Light cone

Inspecial and general relativity,alight cone(or "null cone" ) is the path that a flash of light, emanating from a singleevent(localized to a single point in space and a single moment in time) and traveling in all directions, would take throughspacetime.

Details

If one imagines the light confined to a two-dimensional plane, the light from the flash spreads out in a circle after the event E occurs, and if we graph the growing circle with the vertical axis of the graph representing time, the result is acone,known as the future light cone. The past light cone behaves like the future light cone in reverse, a circle which contracts in radius at the speed of light until it converges to a point at the exact position and time of the event E. In reality, there are three spacedimensions,so the light would actually form an expanding or contracting sphere in three-dimensional (3D) space rather than a circle in 2D, and the light cone would actually be afour-dimensional version of a conewhose cross-sections form 3D spheres (analogous to a normal three-dimensional cone whose cross-sections form 2D circles), but the concept is easier to visualize with the number of spatial dimensions reduced from three to two.

This view of special relativity was first proposed byAlbert Einstein's former professorHermann Minkowskiand is known asMinkowski space.The purpose was to create aninvariantspacetime for all observers. To upholdcausality,Minkowski restricted spacetime tonon-Euclidean hyperbolic geometry.^[1]^{[page needed]}

Because signals and other causal influences cannot travel faster than light (seespecial relativity), the light cone plays an essential role in defining the concept ofcausality:for a given event E, the set of events that lie on or inside the past light cone of E would also be the set of all events that could send a signal that would have time to reach E and influence it in some way. For example, at a time ten years before E, if we consider the set of all events in the past light cone of E which occur at that time, the result would be a sphere (2D: disk) with a radius of ten light-years centered on the position where E will occur. So, any point on or inside the sphere could send a signal moving at the speed of light or slower that would have time to influence the event E, while points outside the sphere at that moment would not be able to have any causal influence on E. Likewise, the set of events that lie on or inside thefuturelight cone of E would also be the set of events that could receive a signal sent out from the position and time of E, so the future light cone contains all the events that could potentially be causally influenced by E. Events which lie neither in the past or future light cone of E cannot influence or be influenced by E in relativity.^[2]

Mathematical construction

Inspecial relativity,alight cone(ornull cone) is the surface describing the temporal evolution of a flash of light inMinkowski 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.^[3]

The light cone is constructed as follows. Taking as eventpa flash of light (light pulse) at timet₀,all events that can be reached by this pulse frompform thefuture light coneofp,while those events that can send a light pulse topform thepast light coneofp.

Given an eventE,the light cone classifies all events in spacetime into 5 distinct categories:

Eventson the future light coneofE.
Eventson the past light coneofE.
Eventsinside the future light coneofEare those affected by a material particle emitted atE.
Eventsinside the past light coneofEare those that can emit a material particle and affect what is happening atE.
All other events are in the(absolute) elsewhereofEand are those that cannot affect or be affected byE.

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.

The above refers to an event occurring at a specific location and at a specific time. To say that one event cannot affect another means that light cannot get from the location of one to the otherin a given amount of time.Light from each event will ultimately make it to theformerlocation of the other, butafterthose events have occurred.

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). However, 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 farther locations will be at later 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 theobservable universe), but only as they looked long ago, when theUniversewas young.

Two events at different locations, at the same time (according to a specific frame of reference), are always outside each other's past and future light cones; light cannot travel instantaneously. Other observers 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 each other's cones.

If using asystem of unitswhere the speed of light in vacuum is defined as exactly 1, for example if space is measured inlight-secondsand time is measured in seconds, then, provided the time axis is drawnorthogonallyto the spatial axes, as the conebisectsthe time and space axes, it will show a slope of 45°, because light travels a distance of one light-second invacuumduring one second. Since special relativity requires the speed of light to be equal in everyinertial frame,all observers must arrive at the same angle of 45° for their light cones. Commonly aMinkowski diagramis used to illustrate this property ofLorentz 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 byangles,and calculated withtrig functions.Space-time tilt is measured byrapidity,and calculated withhyperbolic functions.)

In general relativity

In flat spacetime, the future light cone of an event is the boundary of itscausal futureand its past light cone is the boundary of itscausal past.

In a curved spacetime, assuming spacetime isglobally hyperbolic,it is still true that the future light cone of an eventincludesthe boundary of its causal future (and similarly for the past). Howevergravitational lensingcan cause part of the light cone to fold in on itself, in such a way that part of the cone is strictly inside the causal future (or past), and not on the boundary.

Light cones also 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 ofspacetimefree of matter), this inability to tilt all the light cones so that they are all parallel is reflected in the non-vanishing of theWeyl tensor.

References

^Cox, Brian; Forshaw, J. R. (2009).Why does E=mc2: and why should we care?.Cambridge, MA: Da Capo Press.ISBN 978-0-306-81758-8.OCLC 246894061.
^Curiel, Erik (2019)."Singularities and Black Holes > Light Cones and Causal Structure (Stanford Encyclopedia of Philosophy)".plato.stanford.edu.Stanford Encyclopedia of Philosophy.Retrieved3 March2020.
^Penrose, Roger (2005).The road to reality: a complete guide to the laws of the universe(Nachdr. ed.). London: Vintage Books.ISBN 978-0-09-944068-0.

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

[1] Cox, Brian; Forshaw, J. R. (2009).Why does E=mc2: and why should we care?.Cambridge, MA: Da Capo Press.ISBN 978-0-306-81758-8.OCLC 246894061.

[2] Curiel, Erik (2019)."Singularities and Black Holes > Light Cones and Causal Structure (Stanford Encyclopedia of Philosophy)".plato.stanford.edu.Stanford Encyclopedia of Philosophy.Retrieved3 March2020.

[3] Penrose, Roger (2005).The road to reality: a complete guide to the laws of the universe(Nachdr. ed.). London: Vintage Books.ISBN 978-0-09-944068-0.

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v t e Special relativity
Overviews	Principle of relativity Theory of relativity
History	Galilean relativity Galilean transformation Aether theories
Foundations	Relative motion Inertial frame of reference Speed of light Maxwell's equations Lorentz transformation
Consequences	Time dilation Relativistic mass Mass–energy equivalence Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Relativistic disk Bell's spaceship paradox Ehrenfest paradox
Spacetime	Minkowski spacetime World line Spacetime diagrams Light cone
Dynamics	Proper time Proper mass 4-momentum

Details

Mathematical construction

In general relativity

See also

References

External links