Introduction
Einstein's appropriation of the metric tensor to also represent the gravitational field led to an inevitable, logical conclusion: If you took away the gravitational field, this meant that gmn would be everywhere and for all time equal to zero, but so too would the metric for spacetime. Spacetime would lose its metric, the distance between points in the manifold would vanish, and the manifold itself would disappear into 'nothingness'. Einstein expressed this quality of spacetime as follows,
"Spacetime does not claim existence on its own but only as a structural quality of the [gravitational] field." — Albert Einstein.
Einstein's view point was beginning to sound a lot like the old philosophical discussion of the Void which emphasized that without bodies, "place" and therefore vacuum could not exist. If we consider that all bodies produce gravitational fields, we see that Einstein's general relativity arrives at nearly the same Aristotlean conclusion. The intuitive idea that something must serve as the foundation for space and spacetime for that matter, is powerfully seductive, and one to which virtually all physicists when caught off-guard, swear allegiance. They do so for the simple reason that to do otherwise leaves their mental constructs of the world literally hanging in mid-air. When we write our equations that depend on time and space locations, we consider this coordinate gridwork to exist in some more fundamental way than the particles, fields and energy they are meant to locate in space and time. We think of these coordinates much the way Newton must have in his intuitively compelling world of Absolute Space and Time. The coordinates are thought of as describing some immutable, rigid lattice work that is entirely aloof from the less than perfect matter and energy that moves through the gridwork subject to Nature's physical laws. But Einstein firmly believed that this intuitive view is wrong. If the metric gmn is identical to the gravitational field, which is what experimental evidence has since shown, then the coordinates of the physical spacetime manifold we erect to define place and time must also in some sense be constructs of the gravitational field.
Is spacetime painted on top of something more fundimental, like the frosting on a cake?
Einstein emphatically says 'No', Pre-geometry theory says 'Yes'.
Let's look at these issues one at a time and see how modern-day mathematicians and physicists are trying to resolve them. First, let's examine Einstein's assertion that spacetime is a fundamental field in nature, and then let's have a closer look at the issue of how to interpret the points in the spacetime manifold as physical objects.
Prior-Geometry
Beginning with a landmark paper by Gunnar Nordström of Helsingfors in 1913, there have been many attempts to create what are called "bi-metric" or "prior-geometry" theories for gravity and spacetime. The object is to re-assert the existence of an underlying metric to the world which, like a cake, supports the frosting which we see as the gravitational field, gmn. We might then have the option of "turning off" a gravitational field without Reality flashing out of existence at the same time. But gravity does not behave like light which can be turned on and off at will with a switch. Every erg of energy and scrap of matter produces a gravitational field. So, to turn off a gravitational field you must nullify all forms of matter and energy in the universe. This is hardly a sensible experiment to perform and would certainly not preserve the shape of Reality as we have come to know it.
These approaches always run into other problems as well. Prior-geometry theory sees gmn as being actually a compound object in disguise; one part being the gravitational field, the other part representing a pre-existing and immutable arena of spacetime. To make such a decomposition work, the part of gmn that is prior-geometry cannot be affected by matter or energy; that was the exclusive role to be played by the first component of gmn representing the gravitational field. Prior-geometry would have to play the role of the absolute bedrock of spacetime that both special relativity and Newtonian physics are built-up from.
Can such a decomposition really work? No observation by the time Einstein proposed general relativity, or since, has ever uncovered any physical evidence for some "universal geometric object" or plenum which stands aloof from physics in the manner that prior-geometry would have to. Prior-geometry theory would also require that some preferred universal frame of rest exist against which, like the ether or Newton's Absolute Space and Time, we could gauge our motion. Like the trees in a forest, the coordinate gridwork in prior-geometry would serve as an absolute reference frame against which we could tell where we are as we move through the forest of coordinate labels. Also, no phenomenon had ever been discovered which did not obey the principle of reciprocity; the property that if something can act upon matter, it can also be acted upon by matter. If this sounds like the old argument Maxwell used for believing in the Ether, you are right. It is, after all, rather hard not to consider something like a prior-geometry at work in nature for much the same reason that the ether was such a seductive idea in electrodynamics. Once again science moved along a parallel track, recapitulating the intuitive prejudices of an earlier time.
Attempts were, in fact, made to create a workable prior-geometry theory. Most of them were categorized in 1972 by Caltech physicists Wei-Tou Ni, Clifford Will and Kenneth Nordvedt at Montana State University. There have even been attempts at finding alternate mathematical descriptions to spacetime such as the work by Hans Reichenbach in 1956 described in "The Direction of Time." Reichenbach proposed that gravity is actually not a "universal force" according to his strict definition of such things. Philosopher Roberto Torretti at the University of Puerto Rico, however, commented on Reichenbach's analysis in "Relativity and Geometry" by stating that Reichenbach's universal forces cannot be detected by any means because they modify the shape of the instrument used to measure them in the exact way needed to conceal their presence.
They "...belong to the realm of science fiction and cannot be seriously countenance in real science" [p. 238]. — Philosopher Roberto Torretti.
The fact of the matter is that the experimental tests of general relativity are even now so restrictive that no other interpretation than Einstein's original one survives. Still, bi-metric theories continue to be of interest to some theoreticians because of their tantalizing capacity to offer slightly different solutions to older problems in general relativity. If only it were possible to preserve these beneficial features of prior-geometry theory without violating most experimental evidence for how gravitational fields operate. For example, as recently as 1989, in an article to the Astrophysical Journal, astronomer Nathan Rosen and his colleague Amos Harpaz at the Israel Institute of Technology resurrected bi-metric general relativity and showed how it could modify what happens to a star collapsing to become a black hole. Instead of passing through its so-called event horizon and continuing to collapse to a singularity, it stops collapsing shortly after it arrives at its horizon size. It never evolves further to become a singularity as predicted by Einstein's theory of gravity.
So far as prior-geometry is concerned, Einstein had a strong opinion about this issue. His choice was that the gravitational field represented EVERYTHING, with no pre-existing framework for spacetime.
Einstein once remarked that ...[prior-geometry] is built on the a priori, Euclidean four-dimensional space, the belief in which amounts to something like a superstition"
What will become of the debate over prior-geometry?
It has occasionally been said that the only way that wrong theories actually vanish is that their proponents die-off. They are never replaced by a new generation of students willing to pursue ideas that consistently go against experimental evidence and logical consistency. Bi-metric general relativity may be an unpopular theory whose days are numbered. Having dispatched prior-geometry as being unsupported by the results of any experiment, let's now look at the second part of our question of what spacetime represents physically.
The Physical Nature of Spacetime
Although Einstein defined the association between his gravitational field to be exactly equivalent to what mathematicians had previously called the metric to the manifold, there was one other issue that remained open. In Gauss's surface geometry, and Riemann's manifold geometry, the properties of space were not tied to a particular coordinate system. Physically, this means that if I used "spherical" coordinates and you used "cartesian" coordinates, we would come to identical conclusions about the motion of a planet around the sun. In fact, anyone would do so long as they assigned to every point in the manifold a unique coordinate address expressed as a set of pure numbers, one number for each space dimension. These so-called Gaussian coordinates had absolutely no physicality to them. But now comes Einstein who appropriates the metric to represent the gravitational field. How are we now to interpret the points that make up the mathematical manifold in terms of physical properties of the gravitational field?
Geometrically, a point has no size at all, and manifolds are built up from quite literally an uncountable infinitude of these points. Physically speaking, a point in spacetime is defined as an "event" which has a unique address in the manifold. All observers will agree that such an event occurred, and each will assign it a unique address in their own coordinate system, but in comparing these addresses with other observers, the space and time components to the addresses will be different. An event at its most elementary level could be the collision between two particles or the emission of a photon of light by a particle. An event could be any intersection between two worldlines on the manifold. By filling up the manifold in this way, every mathematical point eventually finds itself near some intersection point in the net of intersecting worldlines described by the energy (light) and matter worldlines that fill-up the spacetime. At some level, one may then disregard the reality of the abstract manifold and focus on the reality of the webwork of worldlines of the real particles which now defines the physical manifold of spacetime. Physicist Robert Dicke expressed it this way in a 1964 article "Experimental Relativity."
"To me the geometry of a physical space is primarily a subjective concept. What is objective is the material content of the space, the photons, electrons [etc]...When particles are present, it becomes possible to add objective elements to the mathematical elements. Thus, the collision between two particles can be used as a definition of a spacetime point...If particles were present in large numbers, for example, as virtual photons or gravitons, collisions with a test particle (e.g. electron) could be so numerous as to define an almost continuous trajectory. It is not [however] necessary that one have a physical definition of all points in our 4-dimensional spacetime...The empty background of space, of which ones knowledge is only subjective, imposes no dynamical conditions on matter." — Physicist Robert Dicke.
Einstein's own interpretation of the reality of the points in the spacetime manifold is best expressed in his own book "Relativity" written in 1952 a few years before his death. First of all, Einstein asserts that [p. 9] we "entirely shun the vague word 'space' of which we must honestly acknowledge we cannot form the slightest conception." It is a perfectly straightforward view point for who among us has not at some point tried to imagine what space is of itself without recourse to some clunky analogy like a "rubber sheet" or soap bubble film. Like a trapeze artist suspended in mid-air, we deftly step over this yawning emptiness enroute to the more concrete security of examining the bodies that fill space like raisins in a bread. We should also be mindful of another comment by Einstein recounted by Alysea Forsee in "Albert Einstein: Theoretical Physicist, "...time and space are modes by which we think and not conditions in which we live." This is a spirit of thinking that is similar to Immanual Kant's point of view that space and time do not belong to the external world, but are instead merely constructs that we humans create internally and project outwards into the external world.
"Space...is the subjective condition of our sensibility...Time is nothing but the form of the internal sense of our internal state" — Immanual Kant, Critique of Pure Reason.
Einstein goes to great pains to define what he considers the physical basis for a "point" in the continuum:
"...the only statements having regard to these points which can claim a physical existence are in reality the statements about their encounters." — Albert Einstein: Theoretical Physicist, p. 95.
The encounters he has in mind are the abstract coordinate values ( x,y,z and t) nearest some "event" in the history of a particle we wish to call out as important to us as observers. "The totality of physical events is thus thought of as being embedded in a four-dimensional, continuous manifold." But the central idea is that not all of these mathematical points are physically significant. Only the ones nearest the string of physical events that make up the individual worldlines of particles (electrons, quarks, photons) in the physical spacetime manifold. Even when we choose to glance at the night sky, we admire the feeble light from the twinkling stars, not the unimaginably empty spaces that separate them. That feeble light consists of innumerable spacetime events, one for each photon that reaches our retinae and causes a signal to be sent to our visual cortex. We receive no such signals from the vast, dark tracts of space between the stars. The spacetime that we perceive is therefore filled with vast empty voids containing no information for us to perceive. Physically, however, even the depths of interstellar spacetime are filled with a cross-hatched network of worldlines from stray particles of gas and dust, and even light rays enroute to us. Spacetime is, thereby, far more complex than that tenuous spiderweb of events and worldlines that we perceive.
Some philosophers prefer to distinguish between events in the abstract manifold and those in the physical manifold by labeling the former "possible events" in comparison to the actual events which make up our physical manifold. Should a real physical particle happen to be located at a specific "possible event," that position in the abstract manifold then corresponds to an actual event [p. 22]. The question now becomes, how densely populated is the physical manifold in comparison to the abstract one? Riemann himself, considered space to be a continuous manifold, however, he did not exclude the possibility that it could be discrete and full of gaps, although at least macroscopically it looked absolutely smooth with no missing points. In their book "General Relativity for Mathematicians," Rainer Sachs and Nung-Hsi Wu at the University of California, Berkeley, write [p.28]:
"To a geometer, that [spacetime] should be a [continuous] manifold is perhaps the most acceptable and the most obvious requirement. However, this is probably the most mystifying requirement on a deeper level. Why should all macroscopic physical phenomena be regarded as occurring on a smooth [manifold]? Off hand, one would think that Nature might use something logically simpler. Perhaps she does. The internal contradictions of present special relativistic quantum theory are severe. These contradictions may stem from trying to force a 'jumpy' quantum world onto a [smooth] manifold." — Rainer Sachs and Nung-Hsi Wu
Schools of Thought
The notion expressed implicitly by Sachs and Wu, that physical phenomena occur ON a smooth manifold, is one we will return to a bit later, because it shows that even theoretical physicists find it hard to think of physical things not occurring on top of some more primitive structure which has some of the intuitive qualities of an ether-like prior-geometry. Philosophers and mathematicians, by the way, were also no slouches to this discussion and a variety of camps have sprung up over the last 50 years. John Earman summarized many of the philosophical issues which spacetime presents in his book "World Enough and Space-Time." Over the years the discussions have revolved around three schools of thought: the Manifold Substantivists, the Manifold Constructivists and the Manifold Essentialists.
- Manifold Substantivists express the point of view that spacetime exists apart from matter, energy and the rubric of the physical world. They take strong exception to Einstein's comment that "spacetime does not claim an existence of its own, but only as a structural quality of the [gravitational] field". The manifold with its continuum of points is absolutely fundamental.
- Manifold Constructivists, on the other hand, believe that spacetime is constructed from physical events and agree whole heartedly with Einstein as to the subordinate role played by spacetime relative to matter and energy. Spacetime is a construct; a synthetic manifold that does not pre-exist, but is synthesized from matter.
- Manifold Essentialism takes the point of view that metrical properties are essential to spacetime points. Spacetime points wouldn't be what they are if they lacked the metrical properties they actually have. The difficulty here is that Manifold Essentialists seem to demand that spacetime points become defined only after more complicated objects which depend on these points such as metric or field have already been brought into existence. One is reminded of the riddle, "Which came first, the chicken or the egg?"
Spacetime and Particle Physics
A piece of chain link serves as an analog for spacetime fashioned from more primitive elements.
A piece of chain link serves as an analog for spacetime fashioned from more primitive elements.
In general relativity, the physical spacetime manifold is the gravitational field, or more correctly, the combined gravitational field of every scrap of matter and energy in the universe. Even the gravitational field itself can serve as its own source of gravity. If we search for an answer to what corresponds to a mathematical point in the real, physical manifold of the gravitational field, we have to come to grips with what the gravitational field is doing at a scale far smaller than an atom.
While particle physicists were obsessed with finding fundamental particles, journeying ever farther down the ladder of particle states, another group of physicists has been quietly pondering the fundamental construction of the entire spacetime continuum. There are two distinct schools to this activity:
- The first school contains the classically-trained general relativists who are at home with N-dimensional topology and Riemann's geometric techniques for studying the structure of manifolds.
- The second school contains cross-over particle physicists who see everything in terms of quantum field theory in which each fundamental force is mediated by the exchange of special particles called the field quanta.
Among physicists, the geometric school has scored many successes over the last century, especially in understanding how to describe the global geometry of the spacetime manifold. Beginning with Lorentz and Einstein, the local metric for the manifold was found to be the one used in special relativity. The relationship between this local approximation and the global metric was specified by Einstein's general relativity. Today, astronomers continue to narrow the observational possibilities for the global metric of spacetime.
The first topologists to consider the underlying geometric structure of the world was Riemann himself, and William Clifford around 1870. Clifford's paper "On the Space Theory of Matter" appeared in 1870 and outlines in non-mathematical language, his notion that matter was not fundamental to nature. All matter and radiation are constructed out of empty, curved space. This was a bolder proposal than the one made by Riemann some years earlier. Riemann's concern was primarily with the topological properties of space, and the possibility that on the smallest scales space may look nothing like a Euclidean, flat manifold. It may have holes and all manner of other topological structure.
Clifford's paper was the landmark paper for what eventually became the subject of Geometrodynamics. Physicist John Wheeler resurrected this idea in 1962 by adding that the relevant scale at which spacetime becomes topologically complex is the so-called Planck scale of 10-33 centimeters; fully 20 powers of ten smaller than the size of an atomic nucleus. A detailed discussion of the early years of geometrodynamics is presented by Adolph Gruenbaum in the 1973 book "Space, Time and Geometry."
By 1972, however, the geometrodynamic school of relativity began to show its age. Its chief spokesperson John Wheeler began to disavow his previous conviction that all of physics could be treated as the geometry of empty space. At a conference on gravity held at Boston University, he announced that spacetime could only be fully understood in terms of elementary particle theory. The reason that the pendulum had now swung towards the particle physicist's direction was that Wheeler ultimately had to confront the failure of general relativity to provide a natural setting for the quantum mechanical property of matter called spin. It didn't distinguish between the behavior of electrons (spin-1/2 particles) and photons (spin-1 particles). The topology of empty curved space did not seem to have the necessary mathematical structure that could be consistently interpreted in terms of this fundamental property of matter. On the other hand, by 1972 it was becoming increasingly well-known to gravity theorists that particle physics could be extended to include even the force of gravity under certain conditions.
We have already come into contact with the particle physicists conception of spacetime expressed by Dicke, but there is actually quite a bit more to this approach. Particle physicists have been extraordinarily successful in describing the fundamental forces in terms of a so-called Standard Model based on the idea of the quantum field. We saw in previous chapters how the electromagnetic interaction may ultimately be described by quantum electrodynamics which shows virtual photons interacting with various species of quarks and leptons.
Graviton Networks
A basic property of a fundamental particle is its spin, so the particle physicist attempting to create a field theory for gravity would reasonably ask what would the spin of the carrier of gravity, the graviton, have to be so that its behavior matched gravity as we know it? In a 1964 paper by Nobel Physicist Steven Weinberg a powerful theorem was announced that related the spin of a force-carrying particle to the conservation laws that it would have to obey to be consistent with special relativity. Spin-1 particles would resemble photons and lead to both attractive and repulsive forces. Gravity was only attractive, so gravitons could not be spin-1 particles. Spin-2 particles, however, would interact in a manner that depends on how much mass was present, and would behave in a consistent, universal fashion between all massive particles. This closely paralleled the way gravity is known to act. For particles with spins higher than 2, no "static" forces will occur which means that the solar system would not be stable over long periods of time. Gravitons have to be exactly massless so that light from distant stars seen near the limb of the sun would be deflected in exactly the way that is observed. Massive gravitons would cause a deflection only 75 % of what is seen. But there was more. Not only are gravitons exchanged in large numbers between all forms of matter and energy, but they are exchanged among themselves. Finally, in 1975, a remarkable paper by David Boulware at the University of Washington and Stanly Deser at Brandeis University who showed how Einstein's equation for gravity is precisely the equation that any good quantum field theorist would be forced to write down based on what we already know about gravity from simple experiences.
Another clue to how particle physicists think about spacetime comes from their efforts in understanding how to wrestle gravity into the same mathematical language used to describe the other three forces; a struggle which has yet to prove successful. Out of this struggle has come some remarkable new insights expressed by Nobel Physicist Abdus Salam writing in "Impact of Quantum Gravity Theory on Particle Physics":
"...The universe and its quantum topology are determined by WHERE gravitons are and what space-time interaction patterns they give rise to. [p. 507] ... The concept of space-like separated points...cannot be defined until AFTER the [theory of quantum gravity] is solved and the [light-cone] structure at each space-time point [is] constructed and specified." — Nobel Physicist Abdus Salam.
The words, where and after, have been emphasized by me to highlight the unusual aspects of quantum gravity that bare directly on the nature of physical spacetime. Abdus Salam's statement is remarkable and would have warmed the hearts of the Manifold Constructivists, but there was a catch to how this program was implemented that would have caught the attention of the Manifold Substantivists as well. It was common for particle physicists ca 1975 to begin developing toy models of a quantum theory of gravity by beginning with some pre-existing flat spacetime manifold. A diagram would then be drawn of, say, an electron emitting and absorbing a single virtual photon accompanied by the simultaneous emission and absorption of billions and trillions of virtual gravitons. The gravitational field would in this way be built-up out of the interactions of innumerable virtual gravitons just like the electromagnetic field is built-up out of virtual photons.
A Reprieve for Prior-Geometry?
But all of this activity would seem to take place on top of some kind of pre-existing, flat spacetime manifold that looks suspiciously like some type of prior-geometry. Salam, however, was quick to point out that "this manifold may have nothing to do with the spacetime of general relativity." This approach was pioneered by W. Thirring in 1968 who also proposed that the underlying prior-geometry manifold upon which the calculations were performed, was totally unobservable just like the "bare mass" of an electron in the theory of quantum electrodynamics.
What has become of the gravitational field and what we might call classical spacetime? In most quantum formulations, it has been replaced by a swarming population of virtual gravitons whose network structure in 4-dimensional spacetime seems to confer upon the gravitational field a measure of graininess. Because each graviton supposedly begets "billions and trillions" of other gravitons in an orgy of self-interaction, physical spacetime is populated by a continuum of collision events between one graviton and another. There are, essentially, no mathematical points left over in the abstract manifold that are not close by some "event" involving the collision or spawning of gravitons. This also means that the prior-geometry-like manifold that particle physicists have to propose as the foundation upon which to carry out their calculations becomes a temporary scaffolding that for all intents and purposes vanishes once physical spacetime has emerged as the end product of the quantum mechanical calculation. All of these graviton interaction events presumably occur at a scale comparable to the Planck limit so that the properties of the innumerable graviton worldlines can be suitably averaged to give the very real illusion that the physical spacetime manifold is smooth.
By 1975, this was about as far as we could have taken this story, however in the next chapter you will see how the battle for a "Theory of Everything" carried the discussion of the nature of spacetime and "empty space" in some unexpected directions.
Related EoC Articles
- Spacetime
- Spacetime: Geometry
- Spacetime: Gravity
- Spacetime: Light Cones
- Spacetime: Special Relativity
External Links
- "Inertial and Gravitational Mass in Relativistic Mechanics" - Gunnar Nordström of Helsingfors, 1913. Boston Studies in the Philosophy of Science, SpringLink.com.
- "Theoretical Frameworks for Testing Relativistic Gravity. I. Foundations" - Astrophysical Journal, vol. 163, p.595 (SAO/NASA Astrophysics Data System (ADS) - Harvard Univeristy.)
Preview Image
An artist's concept of twisted space-time around Earth. (Source: Spacetime Vortex - NASA.)
Citation
Odenwald, Sten, Ph.D. (Contributing Author); Bernard Haisch (Topic Editor). 2009. "Spacetime: Foundations." In: Encyclopedia of the Cosmos. Eds. Bernard Haisch and Joakim F. Lindblom (Redwood City, CA: Digital Universe Foundation). [First published January 13, 2008].
<http://www.cosmosportal.org/articles/view/138079/>
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