Isotropic coordinates

Isotropic coordinates

In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of "nested round spheres". There are several different types of coordinate chart which are "adapted" to this family of nested spheres; the best known is the Schwarzchild chart, but the isotropic chart is also often useful.The defining characteristic of an isotropic chart is that its radial coordinate (which is different from the radial coordinate of a Schwarschild chart) is defined so that light cones appear "round". This means that (except in the trivial case of a locally flat manifold), the angular isotropic coordinates do not faithfully represent distances within the nested spheres, nor does the radial coordinate faithfully represent radial distances. On the other hand, angles in the constant time hyperslices are represented without distortion, hence the name of the chart.

Isotropic charts are most often applied to static spherically symmetric spacetimes in metric theories of gravitation such as general relativity, but they can also be used in modeling a spherically pulsating fluid ball, for example. For isolated spherically symmetric solutions of the Einstein field equation, at large distances, the isotropic and Schwarzschild charts become increasingly similar to the usual polar spherical chart on Minkowski spacetime.

Definition

In an isotropic chart (on a static spherically symmetric spacetime), the line element takes the form:ds^2 = -f(r)^2 , dt^2 + g(r)^2 , left( dr^2 + r^2 , left( d heta^2 + sin( heta)^2 , dphi^2 ight) ight), :-infty < t < infty, , r_0 < r < r_1, , 0 < heta < pi, , -pi < phi < pi

Depending on context, it may be appropriate to regard f,g as undetermined functions of the radial coordinate (for example, in deriving an exact static spherically symmetric solution of the Einstein field equation). Alternatively, we can plug in specific functions (possibly depending on some parameters) to obtain an isotropic coordinate chart on a specific Lorentzian spacetime.

Killing vector fields

The Lie algebra of Killing vector fields of a spherically symmetric static spacetime takes the same form in the isotropic chart as in the Schwarzschild chart. Namely, this algebra is generated by the timelike "irrotational" Killing vector field: partial_t and three spacelike Killing vector fields: partial_phi, ; ; sin( heta) , partial_ heta + cot( heta) , cos(phi) partial_phi, ; ; cos( heta) , partial_ heta - cot( heta) , sin(phi) partial_phiHere, saying that vec{X} = partial_t is irrotational means that the vorticity tensor of the corresponding timelike congruence vanishes; thus, this Killing vector field is hypersurface orthogonal. The fact that the spacetime admits an irrotational timelike Killing vector field is in fact the defining characteristic of a static spacetime. One immediate consequence is that the "constant time coordinate surfaces" t=t_0 form a family of (isometric) "spatial hyperslices" (spacelike hypersurfaces).

Unlike the Schwarzschild chart, the isotropic chart is not well suited for constructing embedding diagrams of these hyperslices.

A family of static nested spheres

The surfaces t=t_0, , r=r_0 appear as round spheres (when we plot loci in polar spherical fashion), and from the form of the line element, we see that the metric restricted to any of these surfaces is: dsigma^2 = g(r_0)^2 , r_0^2 , left( d heta^2 + sin( heta)^2 , dphi^2 ight), ; 0 < heta < pi, -pi < phi < pi That is, these "nested coordinate spheres" do in fact represent geometric spheres, but the appearance of g(r_0) , r rather than r shows that the radial coordinate does not correspond to area in the same way as for spheres in ordinary euclidean space. Compare Schwarschild coordinates, where the radial coordinate does have its natural interpretation in terms of the nested spheres.

Coordinate singularities

The loci phi=-pi, , pi mark the boundaries of the isotropic chart, and just as in the Schwarschild chart, we tacitly assume that these two loci are indentified, so that our putative round spheres are indeed topological spheres.

Just as for the Schwarschild chart, the range of the radial coordinate may be limited if the metric or its inverse blows up for some value(s) of his coordinate.

A metric Ansatz

The line element given above, with f,g, regarded as undetermined functions of the isotropic coordinate r, is often used as a metric Ansatz in deriving static spherically symmetric solutions in general relativity (or other metric theories of gravitation).

As an illustration, we will sketch how to compute the connection and curvature using Cartan's exterior calculus method. First, we read off the line element a coframe field,: sigma^0 = -f(r) , dt: sigma^1 = g(r) , dr: sigma^2 = g(r) , r , d heta: sigma^3 = g(r) , r , sin( heta) , dphiwhere we regard f,g as undetermined smooth functions of r. (The fact that our spacetime admits a frame having this particular trigonometric form is yet another equivalent expression of the notion of an isotropic chart in a static, spherically symmetric Lorentzian manifold). Taking the exterior derivatives and using the first Cartan structural equation, we find the nonvanishing "connection one-forms":{omega^0}_1 = frac{f' , dt}{g}:{omega^1}_2 = -left( 1 + frac{r , g'}{g} ight) , d heta:{omega^1}_3 = -left( 1 + frac{r , g'}{g} ight) , sin( heta) , dphi:{omega^2}_3 = -cos( heta) , dphiTaking exterior derivatives again and plugging into the second Cartan structural equation, we find the "curvature two-forms".

ee also

*static spacetime,
*spherically symmetric spacetime,
*static spherically symmetric perfect fluids,
*Schwarzschild coordinates, another popular chart for static spherically symmetric spacetimes,
*frame fields in general relativity, for more about frame fields and coframe fields.


Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Isotropic radiator — An isotropic radiator is a theoretical point source of waves which exhibits the same magnitude or properties when measured in all directions. It has no preferred direction of radiation. It radiates uniformly in all directions over a sphere… …   Wikipedia

  • Gaussian polar coordinates — In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres. In each of these spheres, every point can be carried to any other by an appropriate rotation about the center of symmetry.There are… …   Wikipedia

  • Schwarzschild coordinates — In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres . In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical… …   Wikipedia

  • Gullstrand-Painlevé coordinates — GullStrand Painlevé (GP) coordinates were proposed by Paul Painlevé [Paul Painlevé, “La mécanique classique et la théorie de la relativité”, C. R. Acad. Sci. (Paris) 173, 677–680 (1921). ] and Allvar Gullstrand [Allvar Gullstrand, “Allgemeine… …   Wikipedia

  • Kruskal-Szekeres coordinates — In general relativity Kruskal Szekeres coordinates, named for Martin Kruskal and George Szekeres, are a coordinate system for the Schwarzschild geometry for a black hole. These coordinates have the advantage that they cover the entire spacetime… …   Wikipedia

  • Deriving the Schwarzschild solution — The Schwarzschild solution is one of the simplest and most useful solutions of the Einstein field equations (see general relativity). It describes spacetime in the vicinity of a non rotating massive spherically symmetric object. It is worthwhile… …   Wikipedia

  • Isotropy — is uniformity in all orientations; it is derived from the Greek iso (equal) and tropos (direction). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy.… …   Wikipedia

  • Static spherically symmetric perfect fluid — In metric theories of gravitation, particularly general relativity, a static spherically symmetric perfect fluid solution (a term which is often abbreviated as ssspf) is a spacetime equipped with suitable tensor fields which models a static round …   Wikipedia

  • List of mathematics articles (I) — NOTOC Ia IA automorphism ICER Icosagon Icosahedral 120 cell Icosahedral prism Icosahedral symmetry Icosahedron Icosian Calculus Icosian game Icosidodecadodecahedron Icosidodecahedron Icositetrachoric honeycomb Icositruncated dodecadodecahedron… …   Wikipedia

  • Metric tensor (general relativity) — This article is about metrics in general relativity. For a discussion of metrics in general, see metric tensor. Metric tensor of spacetime in general relativity written as a matrix. In general relativity, the metric tensor (or simply, the metric) …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”