Spacetime Invariance and Lorentz Equations

[LUphysics]

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?

 

[tiny-tim]

Welcome to PF!

Hi LUphysics! Welcome to PF!

(have a gamma: γ )

Sorry, but this is almost unreadable …

can you type it again, using the X² tag just above the Reply box?

 

[tomcue]

I understand your frustration with this problem. However, it is important to remember that scientific concepts, such as spacetime invariance and Lorentz equations, are not always easy to understand or prove. It takes time and effort to fully grasp and apply these concepts.

In this case, it is important to remember that the Lorentz equations are derived from the fundamental principles of special relativity, which include the constancy of the speed of light and the equivalence of all inertial reference frames. These principles have been extensively tested and have been found to accurately describe the behavior of objects in our universe.

Therefore, while it may seem like spacetime is not invariant in this specific case, it is important to remember that the Lorentz equations are only one way of expressing the relationship between space and time in special relativity. There are other forms of the equations that may not involve gamma, but they still ultimately lead to the same conclusions.

In conclusion, it is important to continue exploring and learning about these concepts, and to not get discouraged by seemingly contradictory results. Your hard work and determination will ultimately lead to a deeper understanding of spacetime and its invariance.