Spherical solutions for stars

10.1 Co o rd i n a t e s f o r s p h e r i c a l l y s y m m e t r i c s p a ce t i m e s

For our first study of strong gravitational fields in GR, we will consider spherically symmetric systems. They are reasonably simple, yet physically very important, since very many astrophysical objects appear to be nearly spherical. We begin by choosing the coordinate system to reflect the assumed symmetry.

Flat space in spher ical coord inates

By defining the usual coordinates (r, θ , φ), the line element of Minkowski space can be written

Each surface of constant r and t is a sphere or, more precisely, a two-sphere – a twodimensional spherical surface. Distances along curves confined to such a sphere are given by the above equation with dt = dr = 0:

which defines the symbol d 2 for the element of solid angle. We note that such a sphere has circumference 2π r and area 4π r2, i.e. 2π times the square root of the coefficient of d 2 and 4π times the coefficient of d 2 respectively. Conversely, any two-surface where the line element is Eq. (10.2) with r2 independent of θ and φ has the intrinsic geometry of a two-sphere.

Two - spheres in a curved spacetime

The statement that a spacetime is spherically symmetric can now be made more precise: it implies that every point of spacetime is on a two-surface which is a two-sphere, i.e. whose line element is

where f (r , t) is an unknown function of the two other coordinates of our manifold r and t. The area of each sphere is 4π f (r , t). We define the radial coordinate r of our spherical

geometry such that f (r , t) := r2. This represents a coordinate transformation from (r , t) to (r, t). Then any surface r = const., t = const. is a two-sphere of area 4π r2 and circumference 2π r. This coordinate r is called the ‘curvature coordinate’ or ‘area coordinate’ because it defines the radius of curvature and area of the spheres. There is no a priori relation between r and the proper distance from the center of the sphere to its surface. This r is defined only by the properties of the spheres themselves. Since their ‘centers’ (at r = 0 in flat space) are not points on the spheres themselves, the statement that a spacetime is spherically symmetric does not require even that there be a point at the center. A simple counter-example of a two-space in which there are circles but no point at the center of them is in Fig. 10.1. The space consists of two sheets, which are joined by a ‘throat’. The whole thing is symmetric about an axis along the middle of the throat, but the points on this axis – which are the ‘centers’ of the circles – are not part of the two-dimensional surface illustrated. Yet if φ is an angle about the axis, the line element on each circle is just r2dφ2, where r is a constant labeling each circle. This r is the same sort of coordinate that we use in our spherically symmetric spacetime.

Meshing the two - spheres i nto a thre e - space for t = const

Consider the spheres at r and r + d r. Each has a coordinate system (θ , φ), but up to now we have not required any relation between them. That is, we could conceive of having the pole for the sphere at r in one orientation, while that for r + d r was in another. The sensible thing is to say that a line of θ = const., φ = const. is orthogonal to the two-spheres. Such a line has, by definition, a tangent er. Since the vectors eθ and eφ lie in the spheres, we require er · eθ = er · eφ = 0. This means grθ = grφ = 0 (recall Eqs. (3.3) and (3.21)). This is a definition of the coordinates, allowed by spherical symmetry. We thus have restricted

Spherical ly symmetric spacetime

Since not only the spaces t = const. are spherically symmetric, but also the whole spacetime, we must have that a line r = const., θ = const., φ = const. is also orthogonal to the two-spheres. Otherwise there would be a preferred direction in space. This means that et is orthogonal to eθ and eφ , or gtθ = gtφ = 0. So now we have

This is the general metric of a spherically symmetric spacetime, where g00, g0r, and grr are functions of r and t. We have used our coordinate freedom to reduce it to the simplest possible form.

10.2 S t a t i c s p h e r i c a l l y s y m m e t r i c s p a ce t i m e s

The metric

Clearly, the simplest physical situation we can describe is a quiescent star or black hole – a static system. We define a static spacetime to be one in which we can find a time coordinate t with two properties: (i) all metric components are independent of t, and (ii) the geometry is unchanged by time reversal, t → −t. The second condition means that a film made of the situation looks the same when run backwards. This is not logically implied by (i), as the example of a rotating star makes clear: time reversal changes the sense of rotation, but the metric components will be constant in time. (A spacetime with property (i) but not necessarily (ii) is said to be stationary.)

Condition (ii) has the following implication. The coordinate transformation (t, r, θ , φ) →

¯

(−t, r, θ , φ) has 00 = −1, ij = δij, and we find

Since the geometry must be unchanged (i.e. since gαβ¯ ¯ = gαβ ), we must have g0r ≡ 0. Thus, the metric of a static, spherically symmetric spacetime is

where we have introduced (r) and (r) in place of the two unknowns g00(r) and grr(r). This replacement is acceptable provided g00 < 0 and grr > 0 everywhere. We shall see below that these conditions do hold inside stars, but they break down for black holes. When we study black holes in the next chapter we shall have to look carefully again at our coordinate system.