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In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take through spacetime.
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 a cone, 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 space dimensions, 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 a four-dimensional version of a cone whose 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 by Albert Einstein's former professor Hermann Minkowski and is known as Minkowski space. The purpose was to create an invariant spacetime for all observers. To uphold causality, Minkowski restricted spacetime to non-Euclidean hyperbolic geometry. [1] [ page needed ]
Because signals and other causal influences cannot travel faster than light (see special relativity), the light cone plays an essential role in defining the concept of causality: 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 the future light 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]
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. [3]
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:
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.
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 other in a given amount of time. Light from each event will ultimately make it to the former location of the other, but after those 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 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 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 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, provided the time axis is drawn orthogonally to the spatial axes, as the cone bisects the time and space axes, it will show 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.)
In flat spacetime, the future light cone of an event is the boundary of its causal future and its past light cone is the boundary of its causal past.
In a curved spacetime, assuming spacetime is globally hyperbolic, it is still true that the future light cone of an event includes the boundary of its causal future (and similarly for the past). However gravitational lensing can 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 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.
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time. In Albert Einstein's 1905 treatment, the theory is presented as being based on just two postulates:
In physics, spacetime is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualizing and understanding relativistic effects such as how different observers perceive where and when events occur.
In the philosophy of space and time, eternalism is an approach to the ontological nature of time, which takes the view that all existence in time is equally real, as opposed to presentism or the growing block universe theory of time, in which at least the future is not the same as any other time. Some forms of eternalism give time a similar ontology to that of space, as a dimension, with different times being as real as different places, and future events are "already there" in the same sense other places are already there, and that there is no objective flow of time.
In standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects. Comoving distance factors out the expansion of the universe, giving a distance that does not change in time due to the expansion of space. Proper distance roughly corresponds to where a distant object would be at a specific moment of cosmological time, which can change over time due to the expansion of the universe. Comoving distance and proper distance are defined to be equal at the present time. At other times, the Universe's expansion results in the proper distance changing, while the comoving distance remains constant.
Physical causality is a physical relationship between causes and effects. It is considered to be fundamental to all natural sciences and behavioural sciences, especially physics. Causality is also a topic studied from the perspectives of philosophy, statistics and logic. Causality means that an effect can not occur from a cause that is not in the back (past) light cone of that event. Similarly, a cause can not have an effect outside its front (future) light cone.
The world line of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept of modern physics, and particularly theoretical physics.
In physics, Minkowski space is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van Stockum in 1937 and later confirmed by Kurt Gödel in 1949, who discovered a solution to the equations of general relativity (GR) allowing CTCs known as the Gödel metric; and since then other GR solutions containing CTCs have been found, such as the Tipler cylinder and traversable wormholes. If CTCs exist, their existence would seem to imply at least the theoretical possibility of time travel backwards in time, raising the spectre of the grandfather paradox, although the Novikov self-consistency principle seems to show that such paradoxes could be avoided. Some physicists speculate that the CTCs which appear in certain GR solutions might be ruled out by a future theory of quantum gravity which would replace GR, an idea which Stephen Hawking labeled the chronology protection conjecture. Others note that if every closed timelike curve in a given spacetime passes through an event horizon, a property which can be called chronological censorship, then that spacetime with event horizons excised would still be causally well behaved and an observer might not be able to detect the causal violation.
The Penrose–Hawking singularity theorems are a set of results in general relativity that attempt to answer the question of when gravitation produces singularities. The Penrose singularity theorem is a theorem in semi-Riemannian geometry and its general relativistic interpretation predicts a gravitational singularity in black hole formation. The Hawking singularity theorem is based on the Penrose theorem and it is interpreted as a gravitational singularity in the Big Bang situation. Penrose was awarded the Nobel Prize in Physics in 2020 "for the discovery that black hole formation is a robust prediction of the general theory of relativity", which he shared with Reinhard Genzel and Andrea Ghez.
In theoretical physics, a Penrose diagram is a two-dimensional diagram capturing the causal relations between different points in spacetime through a conformal treatment of infinity. It is an extension of the Minkowski diagram of special relativity where the vertical dimension represents time, and the horizontal dimension represents a space dimension. Using this design, all light rays take a 45° path . Locally, the metric on a Penrose diagram is conformally equivalent to the metric of the spacetime depicted. The conformal factor is chosen such that the entire infinite spacetime is transformed into a Penrose diagram of finite size, with infinity on the boundary of the diagram. For spherically symmetric spacetimes, every point in the Penrose diagram corresponds to a 2-dimensional sphere .
Two entities are in causal contact if there may be an event that has affected both in a causal way. Every object of mass in space, for instance, exerts a field force on all other objects of mass, according to Newton's law of universal gravitation. Because this force exerted by one object affects the motion of the other, it can be said that these two objects are in causal contact.
Rindler coordinates are a coordinate system used in the context of special relativity to describe the hyperbolic acceleration of a uniformly accelerating reference frame in flat spacetime. In relativistic physics the coordinates of a hyperbolically accelerated reference frame constitute an important and useful coordinate chart representing part of flat Minkowski spacetime. In special relativity, a uniformly accelerating particle undergoes hyperbolic motion, for which a uniformly accelerating frame of reference in which it is at rest can be chosen as its proper reference frame. The phenomena in this hyperbolically accelerated frame can be compared to effects arising in a homogeneous gravitational field. For general overview of accelerations in flat spacetime, see Acceleration and Proper reference frame.
In the mathematical field of Lorentzian geometry, a Cauchy surface is a certain kind of submanifold of a Lorentzian manifold. In the application of Lorentzian geometry to the physics of general relativity, a Cauchy surface is usually interpreted as defining an "instant of time". In the mathematics of general relativity, Cauchy surfaces provide boundary conditions for the causal structure in which the Einstein equations can be solved
In physics, the relativity of simultaneity is the concept that distant simultaneity – whether two spatially separated events occur at the same time – is not absolute, but depends on the observer's reference frame. This possibility was raised by mathematician Henri Poincaré in 1900, and thereafter became a central idea in the special theory of relativity.
The mathematics of general relativity is complicated. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be solved by algebra alone. In relativity, however, an object's length and the rate at which time passes both change appreciably as the object's speed approaches the speed of light, meaning that more variables and more complicated mathematics are required to calculate the object's motion. As a result, relativity requires the use of concepts such as vectors, tensors, pseudotensors and curvilinear coordinates.
In general relativity, an absolute horizon is a boundary in spacetime, defined with respect to the external universe, inside which events cannot affect an external observer. Light emitted inside the horizon can never reach the observer, and anything that passes through the horizon from the observer's side is never seen again by the observer. An absolute horizon is thought of as the boundary of a black hole.
A spacetime diagram is a graphical illustration of locations in space at various times, especially in the special theory of relativity. Spacetime diagrams can show the geometry underlying phenomena like time dilation and length contraction without mathematical equations.
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.
Spacetime topology is the topological structure of spacetime, a topic studied primarily in general relativity. This physical theory models gravitation as the curvature of a four dimensional Lorentzian manifold and the concepts of topology thus become important in analysing local as well as global aspects of spacetime. The study of spacetime topology is especially important in physical cosmology.
In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s.
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