30th Annual Student Research Conference
The Geometry of Minkowski Spacetime
Matthew P. Matheney
Dr. Kevin Easley, Faculty Mentor
This study explores the mathematical arena in which Einstein’s 1905 theory of special relativity is developed. It is first shown that Einstein’s theory supplants Newtonian "Absolute Space" and "Absolute Time" with Spacetime - a set that quite naturally carries the structure of an affine space. With the choice of a single spacetime event O as origin, spacetime may be viewed as a real four-dimensional vector space, V. Einstein’s fundamental result on the invariance of the spacetime interval, -D(ct)^2 + D(x)^2 + D(y)^2 + D(z)^2 , may then be interpreted mathematically as suggesting an indefinite scalar product, g, on V of signature (-,+,+,+). This indefinite scalar product leads to the appearance of hyperbolic geometry and some geometric structures, such as null cones, for which there are no positive-definite analogues. Connections are made through linear algebra to Lorentz Transformations, and some classic "paradoxes" dealing with special relativity are explored.
Keywords: Spacetime, Special Relativity, Hyperbolic Geometry, Lorentz Transformations
Topic(s):Mathematics
Physics
Presentation Type: Oral Paper
Session: 405-4
Location: MG 2050
Time: 3:15