Tensors & Lorentz Transform: Is There a Connection?

[geordief]

Are these two subjects closely related?

It seems a tensor can be invariant when viewed from any **co ordinate system and

The Lorentz Transformation seems to allow 2 moving co ordinate frames to agree on a space time intervals.

Is there some deep connection going on?

**=moving frames of reference?

 

[kent davidge]

Are these two subjects closely related?

Yes. Because Lorentz interval should be invariant in two inertial frames, and can be written as $\eta_{\alpha \beta} dx^\alpha dx^\beta$, then if in frame with coordinate system $x$ the interval is $\eta_{\alpha \beta} dx^\alpha dx^\beta$ and in frame with coord sys $x'$ it is $\eta_{\sigma \kappa} dx'^\sigma dx'^\kappa$ (notice that we have the same $\eta$ in the two frames because the lorentz interval should take the same form in the two inertial frames) then $\eta_{\alpha \beta} dx^\alpha dx^\beta = \eta_{\sigma \kappa} dx'^\sigma dx'^\kappa$, from which one defines the components of the metric tensor as $\eta_{\alpha \beta}$ and the basis as $dx^\alpha dx^\beta$ in the "frame $x$" and by noticing that $x' = \Lambda x + b$ one realizes that the components in $x'$, in terms of the components in $x$, are $\eta_{\sigma \kappa} = \Lambda^\alpha{}_\sigma \Lambda^\beta{}_\kappa \eta_{\alpha \beta}$, but the tensor remains the same.