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 ()() ()()() 22 2 1 tx y c τ 2 z ∆= ∆− ∆+ ∆+ (22.1) and its twin (22.2) () ()()() () 2 2 2 sx 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: ( )( ) 2222 2 x xx x i c ti c t x y z c t =+ =++ GG ii 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 xc t = G G and the dot product defined with the help of a “metric tensor”
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() 100 0 010 0 ,,, 001 0 000 1 T x y xx xgx xyzc t z ct ⎛⎞ ⎜⎟ == ⎝⎠ i GG which produces the same result. There are other variations as well. The metric can be defined with -1 for the space components and +1 for the time component (which gives –s 2 for the dot product which is also Lorentz invariant), and you can also find the four vector defined with the time component as its first element rather than its last. All of these are equivalent, and you just have to make sure you are consistent through your calculation. I’ve chosen just to stick with the way our book defines the four-vectors and their dot products, but you will find other books that do it differently. The Lorentz transformation is now produced by a 4x4 matrix 00 01 10 i i γ βγ λ β γγ = I (22.4) as we can check: x ix yy zz ict i ict γβγ = In fact, a four-vector can be defined as any four components, A µ , that transform in this way A A µν ν = (22.5) There is a three-vector given by the time derivative of the position, i.e. the velocity, but simple derivatives with respect to time won’t give us a four-vector from x G because dt is not a Lorentz invariant. However we can start from the proper time, which is a Lorentz
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Lecture 6 - Physics 325 Lecture 22 Four-Vectors We have...

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