### Spacetime

According to Einstein, our commonsense intuitions about space and time are wrong. Intuitively, we think space and time are independent of each other. We assume it makes sense to talk about distances without mentioning time, and that it makes sense to talk about durations of time without mentioning distance. On Einstein’s view, however, space and time aren’t independent entities; they are interconnected in a way that is not easy to comprehend.

For this reason, it is helpful to think of space and time as inseparable aspects of a four-dimensional reality called spacetime. From the perspective of any observer, spacetime always has three spatial dimensions and one dimension of time; but if two observers are moving relative to each other, they will disagree about the way in which those dimensions are oriented. To understand this idea visually, let’s look at some diagrams. A Spacetime Diagram Event B occurs later than A, at a location to the right of where A occurred.

A spacetime diagram is a graph in which each point represents a specific location at a specific moment in time. Physicists refer to these points in spacetime as events. Technically, an “event” in this sense has zero duration and zero size (since it’s a single point in space and time), and a point on the graph is still called an event even if nothing is happening there. However, in order to make the following discussion a little easier to understand, I’ll use the word “event” in the ordinary sense to mean something that happens. And I’ll represent brief events as points in spacetime, even though real-life events have more than zero duration. In a typical spacetime diagram, time is represented using the vertical axis of the graph, so events near the top of the graph occur later than events at the bottom. The horizontal axis of the graph represents one dimension of space. It may represent the left/right dimension, for example: points on the left side of the graph represent events that occur to your left; points on the right of the graph represent events that occur to your right. (In a 3D graph, a second dimension of space can also be represented, using the horizontal depth of the graph.) Worldlines The vertical red line represents an object at rest 3 meters to the left. The yellow line represents an object moving with a velocity of 1 m/s toward the left. The green line represents an object moving 4 m/s toward the right. The curved blue line represents an accelerating object, with increasing rightward velocity.

The path of an object through spacetime is represented by a worldline—a line connecting the places where the object is located at each moment. An object at rest is represented by a vertical worldline, since it stays at the same location (on the horizontal axis) as time ticks upward along the vertical axis of the graph. An object moving to the right has a worldline slanting to the right; an object moving to the left has a worldline slanting to the left. The slope of the line (or rather, the reciprocal of the slope) represents the speed at which the object is moving: the faster the object moves, the more its worldline slants sideways. Conventional Slopes An object traveling at the speed of light has a worldline exactly 45 degrees from horizontal. The worldline of an object moving slower than light is closer to vertical, and the worldline of an object moving faster than light (if that were possible) would be closer to horizontal.

Spacetime diagrams are especially useful for visualizing the space and time relations between objects moving close to the speed of light. But anything close to the speed of light would be going so fast that its worldline would be practically horizontal, if we drew the graph using ordinary unit scales. In order to solve this problem, physicists have adopted the convention of adjusting the unit scales (on the horizontal and vertical axes of the graph) so that the speed of light corresponds to a slope of exactly 1, that is, a slope 45 degrees from horizontal. This convention also makes it easier to visualize some of the most interesting consequences of special relativity, as we’ll see.

A spacetime diagram represents space and time from the perspective of one specific reference frame. Since different reference frames may disagree about the velocity of an object, two observers may draw their spacetime diagrams differently. In the alien spaceship example, the alien will draw her spaceship’s worldline vertically (since she regards it as being at rest) and the earth’s worldline will be slanted (since she regards the earth as being in motion). In the farmer’s spacetime diagram, it will be the other way around.

Nevertheless, there are some things on which all inertial reference frames agree. For one, all inertial frames agree about whether an object is accelerating. Objects moving with constant velocity will have straight worldlines in the spacetime diagram for any inertial frame (though the slopes of these worldlines will differ from one frame to another). But if an object is accelerating, its worldline will be curved in all inertial reference frames.