In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of 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.
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.
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 Where
Depending on context, it may be appropriate to regard a and b as undetermined functions of the radial coordinate (for example, in deriving an exact static spherically symmetric solution of the Einstein field equation).
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.
is irrotational means that the vorticity tensor of the corresponding timelike congruence vanishes; thus, this Killing vector field is hypersurface orthogonal.
(This is not true for example in the Boyer–Lindquist chart for the exterior region of the Kerr vacuum, where the timelike coordinate vector is not hypersurface orthogonal.)
appear as round spheres (when we plot loci in polar spherical fashion), and from its form, we see that the Schwarzschild metric restricted to any of these surfaces is positive definite and given by Where
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.
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.
In order to compute the proper time interval between two events on the world line of one of these observers, we must integrate
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
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.
People 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.
If the true distances are smaller, we should embed our Riemannian manifold as a spacelike surface in E1,2 instead.
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 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.
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.
(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).
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.
From this we can read off the Bel decomposition with respect to the timelike unit vector field
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.
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).
Some examples of exact solutions which can be obtained in this way include: It is natural to consider nonstatic but spherically symmetric spacetimes, with a generalized Schwarzschild chart in which the metric 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: