Geometry and Physical Reality

©Fernando Caracena 2017

Mysteries of Physical Reality

Our senses tell us that the world around us, the physical world, consists of objects embedded in space and that events happen at measurable times.  A previous post, "On the Nature of Physical Reality—A Mental Construct", discusses some of the philosophy, physical, and spatial aspects of physical reality. There is much more to reality than this aspect of physical reality. Here discussed are the mathematical aspects of spaces, which are characterized by internal relationships that amount to geometries. It turns out that the idea of geometry not only applies to abstract figures drawn on flat pieces of paper, but to the structure of space and time itself and to physical reality even beyond that.

Measuring the Earth

The ancient Egyptians, in the process of carrying out the complex processes of civilization, such as distributing land and constructing buildings, found it convenient to plan their efforts on flat surfaces such as sheets of papyrus. In the process, they discovered a few simple rules governing relationships between aspects of various figures of interest that could be drawn on a flat surface. [See the post, "Geometry and Earth Measure".] Alexander the Great ' s conquest of Egypt brought ancient Greeks in contact with Egyptian thought. The logic of the Greeks and the knowledge of the Egyptians cross fertilised each other resulting in establishing Egypt as a great center of learning in the Greek world. The tangible result of this was the establishment of the famous, ancient library of Alexandria.

From Egyptian rules of thumb, Euclid of Alexandria (a Greek), drew out abstract knowledge of spatial relations, called geometry (now known as plain geometry), putting together all that was then known about geometry in a book called "The Elements", which  became the theoretical basis for the mathematics of figures drawn on a flat surface. For centuries, this work served as the text book for geometry. Spaces

Frans Hals - Portret van René Descartes.jpg

Portrait of René Descartes by Frans Hals painted circa 1649-1700. From Wikicommons, this picture is in the public doamain in the United States.









In more modern times a French mathematician, Rene Descartes (1596 –1650) was able to link Euclid's geometry to the rest of mathematics through a branch of mathematics he developed, called analytic geometry,  the key idea of which is to index each point in space by a unique set of coordinates that refer to a frame of reference. A set of two numbers is required to index each point in a two dimensional space; three numbers, in a three dimensional space; and n numbers,  in an n-dimensional space. Because it takes only two coordinates to specify a point on a plane, Euclid' s geometry was a geometry of two dimensional spaces, but not of all possible two dimensional spaces as is pointed out below. Mathematicians following him, later generalized Euclid's ideas to include three dimensional spaces in a field of study called solid geometry. Nowadays, mathematicians have realized that Euclid' s geometry for two dimensions can be extend to any number of dimensions (n-dimensional spaces). Although figures in the n-dimensional geometry can no longer be drawn on flat surfaces, mathematicians have been able to define the flatness in a way that can be applied to n-dimensional spaces.

The curvature of a space can be diagnosed by geometrical relationships within that space. Curvature effects are more complicated than the effects of bending that space, which could leave a flat surface still "flat". For example, rolling up a flat piece of paper containing all kinds of geometrical figures still leaves those figures invariant within that two dimensional surface.

The concept of  a "flat space" implies that there are also "curved spaces". This is a lesson that was learned by humanity by the hard knocks of experience. [See the post, "Geometry and Earth Measure".] Even so, flat spaces can be quite abstruse. For example, consider Einstein's formulation of the Special Theory of Relativity, which is applied to a flat space representation of space and time that has negative portions of its metric (Riemannian geometry). For reference, see a previous article posted in this blog, "Special Relativity II—standard notation."

Although the idea of of applying Riemannian geometry to physical reality was already way out there, it was a neat trick that allowed Einstein to think still much farther out of the box. Riemannian geometry was essentially "flat." By allowing that geometry to have intrinsic curvature, Einstein was able to automatically include the effects of gravity as geometrical effects of the warping of the space-time manifold.


Coordinates Systems and Geometries

Gaspard monge litho delpech.jpg

A lithograph depicting the French mathematician Gaspard Monge by the lithographer, François Séraphin Delpech (delpech.jpg). A public domain image from Wikicommons.

Rene Descartes's generalization of Euclid' s geometry through coordinate systems resulted in a branch of mathematics called analytic geometry.  Gaspard Monge ,included methods of calculus to further develop another powerful area of mathematics called differential geometry. Methods of differential geometry allowed people to determine the nature of a manifold from relationships determined entirely within its surface.  The ideas of Monge were further elaborated by the famous, great mathematician Carl Friedrich Gauss.

In more modern times, Einstein generalized Descartes' s idea to the description of an event as a point in a four dimensional space that is specified by four coordinates: three for space and one for time.  The set of all possible events constitute a four dimensional manifold, or space, sometimes referred to as space-time, which conforms to a four dimensional geometry. The theory of Special Relativity deals with the geometrical effects of dynamics in a four dimensional space. Einstein's General Theory of Relativity departs from the flat manifold version of Relativity to allow curvature effects of that manifold. [See earlier post, Special Relativity II—standard notation.]

Carl Friedrich Gauss 1840 by Jensen.jpg

An 1840 portrait of Carl Friedrich Gauss rendered by Christian Albrecht Jensen. From Wikimedia, this work is in the public domain in the United States.

What we observe on a microscopic scale are events that are the basic coinage of all physical transactions. An event is characterized by where and when something happens.

Classical physics describes an event as a point in three dimensional space, which is specified by three coordinates plus another number that specifies the time of its happening. Therefore individual events are described classically in terms of four coordinates. A four dimensional space-time point describes all possible events--in total a four dimensional space-time continuum. Classically, a space containing all potential event points satisfies a geometry of four dimensional space-time, which was Einstein's starting point for his General Theory of Relativity. In this theory, gravity is treated as a field of distortions of space-time away from a flat manifold as a its curvature, somewhat like the scale factors of map projections. We ordinary humans cannot see this four dimensional manifold globally as a surface embedded in a higher dimensional space because our observations are made within it; however, using methods of  differential geometry allows us to analyse such spaces and to extract curvature effects from that analysis.

Currently, some physicists are trying to develop a comprehensive theory of physics that uses strings instead of points for specifying events of elementary object of physics. Perhaps this is a mathematical way of allowing events in out four dimensional manifold to connect with higher dimensions not available to direct observation. There is a lot of interesting philosophy here.

String Theories incorporate more than four dimensions in describing the dynamics of fundamental 'particles', which may be 26 dimensions, or 11 dimensions in M-theory. Reading between the lines, we see that this theory suggests that we do not experience the full dimensionality of physical reality because all but four dimensions of the strings are rolled up into very small spaces, perhaps wrapped up entirely in quantum uncertainty. String theory is controversial. Some physicists damn the whole idea by declaring it "not even wrong."


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