Lecture13 - Lecture 13 Notes, 95.658, Spring 2012,...

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Lecture 13 Notes, 95.658, Spring 2012, Electromagnetic Theory II Dr. Christopher S. Baird, UMass Lowell 1. Covariant Geometry - We would like to develop a mathematical framework where special relativity can be applied more naturally. - The Lorentz transformations were derived from Einstein's principle of relativity: c 2 t ' 2 − x ' 2 y ' 2 z ' 2 = c 2 t 2 − x 2 y 2 z 2 - This means that all the terms on the left always equal the same scalar no matter what frame of reference we are in. This value is invariant under Lorentz transformations. - In regular three-dimensional Galilean relativity, the dot product of two position vectors is invariant under transformations. Define the 4-vector (covariant) geometry as the set of rules that lead to the dot product of any two 4-vectors being invariant under Lorentz transformations. - If we designate the column 4-vector A μ as a “covariant” vector (where covariant implies that its dot product does not change under Lorentz transformations), then to form a dot product we must multiply by a row vector. Let us write the row 4-vector as A μ and call it a “contravariant” vector to imply that it is dotted against the covariant vector. - The label μ on the vector is an index that runs from 0 to 3, specifying each component of the 4-vector, so that for instance, x 2 = y . Note the convention that when we are indexing a four-vector, we use Greek letters such as μ , ν, etc. , but when we are indexing a regular three component vector, we use Latin letters such as i , j , k . - Our dot product then looks like this: = 0 3 A A - Note that if we recognize the product of a contravariant vector with a covariant vector always as a dot product, the summation symbol is unnecessary. We drop the summation symbol with the understanding that repeated indices always means summation . A A - Assume we know the contravariant vector. What does its corresponding covariant vector look like? - Let us look at the space-time coordinate vector x μ = ( ct , x , y , z ). We know what its dot product should look like: x x = c 2 t 2 x 2 y 2 z 2 - The only way this is possible is if we define the covariant vector as x μ = ( ct , - x , - y , - z ). - In general then we define the dot product of any two 4-vectors as the product of one in covariant form and the other in contravariant form, where the two forms are related to each other by a sign change of the last three components.
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- Can we mathematically express the relationship between a contravariant vector and its corresponding covariant vector? The metric tensor (similar to a matrix) g μν accomplishes this. -Notice that it has two Greek indices, so it is a two-dimensional tensor with 16 components total.
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Lecture13 - Lecture 13 Notes, 95.658, Spring 2012,...

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