Light cone
From Wikipedia, the free encyclopedia
A Light cone is the path that a flash of light would take through spacetime. As time progresses, the light from the flash spreads out in a circle, and the result is a cone. In reality, there are three space dimensions, so the light actually forms a sphere in space, and the light cone is actually a fourth dimensional shape, but it's much easier to visualize it as a cone.
In spacetime, the time dimension is geometrically different from the three space dimensions. With two space dimensions, you can rotate your frame of reference. If the x-axis moves clockwise, the y-axis also moves clockwise. Eventually, the whole frame of reference can rotate a full circle. This is not possible in spacetime.
With one time dimension and one space dimension, you can analogously change to a moving frame of reference. If the time dimension is rotated clockwise, then the space dimension instead moves counter-clockwise. Time and space units stretch out (see Minkowski Diagrams). This is what causes many of the strange effects in special relativity. If your frame of reference keeps on going faster, eventually the space and time dimensions meet at 45 degrees, and you are going at the speed of light, along the surface of the light cone. Your space and time units stretch to infinite length, and everything slows down to a stop.
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[edit] 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 p a flash of light (light pulse) at time t0, all events that can be reached by this pulse from p form the future light cone of p, while those events that can send a light pulse to p form the past light cone of p.
Given an event E, the light cone classifies all events in spacetime into 5 distinct categories:
- Events on the future light cone of E.
- Events on the past light cone of E.
- Events inside the future light cone of E are those which are affected by a material particle emitted at E.
- Events inside the past light cone of E are those which can emit a material particle and affect what is happening at E.
- All other events are in the (absolute) elsewhere of E and are those that can not affect and can not be affected by E.
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, that 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, each location's past light cone will eventually grow to encompass all locations. The further locations will of course be at far distant times. The earth's past light cone, at its very edges, includes very distant objects, but only what they looked like 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. This is ensured by the Lorentz transformation. 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.)
[edit] Light-cones in general relativity
In general relativity, the future light cone is the boundary of the causal future of a point and the past light cone is the boundary of its causal past.
In a curved spacetime, the light-cones cannot all be tilted so that they are 'parallel'; this reflects the fact that the spacetime is curved and is essentially different from Minkowski space. In vacuum regions (those points of spacetime free of matter), this inability to tilt all the light-cones so that they are all parallel is reflected in the non-vanishing of the Weyl tensor.
[edit] See also
- Absolute future
- Absolute past
- Hyperbolic partial differential equation
- Light cone coordinates
- Method of characteristics
- Minkowski diagram
- Monge cone
- Wave equation
[edit] External links
- The Einstein-Minkowski Spacetime: Introducing the Light Cone
- The Paradox of Special Relativity
- RSS feed of stars in one's personal light cone
