By Kathleen A. Rosser

Kathleen.A.Rosser@ieee.org
Published 29 June  2023
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Also available at ResearchGate

ABSTRACT

A novel general relativistic plane-wave metric is presented with a proposal for possible detection. Called a quasi-inertial oscillation (QIO), the metric is an exact solution to Einstein’s vacuum field equations, yet carries non-Einsteinian longitudinal polarization, permitted by its quasi-inertial status. Although QIOs are metric waves in spacetime, they differ fundamentally from the standard gravitational waves assumed to be detected by LIGO and Virgo. QIOs are an oscillating type of quasi-inertial disturbance (QID), a broader class of exact solutions to Einstein’s equations with varying features that travel at the speed of light. The observable properties of QIDs, and of QIOs in particular, have rarely if ever been studied in the literature. Yet it is shown here that if general relativity accurately models gravity, QIOs should produce test particle acceleration and are thus in principle observable by space-based detectors. Due to Riemann flatness, QIOs were historically dismissed as unphysical, and hence unobservable, by authors such as Taub, McVittie, Weber and others. However, these authors were seeking gravitational waves capable of forming gravitons, a far more stringent requirement than mere observability. Moreover, the claim that Riemann flatness precludes detection does not apply to metrics for which the coordinate system is fixed by a physical structure, such as a space-based platform, an accelerating rocket, or the cosmic microwave background, nor to metrics associated with frame-dependent quantum processes such as the Unruh effect. Nevertheless, subsequent authors continued to dismiss exact longitudinal plane-wave solutions and thus overlooked a real possibility of detection. With the benefit of hindsight, it is proposed here that QIOs may not only be detectable, but that Riemann flatness does not rule out the potential influence of QIOs on astrophysical or cosmological backgrounds, including the stochastic gravitational wave and photon backgrounds. QIOs are presented first in rectangular coordinates, then in Brinkmann coordinates for comparison with pp-waves. It is also shown that the Riemann-flat metric of a uniformly accelerating Rindler frame, theoretically detectable by Unruh radiation, can be constructed as a product of (rectangular) advanced and retarded QIDs, offering further argument for observability through an Unruh-type effect. Astrophysical and cosmological mechanisms are proposed as physical sources of QIOs. It is further suggested that, since QIOs are energy-free exact solutions to Einstein’s equations, there is nothing to obstruct their generation by accidental alignments of matter, and thus nothing to prevent the vacuum from being filled with weak transient random spacetime fluctuations. Spacetime fluctuations are of course predicted in theories of quantum gravity; however QIOs would constitute a classical source of fluctuations. Finally, the role of Riemann curvature in pseudo-Riemannian spacetime is challenged in view of categorical differences between space and time. Overall, questions are raised about space, time, and the foundations of gravitational wave theory, leading to the suggestion that historical assumptions may have been misapplied in the standard approaches of today.

I. INTRODUCTION

Interest in gravitational waves (GWs) of both astrophysical and cosmological origin has flourished since the advent of the Advanced LIGO and Virgo detectors, whose observations have unleashed a new era of multi-messenger astronomy [1]. GWs are believed to interact only weakly with matter and may thus provide a powerful tool for investigating inflation models and the high-energy fields of the early universe [2-6]. It is anticipated that GW detectors in the near future will offer precise tests of quantum gravity and of general relativity (GR) in the strong-field regime [1,7]. Unresolved GWs generated by a variety of mechanisms are assumed to form a possibly anisotropic stochastic gravitational wave background (SGWB) [8-11], observable by current or near-future detectors. The SGWB is predicted to arise from the superposition of unresolved GWs emanating from a broad array of astrophysical and cosmological sources [12-15], including close binary systems, supermassive black hole mergers [16], galactic millisecond pulsars [17], primordial black holes [18-25], quasars [26], cosmic strings [27-30], inflation [2,31-33], primordial GWs [34-36], cosmological phase transitions [37-40], photon graviton conversion from black hole photon spheres [41], superradiant instability of spinning black holes in the presence of massive bosons [42], and magnetohydrodynamic turbulence due to high conductivity at early epochs [43,44].

The SGWB is in some ways a gravitational analog of electromagnetic radiation backgrounds such as the cosmic microwave background (CMB) and the stochastic photon background (SPB). The SPB is defined here as a composite of photon backgrounds arising from or influenced by non-CMB sources. These backgrounds include secondary photons from relatively cold CMB photons scattered by the inverse Compton process off of free hot electrons in the intracluster medium of galaxy groups and clusters, causing a distortion of the CMB spectrum known as the thermal Sunyaev-Zel’dovich effect [45,46]. Other SPB components include the cosmic radio background (CRB), believed to be of extra-galactic or cosmological origin and displaying temperatures substantially greater than those observed in radio-emitting galaxies [47]; and the cosmic infrared background (CIB), comprising photons arising from thermal dust emission in star-forming galaxies [45,46].

The SGWB promises to provide one of the best windows we have into the physics of the early universe [22,48,49], and is a topic of growing importance in the literature [9,50-56]. Indeed, there is evidence the SGWB may have already been detected in pulsar timing array (PTA) data from the North American Nanoherz Observatory for Gravitational Waves (NANOGrav) [15,16,21,43]. However, the NANOGrav data implies a GW polarization that departs from that expected in GR, suggesting a nonstandard form of gravitational oscillation or a modified theory of gravity [15]. This question remains open.

Given the relevance of the SGWB and other radiation backgrounds to this and other fundamental questions in gravitation and cosmology, it is worthwhile to reexamine seldom-explored general relativistic phenomena related to nonstandard spacetime disturbances. The purpose of this paper is to derive and investigate a novel spacetime disturbance, to be called the quasi-inertial disturbance (QID), which could conceivably impact both the SGWB and SPB, and potentially shed light on the NANOGrav question. Defined as a plane-symmetric general relativistic spacetime variation traveling at the speed of light c, the QID is represented by a dynamic metric g_μν(t,x) in rectangular coordinates that exactly solves the vacuum Einstein equations R_μν=0. QIDs may be either periodic or nonperiodic. The periodic variety investigated here are called quasi-inertial oscillations (QIOs).

QIOs possess neither energy nor Riemann curvature and are best described as quasi-inertial rather than as gravitational phenomena. QIDs, of which QIOs form an oscillating subclass, are perhaps the simplest dynamic metrics that exactly solve Einstein’s vacuum field equations (EVFE), and one might wonder why such solutions are not routinely studied. The historical reason for this omission is precisely their curvature-free nature, a property that led earlier researchers such as Taub and McVittie to dismiss previously known QIO solutions as spurious or unphysical, and hence unviable as GW candidates [57-59]. The claim that curvature-free waves are unphysical will be referred to here as Taub’s rule. However the above authors were seeking GWs capable of forming gravitons, a unique requirement beyond mere observability. Hence in Taub’s usage, the term unphysical does not automatically mean unobservable. Indeed, that QIOs are observable forms a central theme of the present work. Here and throughout this paper, a metric is defined as observable if the varying features of the metric can in principle be measured in some physical reference frame, where the term varying features denotes explicit space or time dependent quantities.

It is well known that Riemann flat metrics, i.e. metrics with a vanishing Riemann tensor R_μναβ=0, can be transformed into the Minkowski metric by a suitable change of coordinates. Such a transformation property is often assumed to mean that the metric is also undetectable. However, this assumption does not hold for metrics whose coordinates are defined by a physical platform or structure, such as the surface of a planet, a rotating space station, an accelerating rocket, or the CMB, nor for metrics associated with frame-dependent quantum effects, such as the Unruh effect [60]. Indeed, physical structures defining de facto preferred coordinate frames, a setup unavoidable in practice, offer detection capabilities that are often overlooked in pure mathematical contexts, in which all frames are held to be equally viable even if based on null [61] or other unrealizable coordinates. The existence of a de facto preferred frame, of course, does not violate covariance, but arises simply as an artifact of the indispensable role of the observer. Indeed, when observational aspects are paramount, de facto preferred frames are sometimes employed in the literature to provide deeper insight into GR solutions. An informative example is offered by Hobson [62], who constructs preferred frames for Schwarzschild and Reissner-Nordstrom metrics based on trajectories of massive particles. These frames define coordinates that are well-behaved at the event horizon and useful in clarifying many observable phenomena, including Hawking radiation. Thus specific coordinate frames can be essential for detecting or describing the varying features of a metric. With this understanding, it is shown in Section IV that QIOs are in principle observable by suitable detectors aboard any solid space-based platform, where the platform must be larger than the QIO wavelength to hold the detectors rigid. It follows that, contrary to the historical interpretation of Taub’s rule, QIOs are nontrivial from an observational standpoint.

It is further conjectured in Section VI that Riemann flatness does not rule out detection for metrics with the property that the Riemann tensor vanishes due to cancellation between space and time components. Such a cancellation may be questionable in any case by virtue of manifest differences between space and time, as will be argued. Flat yet observable metrics include the Milne and Rindler metrics. These metrics will be discussed briefly in the next paragraph. The Rindler metric will be explored more thoroughly in Section IV.

The idea that Riemann flatness does not preclude detection can be made intuitive by the following three examples. In the first example, we consider the Rindler frame, as defined by Sugiyama et al. in [63], which corresponds to an observation platform undergoing constant acceleration. The Rindler metric can be made Minkowskian by a coordinate transformation, meaning the metric is Riemann flat. Nevertheless an observer in a Rindler frame can detect the varying features of the metric both by inertial forces and theoretically also by Unruh radiation, the latter a prediction of quantum field theory for non-inertial or curved spacetimes [60,64-66]. The Rindler metric is particularly relevant to QID observability in that Rindler metrics can be expressed as a product of rectangular advanced and retarded QIDs. This result is derived in Section IV. The close relationship between Rindler metrics and QIDs suggests that QIDs might also be detectable through a form of Unruh-type radiation. This is a topic for future research.

The second example involves the metric for a constantly expanding Milne universe, given by

ds² = dt² ─ b²t²(dr² +r² dΩ²)

where b is a constant coefficient. Milne originally derived this metric in a special relativistic framework [67]. However, the metric may also be derived in the GR formalism as a solution to Einstein’s field equations. In the latter case, the Milne metric constitutes a special case of the Friedmann-Robertson-Walker (FRW) metric

ds² = dt²─a²(t)(dr² +r² dΩ²)

for an empty universe, where the comoving coordinates expand at a constant rate with scale factor a(t)=bt [68-72]. The Milne metric can be transformed into a patch of Minkowski space and is therefore Riemann flat. However a hypothetical Milne universe, whose spatial coordinates are defined by small comoving test stars, would be detectable by a linear Hubble redshift relation, where in GR, the redshift is assumed to arise from spatial expansion, a behavior incompatible with Minkowski spacetime. As an interesting aside, the Milne metric, though a solution to Einstein’s field equations for empty space only, is not unrealistic when compared to modern astronomical data on cosmic acceleration. Indeed, some authors maintain that SNIa data, as of 2020, do not convincingly demonstrate acceleration, but instead constitute a fair, or even excellent, fit to the Milne universe [69,71,72].

The third example involves the metric of a rotating coordinate frame. This metric can also be made constant by a change of coordinates, meaning it too is curvature free. Yet such a metric is observable through for example the Coriolis force, an inertial force measurable in any earth-based laboratory. In addition, rotating frames are believed detectable by Unruh radiation [73]. The above three examples support the present postulate that, contrary to a prima facie interpretation of Taub’s rule, the vanishing of the Riemann tensor does not make detection impossible with respect to ordinary rigid observation platforms.

It is important to emphasize that the wave features of QIOs are arguably more real the inertial forces found in Rindler or rotating frames, since the latter forces can be measured only on specific accelerating platforms, while QIOs can in principle be measured on all sufficiently large rigid space-based platforms. Moreover, Rindler and rotating metrics are thought to describe spacetime distortions arising from the motion of the platform, while a QIO metric describes spacetime distortions arising from the motion of distant matter, a phenomenon somewhat reminiscent of Mach’s principle. Due to these distinctions, a QIO is defined here as quasi-inertial rather than inertial. With regard to observation, it therefore seems plausible that broad-spectrum, low-amplitude QIOs populate the universe and exert a hidden influence on the dynamic properties of the vacuum, particularly in the sparse realms of the intergalactic medium. Higher-frequency QIOs would be expected to penetrate galaxies, and could conceivably pierce solid matter in the ultraviolet limit, causing elementary particle fluctuations, perhaps by a process akin to the ponderomotive effect discussed by Deepen Garg et al. in [74].

In the vacuum region of a QIO, the energy-momentum tensor T_μν on the rhs of Einstein’s field equations vanishes by definition, where Einstein’s equations are given by

R_μν − ½ g_μν R = κT_μν

for R_μν the Ricci tensor, R the curvature scalar, and κ=─8πG/c². Thus a QIO does not contribute to its own background energy-momentum field and constitutes an energy-free plane wave. The physical interpretation of a QIO therefore requires an understanding of gravitational field energy which, while common in many GR applications [58], differs from that in the literature on standard GWs. There is of course nothing in the GR formalism that prohibits energy-free wave solutions [74]. Energy-free GWs, sometimes associated with “soft gravitons”, are an active topic of research, and methods of detection are now under discussion [75,76]. Historically, the energy relation E=hν for GWs arose from analogies with electromagnetism or quantum mechanical wave functions. More sophisticated definitions of effective GW energy density have also emerged [9]. However, these definitions are approximate and often ambiguous [58,77]. Moreover, introducing field energy into GR frequently involves ad hoc definitions of quasi-local energy density, which is problematic due to gauge dependence [9,78-80]. These issues are avoided by assuming the gravitational field does not contribute to the energy-momentum tensor in any case, and indeed that the field itself contains no energy. It might be objected that waves cannot propagate without energy, implying spacetime somehow impedes wave motion. However according to the present interpretation, the gravitational field is synonymous with the geometry of spacetime, to which energy, impediments, or other material properties cannot be attributed.

Gravitational radiation in the form of standard GWs is believed to have two possible polarizations in GR, both of them transverse spin-2 tensor modes. These are called Einsteinian polarizations. Modified gravity theories, such as f(R) and other metric theories, permit up to four additional modes, including two spin-1 vector modes and two spin-0 scalar modes, where the latter are the breathing mode and the longitudinal mode [81-92]. These are referred to as non-Einsteinian polarizations. QIOs, although consistent with GR, carry non-Einsteinian longitudinal polarization, allowable due to their quasi-inertial nature. It is often claimed that the detection of longitudinal GWs would prove some form of modified gravity prevails in our present universe [81,89,93,94]. However unless the detector were able to distinguish longitudinal GWs from QIOs, the latter might conceivably be misindentified as a GW carrying non-Einsteinian polarization. Modified gravity theories that permit longitudinal GWs include tensor-vector-scalar theories such as TeVeS, bimetric gravity theories, the Einstein-Ether theory [93], the Lightman-Lee theory [90], f(R) and Horndeski theories [95], higher-order R² scalar gravity [96], and massive gravity theories [50,89], including massive teleparallel Horndeski gravity [97].

Some authors propose that the high-energy regime of the early universe may have been governed by a type of modified gravity that allows emission of non-Einsteinian longitudinal GWs [4]. These primordial longitudinal GWs (PLGWs), should they exist, are expected to traverse the cosmos and be observable by future longitudinal GW detectors. However, if GR accurately describes the universe today, and if non-Einsteinian modes are impossible in GR, it seems evident that PLGWs would be unable to propagate through the regions of our present cosmic neighborhood. PLGWs emitted under a modified gravity regime would therefore decay or change form on their way to our observatories. It is postulated here that PLGWs might in fact change form, losing both energy and curvature as they propagate over cosmic distances, finally evolving into a wavelike imprint whose features are consistent with GR. According to this postulate, QIOs may arise as an end state of PLGW evolution, a scenario to be discussed in Section V.

Should provable sources exist and a realistic interpretation be established, QIOs may lead to new physics and exert measurable effects on the SGWB. It is also conceivable that QIOs could induce oscillations in the free electrons of the intracluster medium, which would in turn emit photons as a component of the SPB. This process would create an additional photon background masquerading as a component of the CMB, but with spectral and isotropy signatures distinct from that of the blackbody radiation emitted at recombination. QIOs might in addition be directly observable as a component of the SGWB by future longitudinal mode gravitational wave detectors [98]. Such effects would be particularly relevant today, in view of the recent emergence of tantalizing evidence for SGWB detection [18,99-101].

There appears an intriguing possibility that colliding QIOs might manifest nonzero Riemann curvature. The idea of colliding gravitational wave fronts was explored historically in [102], and is discussed in [103] in the context of Kundt waves, of which QIOs are perhaps a special case [104]. Kundt waves are defined in [103] as exact wave solutions to Einstein’s field equations for the vacuum with shearfree null hypersurface wave fronts that may or may not be hyperplanes; the QIO would be a case of a hyperplane wave front. However, available references to prove a QIO is a Kundt wave are currently unknown to this author. QIOs, when expressed in lightcone or Brinkmann coordinates [75,104], also resemble the exact gravitational plane waves known as pp-waves [106,107], defined in [108] as plane-fronted GWs with parallel rays. However the pp-wave and QIO classes of metric share only a trivial common subclass. QIOs will be transformed into Brinkmann coordinates and compared with pp-waves in Section III.

Traditionally, GWs were derived based on long-debated conventions about what constitutes a viable wave solution to Einstein’s field equations [57,58,60,109,110]. One such convention requires the waves be described by perturbations on a background metric, where the perturbation term h_μν in the metric g_μν=f_μν+h_μν must be small compared to the background term f_μν [111-115]. Standard GWs are therefore approximations valid only in the weak-field regime, (see however [76,116-118]), if they are valid at all, a position debated in [119]. The background metric is usually chosen to have a high degree of symmetry, and may for example be the Minkowski metric, the Schwarzschild metric [120], the Friedman-Robertson-Walker (FRW) metric [3,18,121-123] or the Newtonian gauge representation of the FRW metric [2,43].

The weak-field approximation to GR is often referred to as linearized gravity [111-115,124]. However, the motivation for linearizing GR to derive GWs is somewhat obscure in the literature. Weinberg drew an analogy with the wave equations of particle physics to conclude that GWs carry energy E=hν,¹ which in turn contributes to the energy-momentum tensor T_μν and prevents solutions to the exact Einstein equations, a dilemma supposedly avoided by linearization [125]. Some authors adopt linearized GR to make the field equations tractable [126], while others work in the weak-field regime without explicit justification [112,114,115]. Still others claim that exact plane wave solutions are unphysical, perhaps due to Riemann flatness or to the energy-free nature of the waves, although the reason is not always stated [79]. These claims led to the belief that only linearized GWs are viable [57,58,79,110]. (See however [108] for discussion of exact plane wave solutions.) It thus appears linearized gravity is adopted due to any of three motivating factors: to simplify Einstein’s equations, to compensate for assumed GW energy, or to avoid supposedly unphysical exact plane wave solutions. These three factors, while potentially relevant to standard GWs, are irrelevant in the context of exact plane waves, a viewpoint highlighted in the next paragraph.

First, regarding simplification of Einstein’s equations, it is important to recognize that in rectangular coordinates, the field equations can be reduced to two elementary wave equations that are easily solved in exact form. Indeed, exact plane wave derivations use fewer assumptions and are often far simpler than linearized GW solutions. Second, with regard to Weinberg’s claim that GW energy feeds back into T_μν, requiring perturbations to make the energy negligible, it should be noted that in other GR applications, gravitational field energy is assumed to make no contribution to the energy-momentum tensor in the first place [114,126], rendering moot any need for perturbations. Moreover, some authors point out that metric perturbations are unphysical and inherently incorrect due to gauge dependence, casting doubt on the validity of all supposed GW detections to date [119]. Third, regarding the viability of exact plane wave solutions, there are contexts in which exact plane waves are considered more viable than linearized waves. In such cases, perturbation methods may be inadequate, again due to gauge dependence [3]. In particular, the GW memory effect, defined as a distortion in a GW detector that persists after the wave has passed [127], involves exact plane wave solutions [105] and is described by Christodoulou as “an inherently nonlinear phenomenon that cannot be captured by perturbation theory [128].” (Interestingly, QIDs may also produce a memory effect, as will be shown in Section II.) In any event, fundamental logic calls into question how a physical process could even exist for which approximate solutions are more viable than exact ones, an enigma not addressed this paper.

Another convention adopted in traditional derivations of GWs is that of the traceless transverse (TT) gauge. This stems from the consensus, based on Taub’s rule, that real GWs must have nonzero curvature, and therefore that the Riemann tensor R_μναβ must not vanish [129]. Accordingly, Taub, Weber, and others obtained GW solutions by identifying nonzero components of the Riemann tensor and showing that these correspond to transverse polarizations [57,79,110]. Standard GWs were thus assumed to be transverse waves much like electromagnetic waves [12,130], a similarity often touted as confirmation of Taub’s method. Transverse polarization of course means that a GW with x-directed flow will cause space in the y,z directions to be alternately stretched and contracted, while space in the x direction remains unchanged. The QIO, in contrast, is a longitudinal plane wave. Hence a QIO with x-directed flow will cause space in the x direction to be stretched and contracted, leaving space in the y,z directions unchanged. Notably, a QIO causes time to be stretched and contracted with magnitude and phase identical to that of space.

This paper is organized as follows. The elementary wave nature of Einstein’s equations will be shown in Section II, where the vacuum field equations are solved for a QIO metric in rectangular coordinates. In Section III, lightcone coordinates are used to derive a transformation that renders the QIO metric Minkowskian. QIOs will also be presented in Brinkmann coordinates and compared with pp-waves. In Section IV it will be shown that the particle Lagrangian L=mds/dt predicts test mass acceleration in the field of a QIO, indicating QIOs can in principle be observed in any non-freely falling frame. In addition, QIO observability will be argued by analogy with the Rindler metric, which can be constructed as a product of rectangular advanced and retarded QIDs. Section V is devoted to generation, propagation and detection of QIOs, offering the hypothesis that QIOs may be an end product of PLGW evolution, and featuring a skeletal design for a space-based QIO detector. Section VI highlights categorical differences between space and time, leading to the conjecture that in pseudo-Riemannian spacetime, Riemann curvature is not a necessary criterion for observability.

Greek indices run from 0 to 3 throughout this paper. Units G=c=1 will be used, although these constants are sometimes inserted for clarity. We will work in the signature (+−−−), using Dirac’s sign convention [114]. The notation ∂_μ designates the partial derivative ∂/∂x^μ.

II. DERIVING THE QIO FROM EINSTEIN’S FIELD EQUATIONS

It is of course true that a curvature-free metric exactly solves EVFE, since if the metric obeys R_μναβ=0, it also obeys the contracted equation R_μν=0. However, it will be instructive to solve EVFE explicitly in order to highlight the simple wave nature of Einstein’s equations, especially as this contrasts with the comparative complexity of standard linearized GWs. Accordingly, it is shown in this section that QIDs, and hence QIOs, are exact solutions to Einstein’s field equations. Physical and intuitive properties of QIDs will also be discussed below Eqs. (2). . . READ MORE>> PDF