Spacetime diagrams

Events and diagrams

A spacetime diagram plots space and time together. Usually one spatial dimension $x$ is horizontal and time is vertical, often using $ct$ instead of $t$ so both axes have units of length.

An event is a point in spacetime: something that happens at a definite place and time. A particle or observer traces a path through spacetime called a worldline.

Light rays

If the vertical axis is $ct$ and the horizontal axis is $x$, light moving in the positive x-direction satisfies

$x=ct$

Light moving in the negative x-direction satisfies

$x=-ct$

These appear as lines at $45^circ$ when the axes use equal scales. The light lines divide spacetime into regions of possible and impossible causal influence.

Worldline slopes

A stationary object has constant $x$, so its worldline is vertical. A moving object has a tilted worldline. If its velocity is $v$, then

$x=vt$

or

x=rac{v}{c}ct

The slope relative to the vertical indicates speed. Massive objects have worldlines inside the light cone because $|v|<c$.

Time axis of a moving frame

The time axis of a moving frame $S'$ is the worldline of its origin $x'=0$. In frame $S$, that origin moves as

$x=vt$

so the $ct'$ axis tilts toward the light line as speed increases.

Space axis of a moving frame

The space axis of $S'$ is the set of events simultaneous with $t'=0$. From the Lorentz transformation,

ight)$$ Setting $t'=0$ gives $$t=rac{vx}{c^2}$$ or $$ct=rac{v}{c}x$$ Thus the $x'$ axis also tilts. Moving frames have tilted time and space axes, symmetric around the light line. ## Simultaneity on diagrams In a spacetime diagram, simultaneity in a given frame is represented by lines parallel to that frame's space axis. Since different inertial frames have different space axes, they disagree about which separated events are simultaneous. This gives a powerful visual explanation of relativity of simultaneity. ## Proper time and worldlines The proper time along a timelike worldline is related to the spacetime interval. For small separations, $$c^2d au^2=c^2dt^2-dx^2$$ for one spatial dimension. A straight inertial worldline between two timelike-separated events maximizes proper time. This fact helps explain the twin paradox. ## Causal regions Events inside the future light cone can be reached by signals traveling at or below light speed. Events inside the past light cone can influence the present event. Events outside the light cone are spacelike separated and cannot be causally connected without faster-than-light signaling. ## The big idea Spacetime diagrams turn relativity into geometry. Events are points, objects trace worldlines, light travels on $45^circ$ lines, and moving frames have tilted axes. These diagrams make time dilation, length contraction, simultaneity, and causal structure easier to visualize.