# 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:

Once upon a time, there was an island called Mledina where there lived a society of people called the Mledinans. Every spring, the men of the island would take a sacred voyage to a sacred island called Serona to learn the sacred truths of the sacred universe. They would stay for a couple weeks, and then return to Mledina, where they were forbidden to discuss the journey with any woman. The women, in a similar fashion, journeyed to Serona every fall to learn sacred truths, and they were forbidden to say anything of their travels to the men. For many years, the Mledinans continued to happily complete their travels and, while each was curious of the experiences of the other gender, nobody dared to break the silence. But all this changed one day in 1905 when a young boy named Albert managed to get ahold of two maps; the map used by the women and the map used by the men in their respective trips to Serona. In addition to the great shame of having their maps exposed, the people of Mledina experienced a great shock: the men and the women disagreed on the location of Serona. The men's map clearly showed that Serona was a distance of 164.5 furlongs east and 164.5 furlongs north, while the women had been travelling 210 furlongs east and 100 furlongs north. How could this be? were they really visiting different islands? They discussed their experiences on Serona (by now the convention of silence was completely out the window) and indeed, the descriptions of that sacred island were identical. Perhaps the island moved between spring and fall? The people of the island tried to dismiss the maps as fakes until, in 1908, a wise man named Hermann had an insight: the men had been navigating with magnetic north, while the women were using true north. It was clear that, by Pythagoreas' theory, everyone agreed that the absolute distance to Serona: 232.6 furlongs. What had happened was that the men's north was a mixture of the women's north and east, while the women's north was a mixture of the men's north and west. Thus, the absolute distance was unambiguous, but the easterly and northerly components of that distance depended on one's orientation.

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