• cites [48] [48] R. Gregory and R. Laflamme, 'Black strings and p-branes are unstable,' Phys. Rev. Lett. 70 (May, 1993) 2837-2840. - sourcePartly supported

  The discussion on p. 8 says compactifying the extra dimensions on a sufficiently small scale can avoid the instability and may make the resulting black-doughnut solutions stable. This supports stability depending on mass and a compactification scale, but the paper does not state a stability range for a ratio M/R or establish the claim for solutions generally; its analysis concerns uncharged black strings and branes. (LLM, conf=0.82)

  Experimental — accuracy not yet measured

If b1>0, the Ricci-flat space M_int of dimension D-4 can be written as a free quotient of T^k × M~^{D-4-k}_int where M~ is also Ricci-flat.

“In the case where b 1 > 0, the Ricci-flat space M int of dimension D -4 can be written as a free quotient of T k × ˜ M D -4 -k int where ˜ M D -4 -k int is also Ricci-flat [57].”

• cites [57] [57] A. E. Fischer and J. A. Wolf, 'The structure of compact Ricci-flat Riemannian manifolds,' Journal of Differential Geometry 10 (1975), no. 2, 277 - 288. - sourcePartly supported

  Theorem 4.1 (§4) gives a free finite quotient of a flat k-torus times a Ricci-flat (n−k)-manifold, with k equal to the first Betti number. It applies to compact connected Ricci-flat manifolds; the citing sentence does not state that restriction, so the source does not support the claim for Ricci-flat spaces generally. (LLM, conf=0.85)

  Experimental — accuracy not yet measured

M_int is a free quotient of a Riemannian product of a torus and a connected Ricci-flat space with vanishing b1.

“However the decomposition theorem of [57] states that M int is a free quotient of a Riemannian product of a torus and a connected Ricci-flat space with vanishing b 1 .”

• cites [57] [57] A. E. Fischer and J. A. Wolf, 'The structure of compact Ricci-flat Riemannian manifolds,' Journal of Differential Geometry 10 (1975), no. 2, 277 - 288. - sourceSupported

A scalar field in four dimensions with null sources does not admit a 1/r expansion near I+ when J ∼ 1/r^2; rather one must include log(r)/r^n terms in the expansion.

“However, this is not the case for a scalar field in four dimensions with null sources [65]. A scalar field φ in Minkowski spacetime satisfying”

• Also stated in: § A.1 Constraints on the asymptotic expansion · p. 62 · plain

  “does not admit a 1 r expansion near I + when J ∼ 1 r 2 , which is a configuration with finite flux through I + . Rather one must include log( r ) r n terms in the expansion.”

• cites [65] [65] G. Satishchandran and R. M. Wald, 'Memory effect for particle scattering in odd spacetime dimensions,' Phys. Rev. D97 (2018), no. 2, 024036, 1712.00873 . - sourceCould not verify

  The Introduction (p. 2) says the paper studies odd spacetime dimensions, and Section II (p. 5) models scattering particles with timelike worldlines. Searches for four-dimensional null sources, a J ∼ 1/r² source, and log(r)/rⁿ terms found no passage addressing the claimed expansion. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

From a four-dimensional perspective, the color charge would therefore appear to change because of a burst of radiation, in agreement with [19].

“From a fourdimensional perspective, the color charge would therefore appear to change because of a burst of radiation, in agreement with [19].”

• cites [19] [19] M. Pate, A.-M. Raclariu, and A. Strominger, 'Color Memory: A Yang-Mills Analog of Gravitational Wave Memory,' Phys. Rev. Lett. 119 (2017), no. 26, 261602, 1707.08016 . - sourceSupported

The Einstein news tensor is equivalent to the trace-free Bondi news tensor.

“Thus the Einstein news tensor is equivalent to the trace-free Bondi news tensor - an Einstein frame observer is insensitive to the overall breathing mode as we expect [24].”

• cites [24] [24] S. Tahura, D. A. Nichols, A. Saffer, L. C. Stein, and K. Yagi, 'Brans-Dicke theory in Bondi-Sachs form: Asymptotically flat solutions, asymptotic symmetries and gravitational-wave memory effects,' 2020. - sourcePartly supported

  The asymptotically flat solutions section (pp. 7–9) defines the Bondi news tensor as the time derivative of a trace-free tensor and treats scalar-field radiation separately. The passages from our copy do not establish that this tensor is equivalent to an Einstein-frame news tensor; the frame-to-frame equivalence is the unsupported part. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

The asymptotic symmetry group G is identical to the asymptotic symmetry group of asymptotically flat Einstein-Maxwell-scalar theory where C is replaced with the asymptotic symmetries of the electromagnetic field.

“We note that this asymptotic symmetry group is identical to the asymptotic symmetry group of asymptotically flat Einstein-Maxwell-scalar theory where C is replaced with the asymptotic symmetries of the electromagnetic field [64].”

• cites [64] [64] B. Bonga, A. M. Grant, and K. Prabhu, 'Angular momentum at null infinity in Einstein-Maxwell theory,' Phys. Rev. D 101 (2020), no. 4, 044013, 1911.04514 . - sourcePartly supported

  Appendix A (pp. 33–34) identifies the asymptotic symmetries of Einstein-Maxwell theory as BMS vector fields together with Maxwell gauge transformations parametrized by smooth functions on S². The source does not discuss Einstein-Maxwell-scalar theory or establish the claimed identity with its symmetry group, so it supports the electromagnetic-symmetry component but not that comparison. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

The asymptotic symmetry group extends the BMS group to include angle-dependent isometries of the internal space.

“This asymptotic symmetry group extends the BMS group to include angle-dependent isometries of the internal space.”

The paper develops a general formalism for treating radiative degrees of freedom near I+ in theories with an arbitrary Ricci-flat internal space.

“We develop a general formalism for treating radiative degrees of freedom near I + in theories with an arbitrary Ricci-flat internal space.”

Radiative modes are encoded in a generalized news tensor which decomposes into gravitational, electromagnetic, and scalar components.

“These radiative modes are encoded in a generalized news tensor which decomposes into gravitational, electromagnetic, and scalar components.”

There is a preferred gauge which simplifies the asymptotic analysis of the full nonlinear Einstein equations and makes the asymptotic symmetry group transparent.

“We find a preferred gauge which simplifies the asymptotic analysis of the full nonlinear Einstein equations and makes the asymptotic symmetry group transparent.”

Solutions to eq. (2.12) are harmonic one-forms.

“Solutions to eq. (2.12) are harmonic one-forms.”

After imposing Einstein's equations, the only non-vanishing components of C^(0)_NPQR is the Riemann tensor R_npqr of the Ricci-flat asymptotic internal space M_int with metric ĝ_mn.

“After imposing Einstein's equations the only non-vanishing components of C (0) NPQR is the Riemann tensor R npqr of the Ricci-flat asymptotic internal space M int with metric ˆ g mn .”

The space of massless linearized perturbations h_mn can be decomposed into a set of d_L+1 scalar fields.

“Therefore, the space of massless linearized perturbations h mn can be decomposed into a set of d L +1 scalar fields”

Memory is determined by terms that decay at O(1/r^2) near I+.

“Memory is determined by terms that decay at O ( 1 r 2 ) near I + .”

Including the overall volume modulus, there are 1/2 (D-4)(D-3) metric moduli for a torus.

“Including the overall volume modulus, there are 1 2 ( D -4)( D -3) metric moduli.”

The Lie algebra for these spin-1 massless gauge-fields is determined by the isometry group of M_int.

“The Lie algebra for these spin-1 massless gauge-fields is determined by the isometry group of M int .”

The asymptotic expansion of the metric (3.5) implies the 1/r expansion for the Weyl tensor.

“The asymptotic expansion of the metric given by (3 . 5) implies the 1 r expansion for the Weyl tensor:”

The linearized Einstein equations imply that the scalars φ and Φ^(i) satisfy the massless wave equation.

“Finally, the linearized Einstein equations imply that the scalars φ and Φ ( i ) satisfy the massless wave equation,”

The number of truly massless volume-preserving scalars is d_L_hat ≤ d_L.

“However, the number of truly massless volume-preserving scalars are ˆ d L ≤ d L .”

The scalar fields φ and Φ^(i) have no diffeomorphism freedom.

“Consequently the scalar fields φ and Φ ( i ) in eq. (2.29) have no diffeomorphism freedom.”

The Riemann tensor for the metric (3.4) decays like 1/r.

“The Riemann tensor for the metric (3 . 4) decays like 1 r .”

From ansatz (3.5) and Appendix A, ∫ M_int ĝ^mn G^(2)_mn = 0.

“From our ansatz (3 . 5) and the analysis found in Appendix A, we see that ∫ M int ˆ g mn G (2) mn = 0.”

• cites Appendix ANot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The number of linearly independent harmonic one-forms on M_int is counted by the first Betti number b1.

“The number of linearly independent harmonic one-forms on M int is counted by the first Betti number, b 1 , which is a topological invariant.”

For an isolated system, the nonvanishing components of E_MN satisfy the decomposition given in Lemma 3.

“2. The nonvanishing components satisfy”

The non-vanishing components of E_MN can be compactly expressed in terms of N_ab.

“Finally, a direct calculation of E MN in terms of the metric implies that the non-vanishing components of E MN can be compactly expressed in terms of N ab :”

Equations (4.5) and (4.6) place strong constraints on the asymptotic behavior of the electric part of the Weyl tensor near null infinity.

“We now show that eqs. (4.5) and (4.6) place strong constraints on the asymptotic behavior of the 'electric part' of the Weyl tensor near null infinity.”

In a theory with extra dimensions, the radiative degrees of freedom near I+ are encoded in a generalized news tensor N_ab, with indices running over S2 and M_int.

“In a theory with extra dimensions, we will show that the radiative degrees of freedom near I + are encoded in a generalized news tensor written as N ab , where the indices a, b now run over both the the asymptotic 2-sphere S 2 and the internal space M int .”

The hybrid stress tensor is conserved.

“This is conserved.”

The space of solutions to eq. (2.12) is the space of Killing vectors on M_int.

“The space of solutions to eq. (2.12) is therefore the space of Killing vectors on M int .”

In the full nonlinear theory, the leading order electric Weyl tensor for any isolated system at null infinity admits a harmonic decomposition in a way analogous to linearized Kaluza-Klein analysis.

“In the full nonlinear theory, we will show that the leading order electric Weyl tensor for any isolated system at null infinity admits a harmonic decomposition in a way analogous to linearized KaluzaKlein analysis.”

A generic internal manifold supports additional radiative modes encoded in N_Am and N_mn, involving fluctuations in the internal manifold M_int.

“However, we will see that a generic internal manifold will support additional radiative modes encoded in N Am and N mn , which involve fluctuations in the directions of the internal manifold M int .”

Massless perturbations h_mn are spanned by tensor fields T_mn satisfying eq. (2.22).

“Therefore massless perturbations h mn are spanned by the tensor fields on T mn ( y m ) which satisfy”

Taking the trace of eq. (2.22) implies that U is a constant.

“which implies that U is a constant.”

G^(2)_mn is orthogonal to every exactly massless scalar fluctuation t_mn, not just the breathing mode of M_int.

“In fact, G (2) mn is orthogonal to every exactly massless scalar fluctuation t mn , not just the breathing mode of M int .”

In Kaluza-Klein reduction near I, F=dA gives rise to massless spacetime fields associated to harmonic forms on M_int.

“In Kaluza-Klein reduction near I , F = dA gives rise to massless spacetime fields associated to harmonic forms on M int as”

The first two terms can be field redefined away.

“However the first two terms can be field redefined away.”

Any contractions of Riemann with metrics will also decay at O(1/r) or faster.

“Any contractions of Riemann with metrics will also decay at O ( 1 r ) or faster.”

Bondi gauge does not appear to be the preferred gauge when asymptotically solving the leading order Einstein equations with compact spatial directions.

“We shall see, however, that Bondi gauge does not appear to be the preferred gauge when asymptotically solving the leading order Einstein equations with compact spatial directions, studied in sections 4.2 and 5.1.”

The stress tensor fall-off (3.7) directly implies an asymptotic expansion of the Schouten tensor.

“This stress tensor fall-off directly implies an asymptotic expansion of the Schouten tensor,”

• AssumptionThe authors state this as an assumption: For massless sources, the stress-energy tensor falls off as specified in eq (3.7), with non-vanishing leading components T^(2)_uu, T^(2)_um and T^(2)_mn.

The leading order electric part of the Weyl tensor can be dimensionally-reduced in exactly the same manner as in section 2, but now in the full nonlinear theory.

“In particular, the leading order electric part of the Weyl tensor can be dimensionally-reduced in exactly the same manner as reviewed in section 2, but now in the full nonlinear theory.”

A harmonic V_m = g^mn V_n is a Killing vector if M_int is Ricci-flat.

“Further a harmonic V m = ˆ g mn V n is a Killing vector if M int is Ricci-flat.”

If D>5, then S is a constant.

“If D > 5 then eq. (2.27) implies that S is a constant.”

On any Ricci-flat manifold, the coordinate Laplacian can be replaced by minus the Hodge Laplacian when acting on one-forms.

“Therefore on any Ricci-flat manifold, the coordinate Laplacian can be replaced by (minus) the Hodge Laplacian when acting on one-forms.”

The components N_AB encode the familiar Bondi news contribution N_AB as well as an additional scalar breathing mode N that gives rise to gravitational radiation in the non-compact directions.

“The components N AB will encode the familiar Bondi news contribution N AB as well as an additional scalar breathing mode N which give rise to gravitational radiation in the non-compact directions.”

For an isolated system whose metric has asymptotic expansion (3.5), the leading order electric Weyl tensor E_MN satisfies E_uM and E_rM vanish for any isolated system.

“1. The components E uM and E rM vanish for any isolated system.”

Any solution to eqs. (2.22) and (2.23) can be uniquely decomposed in the form eq. (2.28), where U is a constant and t_mn is trace-free and satisfies eqs. (2.22) and (2.23).

“Any solution to eqs. (2.22) and (2.23) can be uniquely decomposed in the form,”

One can construct a hydrodynamic model for a D=4 particle that interpolates between the black string and the completely localized black hole.

“Using this observation we can actually construct a model for a D = 4 particle, at the level of hydrodynamics, which interpolates between the black string and the completely localized black hole.”

For a wrapped string with 2πμR=m, the right hand side of eq. (1.27) vanishes, as expected for a model that replicates a standard D=4 black hole.

“with 2 πµR = m . This makes the right hand side of (1 . 27) vanish as we expect for a model that replicates a standard D = 4 black hole.”

Expanding h_μν in eigenfunctions of the Laplacian on M_int yields an infinite tower of massive modes, one for each eigenvalue, with mass scale set by the size of the compact extra dimensions.

“Expanding h µν in terms of eigenfunctions of the Laplacian on M int , eq. (2.8) yields an infinite tower of massive modes (one for each eigenvalue). The mass scale is set by the size of the compact extra dimensions.”

The coefficients A^(i)_μ(x) define a set of b1 graviphoton vector fields on R4.

“The coefficients A ( i ) µ ( x ) define a set of b 1 graviphoton vector fields on R 4 .”

The space of solutions to eqs. (2.22) and (2.23) is the moduli space of infinitesimal deformations that preserve the vanishing of the Ricci tensor.

“The space of solutions to eqs. (2.22) and (2.23) is the moduli space of infinitesimal deformations that preserve the vanishing of the Ricci tensor.”

In theories with these extra modes from compact dimensions, the null stress energy appearing in equation (1.12) receives additional contributions.

“In theories with these extra modes arising from compact dimensions, the null stress energy appearing in equation (1.12) receives additional contributions; one now has”

If the truncation from d_L to d_L_hat had not been done, the ansatz would not be consistent with Einstein's equations.

“As we show in Appendix A, if we had not done this truncation then our ansatz would not be consistent with Einstein's equations.”

• cites Appendix ANot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

For a p-form field strength F, at this order T^(2)_mn = 0.

“As noted in (3 . 2), K is null with respect to the asymptotic metric so T (2) mn = 0 again as in the case of the scalar field.”

For an isolated system satisfying the ansatz and stationarity conditions, there exists a unique diffeomorphism preserving gauge and stationarity such that the leading order expansion coefficients satisfy the relations of Lemma 5.

“Lemma 5. Let ( M,g ) be an isolated system that satisfies both our ansatz (3 . 5) in a gauge compatible with Lemma 4 and our stationarity conditions. There exists a unique diffeomorphism which preserves these gauge and stationarity conditions such that the leading order expansion coefficients satisfy the following relations:”

• AssumptionThe authors state this as an assumption: There exists a gauge in which the metric satisfies the stationarity conditions at asymptotically early and late times.

After contracting both sides with a tensor field t_mn(y) annihilated by Lichnerowicz, the right hand side vanishes.

“After contracting both sides with a tensor field t mn ( y ) which is annihilated by Lichnerowicz, it is straightforward to check that the right hand side vanishes.”

To preserve Lemma 4, ξ^(0)_m must be harmonic on M_int.

“To preserve Lemma 4, ξ (0) m must be harmonic on M int .”

The following components of the nonlinear part of the Ricci tensor vanish in a stationary era and in our gauge at order 1/r^3.

“By a lengthy but straightforward calculation, the following components of the nonlinear part of the Ricci tensor vanish in a stationary era and in our gauge at order 1 r 3 :”

Equation (4.38) implies that the sum q^AB h_AB + φ can have, at most, linear dependence on retarded time u.

“Finally eq. (4.38) implies that the sum q AB h AB + φ can have, at most, linear-dependence on retarded time u .”

q^AB ̃E_AB = ˆg^mn ̃E_mn.

“Finally, we again have that q AB ˜ E AB = ˆ g mn ˜ E mn .”

The volume mode is unobstructed.

“It is straightforward to check that the volume mode, as expected, is unobstructed.”

Equations (4.29), (5.6), (5.7) and (5.9) imply that h^(1)_uu, h^(1)_ur, φ, Φ^(i)_mn and A^(1;i)_u are spherically symmetric.

“Equations (4.29), (5.6), (5.7) and (5.9) imply that h (1) uu , h (1) ur , φ, Φ ( i ) mn and A (1; i ) u are spherically symmetric and”

Equation (B.17) constrains the scalar part of TF[ ̃E_AB] on the 2-sphere.

“Equation (B.17) constrains the scalar part of TF[ ˜ E AB ] on the 2-sphere.”

For such a product, £_V Φ_mn = 0 since V is one of the torus isometries.

“For such a product, £ V Φ mn = 0 since V is one of the torus isometries.”

This truncation also ensures that our ansatz is consistent with Einstein's equations at order 1/r^3.

“As we will see in section A.3, this truncation also ensures that our ansatz is consistent with Einstein's equations at order 1 r 3 .”

F^(i)_μν is gauge invariant.

“This field strength is now gauge invariant and ∂ µ is the derivative operator compatible with the flat metric η µν .”

The composition of a supertranslation and an angle-dependent internal isometry only affects the zero-modes of the leading order metric.

“So the composition of a supertranslation and an angle-dependent isometry only affects the zero-modes of the leading order metric.”

The space of exactly massless modes ˆd_L ≤ d_L is smaller than the kernel of Lichnerowicz.

“As we spelled out in section 2.2, the space of exactly massless modes ˆ d L ≤ d L is smaller than the kernel of Lichnerowicz.”

In this gauge, the news tensor defined in (4.18) is related to the leading order metric by the stated expression.

“In this gauge the news tensor, defined in (4 . 18), is very nicely related to the leading order metric by:”

∆φ^(1)_mn is spanned only by l=0 spherical harmonics.

“We first note that, by Lemma 5, ∆ ϕ (1) mn is clearly spanned only by ℓ = 0 spherical harmonics.”

The non-vanishing components of R^(2)_MN can be written entirely in terms of the news eq. (4.46).

“A direct calculation of R (2) MN in the gauge of Lemma 4 shows that the non-vanishing components of R (2) MN can be written entirely in terms of the news eq. (4.46).”

• AssumptionThe authors state this as an assumption: The metric is in the gauge described by Lemma 4.

The zero modes of R^(2)_uu and R^(2)_um determine the change in the metric due to a burst of radiation.

“More precisley, we find that the zero modes of R (2) uu and R (2) um determine the change in the metric due to a burst of radiation.”

By Lemma 4 the only non-vanishing component of ̃E_μν is ̃E_AB.

“By Lemma 4 the only non-vanishing component of ˜ E µν is ˜ E AB .”

The components of electromagnetic and scalar radiative degrees of freedom are unchanged in Einstein frame.

“The components of electromagnetic and scalar radiative degrees of freedom are unchanged:”

The vector part of the angle-angle components of memory are determined by the magnetic Weyl tensor on R^4.

“The vector part of the angle-angle components of memory are determined by the magnetic Weyl tensor on R 4 given by,”

Since the right hand side of eq (4.37) is in the kernel of the Laplacian D^2, the left and right hand sides must both vanish.

“Since the right hand side of eq. (4.37) is in the kernel of the Laplacian D 2 , the left and right hand sides must both vanish implying”

Any metric which admits an asymptotic expansion (3.5) and which satisfies the Einstein equations can be put into the gauge of Lemma 4.

“We note that any metric which admits an asymptotic expansion (3 . 5), and which satisfies the Einstein equations, can be put into this gauge.”

∆A^(1)_Am is orthogonal to the l=0 spherical harmonics.

“Furthermore, by the form of eq. (5.29), we have that ∆ A (1) Am is orthogonal to the ℓ = 0 spherical harmonics.”

The R^4 components of the linearized electric Weyl tensor satisfy the same equations as the components of the linearized electric Weyl tensor in flat spacetime.

“Therefore, the R 4 components of the linearized electric Weyl tensor satisfy the same equations as the components of the linearized electric Weyl tensor in flat spacetime.”

In the gauge satisfying eq (4.32), eqs (4.26) to (4.28) imply that h^(1)_ur, h^(1)_uA and h^(1)_AB are constant on M_int.

“In this gauge, eqs. (4.26) to (4.28) imply that h (1) ur , h (1) uA and h (1) AB are constant on M int .”

Compactifying the higher-dimensional gravity theory with a D-4 form field strength on S^{D-4} with radius L gives an effective four-dimensional potential for the radius L.

“Compactifying this theory on S D -4 with radius L gives an effective four-dimensional potential for the radius L of the form:”

The internal velocity vector can rotate as higher-dimensional gravitational radiation passes by.

“The novelty in this case, by comparison with the Ricci-flat case, is that the internal velocity vector can rotate as higher-dimensional gravitational radiation passes by.”

Truncating the linearized massless moduli to exactly massless moduli yields a solution consistent with our ansatz and Einstein's equations at order 1/r^2.

“Thus, as in Lemma 4, we truncate the linearized massless moduli to exactly massless moduli and obtain a solution consistent with our ansatz and Einstein's equations at order 1 r 2 .”

Truncating to exactly massless scalar fluctuations guarantees solvability of eq. (A.43).

“As in our discussion of section A.1, we again truncate to exactly massless scalar fluctuations for which the right hand side of eq. (A.43) has no Lichnerowicz zero modes. This guarantees solvability of eq. (A.43).”

In the presence of compact extra dimensions, the tracelessness of the Weyl tensor implies that η^μν ̃E_μν vanishes if and only if ˆg^mn ̃E_mn vanishes.

“In flat spacetime this quantity does vanish but, in the presence of compact extra dimensions, the tracelessness of the Weyl tensor implies that η µν ˜ E µν vanishes if only if ˆ g mn ˜ E mn vanishes.”

The enlarged asymptotic symmetry group G is the semi-direct product of B with the abelian group C.

“Therefore, the enlarged asymptotic symmetry group ( G ) is the semi-direct product of B with the abelian group C :”

There exists a unique diffeomorphism which preserves ansatz eq (3.5) such that the leading order expansion coefficients of the metric have the properties listed in Lemma 4.

“There exists a unique diffeomorphism which preserves our ansatz such that the leading order expansion coefficients of the metric have the following properties:”

The remaining components of R^(2)_MN vanish.

“The remaining components of R (2) MN vanish.”

• AssumptionThe authors state this as an assumption: The metric is in the gauge described by Lemma 4.

The asymptotic symmetry group of the spin-2 degrees of freedom is the BMS group B.

“Therefore, the asymptotic symmetry group of the spin-2 degrees of freedom is the BMS group B .”

For spaces with non-abelian isometry groups, like the sphere, the Christoffel symbols along Killing directions do not vanish.

“For spaces with non-abelian isometry groups, like the sphere, this is no longer true.”

There exists a residual gauge transformation, compatible with Lemma 4, which allows setting c=0.

“We now show that there exists a residual gauge transformation, compatible with Lemma 4, which allows us to set c = 0.”

There is no choice of asymptotic vector field which preserves the asymptotic conditions and the gauge conditions of Lemma 4 that can affect either Φ_mn or φ.

“There is no choice of asymptotic vector field which preserves our asymptotic conditions and the gauge conditions given in Lemma 4 that can affect either Φ mn or φ .”

Using gauge conditions, harmonicity of the spacetime components h^(1)_μν and eq (4.39), the (um;1) and (Am;1) components of Einstein's equation imply that A^(1)_um and A^(1)_Am are harmonic with decomposition.

“Using our gauge conditions, harmonicity of the spacetime components h (1) µν and that eq. (4.39) implies D m ϕ (1) mn = 0, the ( um ; 1) and ( Am ; 1) components of Einstein's equation imply that A (1) um and A (1) Am are harmonic with decomposition”

Our stationarity conditions imply constraints on the angular behavior of the metric at leading order in 1/r.

“Our stationarity conditions turn out to imply constraints on the angular behavior of the metric at leading order in 1 r .”

• AssumptionThe authors state this as an assumption: There exists a gauge in which the metric satisfies the stationarity conditions at asymptotically early and late times.

The trace q^AB ̃E_AB is equivalent to -ˆg^mn ̃E_mn.

“Since the trace q AB ˜ E AB is equivalent to -ˆ g mn ˜ E mn we shall focus on the trace-free part of ˜ E AB on the 2-sphere.”

The tracefree part of ∆h^(1)_AB on S^2 is orthogonal to the l=0,1 spherical harmonics.

“Therefore, it follows that the tracefree part of ∆ h (1) AB on S 2 is orthogonal to the ℓ = 0 , 1 spherical harmonics.”

When one has compact extra dimensions, η^μν ̃E_μν is non-vanishing.

“One major difference is that, when one has compact extra dimensions, η µν ˜ E µν is non-vanishing.”

Elements of the group C of angle-dependent internal isometries do not commute with Lorentz transformations in L.

“We note that elements of this group do not commute with Lorentz transformations in L .”

The leading order linearized electric Weyl tensor has non-vanishing components ̃E_AB and ̃E_Am that are harmonic on M_int.

“Lemma 3 applies to the leading order linearized electric Weyl tensor, which has non-vanishing components ˜ E AB and ˜ E Am that are harmonic on M int .”

Any such ξ^(0)_m is a smooth function S(θ) multiplied by a Killing vector V^m(y) on M_int.

“Any such ξ (0) m is a smooth function S ( θ ) multiplied by a Killing vector V m ( y ) on M int ,”

The system of equations given by eq. (B.20) are identical to their analogous equations in flat spacetime.

“The system of equations given by eq. (B.20) are therefore identical to their analogous equations in flat spacetime.”

The component ̃E_mn satisfies the Lichnerowicz equation on M_int.

“The component ˜ E mn satisfies the Lichnerowicz equation on M int .”

The linearized magnetic Weyl tensor satisfies the same relations as the linearized magnetic Weyl tensor in flat spacetime.

“Therefore, the linearized magnetic Weyl tensor satisfies the same relations as the linearized magnetic Weyl tensor in flat spacetime.”

Aside from special cases like M_int = T^k, neither Ψ^(2) nor Φ^(2)_mn are zero modes on M_int.

“Aside from special cases like M int = T k , neither Ψ (2) nor Φ (2) mn are zero modes on M int .”

The zero mode of eq. (A.20) is manifestly non-vanishing unless N_ab = 0.

“The zero mode of eq. (A.20) is manifestly non-vanishing unless N ab = 0.”

∆Φ^(i) and ∆φ in Theorem 1 are gauge invariant quantities.

“We note that ∆Φ ( i ) and ∆ φ in Theorem 1 are actually gauge invariant quantities and therefore, the derivation of scalar memory is exactly analogous to the derivation in the nonlinear theory:”

The flux F(θ) can be decomposed into gravitational, electromagnetic and scalar contributions.

“In analogy with the decomposition of the news in eqs. (4.20) and (4.21) we can decompose the flux F ( θ ) into gravitational, electromagnetic and scalar contributions to the flux:”

The non-vanishing components of the velocity kick are ∆v_A^(1) and ∆v_r^(1).

“Expanding eq. (6.20) in powers of 1 r and integrating the geodesic equation a straightforward computation yields in the gauge described by Lemma 4 that the non-vanishing components of the velocity kick are ∆ v A (1) and ∆ v r (1) .”

The exactly massless volume-preserving moduli satisfy eq. (A.28).

“The exactly massless volume-preserving moduli satisfy eq. (A.28).”

The diffeomorphisms that act on A^(1)_μm and preserve metric asymptotics (3.5) are generated by ξ^(0)_m(θ), which cannot depend on u.

“The diffeomorphisms that act on A (1) µm and preserve our metric asymptotics (3 . 5) are generated by ξ (0) m ( θ ), which cannot depend on u .”

Applying ĝ^mn to eq (4.31) and using eqs (4.32) and (4.36) implies that the trace-free scalar part of Φ_mn vanishes.

“which, by Proposition 2, implies that the trace-free scalar part of Φ mn vanishes.”

The magnetic parity parts of A^(1;i)_A, h^(1)_uA and h^(1)_AB vanish.

“and therefore the magnetic parity parts of A (1; i ) A , h (1) uA and h (1) AB vanish.”

The dominant contribution to eq. (1.10) is the null memory term.

“Therefore, the dominant contribution to eq. (1.10) is the null memory term.”

This range includes masses light enough that we can simply ignore the mass and treat the scalar as massless for the purposes of detection by a gravitational wave detector.

“This is a huge range of masses but it certainly includes masses light enough that we can simply ignore the mass and treat the scalar as massless for the purposes of detection by a gravitational wave detector.”

The frame-dependence can contribute to the memory in competition with null stress-energy as long as the associated scalar fields can be treated as massless.

“The frame-dependence can therefore contribute to the memory in competition with null stress-energy as long as the associated scalar fields can be treated as massless.”

• AssumptionThe authors state this as an assumption: The stress tensor obeys the dominant energy condition with O(1/r^2) decay for large r.

The scalars from complex structure moduli have a mass in the range of 10^-14 - 10^4 eV for a Kaluza-Klein scale ranging from 1 eV - 1 TeV.

“The scalars then have a mass in the range of 10 -14 -10 4 eV for a Kaluza-Klein scale ranging from 1 eV -1 TeV.”

• AssumptionRests on an assumption: For a string coupling of order one, the string scale and Planck scale are comparable: M_s ∼ M_D=10p.

Memory is insensitive to higher derivative corrections.

“Based on this discussion, it appears that memory is insensitive to higher derivative corrections.”

The decomposition given by Lemma 3 is a consequence of the leading order Bianchi identity and Einstein's equations.

“The decomposition given by Lemma 3 is a consequence of the leading order Bianchi identity and Einstein's equations.”

The discrepancy is captured in the four-dimensional effective stress tensor T(2)uu, which includes electromagnetic and scalar contributions from higher-dimensional gravity modes.

“This discrepancy is captured in the four-dimensional effective stress tensor T (2) uu , which includes the electromagnetic and scalar contributions from the higher-dimensional gravity modes.”

For Calabi-Yau, G2 and Spin(7) spaces, all zero modes seen in a linear analysis survive to the full nonlinear theory.

“For Calabi-Yau, G 2 and Spin (7) spaces, all zero modes seen in a linear analysis survive to the full nonlinear theory [60].”

• cites [60] [60] R. Goto, 'Moduli Spaces of Topological Calibrations, Calabi-Yau, Hyperk¨ ahler, G 2 and Spin(7) Structures,' International Journal of Mathematics 15 (2004), no. 03, 211-257, https://doi.org/10.1142/S0129167X04002296 . -Not checked yet

  Not checked in the free run.

Terms of O(R^3) are already decaying too fast to affect memory.

“This means that terms of O ( R 3 ) are already decaying too fast to affect memory.”

For large R, the localized black hole configuration is the preferred solution.

“For large R , the localized black hole configuration is the preferred solution.”

The luminosity of gravitational waves released in the merger process is about 20% less than the merger of four-dimensional black holes mainly because of scalar radiation produced in the merger.

“The luminosity of gravitational waves released in the merger process is about 20% less than the merger of four-dimensional black holes mainly because of scalar radiation produced in the merger.”

• cites [50] [50] R. Durrer and P. Kocian, 'Testing extra dimensions with the binary pulsar,' Class. Quant. Grav. 21 (2004) 2127-2138, hep-th/0305181 . -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [51] [51] Y. Du, S. Tahura, D. Vaman, and K. Yagi, 'Probing Compactified Extra Dimensions with Gravitational Waves,' 2004.03051 . -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Black holes with a size small compared to the circle radius R look locally like a D=5 Myers-Perry solution.

“Black holes with a size small compared to R look locally like a D = 5 Myers-Perry solution [47].”

• cites [47] [47] R. Myers and M. Perry, 'Black holes in higher dimensional space-times,' Annals of Physics 172 (1986), no. 2, 304-347. - sourceNot checked — no full text of the source

A completely delocalized black string wrapping the y direction is much more likely.

“Much more likely is a completely delocalized black string wrapping the y direction.”

The distance scale corresponding to astrophysical black hole mergers is ten orders of magnitude larger than the best upper bound on the Kaluza-Klein scale.

“This is ten orders of magnitude larger than the best upper bound on the Kaluza-Klein scale.”

• AssumptionRests on an assumption: Astrophysical black hole mergers detectable by LIGO have constituent masses of roughly O(10) solar masses, corresponding to a distance scale of O(10^4) m.

The ordinary electromagnetic and scalar memory effects generate a velocity kick for a suitable charged test particle.

“The ordinary electromagnetic and scalar memory effects generate a velocity kick for a suitable charged test particle.”

Memory can be used to detect new physics.

“In Section 1.1 we saw that memory can be used to detect new physics.”

The mismatch gets worse with more compact dimensions.

“The situation gets worse with more compact dimensions.”

The t^(i)_mn are a basis of the d_L symmetric, rank 2 tensors which satisfy the Lichnerowicz equation on M_int and D_m t^(i)_mn = ĝ^mn t^(i)_mn = 0, where d_L+1 is the dimension of the moduli space.

“The t ( i ) mn are a basis of the d L symmetric, rank 2 tensors which satisfy the Lichnerowicz equation on M int and D m t ( i ) mn = ˆ g mn t ( i ) mn = 0 , where d L + 1 is the dimension of the moduli space.”

The results about asymptotic dimensional reduction of the Weyl tensor, local constraints on the radiative order metric and asymptotic symmetries in sections 4.1 to 4.3 remain valid as long as the metric asymptotes to ĝ_MN at I+.

“In particular, our results about the asymptotic dimensional reduction of the Weyl tensor, the local constraints on the radiative order metric and asymptotic symmetries, found in sections 4.1 to 4.3, remain valid as long as the metric asymptotes to ˆ g MN at I + .”

For circle compactifications, the binary merger of black holes localized at a point was studied using a point particle approximation.

“For circle compactifications, the binary merger of black holes localized at a point was studied in [50, 51] using a point particle approximation.”

• cites [50] [50] R. Durrer and P. Kocian, 'Testing extra dimensions with the binary pulsar,' Class. Quant. Grav. 21 (2004) 2127-2138, hep-th/0305181 . -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [51] [51] Y. Du, S. Tahura, D. Vaman, and K. Yagi, 'Probing Compactified Extra Dimensions with Gravitational Waves,' 2004.03051 . -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

There are many potential stress tensors that could describe black objects in R4×S1 with varying amounts of scalar charge from the D=4 perspective, whose dynamics can be made consistent with current observation.

“The upshot is that there are many potential stress tensors that could describe black objects in R 4 × S 1 with varying amounts of scalar charge from the D = 4 perspective, whose dynamics can be made consistent with current observation.”

Lemma 3 implies that the non-vanishing components of the leading order electric Weyl tensor E_MN can be viewed as a tensor on S^2 × M_int.

“Lemma 3 implies that the non-vanishing components of the leading order electric Weyl tensor, E MN , can be viewed as a tensor on S 2 × M int .”

Einstein's equations imply that the zero mode ∫ M_int ĝ^mn T^(2)_mn vanishes.

“Einstein's equations then imply that the zero mode, ∫ M int ˆ g mn T (2) mn , vanishes.”

If one can measure the trajectory of point particles undergoing geodesic motion relative to a suitably stationary lab frame, one can in principle extract all of N_Am and a part of N_mn.

“Instead, if one can measure the trajectory of point particles - even a single point particle - undergoing geodesic motion, relative to a lab frame which is stationary in an appropriate sense, then one can in principle extract all of N Am and a part of N mn described in section 6.”

This provides a gauge invariant description of radiation, Kaluza-Klein decomposed into spin-0, spin-1 and spin-2 components, in full nonlinear general relativity.

“This provides a gauge invariant description of radiation, Kaluza-Klein decomposed into spin-0, spin-1 and spin-2 components, in full nonlinear general relativity.”

The decomposition of radiative modes given by eq (4.21) corresponds to the exactly massless modes arising from M_int.

“The decomposition of the radiative modes given by eq. (4.21) corresponds to the exactly massless modes arising from M int .”

In string compactifications there can be many scalar fields with masses potentially below the Kaluza-Klein scale.

“The upshot of this stringy top down look at compactified extra dimensions is that there can be many scalar fields with masses potentially below the Kaluza-Klein scale.”

Given a particular model of the stress-energy, gravitational observatories can independently measure arm motion and gravitational memory; disagreement indicates a missing contribution to the stress-energy.

“More precisely, given a particular model of the stress-energy in a theory, gravitational observatories can make independent measurements of arm motion and of gravitational memory, and then compare these measurements; disagreement indicates a missing contribution to the stressenergy.”

Because M is much greater than the range of Kaluza-Klein scales, the generic compact object should be delocalized.

“M is clearly much greater than the range of Kaluza-Klein scales discussed in section 1.2, and therefore one should expect that the generic compact object will be delocalized.”

• AssumptionRests on an assumption: Astrophysical black hole mergers detectable by LIGO have constituent masses of roughly O(10) solar masses, corresponding to a distance scale of O(10^4) m.

This moduli space is finite-dimensional.

“This moduli space is known to be finite-dimensional [58].”

• cites [58] [58] A. Besse, Einstein Manifolds . Classics in mathematics. Springer, 1987. - sourceNot checked — no full text of the source

These additional sources of news are the primary signatures of extra dimensions that might be seen with memory measurements alone.

“These additional sources of news are the primary signatures of extra dimensions we might hope to see with memory measurements alone.”

The V^(i)_m are a basis for the b1 harmonic 1-forms on M_int, where b1 is the first Betti number of M_int.

“The V ( i ) m are a basis for the b 1 harmonic 1 -forms on M int , where b 1 is the first Betti number of M int .”

For a particular model of the null stress energy T(2)uu that should contribute to memory, the presence of extra compact dimensions will generate a discrepancy between predicted and measured memory effects.

“Therefore, for a particular model for the null stress energy T (2) uu that should contribute to memory, the presence of extra compact dimensions will generate a discrepancy between the predicted and measured memory effects.”

In theories with extra dimensions, a particle with velocity in the internal directions will experience a velocity kick in R4 because of gravitational radiation in the internal space.

“Likewise, in theories with extra dimensions, a particle with velocity in the internal directions will experience a velocity kick in R 4 because of the passage of gravitational radiation in the internal space.”

Taken at face value, the mismatch would seem to rule out this simple model of compact extra dimensions.

“Taken at face value, this would seem to rule out this simple model of compact extra dimensions.”

The derivation of scalar memory is exactly analogous to the derivation in the nonlinear theory.

“We note that ∆Φ ( i ) and ∆ φ in Theorem 1 are actually gauge invariant quantities and therefore, the derivation of scalar memory is exactly analogous to the derivation in the nonlinear theory:”

The natural four-dimensional frame corresponding to studying radiation in terms of the D-dimensional metric is not Einstein frame.

“As illustrated in the circle example of section 6.2, the natural four-dimensional frame that corresponds to studying radiation in terms of the D -dimensional metric is not Einstein frame.”

V_A and T_AB can be decomposed uniquely as in eqs. (2.14) and (2.24) on S^2.

“Therefore, V A and T AB can be both be decomposed uniquely as in eqs. (2.14) and (2.24) where the covariant derivative is now the derivative operator D A compatible with metric q AB .”

There is a general obstruction to integrating in from I+ if a specific scalar fluctuation of M_int is obstructed or gets a mass beyond the linearized approximation.

“In our case, there is a general obstruction to integrating in from I + . Namely, if a specific scalar fluctuation of M int is obstructed, or equivalently gets a mass at some order beyond the linearized approximation, then our ansatz is simply not valid for that mode.”

There are b1 Killing vectors on M_int.

“There are b 1 Killing vectors on M int .”

At leading order in 1/r, the geodesic deviation equation can be replaced with an equation using the electric Weyl tensor E_MN.

“Since we are only considering the memory effect at leading order in 1 r , we can replace v M with ( ∂/∂u ) M and R PML N v P v L with the electric Weyl tensor E M N (as defined in eq. (4.11)) in eq. (6.1) which yields”

Certain metric components appearing in Theorem 1 can be directly related to definitions of the Bondi mass aspect and electric charge aspect in R^4.

“We first note that certain metric components appearing in Theorem 1 can be directly related to definitions of the Bondi mass aspect and electric charge aspect in R 4 .”

Thus there is no remaining diffeomorphism freedom for the spin-0 modes.

“Thus there is no remaining diffeomorphism freedom for these modes.”

This difference leads to contributions from the breathing mode of M_int to the observed gravitational memory in this frame.

“This is a crucial difference that leads to contributions from the breathing mode of M int to the observed gravitational memory in this frame.”

If the test masses had some initial displacement in the internal space then, due to a change in scalar charge, the relative displacement in the internal space will change by an amount ∆_mn.

“If the test masses had some initial displacement in the internal space then, due to a change in scalar charge, the relative displacement in the internal space will change by an amount ∆ mn .”

The 1/r terms in the expansion of A_μ and e^{2φ(x)} can be identified with A^(1)_μy and φ^(1)_yy.

“The 1 r terms in the expansion of A µ and e 2 ϕ ( x ) can be identified with A (1) µy and ϕ (1) yy defined in eq. (3.1) and discussed in the preceding sections.”

The second term is the electromagnetic kick.

“The second term is the electromagnetic kick we expect.”

In both pure gravity and Maxwell-Einstein, there exists a large class of solutions which are smooth at I+ in a particular gauge.

“This might seem fairly reasonable because in both pure gravity and Maxwell-Einstein, there exists a large class of solutions which are smooth at I + in a particular gauge [68].”

• cites [68] [68] P. T. Chrusciel and E. Delay, 'Existence of nontrivial, vacuum, asymptotically simple space-times,' Class. Quant. Grav. 19 (2002) L71, gr-qc/0203053 . -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The only non-vanishing components of E_MN are E_ab = -1/2 ∂_u N_ab where a,b are along S^2 × M_int.

“Further, as noted in Lemma 3, the only non-vanishing components of E MN are E ab = -1 2 ∂ u N ab where a, b are along S 2 ×M int .”

C has the natural interpretation as the asymptotic symmetry group of the graviphotons.

“Therefore, C has the natural interpretation as the asymptotic symmetry group of the graviphotons.”