| Blog entry: Einstein's speed limit |
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| Written by Sean Miller | |
| Thursday, 12 October 2006 | |
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The following is an essay on Einstein's theory of special relativity as a conceptual metaphor. In the mid-1800s, James Clerk Maxwell, building on the work of Michael Faraday, proposed a theory to account for what were recognized to be two other forces of nature distinct from gravity, namely, electricity and magnetism. Conceptually, electricity and magnetism were observed to behave in ways that the Newtonian cosmology could not adequately explain, limited as it was to describing the world in terms of rigid bodies and the force of gravity that attracted them. Furthermore, Faraday observed that electricity and magnetism behaved in a remarkably similar fashion. Maxwell was able to unify the behaviour of electricity and magnetism in a set of equations (known, appropriately, as Maxwell’s equations) that mathematically articulated the concept, not of an instantaneous force interanimating rigid bodies, but of a pervasive, disembodied field. Maxwell’s was a theory of electromagnetism, where this one force permeates Newtonian space as a harmonically oscillating field with charge, frequency, and momentum. Oddly, the image of a field does not necessarily evoke harmonic oscillation. Though broad in extent, agricultural fields imply a flat, even static horizon. One might argue that nineteenth-century physicists’ use of the field as a conceptual metaphor held the added inference of the rippling of grain. Maxwell may have been better served to describe electromagnetism as a sea or ocean—with their harmoniously rippling waves. As such, the field in physics would seem to be a conceptual blend of field and sea.
The above narrative paraphrases the typical explanation offered up by physicists in popularizations.[1] Let us now examine two of the formulas canonical to special relativity:
The above equation represents a flat four-dimensional Minkowski space: it describes the differential of distance as an object moves through the three dimensions of space and the one of time. d stands for a differential calculus of infinitesimal distance. s measures that distance. x1,2,and 3 are the point positions along the axese of the spatial dimensions. c represents a constant, the speed of light. t is the point position along the time axis. What distinguishes this Minkowski space from a similar Euclidean space is the incorporation of time into the space itself—notably, in an inverse relationship to the spatial dimensions (this particular Minkowski metric is sometimes short-handed as + + + -). While Newtonian mechanics treats time as a space-like dimension, time nevertheless remains ontologically isolated. In Minkowski space, time forms an integral part of the total fabric of space. In this sense, one could consider a Minkowski space not a physical space, but an informational space: a geometric entity that describes the interrelationship between space and time. I would define informational spaces as imaginary spaces that encapsulate information, fundamental binary oppositions (such as positive and negative in the case of direction or electromagnetic charge) that in aggregate determine physical behaviour. Special relativity dubs its particular informational space, ‘spacetime’. Although x and t are the independent variables in this equation, I would argue that the most significant conceptual metaphor in special relativity is represented metonymically by c, the speed of light constant. The basic-level metaphor at play in special relativity, the metaphor that embodies the radical ‘epistemological rupture’ from Newtonian cosmology, is the object light itself. Consistent with his poetics of rigid bodies, Newton, in his work Opticks, conceived of light as tiny material ‘corpuscles’. Maxwell and his immediate successors, on the other hand, made a compelling case for treating light more as a fluid-like oscillating wave.[2] Einstein’s special relativity revitalized the debate concerning the essence of light. Metaphorically, in special relativity, light is a substance that possesses attributes: most significantly, a fixed speed. Whereas the Newtonian cosmology suggests that the essence of the universe is the substance matter, special relativity persuades us that its essence is the substance energy, and that secondarily, this substance energy manifests in the form of matter.[3] The following famous formula expresses just such an ontological revision:
E stands for energy. γ (gamma) encapsulates a Lorentz factor (explained below). m is rest or invariant mass. c is the speed of light. In effect, energy is mass times the speed of light squared times a Lorentz factor. Mathematically, γ functions as a container that gets expanded out into:
where v stands for velocity. The Lorentz factor γ formalises the asymptotic relationship between energy and momentum with the fixed speed of light, the inviolate law in special relativity that decrees that nothing in this universe, no matter or energy, can travel faster than the speed of light. In a sense, in special relativity, the all-pervasive, inviolate energetic substance light usurps the absolute throne held by the one God-like absolute observer of Newtonian cosmology. As a consequence, a multitude of observers spring up in the theory. These observers possess attributes: each bears a Lorentz frame, a unique point of view that depends on position and momentum. In special relativity, the blind and embracing rigid bodies of Newtonian cosmology sprout eyes. The theory allows for this multitude of bodies to observe each other, and as such, perceive each other uniquely. For instance, if one observer is stationary with respect to another observer moving past her at a given velocity, that first observer will see the second as smaller in extent than she would be at rest. The closer to the speed of light this velocity is, the more noticeable the effect, known as Lorentz contraction. Extent in space dilates proportionately to momentum. In this sense, the bodies of special relativity are not rigid, but extensive and plastic. This is also the case with time. For example, according to special relativity, if an astronaut travels at a velocity near the speed of light away from the relatively stationary earth, time will slow down for the astronaut relative to the flow of time on earth. In other words, if it takes the astronaut four days to reach a star four light-years away from the earth, when the astronaut returns from his eight day journey, she will be shocked to realize that eight years have passed on earth. In the Minkowski four-dimensional space, momentum diverted into movement through space diminishes proportionately movement through time. In this sense, in special relativity, the essence of the universe is also a particular kind of change, namely, a rotation through spacetime. Change is a turning motion, and since no moving body can travel faster than the speed of light, light itself forms an absolute boundary in the Minkowski space: what is often called in the theory a ‘light cone’. Motion within the light cone necessarily includes motion through the space-like time dimension. In the source-path-goal schema of special relativity, the trajectory of an observer-cum-(extensive, plastic) body must stay within the boundaries of the light cone. As such, unlike Newtonian cosmology—with its object-event structure, special relativity reflects an antipodal scenario where figure and ground reverse; it is a location-event structure where spacetime is the ground and the observer-cum-body is the subordinate figure. Causation in the universe of special relativity is therefore an object causation: causation functions as the transfer of a possessible attribute (spatial extent or duration, forms of matter) to or from an observer-cum-body (as energy substance). Spacetime, in this sense, is an absolute totality. The irony in popular readings of special relativity is that the theory is not all that relativistic in the epistemological sense: yes, observers’ perceptions of other bodies change relative to position and momentum, yet all observers and bodies within the spacetime totality are subject to the same universal laws of position and momentum rotation. Matter may seem derivative in this cosmology, but energy, with its panoptic and perhaps more democratic vision, still reigns absolutely. Furthermore, a universe predicated on Minkowski spacetime suggests that time does not flow from past to present to future, but that it too is an all-inclusive totality, and that this spacetime totality is what is ultimately real, not each individual observer’s particular vantage-point and trajectory through spacetime. Brian Greene describes this spacetime totality as a ‘loaf’ and postulates thus: [I]f you buy the notion that reality consists of the things in your freeze-frame mental image right now, and if you agree that your now is no more valid than the now of someone located far away in space who can move freely, then reality encompasses all of the events in spacetime. […] Just as we envision all of space as really being out there, as really existing, we should also envision all of time as really being out there, as really existing, too.[4] Trapped within the totality of the spacetime loaf, energy disguises itself as matter, fractures and scatters its anthropomorphic eyes, and distracts itself through the protean play of rotating form. Yet special relativity addresses only the energy of invariant mass and momentum. Einstein would institutionalize the conceptual metaphor of spacetime to an even greater degree with his proposal, ten years later, to fold the force of gravity into the theory.
[1] See, for example, Chapter 2 ‘Space, Time, and the Eye of the Beholder’ in Brian Greene, The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory (New York: Random House, 1999), pp. 23-52. [2] Light is sometimes conceptualized in everyday language as a fluid, for example: ‘Light flooded into the room.’ It would be interesting to trace the history of the metaphor of light as a fluid. [3] This institutional substantiation of energy over matter nevertheless failed to relegate debates concerning ‘materialism’ to the nineteenth century. [4] Brian Greene, The Fabric of the Cosmos: Space, Time, and the Texture of Reality (New York: Random House, 2004), pp. 138-9 [his italics]. |
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| Last Updated ( Sunday, 15 October 2006 ) |
Maxwell went on to make the prediction that there are electromagnetic waves that travel at the speed of light, and that also possess light’s properties of polarization. Heinrich Hertz confirmed his prediction experimentally in 1888; light has since been considered a particular manifestation of electromagnetism—with distinctly field-like behaviour. In 1905, Einstein, building on the work of Maxwell, 
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