#### 1. Whitehead and Physics

**Whitehead as mathematical physicist**—Whitehead was one among several mathematical physicists (using terminology that arose later in the century) who participated in the emergence of modern relativity theory during the first two decades of the twentieth century, concurrent with his work in mathematical logic and natural philosophy. His principal work in relativity physics (*R*) includes a full general relativity theory, which appears to meet all observational tests (Synge/Coleman, 2005 [1951]) although it was later eclipsed by Einstein’s theory.

**Maxwell’s field theory**—Whitehead chose Maxwell’s *Treatise on Electricity and Magnetism* as the subject of his research dissertation for the Trinity College fellowship at Cambridge University (1880-1884) under Maxwell’s pupil and close friend, W. D. Niven (Lowe, 1985, 95-99). This work was the basis for his first lecture course at Harvard in 1924, which was on Maxwell’s equations. Maxwell’s field theory became a key architectural tool for all of modern physics and inspired Whitehead’s to “electrify” ontology, and to consider ultimate reality as a web of related events.

**Relations between geometry and physics**—The year 1905, already famous for three great papers by Albert Einstein and marker for the international Einstein centenary celebrations of 2005, was also marked by Whitehead’s innovative memoir “On Mathematical Concepts of the Material World” delivered to the Royal Society of England (*MCMW*). At a time when scientists were primarily thinking in terms of the physical and geometrical concepts of Newton, Whitehead symbolized five different concepts of the relation between geometry and physics. This paper is so revealing of the man’s genius and vision that it prompts a reassessment of Whitehead’s place in the history of ideas.

**Systems as complex web of interconnections**—”The general character of its [relativity theory’s] importance arises from the emphasis which it throws upon relatedness. It helps philosophy resolutely to turn its back upon the false lights of the Aristotelian logic. Ultimate fact is not a mere aggregate of independent entities which are the subjects for qualities. We can never get away from an essential relatedness involving a multiplicity of relata” (*IS* 143). Recent recognition of the importance of networks of relationships in modern science was clearly anticipated by Whitehead in his process-relational philosophical synthesis that emerged directly from constraints and suggestions present in field theory, relativity theory, and the logic of relations presented in *Principia Mathematica,* written by Whitehead and Bertrand Russell.

**Ultimate fact as essentially a process**—Quantum field theory and other areas of modern physics have led us to see that at the microscopic level the world is a web of a series of events (Jungerman, 2000). Substance description arises from human encounters with perceptual objects. More fundamentally, systems are complex spatiotemporal societies of events instead of aggregates of substance. And in that process “it belongs to the nature of a ‘being’ that it is a potential for every ‘becoming.’ This is the ‘principle of relativity’” (*PR* 33). However, this principle operates at a metaphysical level in a way very different from its application to contingent features of the world expressed in physical theory (Nobo, in Eastman and Keeton, 2004a).

**Multi-level systems with emergence**—The new sciences of nonlinear systems have shifted the focus away from only specific efficient causes to networks of relationships at multiple organizational levels. Correspondingly, many system properties are no longer treated as substantial and local properties, but as relational and “social” properties arising or “emerging” from relational networks or “societies.” Thus, many systems are now treated as effectively constituted by these sets of relationships in a way that is closely analogous to Whitehead’s treatment for societies of actual entities at multiple levels [Note: Whitehead’s philosophy affirms panexperientialism (experience as very broadly understood) and not panpsychism because of the emergence associated with such societies]. Research results in a broad range of fields (superconductivity, quantum measurement, self-organization in nonlinear systems, plasmas, biological systems; social creations, economics) are yielding numerous examples of emergence. Paradoxically, among physicists who systematically apply methodological reduction, research results have led many to the rejection of ontological “reductionism.” Within the past three decades, the importance of self-organization and emergence in physical systems has become well established.

**Dualities without dualism**—Building on Whitehead’s subtle contrast of being and becoming, Hartshorne developed a comprehensive set of polar relations in philosophy that go beyond the simplistic symmetries of previous work with dualities such as in Hegel’s philosophy (Hartshorne, 1970). Many of these dualities map well into dualities in modern physics such as that of symmetry and asymmetry (Eastman, in Eastman and Keeton, 2004a). However, dualities of our experience should not be construed as representing some Cartesian dualism because they represent complementary perspectives rather than ontologically-separate domains:

both substance and event-oriented descriptions;

both external and internal relations;

both continuity (e.g. continuous time) and quantization (e.g. epochality);

both symmetry and asymmetry;

both space and time—a coupled space-time metric;

both actuality and possibility;

both particles and waves—many dualities.[1]

**Possibility and the importance of approximation**—Whitehead’s philosophy provides metaphysical, non-epistemic foundations for possibility, anticipating results in quantum theory of measurement that suggest an important distinction between actuality space (Boolean algebra) and possibility space (non-Boolean algebra; Bub 1997). Further, approximation has been found to be inevitable and fundamental limits are set through physical relations such as the speed of light limit to information or energy transfer. One fundamental limit arising from Whitehead’s work is that a concrete temporal fact is not global but local (Hansen, in Eastman and Keeton, 2004a) which, Hansen argues, enables coherent meaning for temporal facts.

#### 2. Whitehead’s Theory of Relativity and Comparisons with Einstein

Alfred North Whitehead and Albert Einstein were both fierce critics of classical physics concepts. They were both awakened from the dream of the classical trinity of absolute space, time and matter by the advent of James Clerk Maxwell’s electromagnetism and by their philosophical reading. Maxwell’s theory was pregnant with possibilities to replace the vision of timeless bits of matter in an empty space-container with that of the vibrating activity of an all-pervading field of electromagnetic energy, and both Whitehead and Einstein seized the occasion to help deliver Maxwell’s revolutionary baby. The philosopher that most influenced Whitehead for this purpose was George Berkeley; in the case of Einstein, it was Ernst Mach. Berkeley’s critique of the classical distinction between primary qualities pertaining to matter and secondary qualities arising from the mind inspired Whitehead’s critique of the classical bifurcation of nature into the smooth world of mathematical abstractions and the rough-edged world of sense perception. Mach’s critique of Newtonian concepts, especially of “the conceptual monstrosity of absolute space,” reinforced Einstein’s natural inclination towards deconstruction and subsequent creation—not that Whitehead turned into an idealist and Einstein into a lifelong adherent of Mach. Whitehead moved beyond Berkeley’s attempt to bridge the gap between the world of mathematical physics and the world of human sensations by turning idealism inside out, combining it with a vision of pervasive electromagnetic events (replacing the classical ether of stuff), and arriving at a process philosophy, which takes the universe to be a plenum of experiential events. Einstein’s introduction of space and time in terms of measuring rods and synchronous clocks in the special theory of relativity (STR) shows that the early Einstein was under the spell of Mach’s epistemology. Einstein’s creation of the general theory of relativity (GTR), however, shows that the later Einstein moved beyond Mach’s positivism by turning Mach’s principle (inertia is due to acceleration, not within absolute space, but relative to the universal mass distribution) into his equivalence principle (inertial and gravitational effects are equivalent), which ultimately led him to GTR, a theory far removed from Mach’s immediate sensations and common sense.

Despite the fact that both Whitehead and Einstein were fully aware of the Maxwellian concepts and of the importance of a philosophical approach to revolutionize the concepts of classical physics, the stories of their intellectual development not only run parallel and in agreement, but also intersect and lead to disagreement. Let’s consider a first point of intersection of Whitehead’s intellectual career with Einstein’s. Although Einstein published his STR in 1905 and Hermann Minkowski gave STR its ultimate geometrical framework in 1908 by merging three-dimensional space and one-dimensional time into a quasi-Euclidean four-dimensional space-time manifold, the influence of Einstein’s 1905 paper and of Minkowski’s 1908 paper on Whitehead was postponed approximately ten years. However, during World War I, Whitehead did assimilate the four-dimensional embrace of space and time, and it changed the course of his career. STR was a catalyzing factor in changing Whitehead from being a Cambridge mathematician-logician into a London mathematical physicist and philosopher of science.

Whitehead did not require an Einsteinian or Minkowskian stimulus to become a critic of classical physics even though the majority of Whitehead’s output prior to 1915 consisted of mathematical publications, including the three volumes of the *Principia Mathematica *(*PM*, 1910-1913), written together with Bertrand Russell. This is clear (1) from his 1905 memoir and (2) from his 1914 Parisian lecture.

First, in the year of Einstein’s STR—1905—Whitehead already tried to transcend the classical concept of the material world by means of the logic of relations that forms a substantial part of the first volume of *PM*. In the memoir “On Mathematical Concepts of the Material World,” he presented a number of non-classical concepts. Most significant is that Whitehead took relations as fundamental and considered entities as forming their “field.” In his fifth concept of the material world, space is not absolute, matter is not ultimate, and mass and energy are not properties of matter; instead, relations between non-material geometrical entities are at the base of a relational theory of space and a field theory of material entities.

Second, in the year of Einstein’s early GTR—1914—Whitehead delivered his lecture on “The Relational Theory of Space” in Paris. This lecture was part of the *PM*-project. Whitehead was preparing a fourth volume on the logical foundations of geometry, and he intended to include this lecture. Yet, his 1914 lecture also dealt with building a bridge between the apparent space of immediate perception and the physical space of mathematical abstraction. This lecture was thus a Whiteheadian exercise in closing the gap between the world of perception and the world of science; the gap caused by the unfortunate bifurcation of nature that lay at the root of classical physics. In this lecture Whitehead for the first time used a bridging device of his own making: the method of extensive abstraction. This method is a Whiteheadian combination of the logic of relations with the notion of convergence developed in the second volume of *PM*. Instead of taking the geometrical point—”without parts and without magnitude”—as the primary notion to start building the abstract space of mathematical physics, Whitehead started from extended appearances, defined a whole-part relation between those appearances, as well as the convergence of appearances in terms of this whole-part relation, and finally equated the geometrical point with a class (or abstractive element) of specific convergent series (or abstractive sets) of appearances. Think of the abstractive element P as the class of those abstractive sets of appearances which would normally be said to converge to the point P, including the series of apparent spheres concentric to P. But, of course, in the formal definition of a point P, any reference to P must be omitted, since otherwise the definition would be circular. So instead of taking the geometrical point as a simple starting point, Whitehead considered a geometrical point to be the complex result of a process of abstraction from sets of appearances similar to sets of Russian dolls extrapolated to infinite sets without a smallest doll, but converging to an ideal limit anyway.

Even though Whitehead did not need Einstein and Minkowski to become a critic of classical physics and to work on overcoming the classical bifurcation of nature, their work had a profound impact on him. The merger of space and time catalyzed the expansion of Whitehead’s mathematical search for the logical basis of geometrical space to the philosophical search for the empirical basis of the concepts of space, time and matter; it enforced Whitehead’s philosophic disposition, and it turned the mathematician-logician writing the fourth volume of *PM* into the mathematical physicist and philosopher of science writing *An Enquiry Concerning the Principles of Knowledge *(*PNK* 1919), *The Concept of Nature *(*CN* 1920) and *The Principle of Relativity *(*R* 1922).

So the first intersection of the career-curve of Whitehead with the career-curve of Einstein coincides with Whitehead’s acquaintance of STR in its Minkowskian geometrical clothing. In *PNK*, *CN* and *R* we can witness how this led Whitehead to enthusiasm as well as disagreement with Einstein. Whitehead’s enthusiasm was due to the fact that STR strengthens Berkeley’s argument in favor of a relational worldview. Berkeley had argued that Locke’s dualism of primary and secondary qualities was ill-conceived, since primary and secondary qualities are *all* observer-dependent qualities. But Berkeley’s undermining of the notion of observer-independent matter, opening the door to a relational concept of matter, did not necessarily imply the removal of observer-independent space and time. Einstein, however, not only demonstrated the interdependence of space and time, but also the concurrent observer dependence of space and time measurement. Observers or measuring devices uniformly moving compared to each other can not use the same, supposedly absolute space-time coordinate system; each has to use a space-time system corresponding with his or her uniform motion. Each space-time system represents an observer-dependent perspective of the four-dimensional Minkowskian space-time manifold. Einstein was able to show that measurement in one special space-time system can be related to measurement in another such system by means of the Lorentz transformations, and Minkowski was able to show that those transformations concur with the geometry of “rotations” in a four-dimensional manifold. So the change from one special space-time perspective to another can be performed by means of a “rotation” in the four-dimensional space-time world of STR. Closely connected with this, the length of each interval between two points in that world, as well as expressions for the special relativistic laws of physics, remain invariant while performing such a rotation. This guarantees the congruence of four-dimensional intervals, and hence consistent measurement, as well as a certain absoluteness of the physical laws within a genuine relativistic universe.

Whitehead realized that STR might lead to idealism, just as Berkeley’s anti-Lockian arguments did, but he convincingly showed that observer dependence does *not* necessarily mean mind dependence, and he generalized the observer-dependent space, time and matter into a space-time manifold and into fields of material characters expressing the universal relatedness of experiential events. The existence of multiple space-time systems became Whitehead’s guiding principle when launching his philosophy of science by giving his own derivation of the STR. But, why did Whitehead feel the need to give an alternative derivation of STR? First of all, Einstein’s STR, especially in its Minkowskian format, was a mathematical abstraction and even more abstract than classical physics had ever been. To avoid recreating the classical bifurcation of nature, Whitehead felt he had to start from scratch, or better, from human sense perception, in order to abstract, step by step, the Einsteinian-Minkowskian four-dimensional world of the STR by means of his method of extensive abstraction. Secondly, Einstein’s guiding principle to derive the STR was that the speed of light *in vacuo *must have the same constant value *c* in all special relativistic space-time systems. But Whitehead considered the constant value of *c* to be a contingent fact of the electromagnetic field, and not an essential fact of the space-time structure of the relational web of ultimate events.

It is important to elaborate on both of Whitehead’s reasons to give an alternative rendering of STR in *PNK* and *CN*. These reasons also characterize his philosophy of science in general *and* his alternative rendering of GTR in *R*. First, Whitehead’s requirement to reconstruct STR by means of his method of extensive abstraction clearly shows that he thought a scientific theory can not be justified solely by experimental testing. Sure, according to Whitehead—and the same holds for Einstein—the experimental testing of empirical predictions is the final arbiter of scientific theories, but Whitehead, contrary to Einstein, revalued the role of human sense experience in order to avoid bifurcation, and hence, required scientific theories to pass a broad gauge next to passing the narrow gauge of experimental testing. Whitehead wanted scientific theories to pass the broad gauge of consistency with immediate sense perception. If a theory does not pass this broad gauge, Whitehead considered it to be unjustified. Second, Whitehead’s rejection of Einstein’s guiding principle—the overall constancy of light’s velocity *in vacuo—*illustrates the importance Whitehead attached to the distinction between three layers of reality: (1) the relatedness of ultimate events, also called the ether of events; (2) the essential relatedness of space-time, abstracted from the relatedness of ultimate events; and (3) the contingent relatedness of the physical field, affecting physical characters of ultimate events. Physical characters—e.g., definite masses and charges—are also called adjectives by Whitehead, and as (scientific) objects are nothing more than (patterns of) adjectives with a certain endurance, they are called adjectival objects. For example, the adjectival particles of Whiteheadian physics are characterized by their gravitational mass and/or their electromagnetic charge. Each law concerning the speed of light clearly is a law concerning the electromagnetic field, or, which amounts to the same, concerning the adjectival particles defined by this field. So Einstein’s principle concerns contingent reality, and Whitehead wanted to reconstruct STR starting from ultimate reality. However, prior to talking about Whitehead’s reconstruction of STR, and subsequently about his construction of an alternative GTR, it is crucial to show how Whitehead deals with two possible objections to his approach based on sense perception and the relatedness of nature.

The first objection is the claim that sense perception is too limited for theory building. There seems to be a double limitation: we only discern a limited scope of events, and we only discern physical characters of these events, and not their space-time relatedness. Whitehead’s answer is that we not only perceive the discerned, but also the discernible, and, which amounts to the same, that our perception is not only a cognizance by adjective, but also a cognizance by relatedness. True, we only discern physical characters of a limited scope of events, but human sense perception also includes the immediate awareness of the discernible space-time relatedness of the discerned events, as well as the immediate extrapolation of this space-time structure beyond our limited scope. Whitehead analyzed human sense perception by distinguishing between (1) sense perception of discerned adjectives of events—part of our cognizance by adjective; and (2) sense perception of the discernable space-time structure—part of our cognizance by relatedness. For example, looking at a number of closed boxes, we not only perceive these adjectival objects, we also perceive their spatial relations and, at the same time, apply the same kind of spatial structure to the situation within the boxes, even though we do not discern any of the physical characters within (e.g., we not only speak of the middle between two boxes, but also of the centre of a box, even though we cannot look within).

The second objection is the claim that no knowledge is possible if everything is related because knowledge of some events in a relational web of events presupposes knowledge of all other events in the web. Whitehead’s answer is closely connected to his previous answer. According to him, it is important to first distinguish between the (cognizance of) essential (space-time) relations, and the (cognizance of) contingent (field) relations. Next, there are two reasons why we do not need a detailed knowledge of all events in order to know some. First, we *can* extrapolate our knowledge of space-time relatedness for some events to larger sets of events. In fact, this *is *what we do thanks to our cognizance by relatedness. And, even though this extrapolation presupposes that global space-time structure is uniform, we are *allowed *to do so because, according to Whitehead, the essential space-time relatedness of the universe *is *uniform. Without this uniformity, human cognizance by relatedness would never have evolved. For example, when looking in the mirror, it only makes sense to estimate how far the virtual mirror image is behind the looking glass if we can apply the spatial structure that holds in front of it to the situation behind it, even though we do not actually discern anything behind the mirror; and when looking at a far-away star, it only makes sense to speak of the distance separating the observer from the star if this space-time relation is independent of all events between observer and star which we cannot directly discern. Secondly, we do not necessarily need to know the field relatedness of all physical characters in order to know some because (fields of) adjectives are atomic. According to Whitehead, the atomicity of (the fields of) adjectives means that an adjective of discerned events can be independent of the adjectives of non-discerned events (and of non-discerned adjectives of the discerned events). Without this atomicity, humans would never have been able to isolate and solve problems.

We can only give a rough outline of Whitehead’s detailed reconstruction of the four-dimensional world of STR. It is clear that he applied his method of extensive abstraction to ultimate reality instead of building on the overall constancy of the speed of light, a principle concerning contingent reality. Recalling the application of Whitehead’s method to define abstractive sets of extended appearances and to abstract the notion of geometrical points—the building blocks of geometry—one might assume that Whitehead applied his method to define abstractive sets of events and to abstract the notion of event-particles—supposedly the building blocks of Minkowski’s manifold. To a certain extent, this assumption is correct. However, with such a simplified procedure, the observer perspective would be lost, and hence the multiplicity of space-time systems, essential for STR, would not be reconstructed. Whitehead’s more complex approach reconstructs Minkowski’s manifold via the multiplicity of space-time systems: first, based on our sense of time, Whitehead abstracted the notion of a space-time system; and second, based on our sense of rest and motion, he abstracted the Einsteinian-Minkowskian multiplicity of space-time systems.

First, of course, our sense of time has nothing to do with instantaneous sense experience. An instant of time is as abstract as a geometrical point. An instantaneous moment is not experienced, it is ideal. What we experience always has extension. We experience slabs of nature called durations. Notice that we do not discern durations, only clusters of simultaneous events. However, because of our cognizance by relatedness, we do perceive spatially completed clusters, which is exactly what durations are. A moment, also called an instantaneous space, is an abstractive element consisting of abstractive sets of durations with the “same” convergence. The different moments abstracted by an individual observer (in an unchanged state of motion) are completely separated, and together they form an ordered series. Whitehead called completely separated simultaneous spaces parallel, and he defined a space-time system to be the ordered series of moments resulting from abstraction by an individual observer. Since the abstraction of a space-time system is observer dependent, we are, at least to a certain extent, ready to conclude that multiple space-time systems exist. But at this point, Whitehead also brings in our sense of rest and motion.

Second, our sense of rest and motion results from the relation between the experiential events that form the natural focus of our sense experience, and the associated durations. Whitehead calls this the relation of cogredience. An observer always has a sense of being *here* within the *now* he experiences. In other words an observer has a sense of being at rest within his durations, which is why Whitehead wrote that a percipient event is always cogredient within the associated duration. On the other hand, our sense of motion results from relations with non-associated durations, in other words, from the cogredience-absent inclusion of percipient events within durations. Our sense of rest helps us to integrate durations into a prolonged present, or, returning to the world of abstractions, to identify a space-time system. On the other hand, our sense of motion differentiates durations, and hence leads to the differentiation of space-time systems. For example, an observer in the carriage of a train can have a double consciousness: one of the whole within which he is *here* and *at rest*, and one of the whole within which the trees, houses and bridges he sees through the window are *there*, and within which he is *in motion*.

Equipped with the notion of multiple series of parallel moments, as well as the notion of prolongation based on cogredience, Whitehead was able to define simultaneous planes, lines, and points, respectively called levels, rects, and puncts, as well as prolonged points, lines and planes, which are simply the points, lines and planes of the prolonged space of the space-time system under consideration, constituting a *unique* four-dimensional manifold. For example, consider two moments m_{1} and m_{2} of a space-time system t, and two moments M_{1} and M_{2} of another system T; then the intersection of m_{1} with M_{1} and the intersection of m_{1} with M_{2} define two levels that are parallel. This example at once illustrates how the concept of parallelism was recovered by Whitehead, and we can add that the notion of rest allowed him to define perpendicularity as well (e.g., a straight line in the space of system t perpendicular to a plane due to system T is the track in t of the position of an adjectival particle at rest in T). In Euclidean geometry, parallelism is needed to define shifts of intervals in a plane, perpendicularity to define rotations of intervals in a plane perpendicular to the rotation axis, and shifts and rotations to define congruence of intervals in a plane (two intervals are congruent if the first coincides with the second after having been shifted and/or rotated). Similarly, based on his notions of parallelism and perpendicularity, Whitehead was able to define the congruence of intervals in the unique four-dimensional manifold he abstracted, thus turning it into the Minkowskian manifold with rotations of space-time systems leaving the length of intervals invariant, and thus allowing consistent measurement. Whitehead’s appropriate reduction of the arbitrariness involved in defining congruence also led him to recover the constant *c. *But whereas in Einstein’s STR, *c* expresses an assumed fixed speed of lightfrom the start, in Whitehead’s reconstruction *c *arises as a constant allowing for a notion of congruence that not only embodies the essential uniformity of the space-time relatedness underlying Whitehead’s whole reconstruction, but also enables a recovery of all essential formulae of Einstein’s STR.

Let’s consider a second point of intersection of Whitehead’s intellectual career with Einstein’s. As a member of the Royal Society of London Whitehead was present at the Society’s meeting of November 6, 1919. Arthur Eddington presented the results of his famous expedition to test the prediction of Einstein’s GTR that rays of light bend when passing the sun. This event turned Einstein into a worldwide celebrity, and at the same time it is a symbol of Whitehead’s acquaintance with GTR, an acquaintance that drastically deepened his disagreements with Einstein’s way of dealing with relativity, and that led Whitehead to reject Einstein’s GTR. What were Whitehead’s objections to GTR? To address this question, let’s first explain what Einstein’s GTR is all about.

Soon after developing his STR, Einstein realized (1) that STR was *only *about special space-time systems that were *not *accelerated compared to each other; and (2) that it *only *adjusted Newton’s laws of motion, but *not *his law of gravitation.

With regard to the first point, an observer walking in a train traveling at constant speed will experience no difference from his walking in the station, but when the train makes an emergency stop his situation in the train dramatically changes. Similarly, the coordinate expression of a dynamical law (e.g., one of Maxwell’s laws of electromagnetism) in one space-time system of the STR is not different from its coordinate expression in another such system because the coordinates are Lorentz related, but when a space-time system accelerates, the kinematical effects *absolutely *interfere with the dynamics and destroy this Lorentz covariance. Is there no other way of expressing dynamical laws (such as the laws of gravitation and of electromagnetism) in order for them to assume a more general covariant form in *all *space-time systems, non-accelerating *and* accelerating? Notice that the claim for a more general covariance is Einstein’s general principle of relativity, as opposed to his special principle of relativity that only required laws to be Lorentz covariant.

Second, Newton’s law of gravitation states that the size of the gravitational pull or force (F) by a body with mass M (e.g., the earth) on a body with mass m (e.g., you) is given by

F = GMm/r^{2} (1)

where G is the gravitational constant and r is the distance between M and m. STR informs us that, even though the length of a space-time interval in the Minkowskian manifold is invariant when rotating from one to another space-time system, separate space and time measurements are relative: lengths of separate space and time intervals change from system to system. But lengths of such space intervals define distances. So the r-distance dependent Newtonian law of gravitation is not a special relativistic law: it changes from system to system instead of being Lorentz covariant (let alone that a more general covariant form is needed in the sense just discussed). Is no appropriate adjustment possible?

Both problems were extremely hard nuts to crack, and it took Einstein several years to solve the acceleration and the gravitation problem by generalizing STR into GTR. Einstein’s guiding principle, that ultimately led him to complete GTR in 1915, was his idea that effects due to acceleration (e.g., being pushed to the floor of an upward accelerating elevator) and effects due to gravitation (e.g., being pulled to earth) are equivalent. But Einstein could not conceive this equivalence to be global and uniform (e.g., suppose the earth’s gravitational pull merely is a global push to its surface due to a uniform upward acceleration of the global earth, then not all humans would feel that gravitational pull; humans at the wrong place on earth would simply fall off). What was needed was a non-uniform solution guaranteeing at least a local equivalence of both types of effects. Einstein did find such a solution by identifying the four-dimensional space-time manifold with the gravitational field, and by allowing this manifold to be *non-uniformly* curved, that is, to be *Riemannian*—in honour of Georg Bernhard Riemann—instead of Minkowskian.

Let’s explain Einstein’s solution with an analogy. Suppose Newton and Einstein, both on the roof of a large building, look down and observe two boys playing on the asphalt parking lot next to the building, a good and a bad boy. The good boy plays at marbles; the bad boy lights a pile of dead branches. Our physicists observe that a marble, rolled away by the good boy, deviates from its straight path when passing the burning pile, and they wonder why the path of the rolling marble bends. After a while, Newton mysteriously smiles and gives his explanation: “The bending is due to a thermal force exerted by the burning pile on the rolling marble at infinite speed, a force depending on the distance between pile and marble.” Einstein, however, is not satisfied, but after careful reflection, his face illuminates and he says: “No, there is no such thing as an instantaneous thermal action-at-a-distance. The thermal field of force and the asphalt surface must be identified. Heat spreads in the surface, and the thermal effect on the rolling marble is entirely due to how the spreading heat of the burning pile warps and curves the asphalt surface in its neighbourhood. It’s simply the non-uniform curvature of the surface that determines the marble’s path of motion.”

Now replace the rolling marble with a moving elementary particle, the burning pile with a star, the thermal field of force with the gravitational field of force, and the asphalt surface with the four-dimensional Riemannian space-time manifold of the GTR. Then Einstein’s GTR holds that there is no such thing as an instantaneous gravitational action-at-a-distance. The gravitational field and the space-time manifold must be identified. The gravitational effect of the star on the motion of the elementary particle is entirely due to the fact that the mass of the star warps and curves the space-time manifold. It’s simply the non-uniform curvature of the space-time manifold that determines the path of motion of the elementary particle. Notice that one of the consequences of Einstein’s STR was the equivalence of mass and energy (epitomized in the famous formula E = m*c*^{2}); so we should replace “mass” with “mass-energy” in our account of the GTR. Furthermore, next to the mass-energy of the star, the mass-energy of the elementary particle of course also determines the gravitational field; in fact, it is the universal mass-energy distribution that determines the structure of the Riemannian space-time manifold of the GTR. John Archibald Wheeler once summarized the GTR as follows: “Matter tells space how to curve, and space tells matter how to move.” In our words: “The universal mass-energy distribution tells the space-time manifold how to curve, and the non-uniform curvature of the space-time manifold tells mass-energy objects how to move.” To end this non-mathematical explanation of GTR, suppose the star under consideration is the sun, and the elementary particle is a light-particle (a photon), then it is immediately clear why GTR holds that light bends when passing the sun.

The most important formula of STR is the one expressing the length of a four-dimensional interval in the Minkowskian manifold. Writing x_{1}, x_{2}, and x_{3} for the space coordinates of an arbitrary space-time system in STR, x_{4} for its time coordinates, and ds for the length of a four-dimensional interval, then ds can be expressed in terms of the differences dx_{1}, dx_{2}, dx_{3}, and dx_{4} between the coordinates of two four-dimensional points determining the interval. The formula reads

ds^{2} = dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2} – *c*^{2}dx_{4}^{2} (2)

This formula is a simple generalization of the Pythagorean formula ds^{2} = dx_{1}^{2} + dx_{2}^{2} in the Euclidean plane, and makes clear why the Minkowskian manifold is said to be a quasi-Euclidean four-dimensional manifold. In STR, spatial coordinate differences are measured by rods and temporal coordinate differences by clocks, and the resulting length ds^{2} is invariant when changing systems. This makes clear why formula (2) is said to represent the metric of the Minkowskian manifold.

Einstein’s generalization of STR into GTR, and hence of the Minkowskian manifold into the Riemannian manifold, led to the replacement of metrical formula (2) by

ds^{2} = g_{μν}dx_{μ}dx_{ν} [summed over the indices μ and ν that run from 1 to 4] (3)

This formula has sixteen terms of three factors: each first factor, in general, is not a constant, but a function of the coordinates; each second and third factor, however, are still coordinate differences, but—due to the non-uniform curvature—coordinate differences can no longer signify direct results of measurement with ideal rods and clocks, and ds^{2} can no longer signify the invariant length of an interval. Even so, ds^{2} represents the metric of the Riemannian manifold, and this metric allows mathematical physicists to calculate the non-uniform curvature of the Riemannian manifold. The sixteen factors g_{μν} taken together form what is called a symmetrical tensor (a table of functions; because of the symmetry g_{μν} = g_{νμ}, the tensor contains at most ten different functions). Given the role of these factors in formula (3), this tensor is called the metric tensor. Recall that the Riemannian manifold is equal to the gravitational field and determined by the universal mass-energy distribution; add to this recollection that Einstein expressed this distribution by means of another tensor, the mass-energy tensor T_{μν}, then it is possible to gain some understanding of Einstein’s law of the gravitational field, given by

R_{μν}—½g_{μν}R = –*κ*T_{μν} (4)

On the left hand side of (4) we find a complicated function of the g_{μν}, which is abbreviated by the symbols “R_{μν}” and “R”; on the right hand side, *κ* is a constant, and T_{μν} is the mass-energy tensor. Einstein’s (set of non-linear) equation(s) given by (4) states how the gravitational field g_{μν} is to be calculated from the mass-energy distribution T_{μν}. However, since g_{μν} also represents the metric of the Riemannian manifold (given the role of g_{μν} in formula (3)), Einstein’s equation (4) also states how the metric of the Riemannian manifold is to be calculated by the distribution of matter (mass-energy). In practice, performing this calculation and hence solving the equation is extremely difficult. Yet, Karl Schwarzschild solved it in the simplest but most important case of a mass-energy distribution that is only determined by a single spherical body (e.g., the sun). It will come as no surprise that the Schwarzschild solution is most important in determining how rays of light bend when passing the sun.

When learning all about GTR, Whitehead rejected it. Not because of the transition from a pseudo-Euclidean to a curved, non-Euclidean manifold; as a mathematician he was familiar with non-Euclidean geometries. Not because rays of light bend when passing the sun either: as a philosopher of science Whitehead held that the constancy of the speed of light was contingent, so he was not perplexed by the non-constancy implied by the bend. No, Whitehead rejected Einstein’s GTR because it presupposes that the space-time manifold and the gravitational field *can be simply identified,* and that the curvature of the space-time manifold *can be non-uniform, *whereas, according to Whitehead, this identification and this non-uniformity are *impossible *on fundamental epistemological grounds*. *Whitehead rejected Einstein’s GTR because its two fundamental presuppositions imply its failure to achieve what any relational theory of physics has to achieve: (1) to neutralize the epistemological objection that—if all is related—partial knowledge is impossible without total knowledge; and (2) to pass the broad gauge of human sense perception.

(1) Recall that, according to Whitehead, neutralization of the epistemological objection requires *distinction* between (cognizance of) the essential space-time relatedness or *manifold*, and (cognizance of) the contingent physical relatedness or *field*, as well as space-time *uniformity*. These two Whiteheadian requirements oppose the two fundamental presuppositions of GTR. No wonder Whitehead claimed that GTR fails to neutralize the epistemological objection. (2) Whitehead also claimed that GTR fails to pass the broad gauge of human sense perception. The manifold/field distinction and the cognizance by relatedness/cognizance by adjective distinction are two sides of the same coin (one cannot exist without the other), and as this last distinction results from Whitehead’s analysis of the possibility of sense perception, no sense perception can exist without the manifold/field distinction. But sense perception *does* exist, and, consequently, so must the manifold/field distinction. Furthermore, recall that no cognizance by relatedness can exist without space-time uniformity, and as our sense perception of time, rest and motion is based on cognizance by relatedness, this also implies that no perception of durations and of cogredience relations can exist without space-time uniformity. But according to Whitehead this perception of time, rest and motion *does* exist, and, consequently, so must the space-time uniformity.

To fully understand Whitehead’s rejection of GTR, it is important to add that, according to him, the failure of GTR to neutralize the epistemological objection (and to justify knowledge), concurs with (1) a circularity that prevents total solutions of Einstein’s equation in practical settings; and (2) a confusion about measurement that prevents consistent measurement.

(1) How can Einstein ever give the input to solve equation (4)? In other words, how can he know the entire mass-energy distribution in order to calculate the gravitational field? The entire mass-energy distribution is the distribution of mass and energy across the entire space-time manifold. So to start with, he has to know the whole space-time manifold. But as this manifold equals the gravitational field, he first needs to solve equation (4) in order to know it. The conclusion is that we need to solve Einstein’s equation in order to solve it. How can we escape this circularity? Would Einstein respond that both the mass-energy distribution and—giving its identification with the gravitational field—the space-time distribution are atomic fields, which—following Whitehead—already solves the need to know both of them completely, and which—it is important to notice—justifies Schwarzschild’s limitation of the mass-energy distribution to that of a single spherical object in order to solve equation (4)? Would Einstein also respond that the circularity is only an apparent circularity, an appearance caused by the non-linearity of the equations, but not preventing their solution, as Schwarzschild’s and other solutions clearly prove?

(2) How can Einstein consistently measure? Going from one space-time system to another in STR changes our contingent measurement instruments (rods and clocks), but it does not affect the structure of the space-time environment of our instruments, since the overall structure is uniform. That’s why—in STR—we can rely on congruence and compare measurements, we can use measurements to determine the length of a four-dimensional space-time interval, and we can rest assured that this length is invariant (independent of the choice of space-time system). In GTR, however, the contingent field containing our measurement instruments is identified with space-time relatedness. That’s why—in GTR—going from one space-time system to another—e.g., going from the earth to Mars—*not only *changes our contingent measuring instruments, *but also *the structure of their space-time environment (space-time structure is contingent and heterogeneous instead of essential and uniform). But then how can we ever compare measurements? Is that not impossible? And is this impossibility not consistent with the impossibility of defining congruence in a non-uniformly curved manifold? And is all this not consistent with the fact that the notion of invariant lengths of space-time intervals in terms of separate space and time measurements is missing in GTR? Would Einstein respond that Whitehead did not understand how a Riemannian metric can lead to consistent measurement? But how could that be true, given Whitehead’s familiarity with non-Euclidean geometries? Or would Einstein respond that his GTR must embody a consistent measurement theory because it is a successful theory that has passed the narrow gauge of experimental measurement? But if that would be his response, then what about the following example questioning even the possibility of experimental measurement?

Recall Arthur Eddington’s famous experiment to prove Einstein right. The experimental evidence—presented by Eddington in London on November 6, 1919 to Whitehead and the other Royal Society members—rests on the possibility to compare different measurements, that is, measurements made in different space-time environments, that is, given the non-uniformity of the global space-time structure in the GTR, measurements made in differently structured space-time environments. Indeed, the experimental evidence to support Einstein’s prediction that light bends when passing the sun is that the visual image of a distant star is shifted when the sun intervenes. We can only talk about the shift of the visual image when we compare the visual image (at night) without the intervening sun, with the visual image (at daytime) when the sun intervenes (and when a solar eclipse admits the visual image). But in order to compare these two visual images, we have to locate them in the same space, and which space would that be—the one belonging to the space-time environment without the intervening sun, or the one belonging to the space-time environment with the intervening sun? If GTR is correct, then the space-time structure of these environments will differ because the energy-matter distribution of the first excludes the sun, while that of the second includes the sun. As the space-time structure determines space structure, we do not know which space to choose in order to locate and compare the two visual images. We are left in confusion. Only if the global space-time structure is uniform, then the two spaces will be identical and the two visual images can be located and compared in any of the two spaces in order to prove or disprove GTR. How would Einstein respond to the paradox arrived at: in order to prove GTR and hence the non-uniformity of the space-time manifold (e.g., by means of Eddington’s experiment), we have to rely on the uniformity of this same manifold?

In the three previous paragraphs we repeatedly asked: “How would Einstein respond?” This begs for the question: “Did Einstein respond?” Let us consider a third biographical point of intersection. Einstein’s biographer Phillip Frank claims that, while Einstein was in England early 1920s, Whitehead had long discussions with Einstein. Whitehead repeatedly attempted to convince Einstein to give up his GTR. Whitehead did not succeed. Einstein did not see logical or experimental reasons to drop his theory, nor considerations of simplicity and beauty. This can mean two things: Einstein failed to understand Whitehead’s arguments, or, he undermined them. If only we had a replay-button … And yet we think the first option is closest to the truth. Not only did Einstein confess to Filmer Northrop: “I simply do not understand Whitehead.” But on top of that, Whitehead certainly did not act as a man whose arguments had been undermined. He both maintained his critique *and* revealed the details of his alternative relativistic theory of gravitation in his 1922 book with the (misleading) title *The Principle of Relativity *(*The Principle of Relatedness *would have been better).

When Einstein developed his GTR in the period between 1907 (the equivalence principle) and 1915 (completion of GTR), he tried to explain both the effects due to acceleration and the effects due to gravitation, guided by his principles that both type of effects must be equivalent (the principle of equivalence) and that the expression of the dynamical laws of gravitation and electromagnetism must be more generally covariant than Lorentz covariant (the general principle of relativity). Einstein himself realized that he had only partially succeeded. First, he was very ambivalent about whether or not, with his GTR, he had realized Mach’s dream to account for all effects due to acceleration in terms of the universal distribution of matter (mass-energy) instead of relying on a more ‘absolute’ explanation: sometimes he said it did, sometimes he denied it. Second, Einstein was definitely not at all happy that he failed to include electromagnetism, and that his GTR merely was a general relativistic theory of gravitation. In a sense electromagnetism *is* included: it is part of T_{μν}. The minimum one can say is that the electromagnetic energy field is part of the mass-energy distribution. The maximum Einstein tended to say was that they coincide, implying that matter is purely electromagnetic. But this inclusion did not lead to unification, but to dualism. Einstein sometimes called this dualism the space/matter dualism, sometimes the gravitational field/electromagnetic field dualism. It is the dualism between the two sides of equation (4): on the one hand the space-time manifold and the gravitational field (equal to each other and baptized “the ether of GTR” by Einstein); and, on the other hand, the electromagnetic field (more or less equal to matter). Until he died, Einstein tried to overcome this dualism by unifying the gravitational field and the electromagnetic field, and hence by turning his GTR, summarized simplistically by

ether = space-time manifold = gravitational field ≠ electromagnetic field ≈ matter

into a unified field theory (UFT), summarized by

ether = space-time manifold = gravitational field = electromagnetic field = matter.

By the time Whitehead published his alternative relativistic theory of gravitation in the period between 1920 (*Times *article “Einstein’s theory: An alternative suggestion”—see *ESP*—and Chapter VIII of *CN*) and 1922 (*R *is the most detailed account published) he clearly had learned from Einstein’s shortcomings.

(1) Whitehead, contrary to Einstein, did not hesitate to deny the success of GTR to explain effects due to acceleration. He looked at the most spectacular effects due to the rotation of the earth (the bulge of the earth at its equator, the invariable directions of rotation for cyclones and anti-cyclones, the rotation of the plane of oscillation of Foucault’s pendulum, and the north-seeking property of the gyrocompass) and could not persuade himself that these rotational effects can be accounted for by means of the distribution of matter, whether that matter is Mach’s matter of the fixed stars, or Einstein’s ambiguous matter. When Einstein successively turned down Newton’s account in terms of absolute space *and *his own STR, he lost the necessary absoluteness to give a good explanation, and when he embraced a relational account to do so, his relations were relations in terms of matter, meaning that he actually enlisted a key notion of classical mechanics in an army of concepts intended to fight classical mechanics. Not that Whitehead advocated a return to Newton’s absolute space, but according to him, the Minkowskian space-time manifold of STR, defined as the essential relatedness abstracted from the extensive relatedness of ultimate events, addresses both the minimum requirement of absoluteness, and the requirement to transcend the classical concept of matter. According to Whitehead, his own STR provides the necessary axes to define the rotation of the earth and to account for the associated effects, without reference to other physical bodies. Evidently, contrary to Einstein’s project to create a GTR, Whitehead’s project did *not* include the goal to find a new explanation of the effects due to *acceleration*, while* transcending* the STR. Whitehead *only* wanted to include the effects due to *gravitation, *while *extending *his own STR.

(2) The ultimate result of Einstein’s GTR was a substance dualism. The first substance—the one Einstein called the ether of the GTR—was a confusing unification of the non-material space-time and the contingent gravitational field, turning space-time into a non-uniform and contingent manifold. The second substance—the one Einstein called matter—was a confusing mix between classical matter and electromagnetic field, an unhappy marriage of the material point concept with the field concept. Whitehead agreed with Einstein that substance dualism had to be eliminated, but Whitehead did not agree with Einstein that the solution to overcome this dualism was an even further identification. To identify Einstein’s two substances would mean: (a) to conflate Whitehead’s three layers of reality; and (b) to spice the resulting mix with an ambiguous notion of matter. With his solution, Whitehead wanted to take a step back from GTR, instead of taking it a step further. According to Whitehead, to solve Einstein’s undesirable substance dualism means: (a) to restore the difference between the essential and uniform space-time relatedness and the contingent and atomic physical field relatedness; and (b) to introduce a third and ultimate layer of reality that unambiguously *turns* the classical dogma of material objects—supposed to constitute all space-time, gravitational and electromagnetic relations—*upside down*, by taking relations to be fundamental and objects to be derivative. Recall that Whitehead called this ultimate layer of reality “the ether of events” because it is a relatedness of events, not only constituting the layer of the essential and uniform space-time relatedness and the layer of the contingent and atomic relatedness of the physical field, but also, via the characters or adjectives of the physical field, the adjectival objects. Indeed, adjectival objects are (patterns of) more or less enduring characters or adjectives of events, and hence the object is obtained through the event and not the event through the object, and “the character is gained through the relatedness and not the relatedness through the character” (*R* 19). With this last sentence, Whitehead truly transcends classical physics. Let’s summarize the discussion with

ether of events ≠ uniform space-time ≠ contingent physical field

where

contingent physical field = gravitational field & electromagnetic field & …

Isn’t the introduction of “layers of reality” that are different (“≠”) a road to fragmentation instead of unification? No, language and symbol play a trick on us. There is only one reality: the ether of events. The space-time manifold and the physical field are but two abstractions from this unique and concrete relatedness. In other words, our perception of the relational web of ultimate events is twofold. We perceive its essential aspects in the uniform space-time manifold that forms the background for our perception of its contingent aspects in the atomic field of characters or adjectives. On top of this “principled unification,” Whitehead’s physics, dealing with the contingent and atomic physical field of mass and charge, also treats the gravitational field and the electromagnetic field in a completely similar way, implying a “methodological unification.” To close our paper, leaving out Whitehead’s treatment of electromagnetism, and focusing on gravitation only, we will briefly present Whitehead’s alternative relativistic theory of gravitation, intended to supplement his own STR with a law of gravitation to deal with the dynamical effects of gravitation.

Let’s return to our earlier analogy and put Whitehead with Newton and Einstein on the roof of the large building, watching how the rolling marble on the asphalt parking lot bends in the neighbourhood of the burning pile of dead branches. Whitehead disagrees with Newton that the bend is due to an instantaneous thermal action-at-a-distance, but agrees that space and time are uniform (though not necessarily Euclidean). Being a true Maxwellian, Whitehead agrees with Einstein that the explanation must involve a thermal field, but he disagrees that both this field and the space-time geometry have to be identified with the surface of the asphalt parking lot, implying a non-uniform curvature of space-time. He tries to convince both Newton and Einstein of the superiority of his intermediate position with the following comment: “There’s no such thing as marble objects possessing non-relational properties such as heat, and interacting at infinite speed. Also, there’s no such thing as a heated geometrical abstraction, for I hold on to the distinction between the physics of contingent characters of events, and the geometry of uniform relations of events. In my theory, there is only a web of events, and the physical field of contingent characters of events (including the heat of events) expresses how the characters of events are geometrically distributed, without determining geometry. Events pop in and out of existence, but successions of events manifesting the same characters—also called historical routes—constitute enduring objects. A character is said to *pervade* a historical route when it is a character of every event of the route or, in other words, “when it is an adjective of every stretch of the route” (*R* 32). Conversely, a historical route can be seen as the *pervasion*-history of one or more characters. Consequently, the rolling marble and the burning pile are patterns

of pervasive characters of events (e.g., mass and heat). Now, let’s make, by means of an extensive abstraction, marble and pile infinitely small. Then my explanation for what we see is that the pervasion of the marble-heat H is influenced not only by (the events constituting) its own pervasion-history, but also by (the events constituting) the pervasion-history of the pile-heat h. It is important to add this notion of history, because there’s no *instantaneous *action-at-a-distance, only a *retarded *action or impetus determined by the limited speed *c. *Indeed, a now-element of the pervasion-history of H can only be influenced by itself and by the past-element of the pervasion-history of h that satisfies the condition that its heat-influence can reach the now-element when propagating from the past-element at speed *c*. To conclude, the heat-effect of the pile on the marble is a retarded action in the thermal field, to be described in terms of elements of the uniform geometric surface.”

Let’s now substitute what Whitehead’s GTR is really about. Replace marble heat H with mass M, pile heat h with mass m, and the thermal field with the gravitational field. Finally, identify the surface only with the gravitational field, and not with the four-dimensional Minkowskian space-time manifold of Whitehead’s STR. Then Whitehead’s GTR claims that every stretch XX’ of the historical route of mass M is gravitationally influenced by itself and by those stretches PP’ of all other historical routes of masses m that can exercise a gravitational influence on XX’. In other words, each (event constituting a particular) now-pervasion-element XX’ of the pervasion-history of mass M is gravitationally influenced by itself and by those (events constituting) past-pervasion-elements PP’ of the pervasion-histories of all other masses m whose retarded gravitational field action or impetus can causally influence (the event constituting) XX’.

Let’s turn this into a formula. All elements XX’ and PP’ are infinitesimal intervals in the Minkowskian manifold satisfying formula (2). Whitehead, however, did not use dS^{2} (the square of the length of XX’) and ds^{2} (the square of the length of PP’) in his formula, but the closely related entities given by

dG_{M}^{2} = –dS^{2} and dG_{m}^{2} = –ds^{2} (5)

These entities will represent the uniform space-time background in the formula of the potential gravitational field action (PGFA) now influencing the pervasion of M. This PGFA of M is determined both by the now-element XX’ and by the retarded gravitational field action of all other m (RGFA_{m}) now influencing the pervasion of M. The overall structure of Whitehead’s law of gravitation is given by

PGFA^{2} = dG_{M}^{2} + RGFA_{m}^{2} (6)

Let’s focus on RGFA_{m}. According to Whitehead, RGFA_{m} is determined both by the past-element PP’ and by the retarded gravitational potential of m (RGP_{m}) as follows

RGFA_{m}^{2} = RGP_{m}dG_{m}^{2} (7)

But how to express RGP_{m}? Instead of formula (1)—Newton’s law of gravitation—let’s look at the formula in which the force is a function of a potential

F = f(ψ_{m}) (8)

in which ψ_{m} is given by

ψ_{m} = Gm/r (9)

This helps, for ψ_{m} is the classical gravitational potential of m, and the stroke of genius that led Whitehead to his alternative GTR is the idea to turn this classical, *instantaneous *gravitational potential of m into the non-classical, *retarded *gravitational potential of m (RGP_{m}). In the classical case, we are confronted with r—a separate space-distance that is *not *Lorentz covariant. Indeed, the space-distance r is the reason why Newton’s law of gravitation needs adjustment in order to become a special relativistic law. Whitehead replaced the factor r in formula (9)—in the expression of the classical gravitational potential of m—by another factor in order to obtain an expression for the RGP_{m} that leads to a new law of gravitation satisfying a series of closely connected conditions:

—the retardation condition (= only those past-elements are included that can have causal influence on the now-element);

—the Lorentz covariance condition (= the principle of special relativity);

—the recoverability condition (= Newton’s law of gravitation needs to be a special case of the new law of gravitation);

—the empirical equivalence condition (= ability to derive all the empirical effects that Einstein derived, but that cannot be derived from Newton’s theory: the bending of light rays in the neighbourhood of the sun, the rotation of the perihelion of Mercury’s orbit, and the spectral shifts towards the red in a gravitational field).

Curiously enough, Whitehead found four laws of gravitation satisfying the listed conditions. Let’s stick to the one Whitehead put in the forefront (due attention is given to all four laws in Russell, 1988), and perform the following replacement:

r è -(*c²*/2)*w* (10)

On the right hand side, next to speed *c*, there’s an entity *w* of which we will not give the mathematical expression. Let’s just add that *w* is a complicated entity that ensures that all conditions are satisfied. Applying generalization (10) to formula (9) gives us an expression for the RGP_{m}:

RGP_{m} = -(2/*c²*)Gm/*w* (11)

Substituting this RGP_{m}-expression (11) in the RGFA_{m}-expression (7), and next, the obtained result in the PGFA-expression (6) yields

PGFA^{2} = dG_{M}^{2} – (2/*c²*)(Gm/*w*)dG_{m}^{2} (12)

Finally, we introduce two more notations in order to obtain the most familiar expression of Whitehead’s law of gravitation. The first notation reads:

dJ^{2 }= PGFA^{2} (`13`

)

The second one reads:

ψ_{m} = Gm/w (14)

Be aware that, even though ψ_{m} is the same symbol as the one used for the classical gravitational potential of m, it does not denote this classical gravitational potential of m. Anyway, substituting (13) and (14) into (12) finally leads to the most familiar expression of Whitehead’s law of gravitation:

dJ^{2} = dG_{M}^{2} – (2/*c²*)ψ_{m}dG_{m}^{2} [summed over the indices m] (15)

The remark between brackets on the necessity of summation is made to remind us that instead of only applying to the gravitational influence of one mass m on M, formula (15) applies to the influence of an indefinite number of masses m on M (so the same remark also applies to equations (7) and (12)). It is in accordance with Whitehead’s linear approach that multiple masses are taken into account by summation (due attention to this aspect of Whitehead’s theory is given in Russell, 1988).

Whitehead’s law of gravitation allows us to calculate the *potential gravitational field* action (PGFA) now influencing the pervasion of M against the uniform space-time background (represented in (15) by dG_{M}^{2} and dG_{m}^{2}). Notice that Whitehead’s term for PGFA was the *potential *mass impetus. But what about the *actual *gravitational field action (AGFA), or, using Whitehead’s term, the *realized *mass impetus? Whitehead’s answer to this question is given by

AGFA = M√ PGFA^{2} (16)

Or, using notation (13) and Whitehead’s terminology:

realized mass impetus = M√dJ^{2} (17)

And what about the *gravitational field* itself? Well, Whitehead admired and adopted the Minkowskian space-time manifold of Einstein’s STR, and admired and adopted the tensor field concept of Einstein’s GTR. Whitehead’s gravitational field is defined in terms of sixteen (ten different) functions J_{μν}, together forming a symmetrical tensor field, just as Einstein’s gravitational field is defined by the symmetrical tensor field g_{μν}. Of course, Whitehead’s embrace of Einstein’s *mathematical method* did not interfere with his critical attitude towards Einstein’s *physical theory*. In fact, Whitehead wrote:

My whole course of thought presupposes the magnificent stroke of genius by which Einstein and Minkowski assimilated time and space. It also presupposes the general method of seeking tensor […] expressions for the laws of the physical field, a method due to Einstein. But the worst homage we can pay to genius is to accept uncritically formulations of truths which we owe to it (

R88).

How to relate Whitehead’s gravitational tensor field J_{μν} to the potential gravitational field action (PGFA) now influencing the pervasion of M? Whitehead’s answer, using notation (13), reads:

dJ^{2} = J_{μν}dx_{μ}dx_{ν} [summed over the indices μ and ν that run from 1 to 4] (18)

Although formula (3) of Einstein’s GTR (ds^{2} = g_{μν}dx_{μ}dx_{ν}) and formula (18) of Whitehead’s GTR look exactly alike, Einstein’s g_{μν} express both the gravitational field *and *the Riemannian space-time manifold that replaces the Minkowskian space-time manifold (and hence Einstein’s coordinate differences are related to the Riemannian space-time manifold), while Whitehead’s J_{μν} express *only *the gravitational field against the uniform background provided by the Minkowskian space-time manifold (and hence Whitehead’s coordinate differences are related to the Minkowskian space-time manifold). In fact, Whitehead’s coordinate differences in formula (18) are the coordinate differences of the now-element of M in the Minkowskian space-time manifold, the ones used to calculate dS^{2} = dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2} – *c*^{2}dx_{4}^{2} by means of Minkowski’s metrical formula (2) and dG_{M}^{2} by means of formula (5). Hence:

dJ^{2 }≠ ds^{2} or PGFA^{2 }≠ ds^{2} or PGFA ≠ ds (19)

In other words:

potential mass impetus ≠ space-time measurement (20)

Failing to notice this difference between Whitehead’s and Einstein’s GTR leads to the confusion about measurement discussed earlier. Hence Whitehead wrote:

By identifying the potential mass impetus [PGFA] of a kinematical element [the now-element of M] with a spatio-temporal measurement [ds] Einstein, in my opinion leaves the whole antecedent theory of measurement in confusion […]. The potential impetus shares in the contingency of appearance. It therefore follows that measurement on his theory lacks systematic uniformity… (

R83).

To conclude, does Whitehead’s law of gravitation really satisfy the empirical equivalence condition? Whitehead may have been able to derive light-bending, perihelion-rotation and red-shifts, but are the quantitative predictions of Whitehead’s GTR about light-bending, perihelion-rotation and red-shift identical to the quantitative predictions of Einstein’s GTR? For light-bending and perihelion-rotation (and all other effects that are also caused by the gravitational field of a single spherical body), both theories give the same quantitative predictions (because, as Arthur Eddington already proved in 1924, they share the Schwarzschild solution). For the spectral shifts towards the red in a gravitational field, the discernment of differences and the comparison with experimental data has led to differences of opinion. Let us, however, not examine the red-shift discussion, nor delve into other ongoing discussions about the empirical strength of Whitehead’s theory, and only quote A. John Coleman’s 2005 assessment “that the evidence for the validity of Einstein’s and of Whitehead’s theories of gravitation is roughly of equal value.” Coleman’s assessment is only one among many, but at least it does not offer an excuse to ignore the philosophical importance of Whitehead’s theory of relativity and “the emphasis which it throws upon relatedness” (*IS* 143).

#### 3. History of Relevant Scholarship

Scholarship focused on Whitehead’s relativity theory and related work in mathematical physics has been ongoing since Norbert Wiener’s PhD dissertation of 1913 on Whitehead and Russell’s algebra of relations, and there is an increasing pace of such scholarship. Classics include, among others, Victor Lowe’s superb biography and key works by Code, Finkelstein, Fowler, Griffin, Henry, Herstein, Jungerman, Leclerc, Lowe, Malin, Mays, Palter, Ranke, Russell, Schmidt, Shimony, Stapp, Stolz, and Synge. An extensive bibliography is provided in the *Resource Guide for Physics and Whitehead* (Eastman and Keeton, 2004b). Recent books and special issues of primary relevance include the SUNY volume on *Physics and Whitehead*, edited by Eastman and Keeton (2004a), and special issues in *Process Studies* on process thought and natural science (Fowler, 1981; Eastman, 1997 and 1998). The context of such scholarship within process thought more generally has been provided by Nicholas Rescher (1996 and 2000).

For related work in math and logic, see the “Mathematics and Logic” section of this Encyclopedia, and the “Sciences” section for other areas of science. In particular, we refer you to complementary Encyclopedia entries: “On interpreting Einstein’s Relativity,” “Vector Physics,” “Vector Mathematics,” “Applied Physics,” and “Quantum Mechanics.”

For relativity, the physics community has chosen Einstein’s theory but *not* because of a clear distinction between empirical predictions and a subsequent falsification of Whitehead’s theory. In fact, the studies of Temple and Eddington in the 1920s, and the re-evaluations of Synge, Rayner and Schild in the 1950s, all pointed in the direction of an empirical equivalence of the two theories. Clark in the 1950s, Will in the 1970s, and Tanaka in the 1980s, raised empirical issues for Whitehead’s theory (the double star issue, the earth tides issue, and the redshift issue, respectively). Confronted with experimental uncertainty regarding the validity of Whitehead’s laws of gravitation, two reactions are typical: Fowler’s reaction to defend Whitehead’s physics against an overly hasty rejection in order to safeguard his underlying philosophy of nature (Fowler, 1974); and Tanaka’s reaction to encourage updating the physics, running the risk of ruining the philosophy (Tanaka, 1987). But it’s probably best not to be provoked into either one of these reactions (Herstein, 2006).

*First* and ironically, Whitehead’s theory was no real competitor for Einstein’s theory on the scientific battlefield to start with, exactly *because* originally it was held to be empirically indistinguishable. *Secondly*, in light of current attempts within mathematical physics to unify Einstein’s theory of general relativity with quantum mechanics, and hence to go beyond them by means of some novel theory (string theory, M-theory, loop quantum gravity, etc.), it is anachronistic to ask which of the two theories of relativity—Einstein’s or Whitehead’s—is technically preferable. Yet it will always remain relevant to ask which of the two underlying philosophies—Einstein’s or Whitehead’s—is the best for interpretive needs and can contribute most effectively to recent scientific developments. Further, for pedagogical reasons if no other, it would be fruitful to further investigate why Whitehead’s alternative theory is so closely equivalent and yet is so much more easily derived and applied than Einstein’s theory. Most recently, Coleman (2005) calls on his physics colleagues to re-evaluate Whitehead’s long neglected alternative approach.

#### 4. Personal Assessment

Whitehead’s work anticipated an amazing number of results in contemporary physics (see examples in Eastman and Keeton, 2004a) just as it anticipated many developments in contemporary philosophy (e.g., Lucas, 1989). The fruitfulness of Whitehead’s creative genius considered over the long term thus contrasts starkly with the profound neglect accorded this major philosopher-scientist throughout the last century. Setting aside possibilities for future technical work on Whitehead’s general relativity whose apparent disconfirmation may itself be in doubt, we think that there remains important philosophical work on relativity and quantum theories that can fruitfully build on key issues and options raised by Whitehead’s work.

A first step to turn the profound neglect of Whitehead into a revaluation is to avoid the widespread misunderstanding that his view is just another form of philosophical and anti-scientific idealism. True, Whitehead was inspired by Berkeley, and he tried to bridge the gap between the phenomenal world of human sense experience and the postulated world of mathematical physics by means of his method of extensive abstraction. But nonetheless, as a philosopher, Whitehead was a realist with the intention to overcome the idealist-sceptical-transcendental trio (Berkeley, Hume and Kant). And as a physicist, when trying to avoid the classical fallacy of misplaced concreteness (to take the classical trinity of space, time and matter for the most concrete reality), Whitehead’s intent was not to replace all abstract scientific concepts with concrete experiential events—his formulae are as abstract as Einstein’s! Whitehead only tried to prevent the promotion of abstractions to the status of primary and concrete realities, and the degradation of human sensations to the status of a secondary and illusory reality. He wanted to appropriately link the ideal, mathematical, and timeless world of mathematical physics to the sensory, colourful, and irreversible world of human experience. So what underlies both his critique of classical physics and his critique of Einstein’s theory of relativity is not the intention to eliminate all scientific abstraction, but to have all scientific abstraction pass the broad gauge of human sense experience. If his desire had been to simply deconstruct Einstein’s world of relativity and to withdraw into a world of immediately sensed events (e.g., durations) without abstract notions (e.g., four-dimensional manifolds), then Whitehead would never have reconstructed Einstein’s STR, nor would he have constructed his own relativistic theory of gravitation.

A second step to revalue Whitehead is to avoid the mistake of thinking that Whitehead simply claimed scientific theories are invented by abstraction from human sense experience. He was well aware of the fact that “the true method of discovery is like the flight of an aeroplane” (*PR* 5). But he also stressed that after their imaginative flight in the atmosphere of discovery, physicists have to land on the runway of justification. And this runway must not only consist of the narrow gauge of experimental testing, but also of the broad gauge of human sense experience. Whitehead’s claim that we experience the uniformity of the space-time relatedness of events, or the existence of a multiplicity of perspectives within this space-time manifold, is not intuitive or merely based on direct sense experience, but depends on a philosophical claim about epistemological requirements. Whitehead thought that the uniformity of space-time relatedness of events was *the* answer (next to the atomicity of the fields relating characters of events) to the epistemological difficulty caused by any theory of relatedness, namely, the deadlock of having to know the whole relational web in order to be able to know part of it. But it would be wrong to exclude *alternative *answers. Are there not other ways out of this difficulty? For example, from a philosophical point of view, comparing it with the difficulty of the hermeneutical circle might lead to new ideas; and turning to physics, it could be argued that iterative and nonlinear techniques may be instrumental to avoid this deadlock.

Suppose we were to agree with Einstein that a true relational theory of reality needs a nonlinear approach, and that its concepts cannot all have an immediate link with the experiences of human sense awareness. Even then, we can take a third step in the rehabilitation of Whitehead as a philosopher-scientist. Whitehead has convincingly demonstrated that Einstein’s relativity theory was only a halfway house between the classical, absolutist and materialist physics of the past, and the new, relational and non-materialist physics of the future. Whitehead’s own relational theory and his overall philosophy of nature provide robust tools to overcome problems with classical notions. This was demonstrated when quantum mechanics arose, and Whitehead’s relational theory proved to be able to philosophically accommodate quantum mechanics. Of course, we cannot here elaborate on the relativity/quantum debate but have three short remarks. (1) The Einstein-Bohr debate has once been characterized (by Kurt Hübner) with the statement: “for Einstein relations are defined by substances; for Bohr substances are defined by relations” (Jammer, 1974, p.157). The affinity between this statement and Whitehead’s aphorism that “the character is gained through the relatedness, and not the relatedness through the character,” epitomizes the affinity between Whitehead and Bohr. (2) However, the affinity between Whitehead’s philosophy and quantum mechanics goes far beyond his affinity with Bohr, and has received considerable attention in several recent studies (e.g., see Malin, 2001; Epperson, 2004; and Hättich, 2004). (3) Finally, physics did not stop evolving after the advent of quantum mechanics, and one recent development in physics to compare with is the attempted integration of relativity and quantum mechanics in recent string and membrane theories. Certainly, the evolution from a four-dimensional space-time manifold into a five-dimensional (Kaluza-Klein), a ten-dimensional (string theory) and an eleven-dimensional manifold (M-theory) has led physics farther away from Whitehead’s immediate sense experience. But at the same time, this evolution further confirms Whitehead’s relational world view. For example, the strings of string theory can be conceived as the non-material and geometrical expression of ultimate relatedness; its vibrations can be conceived as linking essential and contingent relations because the physical field of mass and charge characters arises from the endurance of these vibration patterns; and the perplexing evolution of the variety of material particles and forces, and hence of the laws of physics themselves, can be conceived as the confirmation of Whitehead’s idea of the evolution of the laws of societies of ultimate events. Setting aside string theory and other untested abstractions, these themes are also developed by David Finkelstein (1996) in his work at the interface of quantum and relativity.

Our analysis of Whitehead’s thought has implications for science and philosophy more generally and a philosophy of process “must pivot not on a *thinker *but on a *theory*. What is at issue must, in the end, be a philosophical position that has a life of its own, apart from any particular exposition or expositor” (Rescher, 2000). The classic, two-tier substance-property ontology presupposed in most of Western philosophy until well into the twentieth century is now being replaced, in fits-and-starts, by a one-tier process ontology by interpreters of modern science, especially contemporary physics, although not primarily through the direct influence of process philosophers. In particular, the “ether of stuff” that dominated up to the time of Einstein has now been generally replaced (at least among scientists) through quantum field theory by an ether of events, aided in part by Einstein’s “ether of GTR” although the latter is sometimes mistakenly taken as effectively an “ether of stuff” with space-time treated (via a fallacy of misplaced concreteness) as some new substance in spite of warnings from Feynman. “Substance is really an effect of our macroscopic human senses; *at the microscopic level the world is a web of a series of events*” (Jungerman, 2000).

### Notes

[1] “Duality is a profound and far reaching concept in theoretical physics, with origins in electromagnetism and statistical mechanics. The emergence of duality in recent years in several areas of modern physics, ranging from the quantum Hall fluids to string theory, represents a major development in our understanding of quantum field theory” (Zee 2003, 309).

### Works Cited and Further Readings

Bub, Jeffrey. 1997. *Interpreting the Quantum World *(Cambridge, Cambridge University Press).

Eastman, Timothy (ed.). 1997-98. “Process Thought and Natural Science,” *Process Studies*, special issues 26, 3-4 and 27, 3-4.

Eastman, Timothy E. & Hank Keeton (eds.). 2004a. *Whitehead and Physics: Quantum Process and Experience *(Albany, SUNY Press).

Eastman, Timothy E. & Hank Keeton 2004b. “Resource Guide for Physics and Whitehead.” *Process Studies Supplement*, Issue 6. Accessible online at http://www.ctr4process.org/publications/PSS/index.htm.

Epperson, Michael. 2004. *Quantum Mechanics and the Philosophy of Alfred North Whitehead* (New York, Fordham University Press).

Finkelstein, David. 1996. *Quantum Relativity: A Synthesis of the Ideas of Einstein and Heisenberg* (Berlin, Springer-Verlag).

Fowler, Dean. 1974. “Disconfirmation of Whitehead’s Relativity Theory: A Critical Reply,” *Process Studies, *4, 288-90.

Fowler, Dean. (ed.). 1981. “Whitehead and Natural Science,” *Process Studies*, special issue11, 4.

Hartshorne, Charles. 1970. *Creative Synthesis and Philosophical Method *(La Salle IL, Open Court).

Hättich, Frank. 2004. *Quantum Processes: A Whiteheadian Interpretation of Quantum Field Theory *(Münster, Agenda Verlag).

Herstein, Gary L. 2006. *Whitehead and the Measurement Problem of Cosmology *(Frankfurt, Ontos).

Jammer, Max. 1974. *The Philosophy of Quantum Mechanics* (New York, John Wiley & Sons).

Jungerman, John. 2000. *World in Process* (Albany, SUNY Press).

Lowe, Victor. 1985 & 1990. *Alfred North Whitehead: The Man and His Work, Volume I & Volume II *(Baltimore, Johns Hopkins University Press).

Lucas, George. 1989. *The Rehabilitation of Whitehead* (Albany, SUNY Press).

Malin, Shimon. 2001. *Nature Loves to Hide: Quantum Physics and Reality* (Oxford, Oxford University Press).

Rescher, Nicholas. 1996. *Process Metaphysics* (Albany, SUNY Press).

Rescher, Nicholas. 2000. *Process Philosophy: A Survey of Basic Issues* (Pittsburgh, University of Pittsburgh Press).

Synge, John L. & Coleman, A. John. 2005 [1951]. *Whitehead’s Principle of Relativity: Three* *Lectures given at the University of Maryland by John L. Synge*, edited by A. John Coleman. See online http://arxiv.org/abs/physics/0505027.

Tanaka, Yutaka. 1987. “Einstein and Whitehead: the Principle of Relativity Reconsidered.” *Historia Scientiarum,* 32, 43-61.

Whitehead, A. N. 1906. “On Mathematical Concepts of the Material World,” *Philosophical Transactions: Royal Society of London*, series A, Vol. 205, 465-525. Reprinted in *Alfred North Whitehead: An Anthology*, edited by F.S.C. Northrop and M.W. Gross (Cambridge, Cambridge University Press, 1953), 11-82.

Whitehead, A.N. and Russell, B. 1910, 1912, 1913. *Principia Mathematica*, 3 Vols., 1^{st }ed. (Cambridge, Cambridge University Press). 2^{nd }ed. in 1925, 1927, 1927.

Zee, A. 2003. *Quantum Field Theory in a Nutshell *(Princeton, Princeton University Press).

### Author Information

Ronny Desmet

Center for Logic and Philosophy of Science

Vrije Universiteit Brussel

http://www.vub.ac.be/CLWF

ronny-desmet@skynet.be

Timothy E. Eastman

Plasmas International, Silver Spring, Maryland 20910 USA

plasmas@starpower.net

### How to Cite this Article

Desmet, Ronny, and Timothy E. Eastman, “Physics and Relativity”, last modified 2008, *The Whitehead Encyclopedia*, Brian G. Henning and Joseph Petek (eds.), originally edited by Michel Weber and Will Desmond, URL = <http://encyclopedia.whiteheadresearch.org/entries/thematic/sciences/relativity-physics/physics-and-relativity/>.