Curved Space-Time Continuum
The three dimensions of ordinary space span a continuum of points where the exact location of each point can be described by just three coordinates - provided one has chosen a frame of reference. By the advent of Special Relativity physicists found it convenient to treat time as a fourth dimension. The four dimensions of the resulting space span a continuum of points where one now needs four coordinates to describe the exact location of a point. As long as it is not influenced by forces, any body, moving in these spaces, will follow a straight line - regardless of how many coordinates are used to describe its location. Such spaces are called flat and the associated frames of reference are inertial systems.
A flat space remains flat when forces are introduced, but a body, moving in a flat space, does not in general follow a straight line if it is influenced by gravitational forces. However, as a consequence of the generalized Principle of Equivalence, a curved trajectory in a flat space is equivalent to a straight line in a curved space. Since the curvature of the trajectory in the flat space is determined by the magnitude of the gravitational force, it is only a small step to identify this magnitude with the degree of curvature of the curved space. This, essentially, is what General Relativity is all about.
In theoretical considerations cosmologists replace a star, galaxy, etc. by its gravitational radius (a measure for the range of its gravity) which is proportional to its mass. In this sense matter is absent in General Relativity, its role taken over by warps in the space-time continuum. Thus the images below can represent e.g. a medium heavy star to the left and a heavier one to the right:
Be careful not to take the illustrations too literally. Although the mesh represents 4-dimensional space-time, what you see are just dents in 2-dimensional surfaces. According to General Relativity, the act of a source of gravity is to "curl up" the space-time continuum in its neighborhood - not to make dents or holes in it. There are no holes in the fabric of this universe!
Black "holes" have got a misleading name and are often, misleadingly, depicted as bottom-less basket-ball baskets - black knots would be a more appropriate name. Actually, you can imitate a black "hole" by doing as follows: Put one of your bedsheets flat on the floor. The sheet represents a flat space-time continuum - a region of 4-dimensional space void of gravitation. Now, put your hand in the center of the sheet and start to curl it up into your fist. What do you get - a hole or a knot? If you continue until and beyond the point where you, as well as your sheet, are infinitely small you are a black "hole".
This page has only been dealing with local curvatures of space-time in the vicinity of sources of gravity. The next page will deal with the global curvature of the very universe itself.
