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Lecture 6 - Physics 325 Lecture 22 Four-Vectors We have...

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Physics 325 Lecture 22 Four-Vectors We have discussed treating time, or ct , as a fourth dimension. We have also seen (HW) that the Lorentz transformations can be thought of as a rotation in this four-dimensional space. All that remains is to define the vectors and operations between the vectors in this space. We recall that our three-dimensional rotations preserve the length of vectors as defined by the dot product. We therefore would like the rotation in four-dimensional space, give by the Lorentz transformations, to also preserve the length of our “four- vector”. In the last lecture we discovered the Lorentz invariant quantity ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 1 t x y c τ 2 z = ∆ + ∆ + ∆ (22.1) and its twin (22.2) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 s x y z c = + ∆ + ∆ 2 t These suggest that our four-vector has three spatial components and one time component and that the dot product involves a sign difference between the space and time components. We therefore define the position-time four-vector as x x ict = G G (22.3) where I have used the notation with an arrow below the symbol to denote a four-vector. This definition of the position-time four-vector, with an i before the time component, allows us to take dot products the way we are used to: ( )( ) 2 2 2 2 2 x x x x ict ict x y z c t = + = + + G G i i G G Which is the Lorentz invariant s 2 in (22.2). This four-vector has a constant length under Lorentz transformations (rotations). You will often see the four-vector and the dot product defined slightly differently. The four-vector can be defined without the imaginary component as ( ) , x x ct = G G and the dot product defined with the help of a “metric tensor”
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