Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 4 / 12C / a

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- Solution a

Solution

A four-vector is an object whose components transform between inertial frames by a Lorentz transformation,

U^{' μ} = Λ^{μ}_{ν} U^{ν} .

Lorentz transformations are defined by

Λ^{T} η Λ = η,

where

η

is the Minkowski metric. Consequently

U^{'} \cdot U^{'} = (Λ U)^{T} η (Λ U) = U^{T} Λ^{T} η Λ U = U^{T} η U = U \cdot U .

Thus the Minkowski norm of a four-vector is the same in every inertial frame.

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