Relativistic geoids and time dependence: quasilocal frames versus isochronometric surfaces

TL;DR

This paper compares two definitions of the relativistic geoid, showing that the quasilocal frame approach is more general and applicable to dynamic spacetimes, and demonstrates how to operationally define it via redshift measurements.

Contribution

It demonstrates that the quasilocal frame formulation of the relativistic geoid extends the isochronometric surface approach to non-stationary spacetimes and provides an operational definition using redshift.

Findings

Quasilocal frame approach is more general than isochronometric surfaces.

The geoid quasilocal frame can be defined operationally via redshift measurements.

The approach applies to dynamic, non-stationary spacetimes.

Abstract

In a recent paper [D. Philipp, V. Perlick, D. Puetzfeld, E. Hackmann, and C. L\"ammerzahl, Phys. Rev. D 95, 104037 (2017)], a definition of a general-relativistic geoid, restricted to stationary spacetimes, was presented in terms of "isochronometric surfaces". In this note, we explicate how this definition is just a special, highly restrictive case of our earlier formulation of the general-relativistic geoid using quasilocal frames, which is valid for generic, non-stationary spacetimes [M. Oltean, R.J. Epp, P.L. McGrath, and R.B. Mann, Class. Quantum Grav. 33, 105001 (2016)]. Moreover, like the isochronometric surface geoid, we show how our "geoid quasilocal frame" (GQF) can also be defined, simply and operationally, in terms of redshift measurements.

Figures

Equations

$$0={\partial }_{\mathfrak{i}}N,\text{for }\left(\mathcal{M},\mathbf{g}\right)\text{ stationary}.$$

$$\{\begin{array}{l}0={\alpha }_a,\\ 0=\theta .\end{array}$$

$$\{\begin{array}{l}0={\partial }_{\mathfrak{i}}N+{\dot{u}}_{\mathfrak{i}},\\ 0={\sigma }^{\mathfrak{ij}}{\dot{\sigma }}_{\mathfrak{ij}}.\end{array}$$

$${\nabla }_{\mathbf{k}}(\mathbf{u}\cdot \mathbf{k}){|}_{\gamma }$$

$$=\frac{1}{2}\theta +{\alpha }_a{N}^a+{\theta }_{⟨ab⟩}{N}^a{N}^b,$$

Full text

Introduction and literature overview. — The theoretical formulation and applicational exploitation of the geoid in the context of general relativity has lately begun to attract interest from geodesists and relativists alike Kopeikin et al. (2015); Oltean et al. (2016); Philipp et al. (2017). Previously, the geoid had only been well defined and worked with in (post-)Newtonian gravity Hofmann-Wellenhof and Moritz (2006); Kopeikin et al. (2011), wherein it can essentially be thought of as a surface of constant potential, with applications ranging from the calibration of height measurements of GPS satellites to the study of geophysical processes, climate patterns, oceanic tides etc.

Lacking a notion of “potential” in general relativity, two possibilities—dubbed the a-geoid and the u-geoid—have been variously propounded and discussed in the recent literature for defining the geoid in a fully general-relativistic setting: first in ref. Kopeikin et al. (2015), henceforth the KMK paper (after the authors); then in ref. Oltean et al. (2016), henceforth the GQF paper (after the title); and most recently in ref. Philipp et al. (2017), henceforth the ICS paper (after the title).

While a- and u- “relativistic” geoids had previously been defined (see, e.g., refs. Kopeikin (1991); Kopeikin et al. (2011)), until recently analysis of such constructs had been essentially restricted to the low order post-Newtonian context, wherein they reduce to a surface of constant potential for some potential function (the Newtonian gravitational potential in the Newtonian limit). To our knowledge, refs. Kopeikin et al. (2015); Oltean et al. (2016); Philipp et al. (2017) constitute the first (non-approximate) treatments of the geoid in the full theory of general relativity.

Our intent, first, is to clarify the relation between these proposals, as well as the two (a- and u-) geoid definitions appearing therein—the essential ideas of which we begin by summarizing heuristically as follows.

(1) The a-geoid:

One demands that observers have zero acceleration tangential to the geoid; any acceleration is perpendicular to the geoid. (The Newtonian gravity analogue of this condition is that observers lie on a surface of constant potential, which includes a centrifugal potential term in the case of a rotating geoid.)

(2) The u-geoid:

One demands that the clocks of observers on the geoid “run at the same speed”. (This condition has no Newtonian gravity analogue.)

These two definitions are stated side-by-side in general-relativistic language already in the KMK paper Kopeikin et al. (2015). But crucially, there, the authors impose from the beginning the restrictive assumption of spacetime stationarity—under which, as indeed they point out, the two definitions turn out to be mathematically equivalent. They then proceed to derive, in this particular case, the PDEs that a geoid should satisfy assuming a certain simplified model of the Earth.

In contrast to this, we put forth in the GQF paper Oltean et al. (2016) a formulation of the geoid in general relativity based on the first (a-geoid) idea, without imposing any assumptions on the spacetime. In fact, this formulation emerges naturally as a particular choice (or “gauge”) of a previously developed and more general geometrical construction called a quasilocal frame—that is, essentially, a choice of a two-parameter family of timelike worldlines comprising the worldtube boundary of the history of a finite spatial volume. There are three degrees of freedom in the direction of the four-velocity vector tangent to each worldline, and so we can at least in principle—a point we shall elaborate upon below—impose three constraints on the motion of the quasilocal observers, thereby fixing the specific nature of the quasilocal frame. In most of the past work done on this Epp et al. (2009); McGrath et al. (2012); Epp et al. (2012, 2013); McGrath et al. (2014), the quasilocal frames were chosen to be rigid—called Rigid Quasilocal Frames (RQFs), a natural choice when one is interested in fluxes of gravitational energy, momentum, and angular momentum across the quasilocal frame boundary. However, they can just as well be chosen to describe geoids Oltean et al. (2016). Thus, we called them Geoid Quasilocal Frames (GQFs). Moreover, we found solutions for these GQFs in some spacetimes of interest, focusing on the (perturbed) Schwarzschild and Kerr metrics.

Recently, the ICS paper Philipp et al. (2017) developed the idea of the u-geoid in the form of a geometrical conception known as an isochronometric surface (ICS). These are surfaces that foliate a stationary spacetime according to the requirement that one obtains along them a constant “redshift potential”—essentially, a measure of zero redshift between any pair of observers on the ICS. However, just as in the KMK paper, the assumption of stationarity is imposed from the beginning—and is, in fact, necessary for the very definition of an ICS in the first place. In fact, the main inspiration thereof was one of the earliest (more technologically-focused) attempts to formulate relativistic geodesy Bjerhammar (1986), which explored the same u-geoid idea conceptually, albeit in significantly less mathematical profusion and with very restricted forms of the spacetime metric. Then the authors in ref. Philipp et al. (2017) recover many of the same solutions as we found for GQFs, up to rotation terms (appearing if observers are attached to the surface of a “rotating Earth” as opposed to a “non-rotating” one) which the GQF paper did not consider.

Comparison of geoid definitions. — Now, we would like to put into perspective the scope of the overlap, to the degree that it exists, between these approaches insofar as their mathematical formulations are concerned.

It is in fact not difficult to see that the geoid definitions in all three papers are mathematically equivalent provided that the spacetime is stationary (and the geoid observers move along the associated timelike Killing vector field). We stress again however that, unlike the GQF paper, the KMK and ICS papers impose this assumption from the start (and formulate their definitions of the geoid accordingly); and indeed, it is actually trivial to see that the defining (u-)geoid equation in the KMK paper (their Eqn. (39) = constant) is identical to that in the ICS paper (their Eqn. (17) = constant)—which, in turn, agrees with that given in ref. Bjerhammar (1986) (Eqn. (26) therein).

Notations differ, and so let us establish a common one (mirroring ours in the GQF paper) in order to render the discussion here more precise. Let the spacetime be $(\mathscr{M},\bm{g})$, with coordinates $\{x^{a}\}$. In general, we use $\bm{T}$ to denote a $(k,l)$-tensor in $\mathscr{M}$ with abstract index notation $T^{a_{1}\cdots a_{k}}\,_{b_{1}\cdots b_{l}}$. Let $u^{a}$ denote the timelike unit vector field tangent to the congruence of worldlines of the two-parameter family of observers comprising the geoid. This congruence is a submanifold of $\mathscr{M}$ (with topology $\mathbb{R}\times\mathbb{S}^{2}$) that we will call $\mathscr{B}$. Let $n^{a}$ be the outward-pointing unit vector field normal to $\mathscr{B}$—which is uniquely fixed once $\bm{u}$ is specified. Let $\mathscr{H}$ be the two-dimensional “spatial” subspace of the tangent space of $\mathscr{B}$ that is orthogonal to $\bm{u}$. (A pictorial representation of this setup is given in Figure 1, here taken from the GQF paper.) Let $\bm{\sigma}$ denote the two-dimensional “spatial” metric that projects tensor indices into $\mathscr{H}$, and is induced on $\mathscr{B}$ by the choice of $\bm{u}$ (and thus also $\bm{n}$): $\sigma_{ab}=g_{ab}-n_{a}n_{b}+u_{a}u_{b}$. Let $\{x^{\mathfrak{i}}\}_{\mathfrak{i}=1}^{2}$ be “spatial” coordinates on $\mathscr{B}$ that label the observers’ worldlines, and let $t$ be a “time” coordinate on $\mathscr{B}$ such that surfaces of constant $t$ foliate $\mathscr{B}$ by spatial two-surfaces with topology $\mathbb{S}^{2}$. Finally, let $N$ denote the associated lapse function such that $\bm{u}=N^{-1}\partial/\partial t$.

We can think of the (instantaneous) geoid as a surface of constant $t$ (as indicated in Figure 1), but note that in general $\bm{u}$ need not be hypersurface orthogonal within $\mathscr{B}$ (this happens, e.g., when the geoid is rotating), in which case there exists no foliation of $\mathscr{B}$ by surfaces orthogonal to $\bm{u}$. In other words, the geoid observers are, in general, in motion with respect to the surfaces of constant $t$, and they will not be able to agree on a surface of simultaneity because the $\mathscr{H}$ spaces are not integrable. This is important, because we will often deal with projections of tensor components into the $\mathscr{H}$ spaces, which is not the same as projections tangential to a constant $t$ surface.

In this notation, let us temporarily restrict ourselves to stationary spacetimes, with the geoid observers moving along the timelike Killing vector field proportional to $\partial/\partial t$. Then, the defining (u-)geoid equation in the KMK and ICS papers (their Eqns. (39) and (17), respectively) is in both cases simply expressed in terms of the lapse function $N$ as

[TABLE]

The intuition behind this is that $N={\rm d}\tau/{\rm d}t$ is the rate of change of proper time, and so (1) is essentially saying that this should be the same for each of the geoid observers on a given surface of constant $t$ (clocks are seen to “run at the same speed”), and the same for all such surfaces of constant $t$ (since $\partial N/\partial t=0$ by the stationarity restriction). Moreover, it is not difficult to follow the reasoning of both the KMK and ICS papers to the effect that the a-geoid indeed coincides with the u-geoid under this restriction to stationarity.

In contrast, in the GQF paper, we assume nothing about the spacetime a priori. Our defining (a-)geoid equations (Eqn. (4.1) in ref. Oltean et al. (2016)) are completely general for any spacetime $(\mathscr{M},\bm{g})$: let $a^{a}=\nabla_{\bm{u}}u^{a}$ be the acceleration and $\theta_{ab}=\sigma_{a}^{\phantom{a}c}\sigma_{b}^{\phantom{b}d}\nabla_{c}u_{d}$ the strain rate tensor associated with the congruence; then what we require of a geoid is simply the vanishing of the $\mathscr{H}$ component of the acceleration, $\alpha_{a}=\sigma_{a}^{\phantom{a}b}a_{b}$, as well as that of the scalar expansion, $\theta=\sigma^{ab}\theta_{ab}$:

[TABLE]

In coordinate language, these can be equivalently stated, respectively, in the form of the following time-dependent system of PDEs (Eqns. (4.2)-(4.3) in ref. Oltean et al. (2016)):

[TABLE]

The first condition, which captures the essence of the heuristic a-geoid definition given in the Introduction, fixes two of the three degrees of freedom of the geoid observers’ motion.

Of course this is not sufficient for a sensible definition of a geoid; we must also essentially fix the normal component of the observers’ acceleration, $\bm{n}\cdot\bm{a}$. Starting with a “static” geoid, by increasing/decreasing $\bm{n}\cdot\bm{a}$ we could make the geoid expand/contract arbitrarily. In the KMK and ICS papers this is not a consideration, since the spacetime is stationary, with observers moving along the timelike Killing vector field—trivially making the geoid static, and implicitly fixing $\bm{n}\cdot\bm{a}$. In a general, non-stationary spacetime, however, this issue must be explicitly considered, and the obvious, most geometrically natural choice is to demand zero scalar expansion of the congruence (which still allows for a dynamic shear, necessary to carry gravitational radiation through the geoid—more on this below). Then $\bm{n}\cdot\bm{a}$ is implicitly fixed through a Raychaudhuri-like equation. This condition on the scalar expansion ($\theta=0$) fixes an additional one degree of freedom. Together with $\alpha_{a}=0$, this fixes three degrees of freedom, completely specifying the quasilocal frame, i.e., in essence completely determining the three degrees of freedom in specifying the vector field $\bm{u}$ that defines the congruence.

It is obvious by inspection that if a stationary spacetime is assumed (with the geoid observers moving along the timelike Killing vector field), the time-dependent terms in (3) vanish; hence, the second equation becomes trivial and the first simply reduces to (1), that is, the geoid definition of the KMK and ICS papers.

Time dependence. — We wish now to examine in greater depth the role of time in defining the relativistic geoid.

The contention of the ICS paper is that theirs is a “more operational” formulation since in order to prescribe the geoid, one simply needs (according to them) “precise clocks” that “run at the same speed”. This is equivalent to saying that there should be zero redshift between any two observers on the geoid. In particular, then, let us apply this statement to nearest-neighbor pairs of observers, and to keep the analysis completely general, let us consider a general quasilocal frame (one with no conditions, such as the GQF or RQF conditions, imposed). Let $k^{a}$ be the tangent to an affinely-parametrized null geodesic $\lambda$ along which a light ray is assumed to travel, and let $\gamma$ be the worldline of an observer in the congruence with four-velocity $\bm{u}$. Then the nearest-neighbor zero redshift requirement is simply $0=\nabla_{\bm{k}}(\bm{u}\cdot\bm{k})$.

Let us analyze what this is saying. Since $\bm{k}$ is (necessarily) tangent to $\mathscr{B}$ (nearest-neighbors), we can decompose it into $k^{a}|_{\gamma}=u^{a}+N^{a}$ where $\bm{N}$ is a unit vector tangent to $\mathscr{H}$ defining the direction of the light ray towards a nearest neighbor (i.e., one angular degree of freedom). Using this, along with the fact that $\nabla_{\bm{k}}k^{a}=0$ along $\lambda$, a straightforward calculation reveals:

[TABLE]

where $\theta_{\langle ab\rangle}$ is the symmetric trace-free (STF) or “shear” part of $\bm{\theta}$, which has two independent components. The three terms in the last line can be regarded as the $\ell=0,1,2$ spherical harmonic terms (respectively), and so the geoid definition of the ICS paper (i.e. the vanishing of (4) for all $N^{a}$, to ensure zero redshift) holds if and only if, one by one, we have $\theta=0$, $\alpha_{a}=0$ and $\theta_{\langle ab\rangle}=0$. These are five constraints in total.

Now, the three constraints $\theta=0=\alpha_{a}$ (without $\theta_{\langle ab\rangle}=0$) constitute precisely the geoid definition (2) from our GQF paper. Moreover, the three constraints $\theta=0=\theta_{\langle ab\rangle}\Leftrightarrow 0=\theta_{(ab)}$ (without $\alpha_{a}=0$) are actually the equations previously employed to define rigid quasilocal frames (RQFs) Epp et al. (2009); McGrath et al. (2012); Epp et al. (2012, 2013); McGrath et al. (2014). Either set of three constraints is logically consistent with the availability of the (three) degrees of freedom in the motion of observers in an arbitrary spacetime.

However, imposing all five constraints $\alpha^{a}=0=\theta_{(ab)}$ over-determines the available degrees of freedom. In other words, they not only define the geoid (which only requires three of these), but they inherently restrict (as a consequence of the additional two) the geometry of the spacetime within which it can be embedded. In particular, this means that the spacetime must be stationary—as recognized in the ICS paper itself. The authors’ main point of argumentation for doing this is that a “more operational” determination of the geoid can thus be achieved via the use of standard clocks (connected by optical fibers). Yet, this forces the very restrictive presupposition that the geoid will not undergo any time evolution—for if it does, then it is clear that it will not be possible to consistently capture the details of this evolution just by using ICSs.

The two (u- and a-) geoid notions are thus, in general, not equivalent in general relativity. In other words, because of the time-dependent terms in (3), one cannot—except in very special (stationary) situations—equate surfaces composed of clocks that “run at the same speed” (u-geoids) with those relative to which observers accelerate only perpendicularly (a-geoids). Observers on an a-geoid/GQF, determined (only) by $\theta=0=\alpha_{a}$, will in general, according to (5), observe a time-dependent redshift due to the shear (STF) part of the strain rate tensor. This $\ell=2$ degree of freedom represents a “true” (general-relativistic) gravitational degree of freedom, and plays a crucial role vis-a-vis the gravitational radiation passing through the geoid. In ref. McGrath et al. (2012) (see Eqn. (32) and the discussion following it), we identify, in the case of an RQF, $\alpha_{a}{\cal P}^{a}$ as the operationally defined gravitational power density for gravitational radiation passing through the RQF (where ${\cal P}^{a}$ is the quasilocal momentum density, measured with gyroscopes); similarly, in the case of a GQF, $\theta_{\langle ab\rangle}{\cal S}^{\langle ab\rangle}$ is the operationally defined gravitational power density for gravitational radiation passing through the GQF (where ${\cal S}^{\langle ab\rangle}$ is the STF part of the quasilocal stress tensor—two independent components); see ref. Oltean et al. (2016) for details.

We emphasize that a GQF is just as “operationally” defined as an ICS. By measuring nearest-neighbor redshifts, we can, according to (5), measure each of $\theta$, $\alpha_{a}$, and $\theta_{\langle ab\rangle}$—which can, of course, be measured by other means as well. Measuring these, and responding with the appropriate normal acceleration, as necessary, observers can maintain a GQF frame, and even measure the flow of gravitational energy, momentum, and angular momentum through that frame. See refs. Epp et al. (2009); McGrath et al. (2012); Epp et al. (2012, 2013); McGrath et al. (2014) for more details in the case of RQFs, and ref. Oltean et al. (2016) for the case of GQFs.

Existence of GQFs. — We have generally claimed so far that GQFs can be constructed in any arbitrary spacetime. However, the precise question of existence of any sort of (“gauge” fixed) quasilocal frame (be it a GQF, or RQF, or anything else) is in fact much more subtle and elaborate than perhaps we have implied so far, and certainly than can be explored adequately in this note; nevertheless, this point warrants here a bit more qualification.

We have seen that, intuitively, the three degrees of freedom in specifying our quasilocal (two-parameter) family of observers correspond to the freedom of imposing three constraints thereon—achieving a GQF, an RQF etc. Yet, the mathematical problem of the existence of solutions to these (three) equations must be dealt with in each separate case. In other words, in order to guarantee that “GQFs always exist”, for instance, it is actually necessary to prove the following: that for any arbitrary spacetime $(\mathscr{M},\bm{g})$, one can always construct a GQF—that is to say, solutions to the GQF equations (2) exist—on an appropriately fibered timelike worldtube “centered” around any desired timelike worldline $\Gamma$ in $\mathscr{M}$.

This is a nontrivial problem. Existence of RQFs was shown in ref. Epp et al. (2012); the strategy was to construct, using a Fermi normal coordinates approach, the general solution to the RQF equations as a perturbative series in powers of the areal radius (about the trivial, small-sphere RQF), and to show that there is in principle no technical obstruction to extending these solutions to any desired order.

Following an analogous approach, it is also possible to prove this for GQFs. We will present the full analysis in a forthcoming paper Oltean et al. . This will essentially show that one can generically solve for arbitrary (time-dependent) GQFs in any spacetime.

Conclusions. — Defining the relativistic geoid as a u-geoid/ICS suffers from evident and unnecessary deficiencies in comparison with the completely general a-geoid/GQF definition, from both a foundational as well as pragmatic point of view.

Foundationally, it is highly unsatisfactory to have to impose a priori such a restrictive assumption—stationarity—on the spacetime itself in order to even be able to make sense the u-geoid/ICS, especially when the a-geoid/GQF trivially reduces to it thereunder. Indeed, the generic success and usefulness of the broader formalism of quasilocal frames has lain precisely in eliminating the all-too-pervasive need for stipulating special spacetimes (i.e. necessitating the existence of Killing vector fields) in making sense of the mechanisms behind energy, momentum, and angular momentum transfer (conservation laws) in general relativity Epp et al. (2009); McGrath et al. (2012); Epp et al. (2012, 2013); McGrath et al. (2014); in this regard, it is no different—as will be developed, we expect, in much greater depth in the future—with geodetic modeling.

Pragmatically, one of the principal motivations for moving geodesy from the setting of Newtonian gravity to general relativity is increased accuracy—and, as such, it is unclear how a geoid definition based on stationarity can adequately handle the treatment of relevant time-dependent perturbations. With the uncertainty of the latest generation of optical atomic clocks approaching the level of one part in $10^{18}$, time variations in the geoid—corresponding to an accuracy level of mm to cm—will have to be taken into account Voigt et al. (2016); Müller et al. (2017). These arise, for example, from relative perturbations of the solid Earth tides, periodic effects due to ocean tides, non-tidal oceanic and atmospheric effects, variations due to land hydrology, tectonic processes etc.

The ICS paper suggests dealing with these by thinking of the “true” metric $\bm{g}$ as a sum of a stationary part $\bm{g}_{{\rm stat}}$ (which can include the time-averaged effects of those perturbations that are cyclic), and a “small” time-dependent part $\bm{h}$, i.e. $\bm{g}=\bm{g}_{{\rm stat}}+\bm{h}$. However, working out a “perturbative theory” of the geoid (for $\bm{h}$) in this setting seems unnecessarily contrived and potentially confusing. One would need a criterion for constructing the the components of the “stationary” part of the metric $\bm{g}_{{\rm stat}}$ in a time-dependent setting where, by definition, the ICS’s change with time. It is not at all apparent how one could deduce related changes in $\bm{h}$, without having (from the beginning) a theory of time-dependent ICS’s.

Of course, this is not at all an issue if one uses the a-geoid/GQF approach to define the geoid. Being applicable in arbitrary spacetimes, the inclusion and computation of time-dependent perturbations fits naturally into our general formalism—as, indeed, we will illustrate mathematically in our future paper on GQF existence Oltean et al. . Arbitrarily large time variations in strong gravitational fields—unnecessary for cases such as the geoid of the Earth, but potentially relevant for astrophysical applications—can certainly also be accommodated with GQFs.

As mentioned earlier, and emphasized again in closing, a time-dependent problem of particular interest is that of gravitational waves passing through the geoid; the effect thereof is felt in the form of shearing, which can be measured via the $\ell=2$ component of the time-dependent nearest-neighbor redshift. The gravitational energy flux through the geoid is—as we have mentioned, and as discussed at greater length in the GQF paper—directly related to this quantity (contracted with the STF part of the quasilocal stress). In our forthcoming paper, we will additionally offer a more detailed discussion on gravitational waves in the context of GQFs.

Acknowledgements

This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.