## Theses, Dissertations, and Projects

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# The design and implementation of an applet to simulate curved space

2010

Honors Project

Physics

## Keywords

Space and time, Relativity (Physics), Geometry, Non-Euclidean, Curves, Algebraic, Curved space, General relativity, Java applet

## Abstract

One of the fundamental components of Einstein's theory of general relativity is the concept of curved space-time; while classical theories treat gravity as a force, general relativity proposes that objects follow geodesics, or straight lines, through a curved space-time. In addition to being one of the most fundamental aspects of general rel- ativity, curved space-time is also one of the hardest to understand. Many students attempt to understand curved space-time, and the simpler concept, curved space, through embedding diagrams, which represent non-Euclidean geometries by showing them as curved surfaces in higher-dimensional Euclidean spaces. Although embedding diagrams are extremely useful, because they allow students to use their physical in- tuition to understand the functioning of a certain geometry, they can be misleading; embedding diagrams can encourage students to believe that curved space must exist in some higher-dimensional Euclidean space. To combat this misconception, we have cre- ated an applet to simulate two-dimensional space of constant positive curvature, which is mathematically equivalent to the surface of a sphere. The applet includes features that illustrate the curvature of the space: the viewer can draw shapes and then move around within the space to observe how those shapes appear to distort. The applet is available on the internet and is accompanied by several tutorials that guide the user through activies that show how curved space is different from at space.

English