Mathematics Colloquia and Seminars

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Description

Einstein's famous equation, E = mc^2, is one of the most well-known and important discoveries of 20th century physics. Mathematically, the result is very simply derived from the Lorentz transformation on the momentum 4-vector. Unfortunately, this approach yields little physical insight into the properties of spacetime. However, with a little physical intuition, some (Lie) group theory, and a few gedanken (thought) experiments, one can unravel the meaning and most of the work behind Einstein's relation between mass and energy.

I will review some relevant Lie groups, and explain their importance in physics. I will also introduce the notions of time dilation and length contraction through simple gedanken experiments, which will lead us to the Lorentz transformation. I will show that the Lorentz transformations are one possible parameterization of elements of the Lie group SO(1,1) and will introduce the notions of (Lorentz) vectors and scalars. Time permitting, I will show that the momentum 4-vector is Lorentz covariant (hence, a Lorentz vector). Finally, by construction of the momentum 4-vector, I will derive E = mc^2.