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Lorentz Invariants

# Lorentz Invariants - Lorentz Invariants We found above that...

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Lorentz Invariants We found above that for an event ( x′, y′, z′, t′ ) for which the coordinates of the event ( x , y , z , t ) as measured in the other frame S satisfy The quantity is said to be a Lorentz invariant : it doesn’t vary on going from one frame to another. A simple two-dimensional analogy to this invariant is given by considering two sets of axes, Oxy and Ox′ y′ having the same origin O , but the axis Ox′ is at an angle to Ox , so one set of axes is the same as the other set but rotated. The point P with coordinates ( x , y ) has coordinates ( x′ , y′ ) measured on the Ox′ y′ axes. The square of the distance of the point P from the common origin O is x 2 + y 2 and is also x′ 2 + y′ 2 , so for the transformation from coordinates ( x , y ) to ( x′ , y′ ), x 2 + y 2 is an invariant. Similarly, if a point P 1 has coordinates ( x 1 , y 1 ) and ( x 1 ′, y 1 ′) and another point P 2 has coordinates ( x 2 , y 2 ) and ( x 2 ′, y 2 ′) then clearly the two points are the same distance apart as

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