Observer S' on a rocket vs an observer S on Earth

[71GA]

This is a basic question regarding Lorentz transformations. Let's say we have two observers - S on Earth and S' which we put on a rocket headed for Alpha Centauri (A.C) =).

If i choose 2 events like this:

1. rocket leaves Earth
2. rocket arrives on A.C

These two events clearly do not happen on a same place from perspective of an observer S. But what about an observer S'? Would he say that they happen on a same place?

I want to know this so i can use $\Delta x' = 0$ in Lorentz transformations.

 

[tiny-tim]

hi 71GA!

1. rocket leaves Earth
2. rocket arrives on A.C

… what about an observer S'? Would he say that they happen on a same place?

yes

but remember that that only applies if S' is an inertial observer,

ie if his velocity is constant throughout

so this would have to be a spaceship that goes past Earth and ac, without landing!

 

[ghwellsjr]

This is a basic question regarding Lorentz transformations. Let's say we have two observers - S on Earth and S' which we put on a rocket headed for Alpha Centauri (A.C) =).

If i choose 2 events like this:

1. rocket leaves Earth
2. rocket arrives on A.C

These two events clearly do not happen on a same place from perspective of an observer S. But what about an observer S'? Would he say that they happen on a same place?

I want to know this so i can use $\Delta x' = 0$ in Lorentz transformations.

Here are a couple spacetime diagrams depicting your scenario. Earth is shown in blue. A.C. is shown in red. The rocket is shown in black. The inertial observer that tiny-tim mentioned is shown in grey. The dots show one-year intervals of Proper Time for each observer/object. I picked a speed of 0.8c for the rocket to travel from Earth to A.C. The first diagram is for the rest frame of observer S on Earth:

You can clearly see the two events that you specified where the black rocket leaves Earth and where it arrives on A.C.

Now if we transform the coordinates of all the events (the dots) to the rest frame of the grey observer:

You can clearly see that your two events are at the same location. You can do the math to verify this.

 

[71GA]

Nice MDs :). Thank you.

 

[sardar]

I can provide an explanation for the difference in perspective between observer S and observer S'. The concept of simultaneous events is relative and depends on the frame of reference of the observer. In this scenario, the two events, the rocket leaving Earth and arriving at A.C, may be perceived as simultaneous by observer S on Earth, but not by observer S' on the rocket.

This is due to the fact that the rocket is moving at a high velocity towards A.C, causing a time dilation effect. This means that time appears to pass slower for observer S' on the rocket compared to observer S on Earth. Therefore, from the perspective of observer S', the two events do not occur at the same time, but rather the event of the rocket arriving at A.C happens first.

This difference in perception of simultaneous events is taken into account in the Lorentz transformations. The equation $\Delta x' = 0$ is used when the two events occur at the same location in the frame of reference of observer S', but not necessarily in the frame of reference of observer S. This equation helps to convert the measurements of space and time between the two frames of reference.

In conclusion, the difference in perspective between observer S and observer S' is due to the relative motion between the two frames of reference. The Lorentz transformations take into account this difference and allow for accurate measurements and calculations in different frames of reference.