The Principle of Relativity 3

2. Whitehead's principle of Relativity

Einstein's principle of relativity has two components: one is the special principle, and the other is the general one. The former states that all inertial systems are equivalent for the description of natural phenomena, while the other claims that the same equivalence should hold generally in any chosen frame of reference. Whitehead did not rely on either of them. First, he pointed out in The Principles of' Natural Knowledge that the physical content of Einstein's theory can be deduced without relying on Einstein's principles. The special theory of relativity correlates space to time through the Lorentz-Transformation, which Einstein deduced from the combination of the special principle and the principle of the constant velocity of light. Whitehead, on the other hand, deduced the same transformation from the weaker principles of kinematics and geometry, i.e. (1) the uniformity and symmetry of space-time, (2) the symmetry and transitiveness of transformation, etc.⁽⁶⁾

Secondly, he repeatedly laid stress on the inequality between inertial and rotating systems in his book, The Principle of Relativity, the title of which was certainly ambiguous and therefore misleading.

The principle of relativity in Whitehead's sense must be understood in the context in his philosophical thought. This principle plays the central role not only in his physics, but also in his metaphysics. The physical principle of relativity is generalized to the metaphysical one. The more we understand his metaphysics, the more we comprehend his physics. So we may well begin with the definition of this principle in Process and Reality:

"It belongs to the nature of a 'being' that is a potential for every 'becoming'. This is the 'principle of relativity'.⁽⁷⁾

As the above formulation of the principle is the most general characterization on the metaphysical level, it needs some explanation as to how it is embodied within the realm of physics. What we must bear in mind is that two lines of Whitehead's criticism of classical physics are closely connected with the above principle: i.e. his criticism of scientific materialism, and his rejection of Cartesian dualism involving the "bifurcation of nature."

In Whitehead's metaphysics, "Becoming" is more fundamental than "Being" which is the reversal of Aristotelian ontology. The concept of matter as "hypokeimenon" (substratum) of nature, the cornerstone of scientific materialism, presupposes the Aristotelian concept of substance: matter is conceived as the true Being which exists independently of perceivers: the description of the configuration of matter in space-time through the deterministic laws is thought to be the only task of physicists: there remains no place for the perceiving subjects. Nature, as it is perceived by us, is separated from nature as the object of physics. This bifurcation cannot be easily overcome: if we try to bridge them by considering the one as a cause and the other an effect, then we soon find that such a kind of causality is unintelligible on account of the "fallacy of nusplaced concreteness". Whitehead pointed out this fallacy by grasping the most concrete aspect of nature as creative becoming rather than as static, substantial Being. According to Aristotelian ontology, Being precedes Becoming because the former is the actuality of the latter. The opposite is the case with Whitehead. Becoming is the actuality of Being: what has been thought to be substantial Being must be re-interpreted as derivative from Becoming. Therefore the most fundamental category of nature should be found in "events", and not in "substance".

The Principle of Relativity 4

Concerning the concept of event, Whitehead wrote:

"I give the name 'event' to a spatio-temporal happening. An event does not in any way imply rapid change: the endurance of a block of marble is an event Nature presents itself to us as essentially a becoming, and any limited portion of nature which preserves most completely such concreteness as attaches to nature itself is also a becoming and what I call an event. By this I do not mean a bare portion of space-time. Such a concept is a further abstraction. I mean a part of the becomingness of nature, coloured with all the hues of its content. Thus nature is a becomingness of events in terms of space and time Thus space and time are abstractions from this structure."⁽⁸⁾

Whitehead tried to reduce physical entities which were previously considered as substantial Being to the Becomingness of interrelated events. What he means by "event" must not be interpreted as something cut off from the pre-existing continuity of space-time, but the space-time itself is an abstraction from the concrete relatedness of events. What must be noticed here is that the concept of events as four dimensional structures plays the role of mediation between space and time. Both matter as a self-identical substance and space-time as a fixed framework of physics are to be deconstructed to the interrelation of becoming events. Whitehead executed such deconstruction by what may be called the reversal of subject-predicate logic. In classical physics matter is treated grammatically as subject, and its spatio-temporal determinations as adjectives. Whitehead, on the contrary, treats matter as an "adjective" of 'four-dimensional events with specific characters. Material beings are considered by him, not to be causes of perceived qualities, but treated merely as one of many adjectives uniformly modifying events. This does not mean that events occupy the place of substance, for the essence of an event consists in its relatedness

The reason why classical physics had to fix separately the framework of space and that of time was that it lacked necessary means of representation for four-dimensional events. Whitehead, adopting Minkowski's idea that four-dimensional manifold should give the framework of relativity theory, tried to deduce that framework itself from the interrelated structures of events. This procedure was called by him "the method of extensive abstraction", according to which the elements of Minkowski's manifold, event-particles without extension, were mathematically re-constructed from becoming events with spatio-temporal extension

Thus Whitehead endeavoured to reconstruct the fundamental categories of physics after having deconstructed classical physics through the relativistic reduction of Being to Becoming. Einstein's theory was to be assimilated to his own paradigm, and at the same time to be criticized in certain points, especially the relation of matter to space-time. Whitehead was not satisfied with the view of matter presupposed by Einstein, according to which spatio-temporal determinations, depending on the configuration of matter, had to be separated from our perceptual experiences. Whitehead claimed that the condition of perceptual situation, which makes the measurement of spatio-temporal magnitudes possible, should be given independently of matter. According to Einstein's theory of general relativity, the metric properties are decided completely by matter. Space-time is said to be "warped" by matter: The "curvature" of space-time is variable, and it may be said "fiat" only when the gravitational field caused by matter is negligible

Whitehead rejected the very idea of the priority of matter over space-time. As was stated before, matter was considered by him as an "adjective" of events, and it can not exert any influence on the essential characteristic of space-time, which should be determined only on the level of events. The existence of matter only concerns accidental qualities of space-time. On this point the problem arises whether the metric properties are considered to be essential or accidental. Considering the contingency involved in the configuration of matter, Whitehead rejected the effect of matter on space-time metric: the very idea that the curvature of space-time is variable should be irrelevant in Whitehead's theory. In The Concept of Nature he wrote:

"'Space caught bending' appeared on the news-sheet of a well-known evening paper. This rendering is a terse but not inapt translation of Einstein's own way of, interpreting his results. I should say at once that I am a heretic as to this explanation and that I shall expound to you another explanation based upon some work of my own, an explanation which seems to me to be more in accordance with our whole scientific ideas and with the whole body of facts which have to be explained."⁽⁹⁾

The "bending of space" was and is a favorite phrase used by many physicists to explain the meaning of the previously mentioned verification of Einstein's theory at the time of eclipse. It can be paraphrased more exactly by saying that non-Euclidian geometry holds in the neighbourhood of the sun. It was this thesis that Whitehead wanted to replace by his own theory of measurement. Whitehead was convinced that geometry should be distinguished from physics. Geometry represents the uniform elatedness of nature, especially of spatio-temporal relations. Physics treats the contingent properties of nature. These convictions were related to his rejection of scientific materialism and of the bifurcation of nature. The theme of physics, according to him, is not the material things themselves cut off from the perceptual data but the perceived phenomena which show themselves "contingently" in the uniform framework of space-time. The space-time in which material bodies are located, in his view of unified nature, is nothing other than that in which the visual images of them are situated.

Concerning the reason why the uniformity of space-time should be a necessary condition of measurement, Whitehead wrote:

"By identifying the potential mass impetus of a kinematic element with a spatio-temporal measurement Einstein, in my view, leaves the whole antecedent theory of measurement in confusion, when it is confronted with the actual conditions of our perceptual knowledge. The potential impetus shares in the contingency of appearance. It therefore follows that measurement on his theory lacks systematic uniformity and requires a knowledge of the actual contingent physical field before it is possible. For example, we could not say how far the image of a luminous object lies behind a looking-glass without knowing what is actually behind that looking-glass."⁽¹⁰⁾

If we are to locate a material body and the visual image of it in the same space, it is necessary that the space should have a uniform structure independent of matter. For example, we can interpret the aforementioned result of Eddington's experiment in such a way that we need not say, "Space caught bending". The experimental evidence for the idea that rays of light are bent in the neighborhood of the sun is that the visual image of a distant star is shifted on account of the intervening sun. But how can we talk about the shift unless we locate two visual images in the same space? As one is observed in the presence of the intervening sun and the other during its absence, the same space is required to have a uniform character independent of matter.

Thus Whitehead set about constructing a gravitational theory according to which rays of light are bent through the physical (contingent) effects of the gravitational field. Whereas Einstein's theory states that rays of light pass straightly (i.e. along a geodesic line) in the "warped' space, Whitehead's theory states that they pass literally along a crooked curve in the "flat" space. There are two points which we must notice here. First, the requirement of uniformity is not necessarily connected with that of the space-time metric. The theory which Whitehead propounded as an alternative to Einstein's postulated that the metric structure of space-time should be uniform, i.e., it should have the same curvature everywhere. But this is not the only alternative. Adopting Whitehead's paradigm, we can require that only the topological structure of space-time should be independent of matter, and thus a priori relative to measurement. Secondly, as Whitehead himself admitted, the choice of "flat" (Euclidean) space is not an inevitable requirement of Whitehead's theory. Non-Euclidean space, whether it is hyperbolic or elliptic, will do if it has a constant curvature. So we are not concerned with the problem whether space is Euclidean or not.⁽¹¹⁾ The real issue to be discussed is the relation of matter to space-time. For readers who are interested in mathematical physics, I will next explain the difference between Whitehead's and Einstein's theories in mathematical terminology. The reader may omit the next section if he does not want to bother about mathematical technicalities.

The Principle of Relativity 5

3. The Outline of Whitehead's Theory in Mathematical Terminology

The mathematical formulation of Whitehead's theory is, as in Einstein's case, supplied with tensor-analysis. But it is to the physical structure of gravitational field and not to the geometrical metric of space-time, that the Riemanian theory of differentiable manifold with variable curvature is applied in Whitehead's theory Adopting a different interpretation from Einstein's theory which identifies the gravitational and metric fields, Whitehead introduces the concept of impetus as a physical quantity in order to determine the path of light or of a moving particle in the physical field. There are two kinds of impetus: the potential mass impetus and the potential electro-magnetic impetus.

Writing the potential mass impetus as and the potential electro-magnetic impetus as dF, we can integrate the total impetus realised along the time-like world-line AB as follows:



where M is the proper mass as an "adjective" uniformly qualifying the world-line AB,

E is the charge of the mass, c is the velocity of light.⁽¹²⁾

The two kinds of impetus can be expressed in covariant tensors respectively with first and with second orders, as follows:



The potential mass impetus is split up into the difference of two symmetric covariant tensors, and : the former represents the inertial aspect of motion, and the latter the gravitational aspect of the physical field. Thus we get



In order to derive the equations of motion Whitehead applies the variational principle

to the above impetus

and gets a set of differential equations of the Euler-Lagrange type:

The procedure is mathematically similar to Einstein's use of the variational principle but the meanings of mathematical formula are different: Whitehead separates the physical (contingent) component from the geometrical (uniform) one in what Einstein interprets as a space-time interval. When the effects of gravitation and electro-magnetic fields are negligible, we can derive from Whitehead's equations, as from Einstein's, the law of motion



which is nothing but the law of inertia in the special theory of relativity. In the presence of electro-magnetic fields, we get the equations




If we identify with the electric force and with the magnetic force, the above formulas again agree with those of special relativity.

To sum up, as far as the mathematical syntax is concerned, Whitehead's equation of motion can be regarded as a generalization of the special theory of relativity. It is in the theory of gravitation that the difference between Whitehead and Einstein appears sharply on the level of mathematical formulation, to say nothing of physical interpretation. Whitehead treats the gravitational field on a par with other physical fields, as independent of the metric structure of Mincowski's space-time. Therefore, it is required in Whitehead's theory that the system of n mass particles with gravitational interactions should be mathematically similar to the system of n charges moving under their mutual electro-magnetic interaction. Whitehead's theory of gravitation is sometimes referred as "a theory involving action at a distance with the critical velocity c". This characterization of Whitehead's theory is due to Synge, who located Whitehead's theory between the two extremes of Newtonian theory on the one hand and the general theory of relativity on the other. Such a middle-way character comes from the peculiar definition of the physical field in Whitehead's theory. The physical field of an event P modified with mass m is defined as the domain of P's causal future, i.e. the set of world-lines along which the physical signals propagate from P with the critical velocity c. The distance between P and any event X which is under the causal influence of this physical field vanishes into zero in the Mincowski metric. Thus the causal efficacy may be characterized by an action at a distance propagating with c.

To recast Newton's formula of gravitational potential into a Lorentz-invariant form, Whitehead uses the formula



where is the gravitational constant, and w is a Lorentz-invariant quantity which play the role of distance from P to the time-like worldline uniformly qualified by the mass m. The Lorentz-invariant w can be expressed as the inner product of vector PX and the tangent vector of the world line as follows:



Using the spatial distance and the term depending on the velocity of the mass, we may also rewrite the above formula as follows:




where



and



If the mass is at rest, then w becomes identical with the spatial distance r, and we get the Newtonian formula of gravitational potential. Thus Whitehead's theory can give Newton's formula under the special condition. Whitehead's law of gravitation then takes a simple and elegant form as follows:



where dJ is the potential mass impetus of an event X, dGM is the invariant differential of the world-line which passes through X modified with the proper mass M, and dGm is that with proper mass m which exerts causal influence on X.⁽¹³⁾

If we want to find the components of the tensor gu" which represent the gravitational field, we can get them after necessary calculations as follows:⁽¹⁴⁾



The above equations should not be confused with Synge's reformulation of Whitehead's theory:




As this formula is easy to handle, physicists usually mentions it as if it were Whitehead's own. But we must notice that Synge treats the gravitational field on a par with the metric field in Einstein's manner, and the notation has a different meaning from Whitehead's original formula.