The background for this post is *here*.

For space with time (3+1):

The *gamma transformation* (matrix Γ) expresses the time dilation of clocks and length contraction of rods with a relative speed:

Use vector (*t* *x*)^{T}. The gamma transformation is conjugate to the Lorentz boost (matrix Λ) by the Galilean transformations (G, G^{T}), i.e., G^{T}ΓG = Λ:

or

This expands to

The matrix second from the right represents the Tangherlini transformation (or inertial synchronized Tangherlini transformation).

For time with space (1+3):

Use vector (*x* *t*)^{T}. The *dual gamma transformation* (matrix Γ) expresses the time dilation of clocks and length contraction of rods with a relative pace:

The dual gamma transformation is conjugate to the dual Lorentz boost (matrix Λ) by the dual Galilean transformations (G^{T}, G), i.e., G^{T}ΓG = Λ:

or

Note that superluminal motion is excluded in both space-time and time-space.