Sticky for beginners to draw a spacetime diagram

[granpa]

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.

I'm always sending beginners to that post and it often answers their question.

maybe we should have a sticky listing these equations.

maybe throw in something about how to draw a spacetime diagram too. and a note about the lines of simultaneity being different for observers moving at different velocities (which as what beginners always have trouble with). anyone know of any good posts for that?edit:I see there is already a FAQ. maybe the post above should be included in it.
https://www.physicsforums.com/showthread.php?t=104715

 

[Dale]

maybe throw in something about how to draw a spacetime diagram too

Good idea. Most students are familiar with plotting a diagram with position on the vertical axis and time on the horizontal axis. A spacetime diagram is nothing more than that but for historical reasons the axes are reversed so that position is on the horizontal axis and time is on the vertical axis. The Lorentz transform is then just a way of changing those diagrams from one frame to another.