Phy489 Lecture 4
1 Special Relativity: Lorentz Transformations
not accelerating y y' S S' Consider two inertial reference frames S and S' having a relative velocity v (which v we will align along the x axis). If an event occurs in S at space-time coordinates (t,x,y,z) then in S’ we have:
x x' x′ = γ (x − υt) x = γ(x ′ + υ t ′) y′ = y y = y ′ z′ = z OR z = z ′ υ υ z z' t′ = γ (t − x) t = γ(t ′ + x ′) c2 c 2
1 γ ≡ In each case the Lorentz “boost” factor γ is given by 2 2 1−υ c 1 Usual to also define dimensionless velocity β ≡ υ c γ = 1− β 2 2 Problem 3.1 (derives the S’S expressions)
Starting with the expression for the S S' transformation x ′ = γ(x −υ t) y ′ = y These two z ′ = z } are trivial υ t ′ = γ(t − x) c 2 x ′ x ′ ⎛ t ′ υ ⎞ x ′ = γ x − γυ t → x = + υ t = +υ⎜ + 2 x⎟ γ γ ⎝ γ c ⎠ υ t ′ υ t ′ υ ⎛ x ′ ⎞ t ′ = γ t − γ 2 x → t = + 2 x = + 2 ⎜ + υ t⎟ c γ c γ c ⎝ γ ⎠
collecting x terms on the LHS……….. ⎛ υ 2 ⎞ x ′ υ t ′ x x ′ υ t ′ x⎜1 − 2 ⎟ = + = + → x = γ(x ′ + υ t ′) Similarly for t ⎝ c ⎠ γ γ γ 2 γ γ
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