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LUphysics
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Homework Statement
So, I am working on a question that requires me to prove that s^2 = s'^2 from the Lorentz equations. It seemed like it'd be trivial... and then I ended up here a few hours later, not willing to waste any more time.
Homework Equations
By definition: s^2 = x^2 - (ct)^2 & s'^2 = x'^2 - (ct')^2
And the Lorentz equations are x' = y(x - vt) & t' = y(t - vx/(c^2) ) --> y = gamma for the lazy man
The Attempt at a Solution
So I followed this line of algebra:
Start with: s'^2 = x'^2 - (ct')^2
Sub in Lorentz Equations: = [y(x - vt)]^2 - c^2*[y(t - vx/(c^2) )]^2
Factor out y and Expand Brackets: = y^2 { [x^2 - 2xvt + (vt)^2] - c^2*[t^2 -2xvt/c^2 + ((vx)^2)/(c^4))] }
Multiply in -c^2 over on the right: = y^2 [x^2 - 2xvt + (vt)^2 -(ct)^2 + 2xvt - (v^2*x^2)/(c^2)]
Eliminate the 2xvt terms: = y^2 [x^2 + (vt)^2 -(ct)^2 - (v^2*x^2)/(c^2)]
Now, t = x/c, so by applying this to the rightmost term: = y^2 [x^2 + (vt)^2 -(ct)^2 - (vt)^2]
Eliminate the (vt)^2 terms: = y^2 [x^2 - (ct)^2]
And recall that s^2 = x^2 - (ct)^2: Therefore: ==> s'^2 = y^2[s^2]
There's my dilemma. This would mean spacetime is NOT invariant, since it depends on gamma. I'm not quite prepared to call the people who invented this liars, or to say I'm better at math than them... but I thought my algebra was pretty good and it led me to this. So... what's the problem?
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