How to show time ordering is frame independent

In summary, the concept of timelike and spacelike intervals is important in understanding the effects of time and distance measurements in different frames of reference. In particular, it has been shown that the time ordering of two spacetime points is frame independent if their separation is timelike, while two spacetime points with a spacelike separation can have different temporal ordering in different frames. This concept is important in the study of the spin-statistics theorem and the effects of Lorentz transformations.
  • #1
baixiaojian
3
0
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
 
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  • #2
what is the definition of timelike and spacelike intervals?

timelike:
7d40606e17197f22c8343aa6fd354115.png

spacelike:
36c24ff1c4a2d6564ac46e804382e12f.png

Δt=time difference
Δr=spatial difference
s=interval

time and distance measured in different frames:
https://www.physicsforums.com/showthread.php?t=257689
jtbell said:
Let events 1 and 2 occur at [itex](x_1, t_1)[/itex] and [itex](x_2, t_2)[/itex] in frame S. In frame S' they occur at

[tex]x_1^{\prime} = \gamma (x_1 - v t_1)[/tex]

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

[tex]x_2^{\prime} = \gamma (x_2 - v t_2)[/tex]

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

Subtracting pairs of equations gives

[tex]\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)[/tex]

[tex]\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)[/tex]

That is, the Lorentz transformation applies to [itex]\Delta x[/itex] and [itex]\Delta t[/itex] just as it does to x and t, because the transformation is linear.
 
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  • #3
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.
gg.jpg
 
  • #4
you need Δt' to be the opposite sign of Δt
 

Related to How to show time ordering is frame independent

1. What is time ordering in physics?

Time ordering refers to the arrangement of events in a chronological sequence. In other words, it is the way in which events are ordered according to when they occur in time.

2. Why is it important to show that time ordering is frame independent?

It is important to demonstrate that time ordering is frame independent because it confirms that the laws of physics are the same in all inertial reference frames. This is a fundamental principle of relativity.

3. How do we show that time ordering is frame independent?

To show that time ordering is frame independent, we can use the concept of Lorentz transformations. These transformations allow us to convert between different inertial reference frames and show that the sequence of events remains the same regardless of the frame of reference.

4. What evidence supports the idea that time ordering is frame independent?

One major piece of evidence is the measurement of the speed of light. The speed of light is constant in all inertial reference frames, which supports the idea that the laws of physics, including time ordering, are the same in all frames.

5. Are there any exceptions to the rule that time ordering is frame independent?

There are some situations, such as near the event horizon of a black hole, where the effects of gravity can cause time ordering to appear different from different frames of reference. However, in these extreme cases, the laws of physics as we know them may not apply, so the concept of frame independence may not hold.

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