Light Cones

In a spacetime diagram, we can visualize the regions that have spacelike, timelike, or lightlike separation from a given event by drawing the event’s light cones—shapes consisting of all points that have lightlike separation from the event in question. Every event (every point in spacetime) has two light cones:

  1. An event’s past light cone represents the worldlines that light could travel (from other events in the past) to arrive at this event. In other words, it consists of all the other points in the spacetime diagram that are exactly 45 degrees below horizontal from the event in question (since a slope of 45 degrees represents the speed of light).
  2. An event’s future light cone represents the worldlines light could travel if it was emitted from the event in question (to reach other events in the future). In other words, it consists of all the other points that are exactly 45 degrees above horizontal from the specified event.
spacetime diagram of light cones
Light cones in a 3D diagram

The future light cone of event E (the top cone) represents the set of points in spacetime that would be reached by a flash of light emitted at event E. The past light cone of E (bottom) represents the points in spacetime from which light could travel to reach E. The blue horizontal plane represents the set of points—or events—that are simultaneous with E (in this frame of reference).

In a three-dimensional spacetime diagram—which represents time with the vertical axis and two dimensions of space with the horizontal axes—the past and future light cones are shaped like actual cones. (See the figure to the left.) A two-dimensional spacetime diagram, on the other hand, represents only one dimension of space. In a 2D diagram, the future light cone looks like a “V” and the past light cone looks like an upside-down “V.” Together, they form an “X” centered on the event to which they belong. (See the figure below.)

In reality, of course, there are three dimensions of space; so in real-life spacetime, a light cone is a four-dimensional hypercone: the future light cone is a sphere that expands as time goes forward; the past light cone is a sphere that gets smaller until it vanishes at the point where the future light cone begins. But I don’t know how to draw four-dimensional spacetime diagrams, so we’ll have to ignore one or two spatial dimensions and content ourselves with cones or Vs instead.
spacetime diagram of light cones
Light cones in a 2D diagram

Event L is on the future light cone of E, and is therefore lightlike separated from E. (If a flash of light is emitted at event E and travels to the left, the light will eventually arrive at the time and place marked “L.”) Events P and F occur inside the past and future light cones of E, respectively, so they have timelike separation from E. Event S is spacelike separated from E, because it occurs outside of E’s light cones.

All of the points inside the past and future light cones of an event have timelike separation from that event. All points outside the cones have spacelike separation from the event. Thus, light cones divide spacetime into regions of timelike and spacelike separation from a given event.

Light cones also represent a limitation on causality. No physical object or causal process (that we know of) can travel faster than light. There may be some weird exceptions, though no one is entirely sure. A phenomenon called quantum entanglement suggests that subatomic particles somehow communicate with each other faster than light, as we’ll see in chapter 7, but nobody understands how that works. Also, spacetime itself can expand, as we’ll discuss in chapter 8, and this expansion can exceed the speed of light. However, it is unknown whether either of these exceptions really allows for the possibility of faster-than-light causation. Objects with mass can’t be accelerated beyond the speed of light, as we’ll see later in this chapter. Moreover, all of the massless particles that we know of (photons, gluons, and neutrinos) travel at exactly the speed of light. The fastest waves we know of (electromagnetic waves and gravitational waves, which we’ll talk about later in this chapter) also travel at that same speed.

Furthermore, if anything did go faster than light, it would be traveling backwards in time! (I’ll explain why on the next page.) For these reasons, the speed of light is considered to be the maximum speed at which anything can travel. So, no causal influence can get from one event to another faster than light.

If this is correct, then nothing that happens at a given event can affect what happens outside the event’s future light cone, since that would require some physical object or process to travel faster than light. For the same reason, an event can’t be affected by anything that happens outside its past light cone. In other words, an event’s past and future light cones are its causal boundaries: nothing outside them can affect—or be affected by—that event.