Curvature
Space-Time
We readily accept that our universe has 4 dimensions under the framework of space-time. To represent each dimension, however, the term 'space-time' needs to be expanded. X-Y-Z-t is a more accurate depiction of the 3 spatial and one time dimension. In his general theory of relativity, Einstein predicted that, in the presence of matter, space is curved. The 3 dimension space framework, however, simply does not accommodate curvature.
In order for the X-Y-Z framework of space-time to curve, one more 'spatial' dimension, say 'W', is required. Time gets bumped to a 5th dimension in a W-X-Y-Z framework. If W is the dimension under which curvature occurs, the question becomes is the curvature such that space comes back on itself to form a universe whose structure is finite? As a thought experiment consider an object at a distance of 3GM/C^2 from a black hole, i.e. at the photon sphere. At this radius light, although unstable, is able to orbit a black hole. Falling from this, the traditional cartesian coordinate reference frame, oriented with the z axis downward or pointing at the black hole, results in both the x and y axis curving back on themselves. At this location, an observer looking tangentially to the black hole (with his head pointing away from the black hole) would see the back of his head. In fact if he were to remain in his upright position and rotate 360°, in every direction he looked he would appear to be looking straight but would see the back of his head no matter which way he turned.
We readily accept that our universe has 4 dimensions under the framework of space-time. To represent each dimension, however, the term 'space-time' needs to be expanded. X-Y-Z-t is a more accurate depiction of the 3 spatial and one time dimension. In his general theory of relativity, Einstein predicted that, in the presence of matter, space is curved. The 3 dimension space framework, however, simply does not accommodate curvature.
In order for the X-Y-Z framework of space-time to curve, one more 'spatial' dimension, say 'W', is required. Time gets bumped to a 5th dimension in a W-X-Y-Z framework. If W is the dimension under which curvature occurs, the question becomes is the curvature such that space comes back on itself to form a universe whose structure is finite? As a thought experiment consider an object at a distance of 3GM/C^2 from a black hole, i.e. at the photon sphere. At this radius light, although unstable, is able to orbit a black hole. Falling from this, the traditional cartesian coordinate reference frame, oriented with the z axis downward or pointing at the black hole, results in both the x and y axis curving back on themselves. At this location, an observer looking tangentially to the black hole (with his head pointing away from the black hole) would see the back of his head. In fact if he were to remain in his upright position and rotate 360°, in every direction he looked he would appear to be looking straight but would see the back of his head no matter which way he turned.
