This work was mainly driven by the desire to explore, to what extent embedding some given geometry in a higher dimensional flat one is useful for understanding the causal structure of classical fields traveling in the former, in terms of that in the latter. We point out, in the 4D spatially flat FLRW universe, that the causal structure of transverse-traceless (TT) gravitational waves can be elucidated by first reducing the problem to a 2D Minkowski wave equation with a time dependent potential, where the relevant Green's function is pure tail -- waves produced by a physical source propagate strictly within the null cone. By viewing this 2D world as embedded in a 4D one, the 2D Green's function can also be seen to be sourced by a cylindrically symmetric scalar field in 3D. From both the 2D wave equation as well as the 3D scalar perspective, we recover the exact solution of the 4D graviton tail, for the case where the scale factor written in conformal time is a power law. There are no TT gravitational wave tails when the universe is radiation dominated because the background Ricci scalar is zero. In a matter dominated one, we estimate the amplitude of the tail to be suppressed relative to its null counterpart by both the ratio of the duration of the source to the age of the universe $\eta_0$, and the ratio of the observer-source spatial distance (at the observer's time) to the same $\eta_0$. In a universe driven primarily by a cosmological constant, the tail contribution to the background FLRW geometry after the source has ceased, is the conformal factor $a^2$ times a spacetime-constant symmetric matrix proportional to the spacetime volume integral of the TT part of the source's stress-energy-momentum tensor. In other words, massless spin-2 gravitational waves exhibit a tail-induced memory effect in 4D de Sitter spacetime.
Transverse-Traceless Gravitational Waves In A Spatially Flat FLR W Universe: Causal Structure from Dimension Reduction Yi-Zen Chu Department of Physics, University of Minnesota, 1023 University Dr., Duluth, MN 55812, USA Abstract This work was mainly driven by the desire to explore, to what extent embedding some given geometry in a higher dimensional flat one is useful for understanding the causal structure of classical fields traveling in the former, in terms of that in the latter.
We point out, in the 4-dimensional (4D) spatially flat Friedmann-Lemaˆ ıtre-Robertson- Walker universe, that the causal structure of transverse-traceless (TT) gravitational waves can be elucidated by first reducing the problem to a 2D Minkowski wave equation with a time dependent potential, where the relevant Green’s function is pure tail – waves produced by a physical source propagate strictly within the null cone. By viewing this 2D world as embedded in a 4D one, the 2D Green’s function can also be seen to be sourced by a cylindrically symmetric scalar field in 3D.
From both the 2D wave equation as well as the 3D scalar perspective, we recover the exact solution of the 4D graviton tail, for the case where the scale factor written in conformal time is a power law. There are no TT gravitational wave tails when the universe is radiation dominated because the background Ricci scalar is zero. In a matter dominated one, we estimate the amplitude of the tail to be suppressed relative to its null counterpart by both the ratio of the duration of the (isolated) source to the age of the universeη0, and the ratio of the observer-source spatial distance (at the observer’s time) to the same η0.
In a universe driven primarily by a cosmological constant, the tail contribution to the background geometry a[η]2ηµν after the source has ceased, is the conformal factor a2 times a spacetime-constant symmetric matrix proportional to the spacetime volume integral of the TT part of the source’s stress-energy-momentum tensor. In other words, massless spin-2 gravitational waves exhibit a tail-induced memory effect in 4D de Sitter spacetime. 1 arXiv:1504.06337v3 [gr-qc] 2 Nov 2015 I. MOTIV A TION AND INTRODUCTION The geometry of our universe appears to be well described
(2) due to the presence of Π (T) ij is given by Dij[η,⃗ x] = 16πGN ∫ dη′ ∫ R3 d3⃗ x′a[η′]4G+ 4 [η,⃗ x;η′,⃗ x′]Π(T) ij [η′,⃗ x′], (6) with the retarded G+ 4 obeying □η,⃗ xG+ 4 [η,⃗ x;η′,⃗ x′] =□η′,⃗ x′G+ 4 [η,⃗ x;η′,⃗ x′] = δ[η−η′]δ(3)[⃗ x−⃗ x′] a[η]2a[η′]2 . (7) We see that the study of the causal structure of TT gravitational waves (GWs) propagating in our universe is the same as that of a minimally coupled massless scalar field. By causal structure, we are referring to the fact that, in a curved spacetime, particles that are otherwise massless in 4D Minkowski spacetime no longer travel strictly on the null cone – they travel both on and within it.
(A1) takes the generic pure tail form ˆG2[ξ,ξ′] = 1 2Θ[¯σ]J[ξ,ξ′], ¯σ≡ 1 2(ξ−ξ′)2, (A2) where J obeys the homogeneous wave equation ( ∂2 ξ +U[ξ] ) J[ξ,ξ′] = ( ∂2 ξ′ +U[ξ′] ) J[ξ,ξ′] = 0, (A3) and the boundary condition that it is unity on the light cone J[¯σ = 0] = 1. (A4) Here, ¯σ is half the square of the geodesic distance between ξ and ξ′ in 2D Minkowski. Because we are dealing with the symmetric Green’s function, J[ξ,ξ′] = J[ξ′,ξ ]. The step function in eq. (A2) tells us scalar waves in 2D obeying eq. (A1) travel strictly inside the cone of its physical sources.
Caldwell, “Green’s functions for gravitational waves in FRW space-times,” Phys. Rev. D 48, 4688 (1993) [gr-qc/9309025]. [17] Ya.B. Zeldovich and A.G. Polnarev, Sov. Astron. 18, 17 (1974) [18] D. Christodoulou, “Nonlinear nature of gravitation and gravitational wave experiments,” Phys. Rev. Lett. 67, 1486 (1991). [19] M. Favata, “The gravitational-wave memory effect,” Class. Quant. Grav.27, 084036 (2010) [arXiv:1003.3486 [gr-qc]]. [20] L. Bieri and D. Garfinkle, “Perturbative and gauge invariant treatment of gravitational wave memory,” Phys. Rev. D 89, no. 8, 084039 (2014) [arXiv:1312.6871 [gr-qc]]. [21] A. Tolish, L. Bieri, D. Garfinkle and R. M.
We review the tests of general relativity that will become possible with space-based gravitational-wave detectors operating in the ∼ 10<sup>-5</sup> - 1 Hz low-frequency band. The fundamental aspects of gravitation that can be tested include the presence of additional gravitational fields other than the metric; the number and tensorial nature of gravitational-wave polarization states; the velocity of propagation of gravitational waves; the binding energy and gravitational-wave radiation of binaries, and therefore the time evolution of binary inspirals; the strength and shape of the waves emitted from binary mergers and ringdowns; the true nature of astrophysical black holes; and much more. The strength of this science alone calls for the swift implementation of a space-based detector; the remarkable richness of astrophysics, astronomy, and cosmology in the low-frequency gravitational-wave band make the case even stronger.
Living Rev Relativ Living Rev Relativ 365 springeropen Living Reviews in Relativity 1433-8351 pmc-is-collection-domain yes pmc-collection-title Springer PMC5255528 PMC5255528.1 5255528 5255528 28163624 10.12942/lrr-2013-7 7 1 Review Article Testing General Relativity with Low-Frequency, Space-Based Gravitational-Wave Detectors Gair Jonathan R. jgair@ast.cam.ac.uk http://www.ast.cam.ac.uk/~jgair 1 Vallisneri Michele vallis@vallis.org http://www.vallis.org 2 Larson Shane L. s.larson@northwestern.edu 3 Baker John G.
The fundamental aspects of gravitation that can be tested include the presence of additional gravitational fields other than the metric; the number and tensorial nature of gravitational-wave polarization states; the velocity of propagation of gravitational waves; the binding energy and gravitational-wave radiation of binaries, and therefore the time evolution of binary inspirals; the strength and shape of the waves emitted from binary mergers and ringdowns; the true nature of astrophysical black holes; and much more.
The strength of this science alone calls for the swift implementation of a space-based detector; the remarkable richness of astrophysics, astronomy, and cosmology in the low-frequency gravitational-wave band make the case even stronger.
Mid-frequency space-based observatories The DECi-hertz Interferometer Gravitational wave Observatory (DECIGO [ 408 , 256 , 257 ]) is a proposed Japanese mission that would observe GWs at frequencies between 1 mHz and 100 Hz, reaching its best ( h ∼ 10 −23 ) sensitivity between 0.1 and 10 Hz, and thus bridging the gap between LISA-like and
The ultra-compact binaries also provide laboratories for testing waveform evolution, and for GW polarization and dispersion studies, precisely because they evolve slowly and are easily described by post-Newtonian analysis. This is discussed in Sections 5.1.1 and 5.2 . Gravitational-Wave Tests of Gravitational Physics Almost since its inception, GR was understood to possess propagating, undulatory solutions — GWs, described at leading order by the celebrated quadrupole formula [ 258 ]. It took several decades to establish firmly that these waves were real physical phenomena and not merely artifacts of gauge freedom.
The estimated error in the correction parameter, (Γ −1 ) ii , can then be interpreted as the minimal size of a correction that would be detectable with a GW observation. The “classic tests” of general relativity with gravitational waves As Will points out [ 469 , ch. 10], virtually any Lorentz-invariant metric theory of gravity must predict gravitational radiation, but alternative theories will differ in its properties. Will identifies three main properties that can be measured with GW detectors. These are the polarization, speed , and emission multipolarity (monopole, dipole, quadrupole, etc.) of GWs in GR.
However, the multipolarity of GWs at emission and the energy that they carry away can be influenced by strong-field properties in the near zone where waves are generated. Tests of gravitational-wave polarization GR predicts the existence of two transverse quadrupolar polarization modes for GWs (also described as “spin-2” and “tensor” using the language of group theory), usually labeled h + and h × . Alternative metric theories of gravity predict as many as six polarizations [ 469 ] (three transverse and three longitudinal), corresponding to the independent electric-type components of the Riemann curvature tensor, R 0 i0j .
The model was similar in structure to the ppE models which will be discussed in Section 5.2.2 . This model included both the dipolar component of the waveform, at the orbital frequency, and modifications to the gravitational wave phasing of both the quadrupole and dipole waveform components that arise from the additional energy lost into the dipole mode. In [ 26 ], the model was used to determine the constraints on dipole radiation emission that would be possible using ground-based GW detectors. Results for space-based detectors were included in a subsequent review [ 31 ].
The quadrupole formula and loss of energy to gravitational waves In theories that do not satisfy the strong equivalence principle, the internal gravitational binding energies of bodies can create a difference between the inertial dipole moment (i.e., the linear momentum, which is conserved) and the GW-generating gravitational dipole moment. Thus, alternative theories of gravity generally admit dipole radiation, but it is forbidden in GR, where the two moments are identical.
Therefore the bare coupling could be much higher than inferred from laboratory constraints, allowing the theories to explain cosmological acceleration (see [ 146 ] for a full description of the mechanism and complete references). In a similar way, the effective coupling in the vicinity of a compact object could in principle be different from that in the laboratory and so the weak constraints from gravitational-wave observations are still interesting because they probe a different curvature scale.
Gravitational memory, which describes the permanent shift in the strain after the passage of gravitational waves, is directly related to Weinberg’s soft graviton theorems and the Bondi-Metzner-Sachs (BMS) symmetry group of asymptotically flat space-times. In this work, we provide an equivalent description of the phenomenon in local coordinates around gravitational wave detectors, such as transverse-traceless (TT) gauge. We show that gravitational memory is encoded in large residual diffeomorphisms in this gauge, which include time-dependent anisotropic spatial rescalings, and prove their equivalence to BMS transformations when translated to TT gauge. We then derive the associated Ward identities and associated soft theorems, for both scattering amplitudes and equal-time (in-in) correlation functions, and explicitly check their validity for planar gravitational waves. Furthermore, the in-in identities are recognized as the flat-space analog of the well-known inflationary consistency relations.
Everything we examined (3)
This check searched the claim as stated. It did not run a separate search for evidence against it.