Gravitational Wave Memory In dS and 4D Cosmology
We argue that massless gravitons in all even dimensional de Sitter (dS) spacetimes higher than two admit a linear memory effect arising from their propagation inside the null cone. Assume that gravitational waves (GWs) are being generated by an isolated source, and over only a finite period of time . Outside of this time interval, suppose the shear-stress of the GW source becomes negligible relative to its energy-momentum and its mass quadrupole moments settle to static values. We then demonstrate, the transverse-traceless (TT) GW contribution to the perturbation of any dS written in a conformally flat form () – after the source has ceased and the primary GW train has passed – amounts to a spacetime constant shift in the flat metric proportional to the difference between the TT parts of the source’s final and initial mass quadrupole moments. As a byproduct, we present solutions to Einstein’s equations linearized about de Sitter backgrounds of all dimensions greater than three. We then point out there is a similar but approximate tail induced linear GW memory effect in 4D matter dominated universes. Our work here serves to improve upon and extend the 4D cosmological results of Chu:2015yua (), which in turn preceded complementary work by Bieri, Garfinkle and Yau Bieri:2015jwa () and by Kehagias and Riotto Kehagias:2016zry ().
I Introduction and Motivation
Consider two test masses sweeping out their respective (not necessarily geodesic) world lines in a -dimensional spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW)-like universe, namely one described by the geometry
Suppose a gravitational wave passes by, perturbing the geometry to become
Denote the proper geodesic length between the two masses at a fixed time to be . A direct calculation, reviewed in appendix (A), would then reveal that the presence of such a GW would, at leading order, yield a fractional distortion of this proper length proportional to .
where is the unit radial vector and is a straight line in Euclidean space joining one test mass at to the other at . Furthermore, if after the source of the GWs has ceased and the primary GW train has gone by, the perturbation does not die down to zero amplitude, but settles instead to a non-zero constant matrix , we see that the fractional distortion becomes permanent:
This permanent displacement of test masses after the passage of a GW is known as the memory effect.
In this paper we shall focus primarily on background de Sitter (dS) spacetimes of dimensions greater than . Written in conformally flat coordinates the metric is given by
where is the Hubble parameter describing the acceleration of an expanding universe. The main thrust of this work is to argue that, the portion of GWs traveling inside the null cone of its source – commonly known as its tail – would give rise to a memory effect in even dimensions higher than . As we shall see, this phenomenon is intimately tied to the fact that the tail part of the minimally coupled massless scalar Green’s function in dS, for zero or a positive integer, is a spacetime constant.
One may ask why a 4D physicist should be interested in physics of other dimensions. One answer is that , the dimension of spacetime, may be viewed as a parameter in the equations of physics, including that of Einstein gravity. By varying we may gain insight into questions of principle, and even acquire new means to calculate 4D physics. An example is that of the tail effect itself: even though massless fields/particles propagate strictly on the light cone in 4D flat spacetimes, they no longer do so in and odd dimensions. Moreover, it is possible to understand why tails exist in odd dimensional flat spacetime by embedding it in one higher dimensional Minkowski SoodakTiersten (). This has prompted further generalizations to de Sitter spacetime Chu:2013hra (), which can be viewed as a hyperboloid situated in one higher dimensional flat spacetime. Specifically, the causal structure of the de Sitter scalar Green’s function can be related to that of signals generated by an appropriately defined line source in the ambient flat spacetime. In addition, recent work Strominger:2014pwa () has drawn connections between the (better known) GW memory effect in asymptotically Minkowskian spacetime and the low frequency limit of the Ward-Takahashi identities obeyed by GW scattering amplitudes, which in turn is a consequence of the Bondi-van der Burg-Metzner-Sachs (BMS) symmetry at null infinity. Does an analogous relationship hold in cosmological spacetimes, or does it break down because the tail induced memory effect described here is timelike instead of null? These questions have first been raised in Chu:2015yua (), and subsequently in Kehagias:2016zry (); because it is one based on symmetry, it ought to be examined in all possible dimensions in order to understand the breadth of its validity.
In §(II) we work out Einstein’s equations with a non-zero cosmological constant, linearized about a de Sitter background of dimensions greater or equal to four. We then express the metric perturbations as appropriate retarded Green’s functions convolved against the energy-momentum-shear-stress tensor of the source(s). Following that, in §(III) we take a detour to define the mass and pressure quadrupole moments of an isolated GW source in a spatially flat FLRW universe, and then relate linear combinations of their time derivatives to the spatial-volume integral of the shear-stress of the same source. We then employ these results in §(IV), and describe how the solutions laid out in §(II) lead us to a linear GW memory effect exhibited by the tensor mode, after the source has settled down. In §(V) we re-visit some results obtained in Chu:2015yua () and comment on their implications for the linear GW memory effect in 4D cosmologies. We summarize and discuss future directions in §(VI). In appendix §(A) we review the calculation of spatial geodesic distances between a pair of test masses in a perturbed spatially flat FLRW-like universe. In appendix §(B) we delineate the solutions of the partial differential equations arising from General Relativity linearized about a background de Sitter spacetime. Despite being an appendix, this section is the technical heart of the paper. Finally, in appendix (C) we identify the gauge-invariant metric perturbation variables in a -dimensional background spatially flat FLRW geometry.
Ii General Relativity With , Linearized About (de Sitter)
Einstein’s equations for the metric , sourced by the energy-momentum-shear-stress tensor of some matter source, is
is Newton’s gravitational constant in dimensions. We are including a positive cosmological constant because mounting astrophysical evidence points to its existence in the 4D universe we reside in. Moreover, for this paper, we will assume that describes an isolated, compact astrophysical system which does not distort the overall geometry too much. When we neglect its influence (i.e., set ) we recover the pure de Sitter spacetime in eq. (7), which in turn solves
General Perturbation Theory (PT) To this end, it is convenient to work with the barred graviton, namely
When in 4D, this is more commonly known as the “trace-reversed” graviton. Equivalently, from eq. (4), we may define
In eq. (12), the geometric tensors – Einstein , Ricci , Ricci scalar and Riemann – and the covariant derivative are built solely out of the background ; moreover, all indices are moved with it. The symmetrization symbol is defined through the relation .
PT about dS Now, if we do specialize to being the dimensional de Sitter metric in eq. (7), which satisfies eq. (9), the first line on the right hand side of eq. (12) vanishes. We then use the maximally symmetric form that the de Sitter geometric tensors take, namely
Evaluated on a de Sitter background geometry, eq. (12) then bring us to
To solve Einstein’s eq. (8) perturbatively, we need to choose a gauge for the barred graviton field , so that its wave operator can be inverted and its solution written as a convolution of appropriate Green’s functions against the sources .
Gauge fixing We now require to satisfy the gauge condition
where all indices in eq. (18) and in what follows are moved with the flat metric . This gauge in eq. (18) and the ensuing equations below, are really a generalization of the 4D ones in deVega:1998ia (), Ashtekar:2015lxa () and Date:2015kma (). At this point, eq. (16) becomes
where the is the scalar one with respect to the background de Sitter metric, namely
(The is the wave operator in flat spacetime.) The linearized version of eq. (8) is thus
The solution of eq. (21) is the primary technical focus of this work.
Pseudo-trace mode By adding times of the component of the linearized Einstein’s equation (21) to its spatial-trace, we are lead to
An equivalent form of eq. (22) is
Here and below – equations (25), (31) and (36) – because we are faced with partial differential equations (PDEs) of the same form, namely , we devote appendix (B) to solving the relevant Green’s functions. In this section we will merely quote the final results. The pseudo-trace retarded solutions are
with being Synge’s world function in Minkowski spacetime; and
Vector mode The components of eq. (21) reads
The retarded solutions are
Tensor mode The equations for the components of eq. (21) turn out to be that of the minimally coupled massless scalar in de Sitter spacetime,
This translates to
The retarded solutions are
We highlight here that, these solutions in eq. (37) amount to the convolution of against the minimally coupled massless scalar Green’s function in de Sitter spacetime. As we shall witness, the latter’s spacetime constant tail in even ()-dimensional spacetime is responsible for contributing to the linear GW memory effect.
Iii Mass and pressure quadrupole moments; Conservation laws in a spatially flat FLRW spacetime
Before examining this tail induced linear GW memory effect arising from the even solutions in eq. (37), however, we need to first – following Ashtekar:2015lxa () and Date:2015kma () – relate spatial-volume integrals of the shear-stress of the isolated matter source to its mass and pressure quadrupole moments.
Quadrupole moments Throughout this section, unless otherwise indicated, we will suppose our background metric is a spatially flat FLRW universe, i.e., not necessarily de Sitter:
We then note that the -beins of such a spacetime, whose defining property is
are given by
The (upper) index of transforms as a coordinate vector; while its (lower) index transforms as a local Lorentz -form. Therefore we can form from the matrix of coordinate scalar quantities
The can now be interpreted as the mass-energy density measured by a local observer; as its -momentum density; and as its shear-stress/pressure density. Moreover, from eq. (40), since the induced metric on a constant hypersurface is , we may recognize the proper spatial volume on the said hypersurface to be
The physical mass and pressure quadrupole moments are now defined as
where we associate one scale factor to each of the and to form a physical vector.
Conservation At linear order, the energy-momentum-shear-stress tensor of an isolated astrophysical system in a spatially flat FLRW background geometry is conserved
with being the covariant derivative with respect to the . (This statement, of course, neglects backreaction.) A direct calculation reveals
We may use eq. (50) in the following way. Via integration-by-parts and the assumption that the matter distribution is localized in space (so that surface terms are zero) – one may readily see that
Applying eq. (50) to the left hand side allow us to arrive at
where we have inserted the mass and pressure quadrupole definitions from equations (45) and (46). We reiterate that eq. (52) holds in any -dimensional spatially flat FLRW geometry. When we specialize to de Sitter spacetime, where ,
Iv Tail Induced Linear GW Memory Effect in (de Sitter)
We now turn our attention to the tail part of the metric perturbations in even dimensional () de Sitter spacetime, as encoded in equations (26), (32) and (37). Despite experiencing a non-trivial potential in higher dimensions (cf. equations (25) and (31)), the pseudo-trace and vector exhibit no tails. The physical reason is unclear; however, the mathematical reason is that and are respectively polynomials in of degree and . Therefore,
(We have checked the second line against the 6D light cone boundary condition in eq. (130), i.e., it is zero for .) In the gauge of eq. (18), it is thus only the tensor mode that travels inside the light cone of its source. Moreover, the tail of its Green’s function in eq. (37) is a constant because one is differentiating a degree polynomial times,
(This result follows from Rodrigues’ formula for the Legendre polynomials.) This is, of course, equivalent to the fact that the tail of the de Sitter minimally coupled massless scalar Green’s function is a constant in all even dimensions higher than Chu:2013hra (). In even dimensions, the tail part of the tensor mode solution in eq. (37) is therefore
where we have defined the retarded time . For technical convenience, throughout the rest of this paper, we will assume that the coordinate system has been chosen such that is located within the source. If we now invoke eq. (53) derived in the previous section, the linear GW tail can be expressed in terms of the mass and pressure quadrupole moments of the source, at least in the far zone () where and hence :
Negligible shear-stress, settling of quadrupole moments Let us suppose that the isolated source is active only over a finite interval of time, . This means we will assume that outside this interval, its shear-stress is negligible relative to its energy-momentum, so that we may set the former to zero, namely
We will further assume that the mass quadrupole moment is not static only during this active interval:
Altogether, equations (59), (60) and (61) applied to eq. (58) lead us to a tail induced linear GW memory effect. Specifically for late times (),111The retarded time depends on and therefore takes part in the spatial-volume integral of eq. (57). This is why the far zone limit, replacing , was required when transitioning to eq. (58). However, once the source has settled down, equations (59) and (60) allow us to deduce that for , eq. (58) now holds everywhere within the future null cone of the GW source’s world tube. the assumption in eq. (59) allows us to perform the integral in eq. (57) only over and sets to zero in eq. (53); while eq. (61) puts in eq. (53):
Gravitational radiation It is important to record here that the transverse-traceless portion of is gauge-invariant, namely does not change its form under infinitesimal coordinate transformations, and is in fact what is usually meant by gravitational radiation – like itself, it obeys the de Sitter minimally coupled massless scalar wave equation. (For the reader’s reference, in appendix (C) we identify the SO scalar, vector and tensor gauge-invariant metric variables in a -dimensional spatially flat FLRW geometry.) Assuming equations (59) and (61) hold, the assertion that the GW tail in dS remains a non-zero spacetime constant, after its source has ceased, is therefore a coordinate-invariant and physical one.
Because the GW tail here is a constant in spacetime, notice this linear GW memory effect does not decay with distance from the source, unlike the expected fall-off in even ()-dimensional flat spacetime. Moreover, in even dimensional Minkowski spacetimes higher than , linear massless gravitons continue to propagate strictly on the null cone, like their 4D cousins. For these reasons, the dS tail induced linear GW memory effect captured in eq. (IV) really has no counterpart in the asymptotically flat case.
Remark I Before shifting our attention to 4D cosmology, we mention here that Ashtekar:2015lxa () advocates exploiting the Killing vector in de Sitter spacetime
to define what it means for a GW source to settle down. (This is timelike in the region .) Specifically, they required that the Lie derivative of the energy-momentum-shear-stress tensor vanish outside the time interval of GW production, namely
In contrast, we appeared to have made stronger assumptions (equations (59) and (61)), but without appealing to the symmetries enjoyed by de Sitter spacetime. One reason is that we wish to analyze the more general case of spatially flat FLRW-like geometries in arbitrary dimensions, where