Kruskal–Szekeres Coordinates
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In
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, Kruskal–Szekeres coordinates, named after Martin Kruskal and George Szekeres, are a
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
for the Schwarzschild geometry for a
black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
. These coordinates have the advantage that they cover the entire spacetime
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
of the maximally extended Schwarzschild solution and are well-behaved everywhere outside the physical singularity. There is no coordinate singularity at the horizon. The Kruskal–Szekeres coordinates also apply to space-time around a spherical object, but in that case do not give a description of space-time inside the radius of the object. Space-time in a region where a star is collapsing into a black hole is approximated by the Kruskal–Szekeres coordinates (or by the Schwarzschild coordinates). The surface of the star remains outside the
event horizon In astrophysics, an event horizon is a boundary beyond which events cannot affect an outside 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 c ...
in the Schwarzschild coordinates, but crosses it in the Kruskal–Szekeres coordinates. (In any "black hole" which we observe, we see it at a time when its matter has not yet finished collapsing, so it is not really a black hole yet.) Similarly, objects falling into a black hole remain outside the event horizon in Schwarzschild coordinates, but cross it in Kruskal–Szekeres coordinates.


Definition

Kruskal–Szekeres coordinates on a
black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
geometry are defined, from the Schwarzschild coordinates (t,r,\theta,\phi), by replacing ''t'' and ''r'' by a new timelike coordinate ''T'' and a new spacelike coordinate X: : T = \left(\frac - 1\right)^e^\sinh\left(\frac\right) : X = \left(\frac - 1\right)^e^\cosh\left(\frac\right) for the exterior region r>2GM outside the
event horizon In astrophysics, an event horizon is a boundary beyond which events cannot affect an outside 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 c ...
and: : T = \left(1 - \frac\right)^e^\cosh\left(\frac\right) : X = \left(1 - \frac\right)^e^\sinh\left(\frac\right) for the interior region 0. Here GM is the
gravitational constant The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
multiplied by the Schwarzschild mass parameter, and this article is using units where c = 1. It follows that on the union of the exterior region, the event horizon and the interior region the Schwarzschild radial coordinate r (not to be confused with the Schwarzschild radius r_\text = 2GM), is determined in terms of Kruskal–Szekeres coordinates as the (unique) solution of the equation: : T^2 - X^2 = \left(1-\frac\right)e^ \ , T^2 - X^2 < 1 Using the
Lambert W function In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the Branch point, branches of the converse relation of the function , where is any complex number and is the expone ...
the solution is written as: : r = 2GM \left(1 + W_0\left( \frac \right)\right). Moreover one sees immediately that in the region external to the black hole T^2 - X^2 < 0,\ X > 0 : t = 4GM \mathop(T/X) whereas in the region internal to the black hole 0 < T^2 - X^2 < 1, \ T> 0 : t = 4GM \mathop(X/T) In these new coordinates the metric of the Schwarzschild black hole manifold is given by : g = \frace^(-dT^2 + dX^2) + r^2 g_\Omega, written using the (− + + +)
metric signature In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and z ...
convention and where the angular component of the metric (the Riemannian metric of the 2-sphere) is: : g_\Omega\ \stackrel\ d\theta^2+\sin^2\theta\,d\phi^2. Expressing the metric in this form shows clearly that radial null geodesics i.e. with constant \Omega = \Omega(\theta, \phi) are parallel to one of the lines T = \pm X . In the Schwarzschild coordinates, the Schwarzschild radius r_\text = 2GM is the radial coordinate of the
event horizon In astrophysics, an event horizon is a boundary beyond which events cannot affect an outside 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 c ...
r = r_\text = 2GM. In the Kruskal–Szekeres coordinates the event horizon is given by T^2 - X^2 = 0 . Note that the metric is perfectly well defined and non-singular at the event horizon. The curvature singularity is located at T^2 - X^2 = 1.


Maximally extended Schwarzschild solution

The transformation between Schwarzschild coordinates and Kruskal–Szekeres coordinates defined for ''r'' > 2''GM'' and -\infty can be extended, as an analytic function, at least to the first singularity which occurs at T^2 - X^2 = 1. Thus the above metric is a solution of Einstein's equations throughout this region. The allowed values are : -\infty < X < \infty\, : -\infty < T^2 - X^2 < 1 Note that this extension assumes that the solution is analytic everywhere. In the maximally extended solution there are actually two singularities at ''r'' = 0, one for positive ''T'' and one for negative ''T''. The negative ''T'' singularity is the time-reversed black hole, sometimes dubbed a "
white hole In general relativity, a white hole is a hypothetical region of spacetime and Gravitational singularity, singularity that cannot be entered from the outside, although energy, matter, light and information can escape from it. In this sense, it is ...
". Particles can escape from a white hole but they can never return. The maximally extended Schwarzschild geometry can be divided into 4 regions each of which can be covered by a suitable set of Schwarzschild coordinates. The Kruskal–Szekeres coordinates, on the other hand, cover the entire spacetime manifold. The four regions are separated by event horizons. The transformation given above between Schwarzschild and Kruskal–Szekeres coordinates applies only in regions I and II (if we take the square root as positive). A similar transformation can be written down in the other two regions. The Schwarzschild time coordinate ''t'' is given by : \tanh\left(\frac\right) = \beginT/X & \text \\ X/T & \text\end In each region it runs from -\infty to +\infty with the infinities at the event horizons. Based on the requirements that the quantum process of
Hawking radiation Hawking radiation is black-body radiation released outside a black hole's event horizon due to quantum effects according to a model developed by Stephen Hawking in 1974. The radiation was not predicted by previous models which assumed that onc ...
is
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigr ...
, 't Hooft proposed that the regions I and III, and II and IV are just mathematical artefacts coming from choosing branches for roots rather than parallel universes and that the equivalence relation : (T, X, \Omega) \sim (-T, -X, -\Omega) should be imposed, where -\Omega is the antipode of \Omega on the 2-sphere. If we think of regions III and IV as having spherical coordinates but with a negative choice for the square root to compute r, then we just correspondingly use opposite points on the sphere to denote the same point in space, so e.g. : (t^\text, r^\text, \Omega^\text) = (t, r, \Omega) \sim (t^\text, r^\text, \Omega^\text) = (t, -r, -\Omega). This means that r^\text\Omega^\text = r^\text\Omega^\text = r\Omega. Since this is a free action by the group \mathbb/2\mathbb preserving the metric, this gives a well-defined
Lorentzian manifold In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riema ...
(everywhere except at the singularity). It identifies the limit t^\text = -\infty of the interior region II corresponding to the coordinate line segment T = -X ,\ T > 0, X < 0 with the limit t^\text = -\infty of the exterior region I corresponding to T = -X,\ T < 0, X> 0. The identification does mean that whereas each pair (T,X) \sim (-T, -X) \ne (0,0) corresponds to a sphere, the point (T,X) = (0,0) (corresponding to the event horizon r=2GM in the Schwarzschild picture) corresponds not to a sphere but to the
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
\mathbf^2 = S^2/\pm instead, and the topology of the underlying manifold is no longer \mathbb^4 - \mathrm = \mathbb^2 \times S^2. The manifold is no longer
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
, because a loop (involving superluminal portions) going from a point in space-time back to itself but at the opposite Kruskal–Szekeres coordinates cannot be reduced to a null loop.


Qualitative features of the Kruskal–Szekeres diagram

Kruskal–Szekeres coordinates have a number of useful features which make them helpful for building intuitions about the Schwarzschild spacetime. Chief among these is the fact that all radial light-like geodesics (the
world line The world line (or worldline) 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. The concept of a "world line" is distinguished from c ...
s of light rays moving in a radial direction) look like straight lines at a 45-degree angle when drawn in a Kruskal–Szekeres diagram (this can be derived from the metric equation given above, which guarantees that if dX = \plusmn dT\, then the
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time ...
ds = 0). All timelike world lines of slower-than-light objects will at every point have a slope closer to the vertical time axis (the ''T'' coordinate) than 45 degrees. So, a
light cone In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single Event (relativity), event (localized to a single point in space and a single moment in time) and traveling in all direct ...
drawn in a Kruskal–Szekeres diagram will look just the same as a light cone in a
Minkowski diagram 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 with ...
in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
. The event horizons bounding the black hole and white hole interior regions are also a pair of straight lines at 45 degrees, reflecting the fact that a light ray emitted at the horizon in a radial direction (aimed outward in the case of the black hole, inward in the case of the white hole) would remain on the horizon forever. Thus the two black hole horizons coincide with the boundaries of the future light cone of an event at the center of the diagram (at ''T''=''X''=0), while the two white hole horizons coincide with the boundaries of the past light cone of this same event. Any event inside the black hole interior region will have a future light cone that remains in this region (such that any world line within the event's future light cone will eventually hit the black hole singularity, which appears as a
hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, 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, called connected component ( ...
bounded by the two black hole horizons), and any event inside the white hole interior region will have a past light cone that remains in this region (such that any world line within this past light cone must have originated in the white hole singularity, a hyperbola bounded by the two white hole horizons). Note that although the horizon looks as though it is an outward expanding cone, the area of this surface, given by ''r'' is just 16\pi M^2, a constant. I.e., these coordinates can be deceptive if care is not exercised. It may be instructive to consider what curves of constant ''Schwarzschild'' coordinate would look like when plotted on a Kruskal–Szekeres diagram. It turns out that curves of constant ''r''-coordinate in Schwarzschild coordinates always look like hyperbolas bounded by a pair of event horizons at 45 degrees, while lines of constant ''t''-coordinate in Schwarzschild coordinates always look like straight lines at various angles passing through the center of the diagram. The black hole event horizon bordering exterior region I would coincide with a Schwarzschild ''t''-coordinate of +\infty while the white hole event horizon bordering this region would coincide with a Schwarzschild ''t''-coordinate of -\infty, reflecting the fact that in Schwarzschild coordinates an infalling particle takes an infinite coordinate time to reach the horizon (i.e. the particle's distance from the horizon approaches zero as the Schwarzschild ''t''-coordinate approaches infinity), and a particle traveling up away from the horizon must have crossed it an infinite coordinate time in the past. This is just an artifact of how Schwarzschild coordinates are defined; a free-falling particle will only take a finite
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time ...
(time as measured by its own clock) to pass between an outside observer and an event horizon, and if the particle's world line is drawn in the Kruskal–Szekeres diagram this will also only take a finite coordinate time in Kruskal–Szekeres coordinates. The Schwarzschild coordinate system can only cover a single exterior region and a single interior region, such as regions I and II in the Kruskal–Szekeres diagram. The Kruskal–Szekeres coordinate system, on the other hand, can cover a "maximally extended" spacetime which includes the region covered by Schwarzschild coordinates. Here, "maximally extended" refers to the idea that the spacetime should not have any "edges": any
geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
path can be extended arbitrarily far in either direction unless it runs into a
gravitational singularity A gravitational singularity, spacetime singularity, or simply singularity, is a theoretical condition in which gravity is predicted to be so intense that spacetime itself would break down catastrophically. As such, a singularity is by defini ...
. Technically, this means that a maximally extended spacetime is either "geodesically complete" (meaning any geodesic can be extended to arbitrarily large positive or negative values of its 'affine parameter', which in the case of a timelike geodesic could just be the
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time ...
), or if any geodesics are incomplete, it can only be because they end at a singularity. In order to satisfy this requirement, it was found that in addition to the black hole interior region (region II) which particles enter when they fall through the event horizon from the exterior (region I), there has to be a separate white hole interior region (region IV) which allows us to extend the trajectories of particles which an outside observer sees rising up ''away'' from the event horizon, along with a separate exterior region (region III) which allows us to extend some possible particle trajectories in the two interior regions. There are actually multiple possible ways to extend the exterior Schwarzschild solution into a maximally extended spacetime, but the Kruskal–Szekeres extension is unique in that it is a maximal, analytic,
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
vacuum solution in which all maximally extended geodesics are either complete or else the curvature scalar diverges along them in finite affine time.


Lightcone variant

In the literature, the Kruskal–Szekeres coordinates sometimes also appear in their lightcone variant: : U = T - X : V = T + X, in which the metric is given by : ds^ = -\frace^(dU dV) + r^2 d\Omega^2, and ''r'' is defined implicitly by the equation : UV = \left(1-\frac\right)e^. These lightcone coordinates have the useful feature that radially outgoing
null Null may refer to: Science, technology, and mathematics Astronomy *Nuller, an optical tool using interferometry to block certain sources of light Computing *Null (SQL) (or NULL), a special marker and keyword in SQL indicating that a data value do ...
geodesics In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connec ...
are given by U = \text, while radially ingoing null geodesics are given by V = \text. Furthermore, the (future and past) event horizon(s) are given by the equation UV = 0, and curvature singularity is given by the equation UV = 1. The lightcone coordinates derive closely from Eddington–Finkelstein coordinates.


See also

* Schwarzschild coordinates * Lemaître coordinates * Eddington–Finkelstein coordinates * Isotropic coordinates * Gullstrand–Painlevé coordinates


References


Sources

* * * * * * {{DEFAULTSORT:Kruskal-Szekeres Coordinates Coordinate charts in general relativity Lorentzian manifolds