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The Lorentz Transformation

The Lorentz Transformation. Section 4. An event has coordinates. x ,y,z,t in the K system x ’,y’,z’,t ’ in the K’ system What is the formula that transforms from one set to the other?. Transformation must leave interval s unchanged. s = interval between world points (events) in 4 space.

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The Lorentz Transformation

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  1. The Lorentz Transformation Section 4

  2. An event has coordinates • x,y,z,t in the K system • x’,y’,z’,t’ in the K’ system What is the formula that transforms from one set to the other?

  3. Transformation must leave interval s unchanged. • s = interval between world points (events) in 4 space. • Such transformation is a rotation in 4-D x,y,z,ct coordinate system. • Every rotation can be resolved into six rotations in planes xy, yz, zx, tx, ty, tz.

  4. Consider tx plane • y,z, coordinates don’t change. • Transform must leave (ct)2 – x2 unchanged • That is the square of interval from origin in tx plane to point (ct,x) ct x

  5. Lorentz transformation • To obtain the inverse formula, V-> -V and swap primes.

  6. In limit c -> infinity, we recover the Galileo transform with absolute time.

  7. For V>c, coordinates and time are imaginary. • Denominators go to zero if V = c.

  8. Small velocities, V<<c

  9. Lorentz Contraction If rod is moving, it will appear shorter to an observer at rest.

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