# Special & General Relativity Questions and Answers

## Why is the gravitational field of the universe another name for space-time?

The development of any mathematical theory of natural phenomena such as gravity requires that the mathematical symbols defining the theory must be related to qualities of the phenomena such as the symbol T representing temperature, V representing velocity or M representing mass. In general relativity, a similar association had to be made by Einstein. We have seen how Einstein defined the gravitational field to be identical to the so-called metric tensor,

` g mu,nu `

used by Riemann to describe the geometry of a space. This means that where Newtonian gravity dealt with one quantity to measure the gravitational field, Einstein's theory in the guise of "g-mu-nu" required a total of 10 unique quantities to more completely define how the gravitational field behaved. The force of gravity defined as changes in the gravitational field from place to place in Newtonian mechanics, was replaced by changes in the geometry of space from place to place in spacetime measured by the degree of curvature symbolized by "C-mu-nu" at each point. Einstein's minimalist adoption of "g-mu-nu" as the embodiment of the gravitational field was significant and has far-reaching ramifications. Before Einstein, the metric tensor "g-mu-nu" was a purely geometric quantity that expresses how to determine the distances between points in space. Geometers from the time of Gauss knew nothing about forces, mass and momentum, they did however use the metric tensor to uncover new and bizarre spaces resembling nothing that humans have ever experienced.

Einstein's appropriation of the metric tensor so that it also represented the gravitational field led to an inevitable, logical conclusion: If you took away the gravitational field, this meant that "g-mu-nu" 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. In Relativity: The Special and General Theory page 155, 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"

This is such a profound assumption that I have intentionally enlarged the font to emphasize its significance. It will turn out to be the cornerstone to a radically new understanding of the nature of space and the vacuum. But in its radical departure from older ideas about gravity, Einstein's view point sounds 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 Aristotelian 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 world of absolute space and time, 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 comfortable, intuitive view was wrong. If the metric "g-mu-nu" 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. 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 physically interpret the points in the spacetime manifold.

Beginning with a landmark paper by Gunnar Nordstrom 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, "g-mu-nu". 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 "g-mu-nu" 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 "g-mu-nu" that is prior-geometry cannot be affected by matter or energy; that was the exclusive role to be played by the second component of "g-mu-nu" 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. Also, no phenomenon had ever been discovered which did not obey the principle of reciprocity; the property of acting upon matter and in turn being acted upon by matter.

If this argument for the existence of prior-geometry 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 for supporting light waves. Once again science moved along a parallel track, recapitulating the intuitive prejudices of an earlier time. Attempts were, in fact, made to create improved, workable prior-geometry theory and 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 H. 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 a book called 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" . As Sir James Jeans remarked in 1941 about the Ether, perhaps there is nothing to conceal in the first place.

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, 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.

Einstein had a strong opinion about the issue of prior-geometry. His choice was that the gravitational field represented EVERYTHING, with no pre-existing framework for spacetime. This assumption, as provocative as it seems, is the simplest one consistent with all known phenomena. In a quotation from Abraham Pias book Subtle is the Lord page 235, 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"

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 another such 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.

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 "( R, theta, phi)" and you used "cartesian" coordinates, ( x, y, z) 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 pair of triplet numbers. 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 point, 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.

Princeton University 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."

What this means is that so long as a point in the manifold is not occupied by some physical event such as the interaction point of a photon and an electron, it has no effect on a physical process. It is the collective property of physical events which defines the physical spacetime manifold and its geometry. The interstitial space between the events is simply not there so far as the physical world is concerned. A spider is free to crawl around its web, but it cannot crawl around if the web is not there.

Einstein's own interpretation of the reality of the points in the spacetime manifold is best expressed in his own book Relativity: The Special and the general theory written in 1952 a few years before his death. First of all, Einstein asserts that 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 klunky analogy like a 'rubber sheet' or soap bubble film. Like a trapeze artist suspended in mid-air, we deftly step over this yawning emptiness en route 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."

They are free creations of the human mind to use one of Einstein's own expressions.