DANIEL H. WESLEY AND JOHN A. WHEELER

TOWARDS AN ACTION-AT-A-DISTANCE CONCEPTOF SPACETIME'

John Stachel and action at a distance? Direct connection or not as there may bebetween these two themes, physics, with its ever-growing reach, provides one. Vividin the memory of one of us (JAW) is the seminar in which Richard Feynman repor­ted his work with JAW, defending the 1922 argument of Tetrode that "The sun wouldnot radiate if it were alone in space an no other bodies could absorb its radiation .. . "Einstein, Pauli, and Wigner are among those who attended the seminar, but no oneexpected any of them to try a similar replacement of Einstein's field theory of gravita­tion by a Newtonian direct action account. Did Einstein himself, who had given us thestandard field theory of gravity in place of Newtonian direct action, ever attempt thereverse: to sweep out the space-time continuum and replace it by pure direct couplingbetween particle and particle? That is a question of the history of science, of Einsteinthoughts and Einstein records. For an answer to that question , we will look to JohnStachel, immersed as he is in the Einstein Papers, and working with them day afterday. Who does not rejoice in that life work of Stachel 's, witnessed not least on the flyleaf of The Collected Papers 0/Albert Einstein.'

So, trust Stachel to answer our question. What, if anything, did Einstein ever do orsay about such a theme as "sweeping out space and time from between the particles,"and replacing the space-time of general relativity by particle-to-particle interaction?Space and time are not things, but orders a/things. If these words ofold call to a newand deeper conception of space and time than Einstein 's publications give us, then inwhat direction are we to look? For a reply, many today would point to string theory orother new and exotic developments. But if we are not yet ready to accept them, withall of their new elements, then where can we look?

Here, we consider how we may economize today's account of spacetime by drop­ping fields, instead of adding them. The concept of the field is a very old one inphysics . One of the earliest problems faced by physical theory was the explanation ofthe phenomenon of "action at a distance." Issac Newton's theory of universal grav­itation is very successful at predicting the motions of planets and satellites , providedone makes the assumption that every mass can exert a force on distant masses throughsome invisible mechanism, the field. The idea may is expressed mathematically by

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A. Ashtekar et al. (eds.), Revisiting the Foundations ofRelativistic Physics, 421-436.© 2003 Kluwer Academic Publishers.

422 D ANI EL H. W ESLEY AND JOHN A. WHEELER

defining a quantity at every point of space that allows us to calculate the forces on testpart icles.

As physics progressed, the numb er of fields, and their role in physical theory,increased greatly. In the general theory of gravitation, classical field theory finds itsmost powerful expression ; the very properties of space and time are given by themetric field 9Jlv, In this work, we consider whether the field is a necessary part ofthe express ion of physical law, and how, in the case of gravi ty theory, rethinking ournotion of the field involves reconsidering our usual notions of space and time.

It is not new to formulate a physical law without introducing the field as a sep­arate physical entity. Electrostat ics, for example, accustoms us to account for theforces on a family of electric charges either by consideration of the electric field gen­erated by those charges, or equally well in terms of direct action between charge andcharge, without need of any field at all. However, that field-free formulation appearsto fail when anyone of the charges undergoes acceleration, because it experiences thewell-attested force of radiation reaction, which cannot be caused by the other chargespresent. This circumstance appears to demand that we attribute to the field an exist­ence of its own. Or, it appeared to make this demand until , in 1922 , H. Tetrode arguedthat "The sun would not radiate if it were alone in space and no other bodi es couldabsorb its radiati on ," suggesting that it is indeed possible to account for the force ofradiation reaction by considering only distant charges.

In 1945 JAW and Richard Feynman gave a formal treatment of Tetrode's idea . Thisanalysis provided a quantitative "absorber theory of radiati on," extending a 1938 res­ult of Dirac, which accounted for the force of radiati on reaction in terms of both "ad­vanced" and "retarded" electromagnetic fields. Is it possible to do the same for grav­ity, and eliminate all direct reference to spacetime or any comparable field concept,as all direct reference to an "electromagnetic field" disappears in the Fokker-Tetrode­Schwarzschild action principle for electromagnetism (17).

1. V.Narlikar, writing in 1968 on the general correspondence between field theoriesand theories of direct interparticle action, showed that any field-plus-particle theorydescribed by an action principle of the type (13) admits translation to a direct part icleaction theory of the form (15) . But , the Hilbert variational principle for gravitation isnot quadratic in the fields. Thu s, it does not admit an action principle of the form (13) ,and does not admit of any simple translation into a direct-acti on principle in mann ersimilar to that of electromagneti sm.

Lacking any formal means of translating the Einstein-Hilb ert action principle from"field language" to "particle language," we advocate a combined strategy: (1) Useparticle-borne wristwatches plus light flashes or radar pulses between particle andnearby particle, (2) Use the resulting numbers to derive particle-to-particle separa­tion and local Riemann curvature, (3) Put together these local curvatures to make acomplete spacetime, as one fits together the pieces of a smashed porcelain vase to re­construct the original objec t. As for this third , most difficult (and most interestin g!)part of the enterprise, we have no prescripti on to offer. In this paper, we discuss thefirst two part s of this strategy, and make some comments on how the third part may berealized.

TOWARDS AN ACTION-AT-A-DISTANCE CONCEPTOF SPACETIME 423

THE FI ELD CONCEPT

The most elementary application of the field concept is that of a bookkeeping device ,a function that assigns a force to a given charge or mass as a function of its position ,velocity, or other characteristics. The introduction of the field enables the physicistto conceptually separate the action of part icle on particle into their action on, andresponse to, the field .

As the theory offields developed, it became clear that they provide a natural mech­anism for enforcing "local" physics. By this, we mean the idea that the dynamics ofa particle may be described in terms of qualities of its local environment. As the fieldwas used more extensively to enforce these local laws, it became a dynamical quantityin its own right.

LOCAL CAUSALITY

In the classical theories of gravity and electrodynamics, the field is more than just abookkeeping device . These fields help enforce local causality, which circumvents theconceptual problems of action -at-a-distance . Consider the following two equivalentpresentations of Newton's gravitational theory, in which the force per unit mass f isgiven by,

V· F = - 4r.Gp.

(1)

(2)

Both equations imply the same inverse-square law of force? The first formulation re­flects the action-at-a-distance ofone body on another. The second formulation predictsthe same physical result, but also lends itself to an explanation of action-at-a-distancein terms oflaws that act only in local regions of space.

This may be seen even more clearly by an analogy. If we define the potential viathe relation - \I¢ = f, we obtain the Poisson equation,

(3)

The same equation describes the static deformation ¢ of an elasti c membrane withelasticity proportional to G- 1 and mass density p. In the nonstatic case, the de­formations of the membrane obey a wave equation, similar in some respects to theD 'Alembert equation obeyed by the electromagnetic 4-vector potential AI-"

In the "local interaction" picture, a mass influences the field in its immediate vi­cinity, in a manner quantified by (2), much as a mass deform s a local region of ourelastic membrane. The influenced region further influences regions infinitesimallymore distant from the mass, and so on. The influence of the mass is propagated, fromthe location of the source mass to the location of the test mass, through fully localinteractions of the field with itself.

From the point of view ofequation (2), properties of the mass distributions distantf rom the point under consideration need not enter into the picture. It is true that

424 DANIEL H. WESLEY AND JOHN A. WHEELER

the total force on a particle depends on field originating at possibly large distances.The local, differential properties of the field, however, depend only on the sourcedensity at the point of investigation. Furthermore, in the case of gravitational fields,the equivalence principle forbids us to measure "absolute" gravitational accelerationof particles. In this case, everything we may measure about particle trajectories isdetermined by the local mass distribution.

AUTOMATIC CONSERVATION OF THE SOURCE

The introduction of the field as a dynamical entity in its own right has consequencesfor the expression of physical laws . We may generalize equations such as (2) and (3)to obtain a schematic form of classical field equations:

(force on particle) = k . (field)

(field derivatives) = k · (source density).

(4)

(5)

Equations of this type are found in the field theories of classical gravity and electro­magnetism, as well as in general relativity. When the field equations are expressedin the powerful language of differential geometry, physical conservation laws, suchas the conservation of charge, may be "enforced" through geometrical concepts. Asan elementary example, Maxwell's equations, when expressed in differential form,automatically imply the conservation of charge. This may be seen when the Maxwellequations are written- in terms of the Faraday-Maxwell2-fonn F == ~Flwdx/l Adz "and its dual *F,

dF=O d *F = 41f*J . (6)

Here d is the usual exterior differential operator. If we calculate the divergence of *Jby applying d to the second equation, we find that d *J = d 2 *F == O. The vanishingof this divergence is automatically enforced as a consequence of the geometric identity

d 2 == O.This identification has important consequences for conservation laws. For one, it

implies that the naive expression of charge conservation ("total charge of the Universeis constant"),

JQ d3X = const, (7)

is slightly misleading It must be replaced by a different law that enforces local con­servation. Change in the local charge density cannot be compensated for by corres­ponding changes in the distant charge density. Instead, changes in local charge densitymust be accounted for by the divergence of local currents,

J!' i /l = O. (8)

While (8) implies (7) , they are not equivalent. Again the field enforces a local descrip­tion of physics.

The conservation of charge is built-in, as it were, to Maxwell's equations, as ageometrically conserved source. These same equations can be made to imply the

TOWARDS AN ACTION-AT-A-DISTANCE CONCEPT OF SPACETIME 425

local conservation of energy as well , provided we assign a stress-energy field to theelectromagnetic fields . When we search for a tensor that is (1) bilinear in the dynamicquantities (fields) ," and (2) divergent to exactly the extent required to compensate forthe change in the current stress-energy due to interaction with the field ,

(9)

(10)

we find the proper assignment is,

T = ~ (FJUY. F V- ~gJJ,V F Fa f3)

J1V 41T a 4 af3

The loss in energy by charge-carrying bodies ("work against the electromagneticfield") is compensated for by the loealloss offield energy, as defined in (10) . Thi s as­signment leads to experimentally verifiable predictions. The famous force of radiationreaction may be calculated by imposing local energy conservation, and calculating theenergy carried away by the radiation fields of a charge.

We find a similar "automatic conservation of the source" in the theory of generalrelativity. In this case , the principle of automatic conservation of the source is appliedto stress-energy itself. Indeed, the Einstein equations are uniquely defined when onepostulates that

(1) Gravitational effects are caused by curvature in spacetime,

(2) The field equation for gravity are expressed in a form similar to (5), that is,terms containing field derivatives (in this case the connection coefficients of thespacetime geometry) are proportional to the local stress-energy density.

Thus, one begins with the general form ,

(11)

and searches for a tensor GJ1V that has the proper relations to "natural" measures ofcurvature, eg, GJ1V

( I) Vanishes in flat spacetime,

(2) Is built from the Riemann curvature tensor and the metric,

(3) Is linear in the Riemann tensor.

One then imposes the "automatic conservation of the source." In order to make (11)mathematically consistent, our tensor GJ1 V must share some of the properties of thestress-energy tensor, namely GJ1V is

(I) Symmetric,

(2) Has identically vanishing divergence.

These considerations lead directly to the usual form

426 D ANI EL H. W ESLEY AN D JOHN A . WHEELER

(12)1s.; - 2gJ.lv R = KTJ.lv

for the field equations for gravitation.> The ideas of local interaction and local physicsare closely intertwined in equations such as (5) and (12). If interactions were not takento be local in character, we should not be able to write down expressions for the forcesin tenus of the fields at all. In addition if conservation laws were not local, the fieldequations would not be mathematically consistent.

QUANTIZATION

Quantum theory describes part icles in a field-like manner. Conversely, fields maythemselves be described in a particulate, or "quantized," manner. The local, fieldpicture of interaction allows one to start from the Lagrangian density associated witha field equation, and treat the fields as quantum systems. Currently, we believe that thecorrect description of natural forces is found within the framework of quantized fieldtheories. The fact that no satisfactory quantum description of the gravitational fieldhas yet been found is widely believed to indicate a failure in methodology, and nota failure of quantum theory. Such a prescription reinforces the idea of local physicallaw through describing the action of one particle on another by means of a mediumthat spans the space between them.

IS PHYSICS ENTIRELY LOCAL?

A major advantages of the field concept is the natural way in which the fields enforcelocal physical laws. This begs the question: is physics local? One is reminded ofan argument against quantum theory advanced by Einstein, Podolsky, and Rosen ina well-known paper (1935). The authors assert that the quantum-theoretical processof the reduction of the wave packet (collapse of the wavefunction upon measurement)is not physically realistic, since it allows distant measurements to affect local phys­ics. The implicit nonlocality of this process, they argue, is at odds with the idea thatphysics should be fundamentally local.

In a situation such as this, one is also reminded of the words of Ernst Mach,

The most important result of our reflection is, that precisely the apparently simplest mech­anical prin ciples are ofa very complicated character, that these principles are fou ndedon uncompleted experiences. nay on expe riences that never can befully completed. thatpractically. indeed. they are suffi ciently secured. in view of the tolerable stability ofourenvironment. to serve as the fou ndation ofmathematical deduction. but that they can byno means themselves be regarded as mathematica lly established truths but only as prin­cip les that not only admit ofconstant control by experience. but actually require it (Mach1902, 2:VI, 9).

As has been evidenced by many experimental tests, the view of nature espoused byEinstein et. al. is not quite correct. Various experiments have shown that distant meas­urements can affect local phenomena. That is, nature is not described by physica l lawsthat are entirely local . Effects from distant objects can influence local physics. Whilewe do not consider quantum gravitational theories here, this example from quantum

TOWARDS AN ACTION-AT-A-DISTANCE CONCEPT OF SPACETIME 427

theory serves to illustrate that it may be useful to expand our notions regarding whattypes of physical law are "allowed."

DIRECT ACTION-AT-A-DISTANCE

Here, we consider the question of formulating physical laws without introducing thefield as a separate physical entity. The idea that local physics, and electromagneticradiative phenomena in particular, involved interactions with distant bodies was givena formal treatment by Feynman and Wheeler (1945). In their paper, this idea was de­veloped quantitatively into the "absorber theory of radiation ." We review the conceptsinvolved and consider how they may be applied to the description of gravitationalinteractions .

WHEELER-FEYNMAN ELECTRODYNAMICS

Wheeler and Feynman extended an interesting result obtained by Dirac (1938). In hispaper, Dirac observed that the wave equations of electromagnetism admits solutionscorresponding to waves propagating not only forward in time, but backward as well.

Dirac obtained the following prescription for calculating the force of radiationreaction on a moving charge. First , let the motion of the charge be given. Fromthis, calculate the advanced and retarded electromagnetic fields, with the boundaryconditions that the advanced field contains only incoming waves, and the retardedfield only outgoing waves, both of which vanish at infinity. Then, calculate a newfield, equal to one-half the difference between the advanced and retarded fields. Thisfield is everywhere finite, and when evaluated in the neighborhood of the charge, givesprecisely the force of radiation reaction.

Dirac 's prescription is a clearly defined, and yields the correct answer. Despitethis, the mystery of the origin of the "half-advanced," "half-retarded" field is left un­resolved. In their 1945 paper, Wheeler and Feynman considered the action of theadvanced field of the source on a distant absorbing medium. They found that the re­tarded field of the absorber, when added to the retarded field of the source itself, isequal to precisely the field postulated by Dirac.

In effect, they swept the electromagnetic field from between the charged particle sand replaced it with "half-retarded, half-advanced direct interaction" between particleand particle . It was the high point of this work to show that the standard and well­tested force of reaction of radiation on an accelerated charge is accounted for as thesum of the direct actions on that charge by all the charges of any distant completeabsorber. Such a formulation enforces global physical laws, and results in a quantit­atively correct description of radiative phenomena, without assigning stress-energy tothe electromagnetic field.

428 DANIEL H. WESLEY AND JOHN A. WHEELER

CORRESPONDENCE BETWEEN FIELD AND DIRECT INTERACTIONTHEORIES

Dirac's prescription is interesting for several reasons. Conceptually, while this theoryrequires us to rethink some of our notions of causality, it does not violate them. Moreimportantly, the theory starts with the minimum necessary framework upon whichelectrodynamics can be built. There are no entities other than the particles; the fielditself is removed from consideration, either as a "automatic conserver of the source,"or as the conveyor of stress-energy from the accelerating charge.

In addition to the conceptual simplicity of the theory, it is also more convenientmathematically. One need not calculate the dynamics of the field, a complex dynamicquantity with an infinite number of degrees of freedom; only the particles, with theirfinite number ofdegrees of freedom," Can such a procedure be extended to other typesof fields?

Hoyle and Narlikar (1974) have pointed out that there is a general, mathematicalcorrespondence between field theories and direct-action theories. They quote a generalresult obtained by Narlikar (1968) . Let a field be described by a tensor field ¢ of rankN , and let the dynamics of the field in interaction with particles be given by an actionprinciple of the type

JL[¢]yCg d4x + L JI [¢, (i)] sx» ,t

(13)

where L is a bilinear invariant built from ¢ and its first derivatives, and ax» isan appropriate measure of the configuration space of the it h particle. I [¢ , (i)] is aninvariant describing the interaction of the particle with the field , of the form

I[¢ , (i)] = g . (¢, ~ (i ) ) . (14)

Here, ~ ( i ) is a rank-N tensor depending only on the worldline of the it h particle, andg is a coupling constant. To the field theory given by (13) and (14) there exists adirect-action theory of the form

~ z= g2JJ(~(i) , G(X(i ), X U)) . ~U ) ) dX(i )dX(j ). (15)

t ,J

Here, G(, ) is the symmetric? Green's function of the field equation derived from (13) .8Such a direct-action theory gives the direct field due to particle j at the point X as

¢ (X ) = g JG(X, X(j )) . ~(j ) ex» . (16)

We recognize that electrodynamics is a field theory of the form (13) . Radiative effectsmay be calculated using the effects offields on distant particles, and radiation reactionmay be calcuated using the procedure of Wheeler and Feynman.? Electrodynamicshas its field-free expression in the Fokker-Tetrode-Schwarzschild action principle,

TOWARDS AN A CTION-AT-A-DISTANCE CO NCEPT OF SPACETIME

+ L eiej JZf (T)Zj/l(r )D (Zi' Zj) drtii .i,Ni

429

( 17)

Such an action principle yields the half-advanced, half-retarded direct interaction semployed by Wheeler and Feynman in their analysis of electrodynamics .

ANALOGY WITH GRAVITATIONAL THEORY

The description of gravity in general relativity does not admit an expression in theabove manner. The first failure occurs in (13) , where the Hilbert variational principlefor gravitation is not quadratic in the fields. Just the same, the theory of generalrelativity is itself a field theory. There is a formal similarity between the equat ions ofmoti on in the conventional electromagnetic field theory,

jJ/l = eF /l-v" ,

and the equations of moti on of masses under the influence of gravity,

(18)

(19)

The two theories share some of the same conceptual features as well. Might it be pos­sible to introduce a field theory similar to that ofWheeler-Feynman electromagnetismfor gravity? Before we get too involved, we must note that the unique nature of grav­itation among field theori es requires us to discuss several considerations that were notrelevant in the case of electrodynamics.

THE EQUIVALENCE PRINCIPLE

Formally, the metric plays a role in gravitation theory similar to that of the vectorpotential in electrodynamics . The "field" of gravitational theory is, however, funda ­mentall y distinct from the electromagnetic field in several key ways.

In the case of electromagnet ism, the strength of a particle'S interaction with thefield is proportional to the particle's charge , while the acceleration of the particle inresponse to the field is determined by the mass of the particle. The charge q and themass m of a given body are independent, and so the electrodynamics of a system ofparticles is determined when these two numbers are specified for each particle.

We may translate these ideas into gravitational theory, and specify that each masspoint has a "gravitational mass" that determines its contribution to the field , and an"inertial mass" that determines the dynamics of the particle under the influence of agravitational field. Unlike electrodynamics, the gravitati onal mass and inertial massare found to be proportional to an exceedingly high degree of preci sion. Gravitationaltheories must therefore follow one of two routes. The first method is to assume thatthe proportionality is a mere coincidence, and to introduce the aforementi oned gravita­tional and inertial masses. The other possibility is to accept the striking nature of such

430 DANIEL H. WESLEY AND JOHN A. WHEELER

a result as reflecting an important physical principle . Einstein chose the latter path,and incorporated this equivalence principle directly into the structure of the generaltheory of relativity. By his casting the gravitational field as a fundamental geomet­

ric property of spacetime, all masses in a given region of spacetime "feel" the samegeometry and automatically experience the same acceleration . A theory of gravitationthat does not involve the metric field should incorporate this equivalence principle ina similarly elegant way, or risk taking a step backwards and ruining the simplicity ofthis approach .

This equivalence principle, and the association of gravitation with geometry, hasother consequences as well. Electromagnetic theory presupposes that we are able tomeasure times, distances, and other physical quantities in a manner unaffected bythe system we are studying . In general relativity, this is not possible . Gravitationaltheory is intricately connected with our basic ability to measure space and time. Thefield "works" through distortions of the network of "rods and clocks" that we useto parameterize events in spacetime. Gravitational effects, because of their universalnature, are intertwined with our means of measuring them. The interrelationship ofthe field we wish to study and the means of measurement significantly complicates theproblem of formulating a field-free theory of gravitation.

TOWARDS A FIELD-FREE GRAVITATION

Once we have removed spacetime, what are we to replace it with? Theodore Rooseveltused to advise, "Do what you can with what you have where you are." We report herethe line of reasoning to which this injunction has led us. First, having nothing butparticles and their histories to work with, we parameterize the history of each particleby a scalar. Thus a monotonically increasing parameter a distinguishes successivepoints (events) on the worldline of particle a; f3 events on particle b, "( events on c,etc. We have as yet no tool to measure the passage of time on particle a or any otherparticle, still less any tool to measure the distance between particle and particle.

V T

~ o

Figure 1. The role ofobservation in special relativity. Both scientist in laboratory frame (left)

and observer moving with the source (right) observe an emitted pulse oflight to take the form

ofa sphere ofradius CT a time T after emission.

TOWARDS AN ACTION-AT-A-DISTANCE CONCEPT OF SPACETIME 431

A field-free gravitational theory would have to be expressed in terms of whatevertool is chosen to map out events on the observer's worldline , as well as other nearbyworldlines . Of course, this introduces the question of the proper tool to use. Thissituation is similar to that found in special relativity. The theory holds that, whilethere is no canonical reference frame, fig. 1, every reference frame is related to everyother one by a set ofcoordinate transformations (the Lorentz group) . Similarly, a field­free gravitational theory must make reference to a system of measurements betweenevents, and relate different groups of such measurements.

RODS AND CLOCKS FOR MEASURING GRAVITY

The usual means of measuring space and time are already too complex for our use.Mechanical rods and clocks are affected by a host of factors. The analysis of theseeffects (with an aim to "correcting" for them) can take us far afield into quantumtheory or solid-state physics. There is, however, a means of measuring spacetimewhose analysis requires only the relativistically invariant theory of electromagnetism.Furthermore, this method enables one to measure directly intervals in spacetime. Sucha "clock and measuring rod" is provided by the "ideal clock" discussed by Marzke andWheeler (1964) . An observer may use such a device to measure times, distances, orindeed any type of spacetime interval.

The ideal clock may be constructed as follows. Let an observer free-fall along ageodesic /'1. The observer may construct a parallel path /'2 (not necessarily a geodesic)using the Schild's ladder construction, as discussed in (Misner, Thome , and Wheeler1973). This construction enables one to trace out a trajectory parallel to a given tra­jectory, in this case the observer's geodesic. The advantage of the Schild 's ladderconstruction is that it is entirely geometric, and requires only the tracing of lightliketrajectories in space and time.

In order to perform measurements, a beam of light is made to bounce back andforth between the two trajectories. Each bounce of the light at a point on /'1 is con­sidered one "tick" of the clock. Measurements of space and time are made in terms ofmultiples of these clock ticks (see fig. 2).

The gravitational field is measured through so-called tidal effects . The equivalenceprinciple forbids one to measure the absolute gravitational acceleration of a body. In­stead, one may set up two test particles at positions x l' and xl' + ~jJ" and allow themto fall freely. Gravitational acceleration (in geometric language, spacetime curvature)manifests itself as a change in the relative 4-velocities of the particles , ~jJ, . In Rieman­nian geometry, this change may be given as a function of the 4-velocity of the observerand the Riemann curvature tensor RjJ, i/ar »

~jJ, + RjJ, vaTUv~auT = 0, (20)

In the rest frame of one of the particles, this equation takes the form,

~'jJ, + RjJ,Oao~a = 0. (21)

An observer who wishes to measure all of the components of the Riemann tensorneed only send pairs of"ideal clocks" out in various directions, each pair equipped to

43 2 D ANI EL H. W ESLEY AN D JOHN A. WHEELER

B B

N2

- -:i; - -

[ N I

A A

Figure 2. The ideal clock ofMarzke and Wheeler. The interval between the points A and B is

the product N 1 N2 : fo r a proofconsider the right-ha nd diagram . and the relation

(t - x )(t + x) = t2- x 2

.

measure its separation using the interval measurement system described above. Onemember of each pair emits a light signal at a time o "; which reflects from the othermemb er of the pair at a time (3, and is received by the first memb er of the pair at atime a+. The tabul ation of these times forms the fundamental raw data upon whichobservations are built, and against which theory must be tested.

Although a sing le "clock" may infer information about the positi on of a distantparticle, perhaps by a combination of distance and directional measurements, betterinformation may be gleaned by using a set of such devices in comb ination . Eachdevice may be used to measure the interva l between itself and the distan t particleat various points along its traj ectory. Conceptually, such a measurement places theparticle under consideration on one of many hypersurfaces of constant interval whichfoliate the local regions of space time.I? Four such devices, operating together andwith different 4-velocities, will localize the test particle on the intersection offour suchhypersurfaces. The fact that all of the devices will agree on a consistent localizationof the particle is a feature of Riemannian geometry that has not been shown to beviolated thus far.11

In such a measurement scheme, certain observed quantities depend on the ob­server's history during the measurement process. Let our observer introduce a co­ordinate system in which the coordinate lines are formed from the trajectories of freelyfalling masses and light rays (geo des ics). All points connected to a given point C onthe observer worldline , by geodesics whose tangents at C are spacelike, are causallydisconnected from C. In this scheme, however, the measured position of the distantparticle is a function of the observer's traje ctory between the points of emission andreflection of the observer 's "radar signal."

Consider the follo win g situation: The observer on the first particle, traveling alongtrajectory AB, finds the points C and D to be separated by a purely spacelike interval.Now, let us make an identi cal copy of the spacetime geometry and particle trajectories.In the second version, some kind of localized disturbance (perhaps a gravitational

TOWARDS AN ACTION-AT-A-DISTANCE CONCEPT OF SPACETIME 433

wave packet traveling through the area) affects our observer between the points C andB. The observer is kicked on to another geodesic cf3, fig. 3.

Figure 3. Indeterminacy in measurements in space time.

Now, when the observer receives the return signal from the distant particle, heconcludes that the particle occupied a different position at a different time than in theprevious case. The properties of the spacetime metric near the observed point have notchanged, and yet its observed trajectory is altered. Causally, the perturbation ogJ],V can­not causally affect points C and D , and yet the measurement of spacetime propertiesnear D must ultimately refer to the observer trajectory throughout the measurementprocess , a fundamentally nonlocal effect.

INTERACTIONS

In a spirit similar to our method ofanalyzing spacetime via particle-to-particle signals,we may discuss interactions ofparticles in terms of"signals." The analogue ofa lightsignal between an observer and a test particle is the propagated fields, whose influencetravels at a finite speed. The point masses "recognize" each other in this way beforethere can be any talk of interaction. In keeping with the time-symmetric character ofelementary action-at-a-distance electrodynamics, we adopt a time-symmetric interac­tion protocol. This is also suggested by the results of Narlikar on the correspondencebetween field and direct particle-particle theories, which might lead us to expect thatthe final mathematical formulation ofa direct particle-particle theory ofgravity wouldinvolve both advanced and retarded interactions . Also, our considerations of the actof measurement suggests that we should consider the state of the observer at both theretarded and advanced points of action .

434 DANIEL H. WESLEY AND JOHN A. WHEELER

Thus , to every point or event 0: on the worldline of particle a, we associate twoevents on the worldline of particle b, ,8+ (0:) and ,8- (0:), respectively the retardedand advanced points of action of a on b. These associators evidently satisfy trivialidentities of the form

(22)

These ideas may be mathematically expressed through a one-half advanced, one-halfretarded gravitational theory.

Figure 4. Associators

COMMENTS ON A FIELD-FREE FORMULATION OF LINEARIZED GR

Following this line of reasoning, one may construct a "toy theory" of gravity thatincorporates many of the above elements A simple way to do this is to square thevelocity terms in (17) and neglect the initial kinetic term , When a variation is carriedout , one ends up with a field-free theory of gravitation that is formally identical tolinearized general relativity.

Interestingly enough, the equation which results results in an equation of motionwithout additionalconstants . The usual gravitational constant appears as rank-2 tensorwhich depends on the mass and velocity distribution in the distant universe. This typeof theory is similar to ideas discussed by Sciama (1957) and others. The identificationof these terms is found by Wheeler to be reasonable by order-of-magnitude argu­ments (Wheeler 1964). Additional discussions on similar theories that depend on themass distribution "at infinity" may be found elsewhere (Ciufolini and Wheeler 1995,chap. 7; Hoyle and Narlikar 1974) .

The dependence of the theory on the mass distribution at infinity is somewhat ofa difficulty. This is especially problematic when , in the case of this theory, the inter-

TOWARDS AN ACTION-AT-A-DISTANCE CONCEPT OF SPACETIME 435

action lagrangian depends on mass-energy currents instead of scalar densities. Whilethe isotropy of the distant universe (the "cosmological principle") may be invokedto estimate the contribution from the total mass scalar density, the presence of currentterms ofthe form zmzn, requires detailed assumptions about the relationships betweenthe components of the mass-energy currents at infinity. Unlike the Feynman-Wheelerelectrodynamics, which requires only that "infinity" contains a perfect absorber, this"toy" theory depends on a detailed way on the dynamics at infinity.

CONCLUSION

Ifwe aim to "sweep out" spacetime in the conventional sense, it is only so that space­time may be resurrected, not as another field to be studied, but as a network of obser­vations and events. A true field-free expression of gravity will involve more than justan analogue of expressions for field-free lagrangians for other fields. Instead, whatwill be needed is a revolution in our fundamental conception of what spacetime is.John Stachel , with his colleagues, has given us the opportunity to understand the manwho first gave us our modem conception of a gravitational field theory. It is hopedthat, by freeing the theory from the constraints of its field formulation, a deeper un­derstanding of gravitation may be achieved. Such an understanding, however, willrequire a profound rethinking of the meaning of space and time.

Princeton UniversityUniversity ofTexas

NOTES

* In honour of Professor Stachel. He will agree with me on points, if not on substance, but I am indebtedto him for his guidance on both topics; and on many others besides.

I . One there was, to be sure, who contested Princeton University Press's choice of Stachel as editor. Thatwas Otto Nathan , executor of the Einstein estate, who, according to the book of Highfield and Carter,evidently feared what Stachel would tum up. Nathan brought an unsuccessful legal action against thePress, in which one of us (JAW) well remembers testifying on behalf of Stachel and the Press againstthe barrage of disparagement from Nathan 's lawyer.

2. Actually, one must specify appropriate boundary condition s (the vanishing of the field at infinity) in thecase of (2), and so the second equation in some sense contains "less" information than the first.

3. An equivalent statement may be made i!1 the the usual multivariable calculus. Combining the Maxwell

equations ~ . E = p and ~ x fj = E + J, with the identity ~ . ~ x == 0 yields the law ofchargeconservation, !Jff + ~ .J = O.

4. This quadratic dependence is suggested in the quadratic dependence of kinetic energies for other dy­namical quantities , such as particles in classical mechanics .

5. See (Misner, Thome, and Wheeler 1973), chap. 17 for additional details.

6. Of course, in the case of continuous source distributions , there is an infinite number of degrees offreedom.

7. C(, ) obeys the relation C( X(i) , X (j») = C( X(j) , X (i»). Thus , C(,) must also be invariant ("sym­metric") under time reflection.

8. Typically, we may exclude the i = j terms from the summation. A brief discussion of methods tohandle this "self-energy" is given by Barut (1980).

436 DANIEL H. WESLEY AND JOHN A. WHEELER

9. It should be noted that, while stress-energy is not assigned to the field per se, when one considersa combined action [ em + [ gr a v ity, variations in the g p,v yield a divergence-free object identical instructure to the usual electromagnetic stress-energy tensor. This occurs because the geometry along theline of propagation depends on the metric ; when the metric is varied, the G(, ) is varied as well.

10. That is, a region small enough so that the paths of geodesics that begin with different velocities do notcross (a necessary condition for the uniqueness of assigning coordinates).

II . See (Marzke and Wheeler 1964) for add itional discussions on this point , and on the Marzke-Wheelerclock in general.

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Barut, A. O. 1980. Electrodynamics and Classical Theory ofFields and Particles. New York: Dover Pub­lications.

Ciufolini, lgnazio, and John Archiba ld Wheeler. 1995. Gravitation and Inertia . Princeton : Princeton Uni­versity Press .

Dirac, P. A. M. 1938. "Classical theory of radiating electrons ." Proc. Roy. Soc. London, pp. 148- 168.Einstein , Albert , Boris Podolsky, and Nathan Rosen. 1935. "Can the Quantum-Mechanical Description of

Reality be Considered Comp lete?" Phys. Rev. 47 :777-80.Hoyle, F., and J. V. Narlikar. 1974. Action at a Distance in Physics and Cosmology . San Francisco: W. H.

Freeman.Mach , Ernst. 1902. The Science of Mechanics. 2nd ed. London : Open COUlt.Marzke, Robert F., and John A. Wheeler. 1964. "Gravitation as Geometry- I," Pp. 40-64 in Gravitation and

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Freeman .Narlikar, J. V. 1968. "On the genera l correspondence between field theories and the theories of direct inter­

particle action." Proc. Camb. Phil. Soc. 64:1071.Sciama, Dennis . 1957. " Inertia." Sci. Am. 196 (2): 99-109.Whee ler, John A. 1964. "Mach's Principle as Boundary Condition for Einstein 's Equations ." Pp. 303-350 in

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