Index Concordia

Relativity – Week 02

Posted in Relativity by Index Guy on September 17, 2007

We continue our discussion of the conformal group. One way to derive this symmetry is to start with a spacetime with D - 1 spatial dimensions and one temporal dimension, and add one space and one time dimension. Then write these two extra dimensions as linear combinations similar to the lightcone basis. Next we impose the condition

y^2 = 0 and \delta y^{A} = \zeta (y) y^{A}.

The solution of this constraints gives expressions for the two added coordinates in terms of two different sets of D-1 dimensional coordinate vector. The condition y^2 = 0 can be understood once we take the conformal group as the symmetry group of a massless particles. Recall that for a massless particle the spacetime interval vanished.

Next we explore the different transformations from the conformal group, including scaling, translations, some monstrosity called a “conformal boost”, and an additional time reversal called an inversion. A homework problem was to show that a translation sandwiched between two inversions is equivalent to a conformal group. In the end, one can see that the PoincarĂ© group is a subgroup of the conformal group.

  • The conformal group is bigger than Jesus or Satan.

Finally we discuss some properties concerning determinants. On Wednesday we discussed what professor Siegel called “differential geometry”. Basically most of the lecture was about bashing index-free notation. We discussed vectors, dual vectors, and then tensors; and also their transformation laws. We mentioned the Lie derivative and how differential forms considerations are useful when taking integrals, and the generalized Stoke’s Theorem. Next week – Spinors!


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