Help with time-space diagram equations.

[v3ra]

Consider a situation where we want to combine spacetime diagrams of Alice and Bob, where Bob is moving at a speed of 0.4c to the right (positive x direction). If we draw Alice’s xA axis as horizontal and tA axis as vertical, answer the following questions.

What is the equation, written in terms of xA and tA, that describes Bob’s tB axis (the world line where xB = 0) on Alice’s diagram?

My options are the following:

1. tA = γ xA/(0.4c), where γ is the Lorentz factor
2. tA = xA/(0.4c)
3. tA = (0.4c)(xA)
4. tA = γ (0.4c)(xA), where γ is the Lorentz factor

Can someone explain, step by step, how to derive the correct equation?


p.s. this is not homework, it`s just something for fun I found online that I want to understand...

 

[ghwellsjr]

Step 1: Since speed is distance divided by time, 0.4c = xA/tA.

Step 2: Rearrange by multiplying both sides of the equation by tA and dividing by 0.4c to get answer 2.

 

[pervect]

let $t_a, x_a$ be Alice's coordinates
let $t_b, x_b$ be Bob's coordinates
let $\beta = v/c \quad \gamma = 1 / \sqrt{1-\beta^2}$

then write down the Lorentz tranform:

$x_b = 0 = \gamma \left( x_a - \beta t_a \right)$

From which you conclude the equation is $x_a = \beta t_a$
which is option 2

 

[Fredrik]

The $t_B$ axis is the image of Bob's world line (the set of all events in spacetime that he will pass through) in Alice's coordinates. So for all points $(t_A,x_A)$ on that line, $x_A$ and $t_A$ are (according to Alice) respectively the distance he has traveled since the (0,0) event, and the time that has passed since the (0,0) event. So his velocity is $v=x_A/t_A$.

This means that for any point $(t_A,x_A)$ on that line, we have $x_A=0.4c t_A$.

Hm, maybe I should have looked more closely at the previous answers before I wrote this. I thought that no one had pointed out that velocity=distance/time, but it's right there in ghwellsjr's post.

 

[sardar]

The correct equation is 1. tA = γ xA/(0.4c), where γ is the Lorentz factor.

To understand how this equation is derived, we first need to understand the concept of spacetime diagrams and how they represent the relationship between space and time in special relativity.

A spacetime diagram is a graphical representation of an event in spacetime, where space is represented on the horizontal axis and time is represented on the vertical axis. The slope of a line on the diagram represents the relative velocity between two frames of reference.

In this scenario, Alice and Bob are two observers in different frames of reference. Alice is at rest in her frame of reference and Bob is moving at a speed of 0.4c to the right (positive x direction) in his frame of reference. We want to combine their spacetime diagrams to see how their frames of reference are related.

To do this, we first need to understand how time and space are related in special relativity. According to the Lorentz transformation equations, time and space are not absolute quantities and are dependent on the relative velocity between two frames of reference. The Lorentz factor, γ, is a constant that appears in these equations and is given by γ = 1/√(1-v^2/c^2), where v is the relative velocity between the two frames and c is the speed of light.

Now, let's look at Alice's diagram. The xA axis represents space and the tA axis represents time in her frame of reference. We want to find the equation that describes Bob's tB axis (the world line where xB = 0) on Alice's diagram. Since Bob is moving at a constant velocity, his world line will be a straight line with a slope of 0.4c.

We can use the slope formula, m = (y2-y1)/(x2-x1), to find the slope of Bob's world line. In this case, y1 = 0 (since Bob's world line passes through the origin) and x1 = 0 (since Bob's world line is vertical). We also know that y2 = tB and x2 = xB = 0 (since Bob's world line is at xB = 0). Therefore, the slope of Bob's world line is m = (tB - 0)/(0 - 0) = tB/0 = ∞