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theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, a Penrose diagram (named after mathematical physicist
Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus fello ...
) is a
two-dimensional In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as s ...
diagram capturing the
causal relation In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold. Introduction In modern physics (especially general relativity) spacetime is represented by a Lorentzian ma ...
s between different points in
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
through a conformal treatment of infinity. It is an extension of a
Minkowski diagram A spacetime diagram is a graphical illustration of the properties of space and time in the special theory of relativity. Spacetime diagrams allow a qualitative understanding of the corresponding phenomena like time dilation and length contractio ...
where the vertical dimension represents time, and the horizontal dimension represents a space dimension. Using this design, all light rays take a 45° path.(c = 1). The biggest difference is that locally, the
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
on a Penrose diagram is
conformally equivalent In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two di ...
to the actual metric in spacetime. 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 spacetime In physics, spherically symmetric spacetimes are commonly used to obtain analytic and numerical solutions to Einstein's field equations in the presence of radially moving matter or energy. Because spherically symmetric spacetimes are by definition ...
s, every point in the Penrose diagram corresponds to a 2-dimensional sphere (\theta,\phi).


Basic properties

While Penrose diagrams share the same basic
coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensiona ...
system of other spacetime diagrams for local
asymptotically flat spacetime An asymptotically flat spacetime is a Lorentzian manifold in which, roughly speaking, the curvature vanishes at large distances from some region, so that at large distances, the geometry becomes indistinguishable from that of Minkowski spacetime. ...
, it introduces a system of representing distant spacetime by shrinking or "crunching" distances that are further away. Straight lines of constant time and straight lines of constant space coordinates therefore become
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
e, which appear to converge at
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
s in the corners of the diagram. These points and boundaries represent "conformal infinity" for spacetime, which was first introduced by Penrose in 1963. Penrose diagrams are more properly (but less frequently) called Penrose–Carter diagrams (or Carter–Penrose diagrams), acknowledging both
Brandon Carter Brandon Carter, (born 1942) is an Australian theoretical physicist, best known for his work on the properties of black holes and for being the first to name and employ the anthropic principle in its contemporary form. He is a researcher at t ...
and Roger Penrose, who were the first researchers to employ them. They are also called conformal diagrams, or simply spacetime diagrams (although the latter may refer to Minkowski diagrams). Two lines drawn at 45° angles should intersect in the diagram only if the corresponding two light rays intersect in the actual spacetime. So, a Penrose diagram can be used as a concise illustration of spacetime regions that are accessible to observation. The
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
boundary lines of a Penrose diagram correspond to the "infinity" or to singularities where light rays must end. Thus, Penrose diagrams are also useful in the study of asymptotic properties of spacetimes and singularities. An infinite static Minkowski universe, coordinates (x, t) is related to Penrose coordinates (u, v) by: :\tan(u \pm v) = x \pm t The corners of the Penrose diagram, which represent the spacelike and timelike conformal infinities, are \pi /2 from the origin.


Black holes

Penrose diagrams are frequently used to illustrate the causal structure of spacetimes containing
black holes A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...
. Singularities are denoted by a spacelike boundary, unlike the timelike boundary found on conventional spacetime diagrams. This is due to the interchanging of timelike and spacelike coordinates within the horizon of a black hole (since space is uni-directional within the horizon, just as time is uni-directional outside the horizon). The singularity is represented by a spacelike boundary to make it clear that once an object has passed the horizon it will inevitably hit the singularity even if it attempts to take evasive action. Penrose diagrams are often used to illustrate the hypothetical
Einstein–Rosen bridge A wormhole (Einstein-Rosen bridge) is a hypothetical structure connecting disparate points in spacetime, and is based on a special Solutions of the Einstein field equations, solution of the Einstein field equations. A wormhole can be visualize ...
connecting two separate universes in the maximally extended Schwarzschild black hole solution. The precursors to the Penrose diagrams were Kruskal–Szekeres diagrams. (The Penrose diagram adds to Kruskal and Szekeres' diagram the conformal crunching of the regions of flat spacetime far from the hole.) These introduced the method of aligning the
event horizon In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s. In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact obj ...
into past and future horizons oriented at 45° angles (since one would need to travel
faster than light Faster-than-light (also FTL, superluminal or supercausal) travel and communication are the conjectural propagation of matter or information faster than the speed of light (). The special theory of relativity implies that only particles with zero ...
to cross from the
Schwarzschild radius The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteristic ...
back into flat spacetime); and splitting the singularity into past and future horizontally-oriented lines (since the singularity "cuts off" all paths into the future once one enters the hole). The Einstein–Rosen bridge closes off (forming "future" singularities) so rapidly that passage between the two asymptotically flat exterior regions would require faster-than-light velocity, and is therefore impossible. In addition, highly
blue-shift In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...
ed light rays (called a "blue sheet") would make it impossible for anyone to pass through. The maximally extended solution does not describe a typical black hole created from the collapse of a star, as the surface of the collapsed star replaces the sector of the solution containing the past-oriented "
white hole In general relativity, a white hole is a hypothetical region of spacetime and singularity that cannot be entered from the outside, although energy-matter, light and information can escape from it. In this sense, it is the reverse of a black ho ...
" geometry and other universe. While the basic
space-like In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diffe ...
passage of a static black hole cannot be traversed, the Penrose diagrams for solutions representing
rotating Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
and/or
electrically charged Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
black holes illustrate these solutions' inner event horizons (lying in the future) and vertically oriented singularities, which open up what is known as a
time-like In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
"wormhole" allowing passage into future universes. In the case of the rotating hole, there is also a "negative" universe entered through a ring-shaped singularity (still portrayed as a line in the diagram) that can be passed through if entering the hole close to its
axis An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to: Mathematics * Axis of rotation: see rotation around a fixed axis *Axis (mathematics), a designator for a Cartesian-coordinate ...
of rotation. These features of the solutions are, however, not stable and not believed to be a realistic description of the interior regions of such black holes; the true character of their interiors is still an open question.


See also

*
Causality Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
*
Causal structure In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold. Introduction In modern physics (especially general relativity) spacetime is represented by a Lorentzian ma ...
*
Conformal cyclic cosmology Conformal cyclic cosmology (CCC) is a cosmological model in the framework of general relativity and proposed by theoretical physicist Roger Penrose. In CCC, the universe iterates through infinite cycles, with the future timelike infinity (i.e. the ...
*
Weyl transformation :''See also Wigner–Weyl transform, for another definition of the Weyl transform.'' In theoretical physics, the Weyl transformation, named after Hermann Weyl, is a local rescaling of the metric tensor: :g_\rightarrow e^g_ which produces anothe ...


References

* See ''Chapter 17'' (and various succeeding sections) for a very readable introduction to the concept of conformal infinity plus examples. * * See als
on-line version
(requires a subscription to access) * See ''Chapter 5'' for a very clear discussion of Penrose diagrams (the term used by Hawking & Ellis) with many examples. * Really breaks down the transition from simple Minkowski diagrams, to Kruskal-Szekeres diagrams to Penrose diagrams, and goes into much detail the facts and fiction concerning wormholes. Plenty of easy to understand illustrations. A less involved, but still very informative book is his


External links


Conformal diagrams
– Introduction to conformal diagrams, series of minilectures by Pau Amaro Seoane * {{Roger Penrose Diagrams Coordinate charts in general relativity Mathematical methods in general relativity Lorentzian manifolds