Does back reaction enforce the averaged null energy condition in semiclassical gravity?

Classified as probably not a substantial claim. May combine two claims.

Pathological solutions of the linearized version of Eq. (2.6) exist on scales L ∼ L_p,eff, and we will not regard these solutions as being physical.

“As we shall see in de- tail in Appendix A below, pathological solutions of the linearized version of Eq. (2.6) exist on scales L ∼ Lp,eff , and we will not regard these solutions as being physical.”

Probably not a substantial claim: may not pass S3 (load-bearing).

For a fixed metric gab, the state ω is a linear function of ωin.

“Note that for a fixed metric gab, the state ω is a linear function of ωin.”

Probably not a substantial claim: may not pass S4 (based).

For states in the usual Fock space, a perturbed state will be pure to first order if and only if its perturbed two-point function is pure (i.e., its mixed-frequency part vanishes).

“For states in the usual Fock space – and, presumably, also for general algebraic states – a perturbed state will be pure to first order if and only if its perturbed two-point function is pure.”

May combine two claims.

In the classical theory, the stress tensor violates the pointwise null energy condition for ξ ≠ 0, but satisfies the averaged null energy condition for all negative values of ξ, provided that suitable asymptotic fall-off conditions hold for Φ.

“In the classical theory, this stress tensor violates the pointwise null energy condition for ξ 6= 0, but satisfies the averaged null energy condition for all negative val- ues of ξ, provided that suitable asymptotic fall-off con- ditions hold for Φ.”

Probably not a substantial claim: may not pass S3 (load-bearing).

If one introduces a new variable y=y(x;τ), rewrites Eq. (4.18) as an equation for y, and neglects terms of order τ^2 and higher, the resulting reduced order equation for y need not be precisely equivalent to the reduced order equation for x.

“not be precisely equivalent to the reduced order equation”

• cites [87]Not matched

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

May combine two claims.

The modified equation of motion (4.19) is formally equivalent to Eq. (4.4) up to order O(τ), and differs from it at order O(τ^2).

“This modified equation of motion is formally equivalent”

Probably not a substantial claim: may not pass S3 (load-bearing). The quote only partly states this claim.

The final reduced order equation is uniquely determined by the requirements (i) it contains only terms zeroth or first order in τ, (ii) it is formally equivalent to Eq. (4.18) to O(τ^2), and (iii) it is of form (4.18) where the right hand side contains no time derivatives of order higher than n−1.

“should contain only terms that are zeroth order or first”

Probably not a substantial claim: may not pass S3 (load-bearing).

Although h̄ was formally treated as the small parameter, one could equivalently have started from Eq. (3.45) and used 1/α^2 instead.

“although we have formally treated h̄ as the small parameter, we could equivalently have started from Eq. (3.45) and used 1/α2 instead”

Probably not a substantial claim: may not pass S3 (load-bearing).

This series is precisely the expansion in τ of the non-runaway solution, the second term in Eq. (4.5).

“solution [the second term in Eq. (4.5)].”

Probably not a substantial claim: may not pass S3 (load-bearing).

Eq. (4.19) suffers from none of the problems of the original equation.

“However, Eq. (4.19) suffers from none of the prob- lems of the original equation.”

May combine two claims.

Eq. (4.20) is second order in time and has the form of the classical linearized Einstein equation with a given source.

“This modified equation is second order in time and simply has the form of the classical linearized Einstein equation with a given source.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Iterating the reduction of order procedure until the maximum number of time derivatives in Q is reduced to n−1 yields an equation of the same differential order in time as the original equation with τ=0.

“tion Eq. (4.18) with τ = 0.”

Probably not a substantial claim: may not pass S3 (load-bearing). The quote only partly states this claim.

By following the reduction of order steps for the radiation reaction equation, one obtains Eq. (4.19).

“obtains the equation [82,83]”

• cites [82]Not matched

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

• cites [83]Not matched

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

Classified as probably not a substantial claim.

The quantity λ/L_p in Eq. (4.20) is a h̄-independent constant.

“the quantity λ/Lp in Eq. (4.20) is a h̄-independent constant.”

Probably not a substantial claim: may not pass S3 (load-bearing).

For the linearized semiclassical Einstein equation (3.38), the reduction of order procedure could be ambiguous because integration by parts may alter the apparent differential order of terms in an integrand.

“In general, the reduction of order procedure could be ambiguous for such equations, since it may be possible to alter the apparent differential order of terms in an inte- grand via integration by parts.”

Probably not a substantial claim: may not pass S3 (load-bearing). The quote only partly states this claim.

The reduced order equation obtained from Eq. (3.38) to order h̄^2 is Eq. (4.20).

“the resulting reduced order equation is”

Classified as probably not a substantial claim.

Applying the refined reduction of order algorithm to Eq. (3.38) to order h̄^2 corresponds to substituting h̄ s_ab/κ for G^(1)_ab everywhere on the right hand side of Eq. (3.38), where s_ab ≡ D^(0)_ab F^(1)_in.

“This corresponds to substituting h̄sab/κ for G (1) ab everywhere on the right hand side of Eq. (3.38), where sab ≡ D(0) ab F (1) in .”

May combine two claims.

There is no problem with runaway effects in constructing approximate solutions x_J, and for any given order J, an initial position and velocity determine a unique approximate solution x_J, unlike the exact solutions.

“velocity will determine a unique approximate solution xJ”

• cites [75]Not matched

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

The quote only partly states this claim.

Contributions of order τ2^2 to the solution give small corrections to the inspiral of relative magnitude <~ τ/τp, where τp ≫ τ is the initial orbital period.

“to small corrections to the inspiral of relative magnitude”

• AssumptionThe authors state this as an assumption: The exact equation of motion including all higher order corrections is of the form a = a_ext + τ1 ȧ + α1 τ2^2 ä.

May combine two claims.

All solutions of Eq. (4.19) are well behaved, i.e., there are no runaway solutions nor any preacceleration effects.

“All of its solutions are well behaved, i.e, there are no runaway solutions nor any pre- acceleration effects.”

Classified as probably not a substantial claim.

The equation obtained by substituting G^(1)_ab = 0 on the right hand side of Eq. (3.38) is too trivial in that it does not incorporate any effects of the curved spacetime.

“However, the resulting equation is too trivial in that it does not incorporate any of the effects of the curved spacetime.”

Probably not a substantial claim: may not pass S3 (load-bearing).

If the electric field E is of compact support in time, the runaway solutions of Eq. (4.4) have ẍ ∝ exp(t/τ) for large t.

“the property that ẍ ∝ exp(t/τ) for large t.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In Appendix A we obtain all solutions to Eq. (3.38) whose spatial Fourier transforms exist, thereby generalizing the analysis of Horowitz to allow for a non-vacuum incoming quantum state.

“In Appendix A we obtain all solutions to Eq. (3.38) whose spatial Fourier transforms exist, thereby generalizing the analy- sis of Horowitz to allow for a non-vacuum incoming quan- tum state.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The refined reduction of order procedure can be generalized to any finite order in τ.

“Clearly, this procedure can be generalized to any finite order in τ .”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The reduction of order algorithm produces a new equation formally equivalent to Eq. (4.18) to order τ and with the same general form, but where Q contains at most m−1 time derivatives of x.

“the role of Q now contains at most (m− 1) time deriva-”

Classified as probably not a substantial claim.

Since the right hand side of Eq. (3.38) involves only derivatives of the linearized Einstein tensor, there is an obvious procedure for obtaining a reduced order equation valid to order h̄: substitute G^(1)_ab = 0 on the right hand side of Eq. (3.38).

“since the right hand side of Eq. (3.38) involves only derivatives of the linearized Einstein tensor, there is an obvious procedure in this case for obtaining a reduced order equation valid to order h̄”

Probably not a substantial claim: may not pass S3 (load-bearing).

The choice of initial acceleration a0 that makes the first term in Eq. (4.5) vanish is the unique choice generating a non-runaway solution.

“unique choice which generates a non-runaway solution.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Non-runaway solutions of Eq. (4.4) form a 6-dimensional submanifold of the 9-dimensional manifold of solutions, since they have vanishing acceleration at late times for E of compact support in time.

“dimensional submanifold of the 9-dimensional manifold”

Probably not a substantial claim: may not pass S3 (load-bearing).

For a one-parameter family of electric fields E(x,t;α)=α^{-2}E_0(x,t/α) and X(t;α)=x(αt;α), X satisfies Eq. (4.21).

“If we consider a one parameter family of electric fields given by E(x, t;α) = α−2E0(x, t/α) and define X(t;α) = x(αt;α), then X satisfies the differential equation”

The quote only partly states this claim.

If ρ̃(ω) is of compact support in ω, then ginhom(t;ε) is analytic in ε.

“ginhom(t; ǫ) is analytic in ǫ.”

Probably not a substantial claim: may not pass S3 (load-bearing).

F1(k) can be written as the limit of a function G1(k,ω) analytic on the upper half ω plane.

“From Eqs. (6.17) and (A6) we can write F1 as the limit of a function analytic on the upper half ω plane: F1(k) = limǫ→0+ G1(k, ω + iǫ)”

Probably not a substantial claim: may not pass S3 (load-bearing).

The reduced order semiclassical equation (4.20) expresses the linearized Einstein tensor G^(1)_ab[h^(1)] in terms of the source tensor (3.43).

“The reduced order semiclassical equation (4.20) ex- presses the linearized Einstein tensor G (1) ab [h(1)] in terms of the source tensor (3.43).”

Probably not a substantial claim: may not pass S3 (load-bearing).

In the limit Λ→∞, Λ_L,Λ_T→0 of no transverse smearing, expression (5.19) reduces to expression (5.11), since h^(1)e_e vanishes on γ in the chosen gauge.

“in the limit Λ → ∞, ΛL,ΛT → 0 of no transverse smearing, the expression (5.19) reduces to the previously obtained expression (5.11), since h (1) e e vanishes on γ in our chosen gauge.”

Classified as probably not a substantial claim. May combine two claims.

The approximate equation (3.59) has higher derivative terms multiplied by a small parameter 1/α^2 and has unknown correction terms of order O(lnα/α^6).

“The approximate equation (3.59) also has the char- acter of possessing higher derivative terms multiplied by a small parameter (namely, 1/α2), and has unknown cor- rection terms of order O(lnα/α6).”

Probably not a substantial claim: may not pass S3 (load-bearing).

The general Fourier-transformable solution of the semiclassical equation (3.38) has the formal expression (A5).

“Using Eqs. (3.38), (3.33) and (6.17) we find the following formal expression for the general Fourier transformable solution”

May combine two claims.

The locations of the zeros of G1 depend on λ and ωk: when λ≥λcrit there are four complex zeros ±ω,±ω* with ω=α0+iβ0 in the first quadrant; when λ≤λcrit there are three separate cases.

“When λ ≥ λcrit, there are four complex zeros ±ω,±ω∗, where ω = α0 + iβ0 lies in the first quadrant. When λ ≤ λcrit there are three separate cases.”

Probably not a substantial claim: may not pass S3 (load-bearing).

For general algebraic states, the positivity condition is equivalent to the positivity of the operator M_ij = [[g, f],[f*, g*+J]] in the sense of Eq. (6.6).

“For general, alge- braic states ωin the positivity condition is equivalent to the positivity of the operator Mij = [ g f f∗ g∗ + J ] , (6.5)”

May combine two claims.

M is a positive operator on L^2(R^3) ⊕ L^2(R^3), which implies that g is a positive operator on L^2(R^3): ∫∫ g(k,k') u(k)u(k')* ≥ 0.

“Thus, M is a positive operator on L2(R3) ⊕ L2(R3), which implies, in particular, that g is a positive operator on L2(R3):”

Probably not a substantial claim: may not pass S3 (load-bearing).

Since the smearing function S depends only on the magnitude σ_T≡x_T^2 of x_T, the function Θ^(1)_γ is given by Eq. (5.17).

“Since the smearing function S in Eq. (5.2) depends only on the magnitude σT ≡ x2 T of xT , the function Θ (1) γ that appears in Eq. (5.16) is given by”

Classified as probably not a substantial claim.

Inserting these expansions into Eq. (5.2) yields I_s(ε)=εI_s^(1)+ε^2I_s^(2)+O(ε^3).

“Inserting these expansions into Eq. (5.2) yields Is(ε) = εI(1) s + ε2I(2) s +O(ε3), (5.14)”

May combine two claims.

The case Â=B̂=Ĉ=0 can be excluded; if all these coefficients vanish, Q̂ must vanish and |ψ>= ̄ρ^(1)=0.

“Moreover, the case = B̂ = Ĉ = 0 can be excluded in the following way. If all these coefficients vanish, an argument similar to that given in Sec. VI E 2 shows that the operator Q̂ in Eq. (7.10) must vanish, and that con- sequently |ψ〉 = ¯̂ρ (1) = 0.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In this gauge the vector field v^a described in Sec. V B vanishes.

“In this gauge the vector field va described in Sec. VB vanishes.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Combining the alternative representation (6.14) of the two point function with Eqs. (6.25) and (6.26) yields Eq. (6.27).

“Now by combining the alternative representation (6.14) of the two point function with Eqs. (6.25) and (6.26), we obtain”

Classified as probably not a substantial claim.

The generalized ANEC integral depends on arbitrary choices of P, χ, ζ^a, S, etc.

“Clearly this quantity depends on our arbitrary choices of P , χ, ζa, S etc.”

Classified as probably not a substantial claim.

Using Eqs. (1.13), (2.4) and (6.1), the source tensor s̃_ab can be rewritten in terms of the regularized two-point function of the incoming state as Eq. (6.21), where σ_ab is given by Eq. (6.22).

“Next we rewrite the source tensor s̃ab in terms of the regularized two-point function of the incoming state. Us- ing Eqs. (1.13), (2.4) and (6.1), we find that”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

In the large N limit, the original equation (3.38) is formally known to all orders in h̄, whereas the reduced order equation is valid only to order h̄^2.

“in the large N limit, the orig- inal equation (3.38) is formally known to all orders in h̄, whereas the reduced order equation is valid only to or- der h̄2.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The full nonlinear semiclassical Einstein equation is highly nonlocal and is not known to be even of an integro-differential type.

“the full, nonlin- ear semiclassical Einstein equation is a highly nonlocal equation, which is not known to be even of an integro- differential type.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The second-to-last term in Eq. (7.3) is proportional to Eq. (7.7).

“The more interesting term in Eq. (7.3) is the second to last term, which can be shown to be proportional (at the relevant order in 1/α2) to”

Classified as probably not a substantial claim. The quote only partly states this claim.

Inserting Eq. (6.21) into Eq. (6.18) yields the null-null component of the linearized Einstein tensor, Eq. (6.23), with J given by Eq. (6.24).

“Inserting Eq. (6.21) into Eq. (6.18) yields the null-null component of the linearized Einstein tensor”

Probably not a substantial claim: may not pass S3 (load-bearing).

The explicit form of the second order semiclassical Einstein equation (3.5) is not known, since the term <T^(2)_ab[h^(1),h^(1)],ω_in,0> has not been evaluated.

“the explicit form of this equation is not known, since the term 〈T (2) ab [h(1), h(1)], ωin,0〉 has not been evaluated.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Solving Eq. (A1) using Fourier transform methods reproduces the last term in Eq. (A3).

“Hence, solving Eq. (A1) using Fourier transform methods will re- produce the last term in Eq. (A3).”

Classified as probably not a substantial claim.

∆C depends on ω̄(1)_in, ω̄(2)_in, and the freely specifiable incoming piece of the metric perturbation χ(1,0)_ab.

“The coefficient ∆C in Eq. (7.5) is defined to the contribution from the last two terms in Eq. (7.3), and depends both on ω̄ (1) in and ω̄ (2) in as well as the freely specifiable incoming piece of the metric perturbation χ (1,0) ab .”

Classified as probably not a substantial claim. The quote only partly states this claim.

I(ε)=εI^(1)+ε^2I^(2)+O(ε^3), where I^(1) and I^(2) are given by Eqs. (5.10) and (5.11).

“I(ε) = εI(1) + ε2I(2) +O(ε3), (5.9)”

Probably not a substantial claim: may not pass S3 (load-bearing).

In this gauge, the expression for I_s^(2) reduces to Eq. (5.19).

“The expression for I (2) s in this gauge reduces to”

Classified as probably not a substantial claim.

In subsection III E, an explicit approximate form (3.59) of the second order semiclassical Einstein equation was derived, valid for long wavelengths and when the incoming gravitational radiation does not dominate the first order metric perturbation.

“in subsection III E above, we derived the explicit approx- imate form (3.59) of this equation, which is valid for long wavelengths and when the incoming gravitational radia- tion does not dominate the first order metric perturba- tion.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The general solution to Eq. (3.38) can be written as G(1)ab = G(1),inhom_ab + G(1),free_ab + G(1),free,T_ab.

“The general solution to Eq. (3.38) can be written as G (1) ab = G (1), inhom ab +G (1), free ab +G (1), free,T ab . (A12)”

May combine two claims.

IF(xT) vanishes identically if and only if g(1)=0, i.e., if and only if the perturbed two-point function is pure to first order.

“We now show that ĪF (xT ) cannot vanish identically unless g(1) = 0, i.e., unless the perturbed two-point function is pure to first order.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The volume 4-form can be expanded as ǫ_abcd(ε)=ǫ^(0)_abcd+εǫ^(1)_abcd+O(ε^2), where ǫ^(1)=h^(1)e_e ǫ^(0)/2.

“Similarly the volume 4-form can be expanded as ǫabcd(ε) = ǫ (0) abcd + εǫ (1) abcd + O(ε2), where ǫ(1) = h (1) e e ǫ(0)/2.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The poles in the Green's function 1/F1 are the zeros of G1.

“The poles in the Greens function are just the zeros of G1.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Eq. (A5) is not the general solution; the general solution is obtained by adding extra terms corresponding to poles of 1/F1 and 1/(F1F3).

“While Eq. (A5) is not the general solution we are looking for, it is straightforward to write down the gen- eral solution by inspection, essentially by adding extra terms to Eq. (A5) that correspond to poles of the func- tions 1/F1 and 1/(F1F3).”

Probably not a substantial claim: may not pass S3 (load-bearing).

For a one-parameter family of null geodesics γ(ε), to order ε the family is characterized by the zeroth order geodesic γ and by the vector field v^c=(∂/∂ε)^c on γ.

“To order ε, this one parameter family of geodesics is characterized by the zeroth order geodesic γ = γ(0) (a null geodesic in Minkowski space- time), and by the vector field vc = (∂/∂ε)c on γ.”

Classified as probably not a substantial claim.

A = ∫γ <T^(0)_ab, ω̄^(1)_in> λaλb = ĪF(0).

“A = ∫ γ 〈T (0) ab , ω̄ (1) in 〉λaλb = ĪF (0), (6.43)”

Probably not a substantial claim: may not pass S3 (load-bearing).

In the coordinates γ, β, k_T on momentum space, the positive light cone volume element can be written as Eq. (6.13).

“In these coordinates the positive light cone volume ele- ment can be written as”

Probably not a substantial claim: may not pass S3 (load-bearing).

The linearized Einstein tensor (6.23) has the same regularity properties as were proved in Appendix C for the source tensor (3.43).

“By using the techniques of Appendix C, it is possible to show that the linearized Einstein tensor (6.23) has the same regularity properties as were proved in Appendix C for the source tensor (3.43).”

Probably not a substantial claim: may not pass S3 (load-bearing).

For partial differential equations, the reduction of order procedure is guaranteed only to reduce the differential order of the time derivatives, not of the spatial derivatives.

“in the case of partial differential equa- tions, the procedure is guaranteed only to reduce the dif- ferential order of the time derivatives, not of the spa- tial derivatives.”

Probably not a substantial claim: may not pass S3 (load-bearing).

There exists a unique k0>0 depending on λ such that the locus of the zeros of F3(k) is k^2=ωk^2−w^2=k0^2.

“There exists a unique k0 > 0 depend- ing on λ such that the locus of the zeros of F3(k) is k2 = ω2 k − w2 = k2 0 .”

May combine two claims.

The choice of gauge guarantees that χ(1)_ab=0 along γ, and χ(1,0)_ab also vanishes on γ.

“However, our choice of gauge guarantees that χ (1) ab = 0 along γ. In particular this will be true to each order in the 1/α2 expansion of Eq. (3.52), so that χ (1,0) ab also vanishes on γ, and therefore this term vanishes.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The scalar fields λ(ε), ζ(ε), σ_T(ε), and vector field λ^a(ε) are coordinate-independent scalar and tensor fields, though their domain is restricted to a neighborhood of γ.

“although these objects are defined in terms of an algo- rithm to obtain a coordinate system, they are themselves coordinate-independent scalar and tensor fields. Their domain of definition is however restricted to some neigh- borhood of the geodesic γ.”

May combine two claims.

The stress tensor (7.21) does not depend on which solution h(2) of G(1)_ab[h(2)] + G(2)_ab[h(1),h(1)]=0 is chosen, and thus is a function only of h(1).

“Moreover, the stress tensor (7.21) does not depend on which solution h(2) of G (1) ab [h(2)] + G (2) ab [h(1), h(1)] = 0 is chosen, and thus is a function only of h(1).”

May combine two claims.

The coefficients Â, B̂, and Ĉ obey the same positivity conditions as A, B, and C: Â≥0, B̂≥0 whenever Â=0, and Ĉ≥0 whenever Â=B̂=0.

“Hence the coefficients Â, B̂ and Ĉ obey the same pos- itivity conditions as the coefficients A, B and C: Â ≥ 0, B̂ ≥ 0 whenever = 0, and Ĉ ≥ 0 whenever = B̂ = 0.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Eq. (7.7) contains both a contribution quadratic in ω̄(1)_in and a cross term between ω̄(1)_in and incoming classical gravitational radiation.

“Thus, Eq. (7.7) contains both a contribution quadratic in the incoming state perturbation ω̄ (1) in and a cross term between ω̄ (1) in and the incoming classical gravitational ra- diation.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

Reducing order treating 1/α as the small parameter is equivalent to treating τ as the small parameter.

“It is clear that reducing order treating 1/α as the small parameter is equivalent to treating τ as the small param- eter.”

May combine two claims.

The distribution K(xT) is given by a smooth function away from the origin xT=0, but not at the origin.

“Note that it follows from the analysis in Appendix D that the distribution K(xT ) is given by a smooth func- tion away from the origin xT = 0, but not at the origin.”

May combine two claims.

Whenever Â=B̂=0, the last two terms in Eq. (7.3) vanish, and consequently ∆C=0.

“We now show that whenever = B̂ = 0, the last two terms in Eq. (7.3) vanish, and that consequently ∆C = 0.”

May combine two claims.

From Eq. (7.12), all terms on the left-hand side are individually nonnegative, and hence they all vanish.

“From the arguments given in Sec. VI E 1, it is clear that all the terms on the left hand side are individually non- negative, and hence they all vanish.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Eqs. (6.18)-(6.20) could also be obtained by expanding the exact solutions (A5)-(A8) in powers of h̄ and h̄ ln h̄, assuming λ ∝ √h̄.

“Note that Eqs. (6.18) – (6.20) could also be obtained by ex- panding the exact solutions (A5) – (A8) given in Ap- pendix A in powers of h̄ and h̄ ln(h̄) (assuming λ ∝ √ h̄ as discussed in Sec. III E above).”

• AssumptionThe authors state this as an assumption: λ ∝ √h̄.

Probably not a substantial claim: may not pass S3 (load-bearing).

For Fock space states, the positivity condition is the statement <Φ̂(u)Φ̂(u)†> ≥ 0 for all complex smearing functions u.

“which for Fock space states is just the statement that 〈Φ̂(u)Φ̂(u)†〉 ≥ 0 (6.4) for all complex smearing functions u.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

In the example Eq. (4.25), the reduction of order procedure can eliminate the third order time derivative of Φ but does not eliminate the third order spatial derivatives or all third order mixed spatial and time derivatives.

“The reduction of order procedure can be used to eliminate the third order time derivative of Φ from the equation, but it does not eliminate the third or- der spatial derivatives or all of the third order mixed spa- tial and time derivatives.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The method of reduction of order is directly applicable only to local ordinary or partial differential equations, although it was extended to the integro-differential equation (3.38).

“the method is directly applicable only to local, ordinary or partial, differential equations, although we were able to extend it in a natural manner to the integro- differential equation (3.38).”

Classified as probably not a substantial claim. May combine two claims.

At second order, the restriction on g^(2) is Eq. (6.10); in particular, when g^(1)=0 this reduces to Eq. (6.11).

“At second order we obtain the restriction on g(2) that [ P 0 0 1 ] [ g(2) f (1) f (1) ∗ J ] [ P 0 0 1 ] ≥ 0, (6.10) where P is the projection operator onto the kernel of g(1). In particular, when g(1) = 0 this reduces to”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

A generalized ANEC integral is defined as an integral over all of spacetime, and it reduces to the transversely smeared ANEC integral in a certain limit.

“we define a third in- tegral which we call a generalized ANEC integral, which is an integral over all of spacetime, and which reduces to the transversely smeared ANEC integral in a certain limit”

May combine two claims.

When A=B=0, for ΛT>ΛT,crit, C ≥ 0, with equality holding if and only if g(1)=0.

“We now shall show that when A = B = 0, for ΛT > ΛT,crit we have C ≥ 0, with equality holding if and only if g(1) = 0.”

Probably not a substantial claim: may not pass S3 (load-bearing).

For Fock space states, Eq. (6.7) is the statement that <Φ̂_+(u)†Φ̂_+(u)> ≥ 0 for any test function u.

“For Fock space states, Eq. (6.7) is just the statement that 〈Φ̂+(u)†Φ̂+(u)〉 ≥ 0 for any test function u.”

Classified as probably not a substantial claim.

B = ν0(ΛT2 − ΛT,crit2)(∇T2 ĪF)(0).

“B = ν0(Λ 2 T − Λ2 T,crit)(∇2 T ĪF )(0), (6.44)”

Probably not a substantial claim: may not pass S3 (load-bearing).

Solutions to Eq. (4.23) coincide with second order approximate perturbative solutions.

“Note that solutions to Eq. (4.23) also coincide with second order approximate perturbative solutions.”

Probably not a substantial claim: may not pass S3 (load-bearing). The quote only partly states this claim.

Using Horowitz's formula for the Fourier transform of H_λ, H̃_λ(k)=−2π[lnλ^2|k^2|+2γ−1−iπ sgn(k0)Θ(−k2)], and Eqs. (3.33) and (3.36), one obtains Eqs. (6.18)-(6.20).

“We use the following formula given by Horowitz [48] for the Fourier transform of the distri- bution Hλ”

• cites [48]Not matched

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

Probably not a substantial claim: may not pass S3 (load-bearing).

The corresponding classical positivity condition is the stronger condition that the matrix [[g,f],[f*,g*]] ≥ 0.

“the corresponding classical positivity condi- tion, which requires that the two point function be the expected value of Φ(x)Φ(y) with respect to some posi- tive measure on the space of field configurations, is the stronger condition that [ g f f∗ g∗ ] ≥ 0. (6.8)”

Probably not a substantial claim: may not pass S3 (load-bearing).

Combining Eqs. (5.15) and (6.23) yields the first order perturbation to the generalized ANEC integral expressed in terms of f^(1) and g^(1), Eq. (6.25).

“Combining Eqs. (5.15) and (6.23) yields that”

Classified as probably not a substantial claim.

The resulting equation from applying reduction of order to Eq. (3.59) is Eq. (4.23).

“We obtain”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The assumed regularity properties on the incoming state imply that f̂ and ĝ are continuous as functions of (β,k_T,β',k'_T) and satisfy the bound (6.16) for any integer N.

“our assumed regularity properties on the incoming state imply that f̂ and ĝ are continuous as functions of (β,kT , β ′,k′ T ) and satisfy, for any integer N ,”

May combine two claims.

The Fourier transform s̃_ab is continuous everywhere away from the light cone, and is bounded everywhere except for an integrable divergence at the origin in momentum space.

“In Appendix C we show that the Fourier transform s̃ab is actually continuous every- where away from the light cone, and is bounded every- where except for an (integrable) divergence at the origin in momentum space.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The gauge transformation properties (3.31) and (5.7) are consistent with differential equation (5.5).

“The gauge transformation properties (3.31) and (5.7) are consistent with differential equation (5.5).”

Classified as probably not a substantial claim.

In the gauge associated with the coordinate system described in Sec. V A, λ^(1)a=λ^(1)=ζ^(1)=σ_T^(1)=0.

“This yields a choice of gauge in which we have λ(1) a = λ(1) = ζ(1) = σ (1) T = 0. (5.18)”

Probably not a substantial claim: may not pass S3 (load-bearing).

The inhomogeneous part of the solution is given by Eq. (A13).

“The inhomoge- neous part of the solution is given by”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

The quantities I_s^(1) and I_s^(2) are gauge invariant.

“As in the previous subsection, the quantities I (1) s and I (2) s are gauge invariant.”

May combine two claims.

If λ>λcrit then K̃1(kT) is continuous; its Fourier transform K1(xT) is L2 and continuous everywhere away from the origin.

“Note that if λ > λcrit then K̃1(kT ) is a continuous function; its Fourier transform K1(xT ) is L2 and is con- tinuous everywhere away from the origin.”

May combine two claims.

From properties of f and g and the form of Eq. (C12), Fab and hence S̃ab(l,ω) is continuous everywhere away from the lightcone, and |l|Fab(l,ω) is bounded in a neighborhood of l=0.

“From the properties of the functions f and g and the form of Eq. (C12), it is clear that Fab and hence also S̃ab(l, ω) is continuous everywhere away from the lightcone, and that |l|Fab(l, ω) is bounded in a neighborhood of l = 0.”

• AssumptionThe authors state this as an assumption: The paper assumes that the functions F, ∂F/∂t, ∂F/∂t′, and ∂2F/(∂t∂t′) lie in class V: they are smooth and all their spatial derivatives are L1 on Σ×Σ.

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims. The quote only partly states this claim.

For λ<λcrit, the homogeneous term G(1),free_ab has freely specifiable terms for each pole of 1/F1; poles off the real axis give exponentially growing/decaying terms, and poles on the real axis give tachyonlike contributions analogous to Eq. (A2).

“The poles off the real axis will give exponentially grow- ing/decaying terms, and the poles on the real axis will give”

Classified as probably not a substantial claim.

The assumptions imply that f̂≡ωkωk′f and ĝ≡ωkωk′g are at least C0.

“Moreover our assumptions imply the functions f̂ ≡ ωkωk′f and ĝ ≡ ωkωk′g defined before Eq. (6.15) are at least C0, a fact which is used in the body of the paper.”

• AssumptionThe authors state this as an assumption: The paper assumes that the functions F, ∂F/∂t, ∂F/∂t′, and ∂2F/(∂t∂t′) lie in class V: they are smooth and all their spatial derivatives are L1 on Σ×Σ.

May combine two claims.

The positivity result would not hold for any smearing function which falls off more rapidly than |K1|; there is no positivity result for Gaussian transverse smearing.

“We remark that this positivity re- sult would not hold for any smearing function which falls off more rapidly than |K1|. Thus, for example, there is no positivity result for Gaussian transverse smearing.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The freely specifiable piece associated with the second term in Eq. (A13) does not vanish even for conformal coupling.

“However, as we discuss below, the freely specifiable piece of the solution associated with the second term in Eq. (A13) [Eq. (A16) below] does not van- ish even for conformal coupling.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Poles on the real ω axis occur only for λ<λcrit for 1/F1, but occur for all values of λ for 1/F3.

“As discussed above, poles on the real ω axis will occur only for λ < λcrit for the function 1/F1, but will occur for all values of λ for the function 1/F3.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Fab(l,ω) is given by Eq. (C8).

“where la = (l, ω) and Fab(l, ω) = ∫ d3kωkωk−l (1,n)a × [ δ(ω − ωk + ωk−l)g(k,k − l)(1,−n′)b −δ(ω − ωk − ωk−l)f(k, l − k)(1,n′)b ] . (C8)”

May combine two claims.

The freely specifiable transverse piece of the solution does not vanish in the conformally coupled case ξ=1/6, despite the analogous transverse contribution to the inhomogeneous piece vanishing for conformal coupling.

“Also, as remarked above, this freely specifiable transverse piece of the solution does not vanish in the conformally cou- pled case ξ = 1/6, despite the fact that the the analo- gous transverse contribution to the inhomogeneous piece of the general solution [the second term in Eq. (A13) above] does vanish for conformal coupling.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The second term G(1),free_ab in Eq. (A12) is the freely specifiable homogeneous piece associated with 1/F1; for λ>λcrit it can be written as Eq. (A14).

“The second term G (1), free ab in Eq. (A12) is the freely specifiable, homogeneous piece of the general solution as- sociated with the Greens function 1/F1(k). In the case λ > λcrit, it can be written as [92]”

• cites [92]Not matched

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

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

In the conformally coupled case ξ=1/6, the trace of the source tensor sab vanishes, and hence the second term in Eq. (A13) vanishes.

“Note also that in the conformally coupled case ξ = 1/6, the trace of the source tensor sab vanishes, and hence the second term in Eq. (A13) vanishes.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The solution (A14) is purely exponentially growing/decaying and has no oscillatory parts.

“This is purely an exponentially growing/decaying type solution and has no oscillatory parts.”

May combine two claims.

The proof can be extended to any positive smearing function which does not fall off at large xT more rapidly than K1 (∝|xT|^-4); any function falling off slowly enough will be suitable when sufficiently dilated.

“In fact, our proof below can be easily extended to apply to any positive smearing function which does not fall off at large xT more rapidly than K1 does (∝ |xT |−4); in other words, any function which falls off slowly enough will be a suitable smearing function when sufficiently”

Probably not a substantial claim: may not pass S3 (load-bearing).

Exact solutions to the alternative rescaled version (3.45) of the first-order perturbation equation can be obtained from the above analysis using substitutions h(1)ab→α^2ĥ(1)ab, Lp→Lp/α, λ→λ/α, and sab→s̄ab.

“Finally, we remark that exact solutions to the alterna- tive, rescaled version (3.45) of the first order perturbation equation can be obtained from the above analysis using the substitutions”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The third term G(1),free,T_ab in Eq. (A12) is a purely transverse, freely specifiable piece associated with 1/F3; it can be written as Eq. (A16).

“The third term G (1), free,T ab in Eq. (A12) is a purely transverse, freely specifiable piece of the solution which is associated with the Greens function 1/F3. It can be written as [92]”

• cites [92]Not matched

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

Probably not a substantial claim: may not pass S3 (load-bearing).

For the tachyon type solution (A15), the linearized Einstein tensor can be written as Eq. (B13).

“For the tachyon type solution (A15), the linearized Ein- stein tensor can be written as G̃ab(β, γ,kT ) = δ(−2γβ + k2 T − k2 0)Fab(β, γ, k̂T ), (B13)”

Probably not a substantial claim: may not pass S3 (load-bearing).

The proof could be modified to apply to smearing functions that fall off more slowly than |K1|.

“However, as mentioned above, our proof could easily be modified to apply to smearing functions that fall off more slowly than |K1|.”

Classified as probably not a substantial claim.

I(K,x)=12π/x^3+∫ dφ ∆I(K,x,μ).

“it can be shown that I(K,x) = 12π/x3 + ∫ 2π 0 dϕ∆I(K,x, µ). (D7)”

May combine two claims.

Since Fab is freely specifiable away from 2βγ+k0^2=0, I(1) is generally nonvanishing and can be of either sign.

“Since Fab(β, γ, k̂T ) is freely specifiable away from 2βγ + k2 0 = 0, it is clear that in general I(1) is non vanishing and can be of either sign.”

Probably not a substantial claim: may not pass S3 (load-bearing).

F is determined by the restrictions of F, ∂F/∂t, ∂F/∂t′, and ∂2F/(∂t∂t′) to Σ×Σ, where Σ is the hypersurface t=0.

“The function F is determined by the restrictions of the four functions F , ∂F/∂t, and ∂F/∂t′ and ∂2F/(∂t∂t′) to Σ×Σ, where Σ is the hypersurface t = 0.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

It would be sufficient for all results to assume only that all spatial derivatives up to seventeenth order are L1; the sharp falloff requirements have not been investigated.

“It would be sufficient for all of our results to assume only that the all spatial derivatives up to seventeenth order are L1; we have not investigated what the sharp falloff requirements are.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The conditions on F apply to both F_in^(1) and F_in^(2); conclusions about the expected stress tensor in state (C1) apply to source term (3.43) and to the first term in Eq. (3.54).

“Also the conditions on F that we discuss below apply to both the functions F (1) in and F (2) in that appear in the body of the paper, and our conclusions about the expected stress tensor in the state (C1) clearly then will apply to the source term (3.43) and to the first term in Eq. (3.54).”

Probably not a substantial claim: may not pass S3 (load-bearing).

Similar conclusions apply to the entire stress tensor T̃ab(k).

“Similar conclusions apply to the entire stress tensor T̃ab(k).”

May combine two claims.

For λ<λcrit, expression (B1) blows up at some finite |kT| so that K̃1 is not even L2.

“For λ < λcrit, the expression (B1) blows up at some finite value of |kT | so that K̃1 is not even L2.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

Imposing the condition on any other surface of constant time yields the same class of states.

“Note that imposing this condition on any other surface of constant time would yield the same class of states.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The Fourier transform T̃ab(k) exists as a distribution.

“Therefore its Fourier transform T̃ab(k) exists as a distribution.”

• AssumptionThe authors state this as an assumption: The paper assumes that the functions F, ∂F/∂t, ∂F/∂t′, and ∂2F/(∂t∂t′) lie in class V: they are smooth and all their spatial derivatives are L1 on Σ×Σ.

May combine two claims.

The Fourier transform of k^2 log(k^2) consists of a smooth positive function away from the origin, plus distributional contributions at the origin whose effects are unimportant because of condition (D1).

“The essential idea is that the Fourier trans- form of k2 log(k2) consists of a smooth positive function away from the origin, plus distributional contributions at the origin whose effects are unimportant because of the condition (D1).”

May combine two claims.

If the state of the quantum field is the incoming vacuum, the second term on the right-hand side of Eq. (E2) is of order Lp^2/L^4, whereas the first term is of order 1/L^2, where L is the length scale determined by the classical background solution.

“Now if the state of the quantum field is the incoming vac- uum state, then the second term on the right hand side above will be in order of magnitude ∼ L2 p/L4, whereas the first term should be ∼ 1/L2, where L is the length- scale determined by the classical background solution g (0) ab .”

The quote only partly states this claim.

Examples of the type that appear to violate ANEC qualitatively differently from the cases in the body also exist in perturbation theory off Minkowski spacetime.

“examples of this type also exist in the context of perturbation theory off of Minkowski spacetime”

Probably not a substantial claim: may not pass S3 (load-bearing).

S̃ab(la)=Fab(la)+Fab(−la)*.

“S̃ab(la) = Fab(la) + Fab(−la)∗, (C7)”

May combine two claims.

The stress tensor T̃ab(k) is L2, from which it follows that Tab(x) is L2 on Minkowski spacetime.

“Finally, we note that the stress tensor T̃ab(k) is L2, from which it follows that Tab(x) is L2 on Minkowski spacetime.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

In the λ<λcrit case, an appropriate iε regularization prescription yields a well-defined distribution K̃1 with well-defined distributional Fourier transform K1(xT).

“The regularization prescription yields a well defined distribution K̃1, with well defined distributional Fourier transformK1(xT ).”

May combine two claims.

In the subclass S of exact first-order semiclassical solutions, using half advanced plus half retarded Green's function to obtain the linearized Einstein tensor is equivalent to throwing away runaway solutions by hand for exponential type modes, and yields a particular subclass for oscillatory modes.

“In the case of exponen- tial type modes discussed in Sec. IVC above, the use of this Greens function to pick out a class of solutions is equivalent to throwing away the runaway solutions by hand. For the oscillatory type modes, this choice of Greens function yields a particular subclass of solutions.”

Probably not a substantial claim: may not pass S3 (load-bearing). The quote only partly states this claim.

The inequality (C13) is finite for N≥5.

“where C0 is a constant, which is finite for N ≥ 5.”

May combine two claims.

There is a portion of the general solution with the form of an integral over k for which both poles are real; it has the same form as Φ> of Eq. (A2) except A and B are replaced by transverse traceless tensors and m^2 is replaced by either of two independent constants depending only on λ.

“In particular, there will be a portion of the general so- lution which has the form of an integral over the values of k for which both poles are real. This expression will have the same form as Φ> of Eq. (A2), except that A and B are replaced by transverse traceless tensors, and the quantity m2 is replaced by either of two independent constants depending only on λ”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

We shall build upon and extend the work of Horowitz [48], who considered semiclassical gravity in perturbation theory about flat spacetime to linear order, without allowing perturbations to the incoming quantum state.

“We shall build upon and extend the work of Horowitz [48], who considered semiclassical gravity in perturba- tion theory about flat spacetime to linear order, without allowing perturbations to the incoming quantum state.”

• cites [48]Not matched

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

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

From the viewpoint of the first scheme, we would expect there to be modifications to Eq. (2.6) which are of order ∼ L_p^2/L^2, and consequently the domain of validity of the equation is restricted to L ≫ L_p.

“Thus, from the viewpoint of this first scheme we would expect there to be modifications to Eq. (2.6) which are of order ∼ L2 p/L2, and consequently the domain of validity of the equation is restricted to L ≫ Lp, where Lp is the Planck length.”

Probably not a substantial claim: may not pass S2 (checkable).

This class of models should provide a good testing ground as to whether, in the context of perturbation theory, quantum fields can attain violations of ANEC which are classically forbidden.

“Thus, this class of models should provide a good testing ground as to whether, in the context of perturbation the- ory, quantum fields can attain violations of ANEC which are classically forbidden.”

May combine two claims. The quote only partly states this claim.

A purely classical theory, where gravity is coupled to a statistical ensemble of scalar field configurations, would differ from this version of semiclassical gravity only in two respects: first, the term 〈Tab〉ω0 would be absent from Eq. (2.22); second, the quantum mechanical positivity condition on F arising from Eq. (2.16) is less restrictive than the corresponding classical positivity condition.

“We remark that a purely classical theory, where gravity is coupled to a statistical ensemble of scalar field con- figurations, would differ from this version of semiclas- sical gravity only in the following two respects. First, in the classical theory, the term 〈Tab〉ω0 would be ab- sent from Eq. (2.22).”

Classified as probably not a substantial claim.

Since the expected stress tensor 〈Tab〉ω has a linear dependence on the two-point distribution, G, associated with ω, we have, for any Hadamard state, ω, 〈Tab〉ω = 〈Tab〉ω0 + 〈Tab〉F.

“Since the expected stress tensor 〈Tab〉ω has a linear dependence on the two-point distribution, G, associated with ω, we have, for any Hadamard state, ω, 〈Tab〉ω = 〈Tab〉ω0 + 〈Tab〉F .”

Probably not a substantial claim: may not pass S3 (load-bearing).

The exact status and domain of applicability of the semiclassical equation (2.6) is far from clear, due mainly to the fact that we do not have a complete, more fundamental theory of quantum gravity from which Eq. (2.6) could be derived.

“However, as is well known, the exact status and domain of applicability of this equation is far from clear, due mainly to the fact that we do not have a complete, more fundamental the- ory of quantum gravity from which Eq. (2.6) could be derived.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In the second scheme, the semiclassical Einstein equation (2.6) is formally derived from a fully quantum treatment by imagining that there are N decoupled scalar fields present - all of which are in the same quantum state - and then taking the limit N → ∞, with GN = (constant) [58].

“In the second scheme the semiclassical Einstein equa- tion (2.6) is formally derived from a fully quantum treat- ment by imagining that there are N decoupled scalar fields present — all of which are in the same quantum state — and then taking the limit N → ∞, with GN = (constant) (see, e.g., Ref. [58]).”