How to show time ordering is frame independent

[baixiaojian]

Hello, everyone

I am studying Srednicki's "Quantum Field Theory", section 4 "The spin-statistics Theorem".
Does anyone know how to show that "The time ordering of two spacetime points x and x' is frame independent if their separation is timelike."(P32), explicitly?
And "Two spacetime points whose separation is spacelike, can have different temporal ordering in different frames." How to show this?

Thanks

 

[granpa]

what is the definition of timelike and spacelike intervals?

timelike:

spacelike:

Δt=time difference
Δr=spatial difference
s=interval

time and distance measured in different frames:

https://www.physicsforums.com/showthread.php?t=257689

Let events 1 and 2 occur at $(x_1, t_1)$ and $(x_2, t_2)$ in frame S. In frame S' they occur at

$$x_1^{\prime} = \gamma (x_1 - v t_1)$$

$$t_1^{\prime} = \gamma (t_1 - v x_1 / c^2)$$

$$x_2^{\prime} = \gamma (x_2 - v t_2)$$

$$t_2^{\prime} = \gamma (t_2 - v x_2 / c^2)$$

Subtracting pairs of equations gives

$$\Delta x^{\prime} = x_2^{\prime} - x_1^{\prime} = \gamma ((x_2 - x_1) - v (t_2 - t_1)) = \gamma (\Delta x - v \Delta t)$$

$$\Delta t^{\prime} = t_2^{\prime} - t_1^{\prime} = \gamma ((t_2 - t_1) - v (x_2 - x_1) / c^2) = \gamma (\Delta t - v \Delta x / c^2)$$

That is, the Lorentz transformation applies to $\Delta x$ and $\Delta t$ just as it does to x and t, because the transformation is linear.

 

[baixiaojian]

Thank you for your reply. But my question is how to show "the time ordering operation" of two spacetime points x and x' is frame independent if their separation is timelike.
I attached the following paragraph from Srednicki's book for your reference. Please click the attachment for clarity.

 

[granpa]

you need Δt' to be the opposite sign of Δt