If such terms were backreaction terms, then in gravitational perturbation theory every term of order higher than 0 should be considered as a backreaction term for the assumed background.
We will give an argument in subsection 4. We now illustrate the above issues in terms of examples. Examples using the GW framework Green and Wald provide an example of a family of vacuum spacetimes in [11] section 3 , discussed in subsection 4.
We refer the reader also to Appendix C, where we comment on the example provided by Szybka et al. We are not aware of any example satisfying the GW conditions that does satisfactorily include matter inhomogeneities. We first start with a classical example given by Geroch to demonstrate the coordinate—dependence of the limit of a one—parameter family of metrics.
Moreover, it can be argued that instead of fixing the background metric, it should emerge from a coordinate—independent averaging or rescaling procedure. Such ideas could work whenever there is some identifiable way of taking a weak—limit of the set of curvature invariants. The issue of gauge dependence has been addressed in Ref. A related discussion can be found in [50]. In this latter paper it is demonstrated that even the spacetime integrals of 4D scalars are in general gauge—dependent if the domain of integration is fixed.
This will then imply a non—commutativity of the two operations and would give rise to backreaction terms for the consequences of non—commutativity see Ref. We move now to another issue.
The Geroch coordinate transformations would be excluded with this condition. Path—dependence of the weak—limit procedure In Ref. This is what is called in [3] an ultra—local limit: the oscillations must not even stop at atomic scales. The result obtained from Eq. This procedure does not.
It is non—zero only for very special choices of f N. We repeat that the problem is the ultra—local requirement that perturbations in the envisaged family continue down to indefinitely small wavelengths without limit; otherwise one gets a zero answer.
Then, the quantity tab 0 will be zero. Thus, Equations 2. The backreaction term turns on in a delta—function way, in the exact limit only. Indeed, as we show in Appendix B, the quantity tab 0 is only non—vanishing if we impose further restrictions beyond the conditions imposed by GW and, especially, if we require non—uniform convergence so that this term only arises in the singular limit see the discussion surrounding Eq.
It is worth noting that any uniformly convergent limit of a family of spacetimes will inherit some of the properties of the family. These properties are called hereditary [48]. These works may be found in the reviews mentioned in the introduction.
The procedure is closely related to the idea of metric smoothing by removing density ripples. In line with this he also has shown in Ref. The important issue there is the clustering properties of matter. No proof that backreaction is irrelevant 17 5. Objections to the scalar averaging approach The scalar averaging approach [31,54,55] referred to here is a realisation of the thoughts advanced in Refs.
In this section we address some misinterpretations of the scalar averaging framework that appear in Section 3 of [12]. This which we believe is the only relevant issue in Section 3 of [12] has to be emphasized: even if one included a further evolution equation for the backreaction variable, as GW suggest, the system of averaged equations would not be closed, unless a dynamical equation of state is assumed or derived [60] This issue has been spelled out at various places, e.
Rather, these are conditions on averages similar to, e. We start with a common misunderstanding that also appears in [12]. Can there be average acceleration with local deceleration everywhere? Yes, because backreaction is non—local.
Our emphasis. This situation is akin to the infinite moment hierarchies as they appear, e. A recent paper addressing closure illustrates this issue [61]. By construction, such balance relations cannot provide the local solutions. Taken literally, topologically disconnected FLRW universes are physically less relevant.
Here, we discuss the more relevant case of disjoint domains whose union constitutes the whole spatial section. This remark applies to further citations from GW as in subsection 5. Putting the cosmological term and the vorticity which is active on small scales only to zero, at first sight it seems implausible that a collection of such decelerating fluid elements can lead to acceleration of some patch of the Universe.
Clearly, by shrinking the averaging domain to a point, the two equations agree. Thus, the time—derivative of an averaged expansion may be positive even if the time—derivative of the expansion is negative at every point in D. This is, technically, a consequence of the non—commutativity of averaging and time—evolution and, physically, of the non—local nature of averaging that takes correlations into account. Can energy conditions be violated for the average dynamics?
Again, the answer is yes. It is a possible consequence of what has been said above. Green and Wald present in Ref. We comment on this example in Appendix C. While it is known that a fundamental scalar field e. Since, as we have shown in subsection 3. Addressing the real averaging problem would not deliver a local expression. We remind the reader that there is currently no agreed way to average tensors in GR.
Such a result would be fine but irrelevant for backreaction. Confusion between the scalar averaging framework and metric approaches Green and Wald question the framework where backreaction is discussed in terms of spatial averages of scalar quantities. However, this criticism is based on a fundamental misinterpretation of the scalar averaging formalism [31, 54]: it does not involve any notion of average metric and does not refer to geodesic deviations [22] ; only averaged scalars are considered.
In Ref. No proof that backreaction is irrelevant 20 spacetime split of the 4-metric on which we comment below. As explained at the beginning of this section, the formalism defines spatial average properties among scalar variables that depend on second derivatives of the metric. The metric itself is not specified. We note that there are indeed investigations in the literature that study so—called template metrics [57], [66], [67], [68], [69], that are intended to be compatible with the exact average properties although these are not the papers referred to by GW.
The FLRW metric itself can be viewed as a global template metric. If we find that the global average spatial curvature today is not zero, then the flat FLRW template will constitute a poor approximation of the spatial sections, and small deviations thereof will represent large deviations with respect to the physical background [73]. This is exactly the problem that a template metric, even a single global template, is supposed to correct for. Moreover, as we argued in the context of the steel ball model analogy and the weak—limit framework: curvature inhomogeneities are not required to average out on an assumed background, as GW a priori impose to be true.
Compare here our discussion of the backreaction term QD as it is evaluated in Newtonian cosmology in subsection 3. It is well—known that the average expansion rate depends on the choice of hypersurface, and the issue has been discussed at length in the literature, where it has been argued that the physically relevant averaging hypersurface is the one of statistical homogeneity and isotropy, see e.
It is irrelevant that there are hypersurfaces that are not physically interesting, it only matters that averages on some hypersurfaces give physically meaningful results and can be formulated in a covariant way. For discussion of covariance and gauge—invariance of scalar averaging, see Refs.
The average expansion rate evaluated on some hypersurface is a useful quantity so far as it gives an approximate description of what is observed, which is indeed the case for the hypersurface of statistical homogeneity and isotropy. We emphasize that, in practice, observations of quantities such as the expansion rate and density always involve spatial averages or averages over null geodesics, which can be related to spatial averages [79, 80, 83, 84].
Averaging is not a mathematical artifact, it is a feature of real observations that has to be properly modelled. It may be demonstrated that such an argument is coordinate dependent.
Comparing two metrics in GR is quite an involved task: we can employ Cartan scalars to test for isometry of metrics, see, e. In four dimensional spacetimes, one only needs at most seven derivatives [87]. Such a test is justified by Theorem 9. Appendix B , to those of some metric g 0 , then g is close to g 0 again, in some suitable topological space. Note in this context compare subsections 4. Finally, we turn to the issue of whether there is any reason to think that the standard cosmological model, which is based on work nearly a century old [89, 90] and unlike the standard particle physics model is the simplest conceivable cosmological solution , might not be the best description of our Universe.
Models and observations An implicit assumption of the FLRW model is that the dynamics and observations of the inhomogeneous Universe can be modelled by the dynamics and evolution of a spatially homogeneous and isotropic universe model.
This procedure seemed to be completely adequate for many decades; indeed, unanticipated discrepancies between observations and the FLRW model such as the need for an accelerated homogeneous universe model [91, 92], compare here Ref. In this paper we remind the reader of another possibility. We assume that the Einstein equations hold locally, but because of the inhomogeneous distributions of matter and geometry the FLRW model fails to adequately describe the observations.
We turn now to observational issues in dealing with the inhomogeneous Universe. No proof that backreaction is irrelevant 23 6. Observational issues It is important to take into account large metric derivatives and corresponding changes of observational data interpretation. Green and Wald discuss this, underpinned by the claim that Newtonian notions fully capture these changes with the help of their dictionary, which is restricted to an assumed near—FLRW situation.
This should be regarded as an open question until backed by concrete general calculations. Green and Wald argue that a mapping of solutions of Newtonian cosmological equations with periodic boundary conditions and certain properties to GR solutions given in Ref. One is whether certain Newtonian solutions are the limit of some GR solutions; the other is whether the GR solution that describes the real Universe is at all times close to a corresponding Newtonian solution15 see also Ref.
Even if a particular GR solution starts from initial conditions close to a Newtonian solution, this does not imply that the GR solution would remain close to the Newtonian solution. As a simple example, in Newtonian gravity an isolated two—body system with an elliptic orbit is a stable configuration, whereas in GR the orbit will decay, and the system will be driven far from the Newtonian solution A general GR solution even if the matter is dust does not correspond to any Newtonian solution.
In contrast to these instabilities the Newtonian inhomogeneities average out on the chosen background model by construction and this latter is stable [28].
No proof that backreaction is irrelevant 24 the magnetic part of the Weyl tensor is zero. In GR, both are in general non—zero and have evolution equations, see Refs. In [10] the arbitrariness of the tidal tensor is fixed by the assumption of periodic boundary conditions. As the Newtonian equations are elliptic and do not have a well—defined initial value problem, the boundary conditions are essential [28].
Changing the evolution of the boundary at distances much larger than the GR particle horizon would not impact the GR solutions, but can completely change the Newtonian solutions. A lack of backreaction in GR cannot be established by starting from the assumption that the Universe is well—described by Newtonian theory. Actually, this is incorrect. A counterexample is provided by Enqvist et al. It is also straightforward to construct exact spherically symmetric counterexamples to the flux claim, such as the one in Ref.
This issue has been discussed in Section 2. Even on large scales statistical isotropy and homogeneity is not enough to reduce the luminosity distance to its FLRW value, as shown in Ref. Indeed, the violation of the FLRW relationship between the expansion rate and the luminosity distance can be used as a test of the importance of inhomogeneities [79,80,83,—]. The quint essence of the backreaction approach As Green and Wald stress, second derivatives of the metric i.
Hence, this furnishes an argument for the potential importance of backreaction. The Einstein equations dictate inhomogeneous curvature for inhomogeneous sources. A physical cosmology has to capture both inhomogeneities in the sources and inhomogeneities in the geometry. Idealising the latter by assuming a homogeneous FLRW metric globally, leads to a missing geometrical piece on the left—hand side of the Einstein equations that shows up as missing sources on the right—hand side of the Einstein equations in the standard model.
The oversimplified standard model keeps curvature at this non—generic value, while backreaction models evolve curvature, even if it is set to zero at some initial time [31]. We have to distinguish average properties and fluctuations. Fluctuations may remain small on large scales, but that is not the issue: the issue is the background about which these fluctuations are small, and this is what is addressed by backreaction models.
Metrical deviations from an unknown average metric may be small, but it is an assumption that this average metric is dynamically equivalent to a homogeneous and flat FLRW solution. In other words, small metric perturbations do not imply small distortions of a flat geometry. In , Einstein already clarified this and explained that the perturbations could equally well be small deviations from a large—scale curved space.
Einstein, however, reached this conclusion with calculations on which we comment, below his quote, from a modern perspective. This can be considered a variant of the steel ball model of GW. We might imagine that, as regards geometry, our Universe behaves analogously to a surface which is irregularly curved in its individual parts, but which nowhere departs appreciably from a plane: something like the rippled surface of a lake.
Such a Universe might fittingly be called a quasi-Euclidean Universe. As regards its space it would be infinite. But calculation shows that in a quasi—Euclidean Universe the average density of matter would necessarily be nil. Thus such a Universe could not be inhabited by matter everywhere; it would present to us that unsatisfactory picture On the contrary, the results of calculation indicate that, if matter be distributed uniformly, the Universe would necessarily be spherical or elliptical.
Since in reality the detailed distribution of matter is not uniform, the real Universe will deviate in individual parts from the spherical, i. But it will be necessarily finite. Einstein emphasized that the small deviations require a more careful description of the global spatial curvature. Einstein mentions a positively curved background, but it could as well be negatively curved. We argue that the deviations should be studied with respect to a background that is defined by the actual average distribution the physical background , as it is done in other physical disciplines that investigate fluctuation theories, e.
In Newtonian cosmology the a priori assumption of a background that is equivalent with the average distribution can be realised: the spatially averaged model can be assumed to be an FLRW model on the imposed scale of the 3—torus due to the globally vanishing backreaction, which assures that the inhomogeneities average out to zero, cf.
Even in homogeneous cosmology, the Einstein cosmos [89] provides an example of a curvature—dominated model, its curvature radius being smaller than the Schwarzschild radius if the former is greater than about 3.
In other words, this example shows that if the curvature is cosmologically significant small metric deviations from a flat metric would run into contradictions. More generally, even if fields are weak then an important consideration is the calibration of the asymptotic rulers and clocks of the close to spatially flat metric around bound structures relative to a generally non—FLRW cosmological average.
The problems of averaging and coarse—graining may therefore be intimately related to fundamental unsolved problems concerning gravitational mass—energy [6]. Conclusion There is no proof that backreaction of inhomogeneities is irrelevant for the dynamics and observables of our Universe. The most detailed study claiming to show that backreaction is irrelevant is in a series of papers by Green and Wald [9—12] and a precursor paper by Ishibashi and Wald [13], so we have examined those papers in detail in particular the review in Ref.
In particular, we have demonstrated that the claimed trace—free nature of backreaction is unphysical and is not relevant within the backreaction framework. While the GW framework is not applicable to studying backreaction, it is possible that further developments of their framework may provide useful mathematical results. We have explicitly demonstrated that the definition of backreaction assumed by Green and Wald is too narrow to address this question.
We also thank all of the referees for their helpful comments and acknowledge Bob Wald for discussions. In response to Ref. These additions do not change any of our conclusions. No proof that backreaction is irrelevant 28 Appendix A. On the intrinsic realisation of the steel ball analogy In order to address the analogy given by GW, we have to investigate a well—defined intrinsic geometrical mapping between S2 , gpol and S2 , gcan.
This is not a perturbation argument, but rather a deformation argument in the space of metric geometries over S2. Both are mathematically well—defined and constructive procedures. They provide an explicit mapping typically a conformal transformation between S2 , gpol and S2 , gcan. These mapping techniques are currently applied in the modelling of two—dimensional discrete structures in fields ranging from image analysis to biology and medicine.
In particular, both approaches would map S2 , gpol to a well—defined S2 , gcan without any conical defects whatsoever. We would further like to correct the statement by Green and Wald that the metric of the polyhedron fails to be smooth also at the edges. The metric of S2 , gpol is perfectly smooth at the edges. Appendix B. There are subtle assumptions and hypotheses underlying their claimed results that are unclear and not explicitly stated. Furthermore, as we shall demonstrate below, under ii — iv there is a very delicate regularity issue in taking a weak—limit, with the consequence that one cannot prove that the trace of tab 0 vanishes without further hypotheses additional to those in ii — iv.
We use Sobolev spaces to get to the point as quickly as possible and to pinpoint the origin of the problem lurking in the background.
We also introduce the class of functions in Lploc U whose weak first derivatives, see below for definitions , are also Lploc U —functions. Their definition naturally extends to the appropriate spaces of tensor fields. For further details see, e. It is also appropriate to recall the definitions of distributional and weak derivatives.
Entering into such mathematical detail may appear pedantic. Higher—order distributional derivatives are defined in a similar way. In nonlinear problems, like the one we are discussing here, it is more useful and sometimes indeed necessary to consider, rather than the distributional derivative, the more restrictive concept of weak derivative. The two notions agree for smooth functions, and often the notion of weak derivative is tacitly traded for the definition of derivative in the sense of distributions.
In particular, the weak derivative may not exist: two well—known examples are the Heaviside step function and the Cantor function. Neither admits a weak derivative whereas they both do have a distributional derivative: the Dirac measure supported at the origin, and the Lebesgue- Stieltjes measure supported on the Cantor set, respectively.
Connect your Spotify account to your Last. Connect to Spotify. A new version of Last. Replace video. Add lyrics on Musixmatch. Do you know any background info about this track? Start the wiki. Don't want to see ads?
Upgrade Now. Scrobbling is when Last. Learn more. Javascript is required to view shouts on this page. Go directly to shout page. View full artist profile.
View all similar artists. View all trending tracks. Loading player…. Scrobble from Spotify? Connect to Spotify Dismiss. Search Search.
Join others and track this song Scrobble, find and rediscover music with a Last. Sign Up to Last. Play album. Length Lyrics Add lyrics on Musixmatch. Related Tags jrock rock j-rock japanese rock Add tags View all tags. Featured On Play album. C 16, listeners. C 1 listener. Play track. Artist images more.
C 76, listeners Related Tags j-rock visual kei japanese LM. C is a rock band from Japan. Members: Vo. C Support: Ba. C was founded by Maya in Tokyo,Japan , a support guitarist for musician Miyavi in the support band Ishihara Gundan Ishihara being Miyavi's real last name, Gundan means 'brigade' or 'army' in Japanese and a guitarist in his own band, Sinners.
While still with Miyavi, maya and other support members also played live show… read more. DENKI-… read more.
Similar Artists Play all. Trending Tracks 1. Thursday 15 July Friday 16 July Saturday 17 July Sunday 18 July Monday 19 July Tuesday 20 July Wednesday 21 July Thursday 22 July Friday 23 July Saturday 24 July Sunday 25 July Monday 26 July Tuesday 27 July Wednesday 28 July Thursday 29 July Friday 30 July Saturday 31 July Sunday 1 August Monday 2 August Tuesday 3 August Wednesday 4 August Thursday 5 August Friday 6 August Saturday 7 August Sunday 8 August Monday 9 August Tuesday 10 August Wednesday 11 August Thursday 12 August Friday 13 August Saturday 14 August Sunday 15 August Monday 16 August Tuesday 17 August Wednesday 18 August Thursday 19 August Friday 20 August Saturday 21 August Sunday 22 August Monday 23 August Tuesday 24 August Wednesday 25 August Thursday 26 August Friday 27 August Saturday 28 August Sunday 29 August Monday 30 August Tuesday 31 August Wednesday 1 September Thursday 2 September Friday 3 September Saturday 4 September Sunday 5 September Monday 6 September Tuesday 7 September Wednesday 8 September Thursday 9 September Friday 10 September Saturday 11 September Sunday 12 September Monday 13 September Tuesday 14 September Wednesday 15 September Thursday 16 September Friday 17 September Saturday 18 September Sunday 19 September Monday 20 September Tuesday 21 September Wednesday 22 September Thursday 23 September
0コメント