Open accessJournal Article

Cutting out the cosmological middle man: general relativity in the light-cone coordinates

04 Mar 2021-Classical and Quantum Gravity (IOP Publishing)-Vol. 38, Iss: 5, pp 055011
Abstract: Analytical computations in relativistic cosmology can be split into two sets: time evolution relating the initial conditions to the observer's light-cone and light propagation to obtain observables. Cosmological perturbation theory in the Friedmann–Lemaitre–Robertson–Walker (FLRW) coordinates constitutes an efficient tool for the former task, but the latter is dramatically simpler in light-cone-adapted coordinates that trivialize the light rays toward the observer world-line. Here we point out that time evolution and observable reconstruction can be combined into a single computation that relates directly initial conditions to observables. This is possible if one works uniquely in such light-cone coordinates, thus completely bypassing the FLRW 'middle-man' coordinates. We first present in detail these light-cone coordinates, extending and generalizing the presently available material in the literature, and construct a particularly convenient subset for cosmological perturbation theory. We then express the Einstein and energy–momentum conservation equations in these coordinates at the fully non-linear level. This is achieved through a careful 2 + 1 + 1 decomposition which leads to relatively compact expressions and provides good control over the geometrical interpretation of the involved quantities. Finally, we consider cosmological perturbation theory to linear order, paying attention to the remaining gauge symmetries and consistently obtaining gauge-invariant equations. Moreover, we show that it is possible to implement statistical homogeneity on stochastic fluctuations, despite the fact that the coordinate system privileges the observer world-line.

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Open accessJournal Article
Abstract: The linearized dynamical equation for metric perturbations in a fully general, non-vacuum, background geometry is obtained from the Hamilton variational principle applied to the action up to second order. We specialize our results to the case of traceless and transverse metric fluctuations, and we discuss how the intrinsic properties of the matter stress tensor can affect (and modify) the process of gravity wave propagation even in most conventional geometric scenarios, like (for instance) those described by a FLRW metric background. We provide explicit examples for fluid, scalar field and electromagnetic field sources.

1 Citations

Open accessPosted Content
Abstract: We present a fully nonlinear and relativistically covariant expression for the observed galaxy density contrast. Building on a null tetrad tailored to the cosmological observer's past light cone, we find a decomposition of the nonlinear galaxy over-density into manifestly gauge-invariant quantities, each of which has a clear physical interpretation as a cosmological observable. This ensures that the monopole of the galaxy over-density field is properly accounted for. We anticipate that this decomposition will be useful for future work on nonlinearities in galaxy number counts, for example, deriving the relativistic expression for the galaxy bispectrum. We then specialise our results to conformal Newtonian gauge, with a Hubble parameter either defined globally or measured locally, illustrating the significance of the different contributions to the observed monopole of the galaxy density.

Topics: Density contrast (61%), Galaxy (59%), Newtonian gauge (53%) ... show more

Open accessPosted Content
Markus B. Fröb1, William C. C. Lima2Institutions (2)
Abstract: We consider a recent approach to the construction of gauge-invariant relational observables in gravity in the context of cosmological perturbation theory. These observables are constructed using a field-dependent coordinate system, which we take to be geodesic lightcone coordinates. We show that the observables are gauge-independent in the fully non-linear theory, and that they have the expected form when one adopts the geodesic lightcone gauge for the metric. We give explicit expressions for the Sasaki-Mukhanov variable at linear order, and the Hubble rate -- as measured both by geodesic observers and by observers co-moving with the inflaton -- to second order. Moreover, we show that the well-known linearised equations of motion for the Sasaki-Mukhanov variable and the scalar constraint variables follow from the gauge-invariant Einstein equations.

Open accessPosted Content
Abstract: In single-field inflationary models the bispectra are usually given in the $\zeta$-gauge, because its temporal part leads to the super-horizon conservation of fluctuations. However, this property is independent of the choice of {\it spatial} gauge, so in this letter we explore this freedom. We compute the variation of the bispectra under the most general spatial gauge transformation that is globally defined and privileges no point, direction or scale. In the squeezed configuration we then obtain a generalization of the classic $\zeta$-gauge consistency relation when the long mode is a scalar, a result which we also derive through the `large diffeomorphism' approach. The first effect is a shift of the tilt factor, so that one can significantly reduce the amplitude of that contribution. Secondly, there is now an extra term depending on the triangle shape, the same as in solid inflation, which is due to the fact that the 3-metric has a scalar anisotropy in generic spatial gauge.

Open accessJournal Article
Abstract: The linearized dynamical equation for metric perturbations in a fully general, non-vacuum, background geometry is obtained from the Hamilton variational principle applied to the action up to second order. We specialize our results to the case of traceless and transverse metric fluctuations, and we discuss how the intrinsic properties of the matter stress tensor can affect (and modify) the process of gravity wave propagation even in most conventional geometric scenarios, like (for instance) those described by a FLRW metric background. We provide explicit examples for fluid, scalar field and electromagnetic field sources.

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Open accessJournal Article
Abstract: This article--summarizing the authors' then novel formulation of General Relativity--appeared as Chapter 7 of an often cited compendium edited by L. Witten in 1962, which is now long out of print. Intentionally unretouched, this posting is intended to provide contemporary accessibility to the flavor of the original ideas. Some typographical corrections have been made: footnote and page numbering have changed--but not section nor equation numbering etc. The authors' current institutional affiliations are encoded in: arnowitt@physics.this http URL, deser@brandeis.edu, misner@physics.this http URL .

Topics: Numbering (51%)

2,193 Citations

Open accessJournal Article
Abstract: This article—summarizing the authors’ then novel formulation of General Relativity—appeared as Chap. 7, pp. 227–264, in Gravitation: an introduction to current research, L. Witten, ed. (Wiley, New York, 1962), now long out of print. Intentionally unretouched, this republication as Golden Oldie is intended to provide contemporary accessibility to the flavor of the original ideas. Some typographical corrections have been made: footnote and page numbering have changed–but not section nor equation numbering, etc. Current institutional affiliations are encoded in: arnowitt@physics.tamu.edu, deser@brandeis.edu, misner@umd.edu.

1,906 Citations

Journal Article
Abstract: Fermi coordinates, where the metric is rectangular and has vanishing first derivatives at each point of a curve, are constructed in a particular way about a geodesic. This determines an expansion of the metric in powers of proper distance normal to the geodesic, of which the second‐order terms are explicitly computed here in terms of the curvature tensor at the corresponding point on the base geodesic. These terms determine the lowest‐order effects of a gravitational field which can be measured locally by a freely falling observer. An example is provided in the Schwarzschild metric. This discussion of Fermi Normal Coordinate provides numerous examples of the use of the modern, coordinate‐free concept of a vector and of computations which are simplified by introducing a vector instead of its components. The ideas of contravariant vector and Lie Bracket, as well as the equation of geodesic deviation, are reviewed before being applied.

Topics: Fermi coordinates (68%), Geodesic (67%), Geodesic deviation (63%) ... show more

308 Citations

Journal Article
George F. R. Ellis1, S.D. Nel2, Roy Maartens3, William R. Stoeger  +1 moreInstitutions (4)
01 Jul 1985-Physics Reports
Abstract: Following Kristian and Sach's direct observational approach to cosmology, this paper analyses in detail the information that can be obtained from idealised astronomical observations, firstly in the cosmographic case when no gravitational field equations are assumed, and secondly in the cosmological case when Einstein's field equations of General Relativity are taken to determine the space-time structure. It is shown that if ideal observations are available, in the cosmographic case they are insufficient to determine the space-time structure on the past light cone of the observer; however in the cosmological case they are precisely necessary and sufficient to determine the space-time geometry on the light cone and in its causal past (at least down to where caustics or curps first occur). The restricted case of spherically symmetric space-times is analysed in detail, and necessary and sufficient observational conditions that such a space-time be spatially homogeneous are proven. A subsequent paper will examine the situation of realistic observational data.

Topics: General relativity (54%), Light cone (53%), Gravitational field (51%) ... show more

268 Citations

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