Introduction
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.
Measuring Spacetime Curvature
In general relativity, a similar association had to be made by Einstein. Einstein defined the gravitational field to be identical to the so-called metric tensor, gij, used by Riemann to describe the geometry of a space. 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 spacetime from place to place measured by the degree of spacetime curvature at each point.
Einstein's minimalist adoption of gmn as the embodiment of the gravitational field was significant and has far-reaching ramifications. Before Einstein, the metric tensor gmn 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 gmnwould 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.
This is such a profound conclusion that I have intentionally highlighted it to emphasize its significance. Einstein's viewpoint 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. But if spacetime exists, then surely there must be a larger arena that, by some means, supports it. The intuitive idea that something must serve as the foundation for spacetime 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 our 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 that labels each Event 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 four coordinate labels much the way Newton must have thought of absolute space and time. Coordinates describe 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 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. When we try to look behind, or underneith, spacetime to catch a glimpse of something more fundamental, all we see is nothingness.
Related EoC Articles
- Spacetime
- Spacetime: Foundations
- Spacetime: Geometry
- Spacetime: Light Cones
- Spacetime: Special Relativity
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: Gravity." In: Encyclopedia of the Cosmos. Eds. Bernard Haisch and Joakim F. Lindblom (Redwood City, CA: Digital Universe Foundation). [First published February 13, 2008].
<http://www.cosmosportal.org/articles/view/138097/>
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