Light Cone - Mathematical Construction

Mathematical Construction

In special relativity, a light cone (or null cone) is the surface describing the temporal evolution of a flash of light in Minkowski spacetime. This can be visualized in 3-space if the two horizontal axes are chosen to be spatial dimensions, while the vertical axis is time.

The light cone is constructed as follows. Taking as event a flash of light (light pulse) at time, all events that can be reached by this pulse from form the future light cone of, while those events that can send a light pulse to form the past light cone of .

Given an event, the light cone classifies all events in space+time into 5 distinct categories:

• Events on the future light cone of .
• Events on the past light cone of .
• Events inside the future light cone of are those affected by a material particle emitted at .
• Events inside the past light cone of are those that can emit a material particle and affect what is happening at .
• All other events are in the (absolute) elsewhere of and are those that cannot affect or be affected by .

The above classifications hold true in any frame of reference; that is, an event judged to be in the light cone by one observer, will also be judged to be in the same light cone by all other observers, no matter their frame of reference. This is why the concept is so powerful.

Keep in mind, we're talking about an event, a specific location at a specific time. To say that one event cannot affect another means that there isn't enough time for light to get from one to the other. Light from each event will eventually (after some time) make it to the old location of the other event, but since that's at a later time, it's not the same event.

As time progresses, the future light cone of a given event will eventually grow to encompass more and more locations (in other words, the 3D sphere that represents the cross-section of the 4D light cone at a particular moment in time becomes larger at later times). Likewise, if we imagine running time backwards from a given event, the event's past light cone would likewise encompass more and more locations at earlier and earlier times. The further locations will of course be at more distant times, for example if we are considering the past light cone of an event which takes place on Earth today, a star 10,000 light years away would only be inside the past light cone at times 10,000 years or more in the past. The past light cone of an event on present-day Earth, at its very edges, includes very distant objects (every object in the observable universe), but only as they looked long ago, when the universe was young.

Two events at different locations, at the same time (according to a specific frame of reference), are always outside of each other's past and future light cones; light cannot travel instantaneously. Other observers, of course, might see the events happening at different times and at different locations, but one way or another, the two events will likewise be seen to be outside of each other's cones.

If using a system of units where the speed of light in vacuum is defined as exactly 1, for example if space is measured in light-seconds and time is measured in seconds, then the cone will have a slope of 45°, because light travels a distance of one light-second in vacuum during one second. Since special relativity requires the speed of light to be equal in every inertial frame, all observers must arrive at the same angle of 45° for their light cones. Commonly a Minkowski diagram is used to illustrate this property of Lorentz transformations. Elsewhere, an integral part of light cones, is the region of spacetime outside the light cone at a given event (a point in spacetime). Events that are elsewhere from each other are mutually unobservable, and cannot be causally connected.

(The 45° figure really only has meaning in space-space, as we try to understand space-time by making space-space drawings. Space-space tilt is measured by angles, and calculated with trig functions. Space-time tilt is measured by rapidity, and calculated with hyperbolic functions.)