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 coordinate chart on a static and spherically symmetric spacetime, 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.
Contents
- 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
- References
These charts have many applications in metric theories of gravitation such as general relativity. They are most often used in static spherically symmetric spacetimes. In the case of general relativity, Birkhoff's theorem 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 solution inside the event horizon of a spherically symmetric black hole 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 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 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 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 of Killing vector fields is generated by the timelike irrotational Killing vector field
and three spacelike Killing vector fields
Here, saying that
A family of static nested spheres
In the Schwarzschild chart, the surfaces
That is, these nested coordinate spheres do in fact represent geometric spheres with
- surface area
A = 4 π r 0 2 - Gaussian curvature
K = 1 / r 0 2
That is, they are geometric round spheres. Moreover, the angular coordinates
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
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
In order to compute the proper time interval between two events on the world line of one of these observers, we must integrate
Coordinate singularities
Looking back at the coordinate ranges above, note that the coordinate singularity at
When we said above that
Possibly, of course,
Visualizing the static hyperslices
To better understand the significance of the Schwarzschild radial coordinate, it may help to embed one of the spatial hyperslices
To embed this surface (or at an annular ring) in E3, 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 E1,2 instead. For example, we might have
The point is that the defining characteristic of a Schwarzschild chart 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 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
(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. We obtain:
Fifth, we can lower indices and organize the components
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 with respect to the timelike unit vector field
The magnetogravitic tensor vanishes identically, and the topogravitic tensor, from which (using the fact that
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 of our coframe field is
The fact that the factor
Some exact solutions admitting Schwarzschild charts
Some examples of exact solutions which can be obtained in this way include:
Generalizations
It is natural to consider nonstatic but spherically symmetric spacetimes, with a generalized Schwarzschild chart in which the 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: