Boundaries and Symmetries

7.7  Boundaries and Symmetries

                                John Milton, 1667

Each point on the surface of an ordinary sphere is perfectly symmetrical with every other point, but there is no difficulty imagining the arbitrary (random) selection of a single point on the surface, because we can define a uniform probability density on this surface. However, if we begin with an infinite flat plane, where again each point is perfectly symmetrical with every other point, we face an inherent difficulty, because there does not exist a perfectly uniform probability density distribution over an infinite surface. Hence, if we select one particular point on this infinite flat plane, we can't claim, even in principle, to have chosen from a perfectly uniform distribution. Therefore, the original empty infinite flat plane was not perfectly symmetrical after all, at least not with respect to our selection of individual points. This shows that the very idea of selecting a point from a pre-existing perfectly symmetrical infinite manifold is, in a sense, self-contradictory. Similarly the symmetry of infinite Minkowski spacetime admits no distinguished position or frame of reference, but the introduction of an inertial particle not only destroys the symmetry, it also contradicts the premise that the points of the original manifold were perfectly symmetrical, because the non-existence of a uniform probability density distribution over the possible positions and velocities implies that the placement of the particle could not have been completely impartial.

Even if we postulate a Milne cosmology (described in Section 7.5), with dust particles emanating from a single point at uniformly distributed velocities throughout the future null cone (note that this uniform distribution isn't normalized as a probability density, so it can't be use make a selection), we still arrive at a distinguished velocity frame at each point. We could retain perfect Minkowskian symmetry in the presence of matter only by postulating a "super-Milne" cosmology in which every point on some past spacelike slice is an equivalent source of infinitesimal dust particles emanating at all velocities distributed uniformly throughout the respective future null cones of every point. In such a cosmology this same condition would apply on every time-slice, but the density would be infinite, because each point is on the surface of infinitely many null cones, and we would have infinitely dense flow of particles in all directions at every point. Whether this could correspond to any intelligible arrangement of physical entities is unclear.

The asymmetry due to the presence of an infinitesimal inertial particle in flat Minkowski spacetime is purely circumstantial, because the spacetime is considered to be unaffected by the presence of this particle. However, according to general relativity, the presence of any inertial entity disturbs the symmetry of the manifold even more profoundly, because it implies an intrinsic curvature of the spacetime manifold, i.e., the manifold takes on an intrinsic shape that distinguishes the location and rest frame of the particle. For a single non-rotating uncharged particle the resulting shape is Schwarzschild spacetime, which obviously exhibits a distinguished center and rest frame (the frame of the central mass). Indeed, this spacetime exhibits a preferred system of coordinates, namely those for which the metric coefficients are independent of the time coordinate.

Still, since the field variables of general relativity are the metric coefficients themselves, we are naturally encouraged to think that there is no a priori distinguished system of reference in the physical spacetime described by general relativity, and that it is only the contingent circumstance of a particular distribution of inertial entities that may distinguish any particular frame or state of motion. In other words, it's tempting to think that the spacetime manifold is determined solely by its "contents", i.e., that the left side of G_uv = 8T_uv is determined by the right side. However, this is not actually the case (as Einstein and others realized early on), and to understand why, it's useful to review what is involved in actually solving the field equations of general relativity as an initial-value problem.

The ten algebraically independent field equations represented by G_uv = 8T_uv involve the values of the ten independent metric coefficients and their first and second derivatives with respect to four spacetime coordinates. If we're given the values of the metric coefficients throughout a 3D spacelike "slice" of spacetime at some particular value of the time coordinate, we can directly evaluate the first and second derivatives of these components with respect to the space coordinates in this "slice". This leaves only the first and second derivatives of the ten metric with respect to the time coordinate as unknown quantities in the ten field equations. It might seem that we could arbitrarily specify the first derivatives, and then solve the field equations for the second derivatives, enabling us to "integrate" forward in time to the next timeslice, and then repeat this process to predict the subsequent evolution of the metric field. However, the structure of the field equations does not permit this, because four of the ten field equations (namely, G_0v = 8T_0v with v = 0,1,2,3) contain only the first derivatives with respect to the time coordinate x⁰, so we can't arbitrarily specify the g_uv and their first derivatives with respect to x⁰ on a surface of constant x⁰. These ten first derivatives, alone, must satisfy the four G_0v conditions on any such surface, so before we can even pose the initial value problem, we must first solve this subset of the field equations for a viable set of initial values. Although these four conditions constrain the initial values, they obviously don't fully determine them, even for a given distribution of T_uv.

Once we've specified values of the g_uv and their first derivatives with respect to x⁰ on some surface of constant x⁰ in such a way that the four conditions for G_0v are satisfied, the four contracted Bianchi identities ensure that these conditions remain satisfied outside the initial surface, provided only that the remaining six equations are satisfied everywhere. However, this leaves only six independent equations to govern the evolution of the ten field variables in the x⁰ direction. As a result, the second derivatives of the g_uv with respect to x⁰ appear to be underdetermined. In other words, given suitable initial conditions, we're left with a four-fold ambiguity. We must arbitrarily impose four more conditions on the system in order to uniquely determine a solution. This was to be expected, because the metric coefficients depend not only on the absolute shape of the manifold, but also on our choice of coordinate systems, which represents four degrees of freedom. Thus, the field equations actually determine an equivalence class of solutions, corresponding to all the ways in which a given absolute metrical manifold can be expressed in various coordinate systems. In order to actually generate a solution of the initial value problem, we need to impose four "coordinate conditions" along with the six "dynamical" field equations. The conditions arise from any proposed system of coordinates by expressing the metric coefficients g_0v in terms of these coordinates (which can always be done for any postulated system of coordinates), and then differentiating these four coefficients twice with respect to x⁰ to give four equations in the second derivatives of these coefficients.

Notwithstanding the four-fold ambiguity of the dynamical field equations, which is just a descriptive rather than a substantive ambiguity, it's clear that the manifold is a definite absolute entity, and its overall characteristics and evolution are determined not only by the postulated T_uv and the field equations, but also by the conditions specified on the initial timeslice. As noted above, these conditions are constrained by the field equations, but are by no means fully determined. We are still required to impose largely arbitrary conditions in order to fix the absolute background spacetime. This state of affairs was disappointing to Einstein, because he recognized that the selection of a set of initial conditions is tantamount to stipulating a preferred class of reference systems, precisely as in Newtonian theory, which is "contrary to the spirit of the relativity principle" (referring presumably to the relational ideas of Mach). As an example, there are multiple distinct vacuum solutions of the field equations, some with gravitational waves and even geons (temporarily) zipping around, and some not. Even more ambiguity arises when we introduce mass, as Gödel showed with his cosmological solutions in which the average mass of the universe is rotating with respect to the spacetime background. These examples just highlight the fact that general relativity can no more dispense with the arbitrary stipulation of a preferred class of reference systems (the inertial systems) than could Newtonian mechanics or special relativity.

This is clearly illustrated by Schwarzschild spacetime, which (according to Birkhoff's theorem) is the essentially unique spherically symmetrical solution of the field equations. Clearly this cosmological model, based on a single spherically symmetrical mass in an otherwise empty universe, is "contrary to the spirit of the relativity principle", because as noted earlier there is an essentially unique time coordinate for which the metric coefficients are independent of time. Translation along a vector that leaves the metric formally unchanged is called an isometry, and a complete vector field of isometries is called a Killing vector field. Thus the Schwarzschild time coordinate t constitutes a Killing vector field over the entire manifold, making it a highly distinguished time coordinate, no less than Newton's absolute time. In both special relativity and Newtonian physics there is an infinite class of operationally equivalent systems of reference at any point, but in Schwarzschild spacetime there is an essentially unique global coordinate system with respect to which the metric coefficients are independent of time, and this system is related in a definite way to the inertial class of reference systems at each point. Thus, in the context of this particular spacetime, we actually have a much stronger case for a meaningful notion of absolute rest than we do in Newtonian spacetime or special relativity, both of which rest naively on the principle of inertia, and neither of which acknowledges the possibility of variations in the properties of spacetime from place to place (let alone under velocity transformations).

The unique physical significance of the Schwarzschild time coordinate is also shown by the fact that Fermat's principle of least time applies uniquely to this time coordinate. To see this, consider the path of a light pulse traveling through the solar system, regarded as a Schwarzschild geometry centered around the Sun. Naturally there are many different parameterizations and time coordinates that we could apply to this geometry, and in general a timelike geodesic extremizes d (not dt for whatever arbitrary time coordinate t we might be using), and of course a spacelike geodesic extremizes ds (again, not dt). However, for light-like paths we have d = ds = 0 by definition, so the path is confined to null surfaces, but this is not sufficient to pick out which null path will be followed. So, starting with a line element of the form

where  and  represent the usual Schwarzschild coordinates, we then set d = 0 for light-like paths, which reduces the equation to

This is a perfectly good metrical (not pseudo-metrical) space, with a line element given by dt, and in fact by extremizing (dt)² we get the paths of light. Note that this only works because g_tt, g_rr , g_, g_  all happen to be independent of this time coordinate, t, and also because g_tr = g_t = g_t = 0. If and only if all these conditions apply, we reduce to a simple line element of dt on the null surfaces, and Fermat's Principle applies to the parameter t. Thus, in a Schwarzschild universe, this works only when using the essentially unique Schwarzschild coordinates, in which the metric coefficients are independent of the time coordinate.

Admittedly the Schwarzschild geometry is a highly simplistic and symmetrical cosmology, but it illustrates how the notion of an absolute rest frame can be more physically meaningful in a relativistic spacetime than in Newtonian spacetime. The spatial configuration of Newton's absolute space is invariant and the Newtonian metric is independent of time, regardless of which member of the inertial class of reference systems we choose, whereas Schwarzschild spacetime is spherically symmetrical and its metric coefficients are independent of time only with respect to the essentially unique Schwarzschild system of coordinates. In other words, Newtonian spacetime is operationally symmetrical under translations and uniform velocities, whereas the spacetime of general relativity is not. The curves and dimples in relativistic spacetime automatically destroy symmetry under translation, let alone velocity. Even the spacetime of special relativity is (marginally) less relational (in the Machian sense) than Newtonian spacetime, because it combines space and time into a single manifold that is only partially ordered, whereas Newtonian spacetime is totally ordered into a continuous sequence of spatial instants. Noting that Newtonian spacetime is explicitly less relational than Galilean spacetime, it can be argued that the actual evolution of spacetime theories historically has been from the purely kinematically relational spacetime of Copernicus to inertial relativity of Galileo and special relativity to the purely absolute spacetime of general relativity. At each stage the meaning of relativity has been refined and qualified.

We might suspect that the distinguished "Killing-time" coordinate in the Schwarzschild cosmology is exceptional - in the sense that the manifold was designed to satisfies a very restrictive symmetry condition - and that perhaps more general spacetime manifolds do not exhibit any preferred directions or time coordinates. However, for any specific manifold we must apply some symmetry or boundary conditions sufficient to fix the metrical relations of the manifold, which unavoidably distinguishes one particular system of reference at any given point. For example, in the standard Friedmann models of the universe there is, at each point in the manifold, a frame of reference with respect to which the rest of the matter and energy in the universe has maximal spherical symmetry, which is certainly a distinguished system of reference. Still, we might imagine that these are just more exceptional cases, and that underneath all these specific examples of relativistic cosmologies that just happen to have strongly distinguished systems of reference there lies a purely relational theory. However, this is not the case. General relativity is not a relational theory of motion. The spacetime manifold in general relativity is an absolute entity, and it's clear that any solution of the field equations can only be based on the stipulation of sufficient constraints to uniquely determine the manifold, up to inertial equivalence, which is precisely the situation with regard to the Newtonian spacetime manifold.

But isn't it possible for us to invoke general relativity with very generic boundary conditions that do not commit us to any distinguished frame of reference? What if we simply stipulate asymptotic flatness at infinity? This is typically the approach taken when modeling the solar system or some other actual configuration, i.e., we require that, with a suitable choice of coordinates, the metric tensor approaches the Minkowski metric at spatial infinity. However, as Einstein put it, the specifications of "these boundary conditions presuppose a definite choice of the system of reference". In other words, we must specify a suitable choice of coordinates in terms of which the metric tensor approaches the Minkowski metric, but this specification is tantamount to specifying the absolute spacetime (up to inertial equivalence, as always) in Newtonian physics.

The well-known techniques for imposing asymptotic flatness at "conformal infinity", such as discussed by Wald, are not exceptions, because they place only very mild constraints on the field solution in the finite region of the manifold. Indeed, the explicit purpose of such constructions is to establish asymptotic flatness at infinity while otherwise constraining the solution as little as possible, to facilitate the study gravitational waves and other phenomena in the finite region of the manifold. These phenomena must still be "driven" by the imposition of conditions that inevitably distinguish a particular frame of reference at one or more points. Furthermore, to the extent that flatness at conformal infinity succeeds in imposing an absolute reference for gravitational "potential" and the total energy of an isolated system, it still represents an absolute background that has been artificially imposed.

Since the condition of flatness at infinity is not sufficient to determine a solution, we must typically impose other conditions. Obviously there are many physically distinct ways in which the metric could approach flatness as a function of radial spatial distance from a given region of interest, and one of the most natural-seeming and common approaches, consistent with local observation, is to assume a spherically symmetrical approach to spatial infinity. This tends to seem like a suitably frame-independent assumption, since spatial spherical symmetry is frame-independent in Newtonian physics. The problem, of course, is that in relativity the concept of spherical symmetry automatically distinguishes a particular frame of reference - not just a class of frames, but one particular frame. For example, if we choose a system of reference that is moving toward Sirius at 0.999999c, the entire distribution of stars and galaxies in the universe is drastically shrunk (spatially) along that direction, and if we define a spherically symmetrical asymptotic approach to flatness at spatial infinity in these coordinates we will get a physically different result (e.g., for solar system calculations) than if we define a spherically symmetrical asymptotic approach to flatness with respect to a system of coordinates in which the Sun is at rest. It's true that the choice of coordinate systems is arbitrary, but only until we impose physically meaningful conditions on the manifold in terms of those coordinates. Once we do that, our choice of coordinate systems acquires physical significance, because the physical meaning of the conditions we impose is determined largely by the coordinates in terms of which they are expressed, and these conditions physically influence the solution. Of course, we can in principle define any boundary conditions in conjunction with any set of coordinates, i.e., we could take the rest frame of a near-light-speed cosmic particle to work out orbital mechanics of our Solar system by (for example) specifying an asymptotic approach to flatness at spatial infinity in a highly elliptical pattern, but the fact remains that this approach give a uniquely spherical pattern only with respect to the Sun's rest frame.

Whenever we pose a Cauchy initial-value problem, the very act of specifying timeslices (a spacelike foliation) and defining a set of physically recognizable conditions on one of these surfaces establishes a distinguished reference system at each point. These individual local frames need not be coherent, nor extendible, nor do we necessarily require them to possess specific isometries, but the fact remains that the general process of actually applying the field equations to an initial-value problem involves the stipulation of a preferred space-time decomposition at each point, since the tangent plane of the timeslice at each point singles out a local frame of reference, and we are assigning physically meaningful conditions to every point on this surface in terms that unavoidably distinguish this frame.

More generally, whenever we apply the field equations in any particular situation, whether in the form of an initial-value problem or in some other form, we must always specify sufficient boundary conditions, initial conditions, and/or symmetries to uniquely determine the manifold, and in so doing we are positing an absolute spacetime just as surely (and just as arbitrarily) as Newton did. It's true that the field equations themselves would be compatible with a wide range of different absolute spacetimes, but this ambiguity, from a predictive standpoint, is a weakness rather than a strength of the theory, since, after all, we live in one definite universe, not infinitely many arbitrary ones. Indeed, when taken as a meta-theory in this sense, general relativity does not even give unique predictions for things like the twins paradox, etc, unless the statement of the question includes the specification of the entire cosmological boundary conditions, in which case we're back to a specific absolute spacetime. It was this very realization that led Einstein at one point to the conviction that the universe must be regarded as spatially closed, to salvage at least a semblance of unique for the cosmological solution as a function of the mass energy distribution. (See Section 7.1.) However, the closed Friedmann models are not currently in favor among astronomers, and in any case the relational uniqueness that can be recovered in such a universe is more semantic than substantial.

Moreover, the strategy of trying to obviate arbitrary boundary conditions by selecting a topology without boundaries generally results in a topologically distinguished system of reference at any point. For example, in cylindrical coordinates (assuming the space is everywhere locally Lorentzian) there is only one frame in which the surfaces of simultaneity of an inertial observer coherent. In all other frames, if we follow a surface of simultaneity all the way around the closed dimension we find that it doesn't meet up with itself. Instead, we get a helical pattern (if we picture just a single cylindrical spatial dimension versus time).

It may seem that we can disregard peculiar boundary conditions involving waves and so on, but if we begin to rule out valid solutions of the field equations by fiat, then we're obviously not being guided by the theory, but by our prejudices and preferences. Similarly, in order to exclude "unrealistic" cosmological solutions of the field equations we must impose energy conditions, i.e., we find that it's necessary to restrict the class of allowable T_uv tensor fields, but this again is not justified by the field equations themselves, but merely by our wish to force them to give us "realistic" solutions. It would be an exaggeration to say that we get out of the field equations only what we put into them, but there's no denying that a considerable amount of "external" information must be imposed on them in order to give realistic solutions.