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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 ...
, Eddington–Finkelstein coordinates are a pair of
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 ...
s for a Schwarzschild geometry (e.g. a spherically symmetric
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 ...
) which are adapted to radial null geodesics. Null geodesics are the worldlines of
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
s; radial ones are those that are moving directly towards or away from the central mass. They are named for Arthur Stanley Eddington and David Finkelstein. Although they appear to have inspired the idea, neither ever wrote down these coordinates or the metric in these coordinates.
Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, Philosophy of science, philosopher of science and Nobel Prize in Physics, Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics i ...
seems to have been the first to write down the null form but credits it to the above paper by Finkelstein, and, in his Adams Prize essay later that year, to Eddington and Finkelstein. Most influentially, Misner, Thorne and Wheeler, in their book ''
Gravitation In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
'', refer to the null coordinates by that name. In these coordinate systems, outward (inward) traveling radial light rays (which each follow a null geodesic) define the surfaces of constant "time", while the radial coordinate is the usual area coordinate so that the surfaces of rotation symmetry have an area of . One advantage of this coordinate system is that it shows that the apparent singularity at the Schwarzschild radius is only a coordinate singularity and is not a true physical singularity. While this fact was recognized by Finkelstein, it was not recognized (or at least not commented on) by Eddington, whose primary purpose was to compare and contrast the spherically symmetric solutions in Whitehead's theory of gravitation and Einstein's version of the theory of relativity.


Schwarzschild metric

The Schwarzschild coordinates are (t,r,\theta,\varphi), and in these coordinates the Schwarzschild metric is well known: :ds^2 = \left(1-\frac \right) \, dt^2 - \left(1-\frac\right)^ \, dr^2- r^2 d\Omega^2 where :d\Omega^2\equiv d\theta^2+\sin^2\theta\,d\varphi^2. is the standard Riemannian metric of the unit 2-sphere. Note the conventions being used here are 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 ...
of ( + − − − ) and the
natural units In physics, natural unit systems are measurement systems for which selected physical constants have been set to 1 through nondimensionalization of physical units. For example, the speed of light may be set to 1, and it may then be omitted, equa ...
where ''c'' = 1 is the dimensionless speed of light, ''G'' 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 ...
, and ''M'' is the characteristic mass of the Schwarzschild geometry.


Tortoise coordinate

Eddington–Finkelstein coordinates are founded upon the tortoise coordinate – a name that comes from one of
Zeno of Elea Zeno of Elea (; ; ) was a pre-Socratic Greek philosopher from Elea, in Southern Italy (Magna Graecia). He was a student of Parmenides and one of the Eleatics. Zeno defended his instructor's belief in monism, the idea that only one single en ...
's paradoxes on an imaginary footrace between "swift-footed" Achilles and a tortoise. The tortoise coordinate r^* is defined: :r^* = r + 2GM\ln\left, \frac-1 \. so as to satisfy: :\frac = \left(1-\frac\right)^. The tortoise coordinate r^* approaches -\infty as r approaches the Schwarzschild radius 2GM. When some probe (such as a light ray or an observer) approaches a black hole event horizon, its Schwarzschild time coordinate grows infinite. The outgoing null rays in this coordinate system have an infinite change in ''t'' on travelling out from the horizon. The tortoise coordinate is intended to grow infinite at the appropriate rate such as to cancel out this singular behaviour in coordinate systems constructed from it. The increase in the time coordinate to infinity as one approaches the event horizon is why information could never be received back from any probe that is sent through such an event horizon. This is despite the fact that the probe itself can nonetheless travel past the horizon. It is also why the space-time metric of the black hole, when expressed in Schwarzschild coordinates, becomes singular at the horizon – and thereby fails to be able to fully chart the trajectory of an infalling probe.


Metric

The ingoing Eddington–Finkelstein coordinates are obtained by replacing the coordinate ''t'' with the new coordinate ''v=t+r^*''. In these coordinates, the Schwarzschild metric can be written as :ds^2 = \left(1-\frac \right) dv^2 - 2 \, dv \, dr - r^2 d\Omega^2. where again d\Omega^2 = d\theta^2+\sin^2\theta \, d\varphi^2 is the standard Riemannian metric on the unit radius 2-sphere. Likewise, the outgoing Eddington–Finkelstein coordinates are obtained by replacing ''t'' with the null coordinate ''u=t-r^*''. The metric is then given by :ds^2 = \left(1-\frac \right) du^2 + 2 \, du \, dr - r^2 d\Omega^2. In both these coordinate systems the metric is explicitly non-singular at the Schwarzschild radius (even though one component vanishes at this radius, the determinant of the metric is still non-vanishing and the inverse metric has no terms which diverge there.) Note that for radial null rays, ''v=const'' or ''v-2r^*=const'' or equivalently ''u+2r^*=const'' or ''u=const'' we have ''dv/dr'' and ''du/dr'' approach 0 and ±2 at large ''r'', not ±1 as one might expect if one regarded ''u'' or ''v'' as "time". When plotting Eddington–Finkelstein diagrams, surfaces of constant ''u'' or ''v'' are usually drawn as cones, with ''u'' or ''v'' constant lines drawn as sloping at 45 degree rather than as planes (see for instance Box 31.2 of MTW). Some sources instead take t' = t \pm (r^* - r)\,, corresponding to planar surfaces in such diagrams. In terms of this t' the metric becomes :ds^2 = \left( 1-\frac \right) dt'^2 \pm \frac \,dt' \,dr - \left( 1 + \frac \right) \,dr^2 - r^2 d\Omega^2 =(dt'^2 - dr^2 - r^2 d\Omega^2)+\frac (dt'\pm dr)^2 which is Minkowskian at large ''r''. (This was the coordinate time and metric that both Eddington and Finkelstein presented in their papers.) The Eddington–Finkelstein coordinates are still incomplete and can be extended. For example, the outward traveling timelike geodesics defined by (with ''τ'' the proper time) : r(\tau)= \sqrt : \begin v(\tau) & = \int \frac \, d\tau \\ & = C+\tau +2\sqrt +4GM\ln\left(\sqrt-1 \right) \end has ''v''(''τ'') → −∞ as ''τ'' → 2''GM''. Ie, this timelike geodesic has a finite proper length into the past where it comes out of the horizon (''r'' = 2''GM'') when ''v'' becomes minus infinity. The regions for finite ''v'' and ''r'' < 2''GM'' is a different region from finite ''u'' and ''r'' < 2''GM''. The horizon ''r'' = 2''GM'' and finite ''v'' (the black hole horizon) is different from that with ''r'' = 2''GM'' and finite ''u ''(the
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 ...
horizon) . The metric in Kruskal–Szekeres coordinates covers all of the extended Schwarzschild spacetime in a single coordinate system. Its chief disadvantage is that in those coordinates the metric depends on both the time and space coordinates. In Eddington–Finkelstein, as in Schwarzschild coordinates, the metric is independent of the "time" (either ''t'' in Schwarzschild, or ''u'' or ''v'' in the various Eddington–Finkelstein coordinates), but none of these cover the complete spacetime. The Eddington–Finkelstein coordinates have some similarity to the Gullstrand–Painlevé coordinates in that both are time independent, and penetrate (are regular across) either the future (black hole) or the past (white hole) horizons. Both are not diagonal (the hypersurfaces of constant "time" are not orthogonal to the hypersurfaces of constant ''r''.) The latter have a flat spatial metric, while the former's spatial ("time" constant) hypersurfaces are null and have the same metric as that of a null cone in Minkowski space (t=\pm r in flat spacetime).


See also

* Schwarzschild coordinates * Kruskal–Szekeres coordinates * Lemaître coordinates * Gullstrand–Painlevé coordinates * Vaidya metric


References

{{DEFAULTSORT:Eddington-Finkelstein Coordinates Coordinate charts in general relativity