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This paper provides a pedagogical introduction to the physics of extra dimensions by examining the behavior of scalar fields in three landmark models: the ADD, Randall-Sundrum and DGP spacetimes. Results of this analysis provide qualitative insights into the corresponding behavior of gravitational fields and elementary particles in each of these models. In these "brane world" models the familiar four dimensional spacetime of everyday experience is called the brane and is a slice through a higher dimensional spacetime called the bulk. The particles and fields of the standard model are assumed to be confined to the brane while gravitational fields are assumed to propagate in the bulk. For all three spacetimes we calculate the spectrum of propagating scalar wave modes and the scalar field produced by a static point source located on the brane. For the ADD and Randall-Sundrum models, at large distances the field looks like that of a point source in four spacetime dimensions, but at short distances it crosses over to a form appropriate to the higher dimensional spacetime. For the DGP model the field has the higher dimensional form at long distances rather than short. The behavior of these scalar fields, derived using only undergraduate level mathematics, closely mirror the results that one would obtain by performing the far more difficult task of analyzing the behavior of gravitational fields in these spacetimes.

Understanding how physicists solve problems can guide the development of methods that help students learn and improve at solving complex problems. Leveraging the framework of cognitive task analysis, we conducted semi-structured interviews with theoretical physicists (N=11) to gain insight into the cognitive processes and skills that they use in their professional research. Among numerous activities that theorists described, here we elucidate two activities that theorists commonly characterized as being integral to their work: making assumptions and using analogies. Theorists described making assumptions throughout their research process, especially while setting their project's direction and goals, establishing their model's interaction with mathematics, and revising their model while troubleshooting. They described how assumptions about their model informed their mathematical decision making, as well as instances where mathematical steps fed back into their model's applicability. We found that theorists used analogies to generate new project ideas as well as overcome conceptual challenges. Theorists deliberately sought out or constructed analogous, indicating this is a skill students can practice. When mapping knowledge from one system to another, theorists sought to use systems that shared a high degree of mathematical similarity; however, these systems did not always share similar surface features. We conclude by discussing connections between the ways theorists use assumption and analogy and offering potential applications to instruction.

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