## 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** (x_{0}). Space and time intervals in co-moving frames are linked by
the same Lorentz transformation, applied in both spaces together:-

cδx_{0a} |

δx_{a} |

= β |

1 | -^{v}/c |

-^{v}/c |
1 |

cδx_{0b} |

δx_{b} |

with β=(1-^{v2}/_{c2})^{-1/2} and

δx_{ta} |

cδt_{a} |

= β |

1 | -^{v}/c |

-^{v}/c |
1 |

δx_{tb} |

cδt_{b} |

δx_{0} is a measure of the time difference between simultaneous events judged
by co-moving observers. δx_{t} is a (vector) measure of empty space.

The frame of twin B is described by x_{0b}=0 and x_{tb}=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:-

Δx_{a} → ∑(δx_{a} + δx_{ta} )
= β(Δx_{b} - v Δt_{b} )

cΔt_{a} → ∑c(δt_{a} + δx_{0a})
= β(cΔt_{b} - ^{v}/_{c} Δx_{b})

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)

Robert Herrod

Örkelljunga, Sweden, March 2019

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