The Tri-space Laboratory
- still committed to reversing global warming
Twin space travellers, with relative velocity v
The twin paradox according to special relativity (click to view)
The twin paradox in Tri-space is explained in just the same way, but the Lorentz transformation is nowhere and never used to mix length and time intervals directly together.
Time and empty space form a signed metric space, as do length and an empty time dimension (x0). Space and time intervals in co-moving frames are linked by the same Lorentz transformation, applied in both spaces together:-
with β=(1-v2/c2)-1/2 and
δx0 is a measure of the time difference between simultaneous events judged by co-moving observers. δxt is a (vector) measure of empty space.
The frame of twin B is described by x0b=0 and xtb=0 The frame of twin A is related by interchanging the subscripts a<->b and changing the sign of v.
The transition of A to a frame co-moving with B is described by the net displacements:-
Δxa → ∑(δxa + δxta )
= β(Δxb - v Δtb )
cΔta → ∑c(δta + δx0a) = β(cΔtb - v/c Δxb)
This is equivalent to the Lorentz transformation that mixes the space and time coordinates together, but it does not happen until an observer changes frames.
Conclusions (click to view)
Örkelljunga, Sweden, March 2019