Confused about Lorentz Transformations

[BWV]

Main Question or Discussion Point

I thought Lorentz Transformations left Δt²-Δx² invariant
but, for example a frame moving at .5C for Δt =1 has Δx = .5 so
Δt²-Δx² = .75

If this is transformed by :

to a rest frame
Δt =0.65 has Δx = 0 and Δt²-Δx²≠ 0.75

not sure where I have gone wrong here, any help would be appreciated

 

[Samshorn]

I thought Lorentz Transformations left Δt²-Δx² invariant but, for example a frame moving at .5C for Δt =1 has Δx = .5 so
Δt²-Δx² = .75. If this is transformed by [Lorentz Transformation] to a rest frame [then] Δt =0.65 has Δx = 0 and Δt²-Δx²≠ 0.75
Not sure where I have gone wrong here, any help would be appreciated

You made a mistake with your arithmetic. If you plug your numbers into the Lorentz Transformation equation that you quoted, the result is Δt' = sqrt(0.75) and Δx' = 0, so the quantity Δt'²-Δx'² equals 0.75, just as before.