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In the theory of Lorentzian manifolds, spherically symmetric spacetime

Spherically symmetric spacetime

A spherically symmetric spacetime is a spacetime whose isometry group contains a subgroup which is isomorphic to the group SO and the orbits of this group are 2-dimensional spheres . The isometries are then interpreted as rotations and a spherically symmetric spacetime is often described as one...

s 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 coordinate chart on a static

Static spacetime

In general relativity, a spacetime is said to be static if it admits a global, non-vanishing, timelike Killing vector field K which is irrotational, i.e., whose orthogonal distribution is involutive...

and spherically symmetric

Spherically symmetric spacetime

spacetime

Spacetime

In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

, which is adapted to these nested round spheres. The defining characteristic of Schwarzschild chart is that the radial coordinate possesses a natural geometric interpretation in terms of the surface area and Gaussian curvature of each sphere. However, radial distances and angles are not accurately represented.

These charts have many applications in metric theories of gravitation such as general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

. They are most often used in static

Static spacetime

spherically symmetric spacetimes. In the case of general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, Birkhoff's theorem

Birkhoff's theorem (relativity)

In general relativity, Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. This means that the exterior solution must be given by the Schwarzschild metric....

states that every isolated spherically symmetric vacuum or electrovacuum solution of the Einstein field equation is static, but this is certainly not true for perfect fluids. We should also note that the extension of the exterior region of the Schwarzschild vacuum

Schwarzschild metric

In Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...

solution inside the event horizon

Event horizon

In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...

of a spherically symmetric black hole

Black hole

A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

is not static inside the horizon, and the family of (spacelike) nested spheres cannot be extended inside the horizon, so the Schwarzschild chart for this solution necessarily breaks down at the horizon.

Definition

Specifying a metric tensor

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

is part of the definition of any Lorentzian manifold. The simplest way to define this tensor is to define it in compatible local coordinate charts and verify that the same tensor is defined on the overlaps of the domains of the charts. In this article, we will only attempt to define the metric tensor in the domain of a single chart.

In a Schwarzschild chart (on a static spherically symmetric spacetime), the line element

Line element

A line element ds in mathematics can most generally be thought of as the change in a position vector in an affine space expressing the change of the arc length. An easy way of visualizing this relationship is by parametrizing the given curve by Frenet–Serret formulas...

takes the form

Depending on context, it may be appropriate to regard f and 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 a Schwarzschild coordinate chart on a specific Lorentzian spacetime.

If this turns out to admit a stress-energy tensor

Stress-energy tensor

The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...

such that the resulting model satisfies the Einstein field equation (say, for a static spherically symmetric perfect fluid obeying suitable energy conditions and other properties expected of reasonable perfect fluid), then, with appropriate tensor fields representing physical quantities such as matter and momentum densities, we have a piece of a possibly larger spacetime; a piece which can be considered a local solution of the Einstein field equation.

Killing vector fields

With respect to the Schwarzschild chart, the Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

of Killing vector field

Killing vector field

In mathematics, a Killing vector field , named after Wilhelm Killing, is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold...

s is generated by the timelike irrotational Killing vector field

and three spacelike Killing vector fields

Here, saying that

is irrotational means that the vorticity tensor of the corresponding timelike congruence vanishes; thus, this Killing vector field is hypersurface orthogonal. The fact that our spacetime admits an irrotational timelike Killing vector field is in fact the defining characteristic of a static spacetime

Static spacetime

. One immediate consequence is that the constant time coordinate surfaces

form a family of (isometric) spatial hyperslices. (This is not true for example in the Boyer-Lindquist chart

Boyer-Lindquist coordinates

A generalization of the coordinates used for the metric of a Schwarzschild black hole that can be used to express the metric of a Kerr black hole.The coordinate transformation from Boyer–Lindquist coordinates r, \theta, \phi to cartesian coordinates x, y, z is given bywhereThe Hamiltonian for test...

for the exterior region of the Kerr vacuum

Kerr metric

The Kerr metric describes the geometry of empty spacetime around an uncharged axially-symmetric black-hole with an event horizon which is topologically a sphere. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which...

, where the timelike coordinate vector is not hypersurface orthogonal.)

A family of static nested spheres

In the Schwarzschild chart, the surfaces

appear as round spheres (when we plot loci

Locus (mathematics)

In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....

in polar spherical fashion), and from the form of the line element, we see that the metric restricted to any of these surfaces is

That is, these nested coordinate spheres do in fact represent geometric spheres with

surface area
Surface area
Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces...
Gaussian curvature
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...

That is, they are geometric round spheres. Moreover, the angular coordinates

are exactly the usual polar spherical angular coordinates:

is sometimes called the colatitude and

is usually called the longitude. This is essentially the defining geometric feature of the Schwarzschild chart.

It may help to add that the four Killing fields given above, considered as abstract vector fields on our Lorentzian manifold, give the truest expression of both the symmetries of a static spherically symmetric spacetime, while the particular trigonometric form which they take in our chart is the truest expression of the meaning of the term Schwarzschild chart. In particular, the three spatial Killing vector fields have exactly the same form as the three nontranslational Killing vector fields in a spherically symmetric chart on E³; that is, they exhibit the notion of arbitrary Euclidean rotation about the origin or spherical symmetry.

However, note well: in general, the Schwarzschild radial coordinate does not accurately represent radial distances, i.e. distances taken along the spacelike geodesic congruence which arise as the integral curves of

. Rather, to find a suitable notion of 'spatial distance

Proper length

In relativistic physics, proper length is an invariant measure of the distance between two spacelike-separated events, or of the length of a spacelike path within a spacetime....

' between two of our nested spheres, we should integrate

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

along some coordinate ray from the origin:

Similarly, we can regard each sphere as the locus of a spherical cloud of idealized observers, who must (in general) use rocket engines to accelerate radially outward in order to maintain their position. These are static observers, and they have world lines of form

, which of course have the form of vertical coordinate lines in the Schwarzschild chart.

In order to compute the proper time

Proper time

In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...

interval between two events on the world line

World line

In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" by the time dimension, and typically encompasses a large area of spacetime wherein...

of one of these observers, we must integrate

along the appropriate coordinate line:

Coordinate singularities

Looking back at the coordinate ranges above, note that the coordinate singularity at

marks the location of the North pole of one of our static nested spheres, while

marks the location of the South pole. Just as for an ordinary polar spherical chart on E³, for topological reasons we cannot obtain continuous coordinates on the entire sphere; we must choose some longitude (a great circle) to act as the prime meridian

and cut this out of the chart. The result is that we cut out a closed half plane from each spatial hyperslice

including the axis

and a half plane extending from that axis.

When we said above that

is a Killing vector field, we omitted the pedantic but important qualifier that we are thinking of

as a cyclic coordinate, and indeed thinking of our three spacelike Killing vectors as acting on round spheres.

Possibly, of course,

, in which case we must also excise the region outside some ball, or inside some ball, from the domain of our chart. This happens whenever f or g blow up at some value of the Schwarzschild radial coordinate r.

Visualizing the static hyperslices

To better understand the significance of the Schwarzschild radial coordinate, it may help to embed one of the spatial hyperslices

(they are of course all isometric to one another) in a flat Euclidean space. Humans who find it difficult to visualize four dimensional Euclidean space will be glad to observe that we can take advantage of the spherical symmetry to suppress one coordinate. This may be conveniently achieved by setting

. Now we have a two-dimensional Riemannian manifold with a local radial coordinate chart,

To embed this surface (or at an annular

Annulus (mathematics)

In mathematics, an annulus is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. Or, it is the area between two concentric circles...

ring) in E³

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

, we adopt a frame field in E³ which

is defined on a parameterized surface, which will inherit the desired metric from the embedding space,
is adapted to our radial chart,
features an undetermined function h(r).

To wit, consider the parameterized surface

The coordinate vector fields on this surface are

The induced metric inherited when we restrict the Euclidean metric on E³ to our parameterized surface is

To identify this with the metric of our hyperslice, we should evidently choose h(r) so that

To take a somewhat silly example, we might have

.

This works for surfaces in which true distances between two radially separated points are larger than the difference between their radial coordinates. If the true distances are smaller, we should embed our Riemannian manifold as a spacelike surface in E^1,2 instead. For example, we might have

. Sometimes we might need two or more local embeddings of annular rings (for regions of positive or negative Gaussian curvature). In general, we should not expect to obtain a global embedding in any one flat space (with vanishing Riemann tensor).

The point is that the defining characteristic of a Schwarzschild charts in terms of the geometric interpretation of the radial coordinate is just what we need to carry out (in principle) this kind of spherically symmetric embedding of the spatial hyperslices.

A metric Ansatz

The line element given above, with f,g regarded as undetermined functions of the Schwarzschild radial coordinate r, is often used as a metric ansatz

Ansatz

Ansatz is a German noun with several meanings in the English language.It is widely encountered in physics and mathematics literature.Since ansatz is a noun, in German texts the initial a of this word is always capitalised.-Definition:...

in deriving static spherically symmetric solutions in general relativity (or other metric theories of gravitation).

As an illustration, we will indicate how to compute the connection and curvature using Cartan's exterior calculus method. First, we read off the line element a coframe field,

where 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 a Schwarzschild chart in a static, spherically symmetric Lorentzian manifold).

Second, we compute the exterior derivatives of these cobasis one-forms:

Comparing with Cartan's first structural equation (or rather its integrability condition),

we guess expressions for the connection one-forms. (The hats are just a notational device for reminding us that the indices refer to our cobasis one-forms, not to the coordinate one-forms

.)

If we recall which pairs of indices are symmetric (space-time) and which are antisymmetric (space-space) in

, we can confirm that the six connection one-forms are

(In this example, only four of the six are nonvanishing.)
We can collect these one-forms into a matrix of one-forms, or even better an SO(1,3)-valued one-form.
Note that the resulting matrix of one-forms will not quite be antisymmetric as for an SO(4)-valued one-form; we need to use instead a notion of transpose arising from the Lorentzian adjoint.

Third, we compute the exterior derivatives of the connection one-forms and use Cartan's second structural equation

to compute the curvature two forms. Fourth, using the formula

where the Bach bars indicate that we should sum only over the six increasing pairs of indices (i,j), we can read off the linearly independent components of the Riemann tensor with respect to our coframe and its dual frame field

Frame fields in general relativity

In general relativity, a frame field is a set of four orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime...

. We obtain:

Fifth, we can lower indices and organize the components

into a matrix

where E,L are symmetric (six linearly independent components, in general) and B is traceless (eight linearly independent components, in general), which we think of as representing a linear operator on the six dimensional vector space of two forms (at each event). From this we can read off the Bel decomposition

Bel decomposition

In semi-Riemannian geometry, the Bel decomposition, taken with respect to a specific timelike congruence, is a way of breaking up the Riemann tensor of a pseudo-Riemannian manifold into four pieces. It was introduced in 1959 by the physicist Lluis Bel....

with respect to the timelike unit vector field

. The electrogravitic tensor

Electrogravitic tensor

In general relativity, the tidal tensor or gravitoelectric tensor is one of the pieces in the Bel decomposition of the Riemann tensor. It is physically interpreted as giving the tidal stresses on small bits of a material object , or the tidal accelerations of a small cloud of test particles in a...

The magnetogravitic tensor

Magnetogravitic tensor

In general relativity, the magnetogravitic tensor is one of the three pieces appearing in the Bel decomposition of the Riemann tensor.The magnetogravitic tensor can be interpreted physically as a specifying possible spin-spin forces on spinning bits of matter, such as spinning test particles....

vanishes identically, and the topogravitic tensor

Topogravitic tensor

In general relativity, the topogravitic tensor is one of the three pieces of the Bel decomposition of the Riemann tensor.The topogravitic tensor can be interpreted as representing the sectional curvatures for the spatial part of a frame field....

, from which (using the fact that

is irrotational) we can determine the three-dimensional Riemann tensor of the spatial hyperslices, is

This is all valid for any Lorentzian manifold, but we note that in general relativity, the electrogravitic tensor controls tidal stresses on small objects, as measured by the observers corresponding to our frame, and the magnetogravitic tensor controls any spin-spin forces on spinning objects, as measured by the observers corresponding to our frame.

The dual frame field

Frame fields in general relativity

of our coframe field is

The fact that the factor

only multiplies the first of the three orthonormal spacelike vector fields here means that Schwarzschild charts are not spatially isotropic (except in the trivial case of a locally flat spacetime); rather, the light cones appear (radially flattened) or (radially elongated). This is of course just another way of saying that Schwarzschild charts correctly represent distances within each nested round sphere, but the radial coordinate does not faithfully represent radial proper distance.

Some exact solutions admitting Schwarzschild charts

Some examples of exact solutions which can be obtained in this way include:

the exterior region of the Schwarzschild vacuum
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...

,
ditto, for the Reissner-Nordström electrovacuum, which includes the previous example as a special case,
ditto, for the Reissner-Nordström-de Sitter electrolambdavacuum, which includes the previous example as a special case,
the Janis-Newman-Winacour solution (which models the exterior of a static spherically symmetric object endowed with a massless minimally coupled scalar field),
stellar models obtained by matching an interior region which is a static spherically symmetric perfect fluid
Static spherically symmetric perfect fluid
In metric theories of gravitation, particularly general relativity, a static spherically symmetric perfect fluid solution is a spacetime equipped with suitable tensor fields which models a static round ball of a fluid with isotropic pressure.Such solutions are often used as idealized models of...

solution across a spherical locus of vanishing pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

to an exterior region, which is locally isometric to part of the Schwarzschild vacuum region.

Generalizations

It is natural to consider nonstatic but spherically symmetric spacetimes, with a generalized Schwarzschild chart in which the line element

Line element

takes the form

Generalizing in another direction, we can use other coordinate systems on our round two-spheres, to obtain for example a stereographic Schwarzschild chart which is sometimes useful:

Schwarzschild coordinates

Definition

Killing vector fields

A family of static nested spheres

Coordinate singularities

Visualizing the static hyperslices

A metric Ansatz

Some exact solutions admitting Schwarzschild charts

Generalizations

See also