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How do we prove that the spacetime interval is invariant? Also why is it so important?
Have a look please atHow do we prove that the spacetime interval is invariant? Also why is it so important?
Use the Lorentz transforms for x',t' in the expression of the spacetime interval [tex]ds'^2=c^2t'^2-x'^2[/tex]How do we prove that the spacetime interval is invariant? Also why is it so important?
The space-time interval serves the same role in (the geometry of) Special Relativity as the distance formula serves in Euclidean geometry.Also why is it so important?
Do you teach or only use special relativity. If you teach I would send you a story..The Lorentz transformation is defined so as to keep the spacetime interval invariant. More precisely, any [itex]\Lambda[/itex] such that
[tex]\Lambda \eta \Lambda = \eta[/tex]
where [itex]\eta = \mbox{diag}(1,-1,-1,-1)[/itex] is a transformation which keeps the spacetime interval invariant.
EDIT: in component form, using Einstein summation
[tex]\eta_{a'b'} = \eta_{ab}\Lambda^a\mbox{}_{a'}\Lambda^b\mbox{}_{b'}[/tex]
Great find. Ed's site must have been recently updated.The first chapter of Spacetime Physics deals with invariant interval; the exposition is enlightening. You can download the first chapter of the first edition from Edwin Taylor's website: http://www.eftaylor.com/download.html#special_relativity
I learn & use only, I'm afraid.Do you teach or only use special relativity. If you teach I would send you a story..
Although the transforms upon which the invariance of the interval is based were developed by lorentz and Einstein - it was Minkowski that first pointed out the physics - the fact that in our universe, space and time can be unified and the unification is easy to visualize - any two events in spacetime are separated by an interval which has the same spacetime magnitude in every possible uniformly moving frame which can be imagined. Almost all problems in SR can be quickly solved by using this fundamental concept.How do we prove that the spacetime interval is invariant? Also why is it so important?