- #1

daselocution

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## Homework Statement

Prove that for a timelike interval, two events can never be considered to occur simultaneously.

## Homework Equations

Δs'

^{2}=∆s'

^{2}

(∆s is invariant)

s

^{2}=x

^{2}- (ct)

^{2}

s'

^{2}=x'

^{2}- (ct')

^{2}

## The Attempt at a Solution

I first imagined a reference frame K in which two events happened at (x

_{1}, t

_{1}) and (x

_{2}, t

_{2})

Here, ∆s

^{2}= x

_{2}

^{2}- x

_{1}

^{2}- c

^{2}t

_{2}

^{2}+ c

^{2}t

_{1}

^{2}= s'

^{2}=∆x'

^{2}- ∆(ct')

^{2}where ∆(ct')

^{2}= 0 (if I'm trying to prove that this CAN'T happen I thought it would be best to show what would happen if it were indeed zero)

...which can be rewritten as:

∆s

^{2}= x

_{2}

^{2}- x

_{1}

^{2}- c

^{2}t

_{2}

^{2}+ c

^{2}t

_{1}

^{2}= s'

^{2}=∆x'

^{2}= x

_{2}'

^{2}- x

_{1}'

^{2}< 0 (my book says that for events with timelike separation that ∆s

^{2}<0)

I guess I'm not really sure where to head from here. Should I substitute in x'=x-vt...(the lorentz transformations for velocity) and try and solve for some value of v such that (ideally) it might be v>c and thus false? I tried doing that but saw that the algebra seemed pretty long and so I thought that I might not be on the right track...I'm not exactly sure what I'm looking for here, I'm quite new to proof-based math (or physics or really anything). Any guidance would be appreciated.