Navigating Spacetime

In this chapter, Prof. Banchoff briefly mentions spacetime, a view of our universe in which time is a fourth dimension. This view, which comes out of Hermann Minkowski's interpretation of Albert Einstein's theory of special relativity, unites space and time, commonly thought to be separate entities, into a space-time continuum, which behaves similarly to a four-dimensional euclidean system. The key difference is that, in order to find the spacetime distance between two events, one uses a specialized version of the generalized pythagorean formula, in which the square of value of the time dimension is subtracted, rather than added, to the squares of the space dimensions. I have yet to find a satisfactory answer to the question of why this would be true, but I plan to find out.

An interesting consequence of the spacetime view of the universe is that, as special relativity tells us, the dimensions are not exactly distinct. Indeed, "the length of an object and flow of time are different when viewed from different reference frames. My time differs from yours if I move relative to you, and my space differse from yous. My time is a mixture of your time and your space; my space is a mixture of your space and your time" (from Black Holes and Time Warps by Kip S. Thorne). What can it possibly mean, that my space is a mixture of your time and your space? E. Taylor and J. Wheeler (1992) provide the following useful, if goofy, analogy:

Within our universe, the absolute spacetime difference is unambiguous, the components depend on one's orientation, or, more accurately, one's motion. Minkowski's conception of spacetime (which Einstein in fact ridiculed for many years before acknowledging its necessity) became a fundamental precept of general relativity, Einstein's 1916 theory which continues to be the best model for the large-scale behavior of the universe.

-david stanke