Lorentz transformation Problem, Confused on setting it up

[sealsix14]

So this is my first course in upper level physics, and I'm still trying to figure out special relativity.

The problem:

Two events occur at the same place in a certain inertial frame and are separated by a time interval of 4sec. What is the spatial separation between these two events in an inertial frame in which the events are separated by a time interval of 6 sec?

I know the problem states that the two events occur in a certain inertial frame (frame S), and have a dt = 4. but I'm confused as to what It is exactly asking, is it asking what the d would be in another inertial frame where the dt = 6? I'm Just confused on setting up the problem and feel stupid.

 

[jfy4]

I'm always rusty with this stuff too. SR is the hardest thing, don't believe people if they tell you otherwise ;) I would think that the total spacetime interval in both frames has to be the same, that is $(\Delta s)^2$ is the same in all frames. so let's use
$$(\Delta s)^2 = (\Delta t)^2 - (\Delta \vec{x})^2$$
In the first frame we know that
$$(\Delta s)^2 = (4)^2\$$
since there is no spatial change. But in the other frame it is
$$(\Delta s)^2 = (6)^2 - (\Delta \vec{x})^2$$
Now the kicker is that $(\Delta s)^2$ is the same for both frames. The question is then, what does $(\Delta \vec{x})^2$ have to be to make that true? I hope this helps/is right.

 

[Chestermiller]

This problem seems to be designed to give you a little experience in applying the Lorentz Transformation. In addition to the method that jfy4 described, you can also solve this directly by using the Lorentz Transformation. Let's say that the two events occur in the S frame of reference. According to the problem description, what is the difference in spatial positions between the two events (Δ x) and what is the difference in times of the two events (Δt)? Now, from the Lorentz Transformation, what are the corresponding quantities (Δx') and (Δt') for the S' frame of reference (expressed in terms of the relativity factor γ and v/c)? Since the problem statement already tells you what Δt' is in the S' frame of reference(6 sec), you can use one of the LT equations to solve for γ, and then v/c. You can then substitute into the other equation to get Δx'. The result you get should match what jfy4's method gives.

 

[sealsix14]

Thanks, I'll give this a try and see If I get anywhere with it.

 

[sardar]

Dear student,

Thank you for reaching out with your question. It is completely normal to feel confused when learning about new concepts and theories, especially in a complex subject like special relativity. Let me try to break down the problem for you and provide some guidance on how to approach it.

First, it is important to understand what an inertial frame is. An inertial frame is a reference frame in which an object or system is at rest or moving with constant velocity. In other words, there is no acceleration or deceleration in an inertial frame. This concept is essential in understanding special relativity.

Now, let's look at the problem. It states that two events occur at the same place in a certain inertial frame (frame S) and are separated by a time interval of 4 seconds. This means that in frame S, the two events occur at the same location but at different times, with a difference of 4 seconds.

The question is asking for the spatial separation between these two events in another inertial frame (frame S'). This new frame has a time interval of 6 seconds between the same two events. So, essentially, we are looking for the distance between the two events in frame S' using the given information about frame S.

To solve this problem, you can use the Lorentz transformation equations, which describe how measurements of space and time in one inertial frame relate to another inertial frame. Specifically, you will need to use the equation for length contraction, which is:

L' = L * √(1 - v^2/c^2)

Where L is the length in the original frame (S), L' is the length in the new frame (S'), v is the relative velocity between the two frames, and c is the speed of light.

In this problem, the length in frame S is the distance between the two events, which we will call L. The length in frame S' is what we are looking for, which we can call L'. The relative velocity between the two frames is 0 since they are both inertial frames. And the speed of light is a constant value, so we can simply use c.

Now, plug in the values you have into the equation and solve for L'. Once you have the value for L', that will be the spatial separation between the two events in frame S'. I hope this helps and please don't feel stupid for asking questions. It takes time and effort