1.2 Special Relativity
1.3.5 Exact Solutions to the Einstein Equations
1.3.5.1 Flat Space
The Minkowski metric, = diag( 1;1;1;1), is an exact solution to G = 0; but by virtue of its corresponding Riemann-Christo¤el tensor,R 0, it may be considered to be the trivial solution. It is Lorentz invariant and applies to spacetime without sources. Flat spacetime is an idealisation, which …nds use in the construction of a linearised model for gravitation [46, 45].
1.3.5.2 The Schwarzschild Metric - a spherically symmetric solution
In 1915, Karl Schwarzschild, inspired by Einstein’s theory of general relativity, sought a solution to the Einstein equations that described the curvature of spacetime outside a spherically symmetric star with no angular rotation [7]. His result:
d~2 = 1 2 ~ r dt~ 2+ 1 2 ~ r 1 dr~2+ ~r2d 2; (1.28) where d 2 =d 2+ sin2( )d 2;
was presented on his behalf by Einstein in January, 1916. The parameters in equation (1.28) have been normalised with respect to the mass, M, of the gravitation body (i.e.
~
r =r=M, ~t =t=M, and ~ = =M); no normalisation is required for the angles and . It is important to recognise that although in modern times the Schwarzschild geometry is associated with non-rotating black holes, it was originally derived as a solution for stars; indeed, any non-rotating object has the Schwarzschild geometry as its external solution. Later, in 1916, Schwarzschild derived a solution for the interior of a star. Such solutions are complicated [29], and lie beyond the scope of this work.
The exterior Schwarzschild solution describes the gravitational …eld of an isolated particle of mass, M, in a region of free space, therefore, R = 0. The metric is autonomous (not explicitly dependent on time) and it is unchanged under time reversal (see sections 10.1 and 10.2 in [29]).
In the Schwarzschild line element (equation (1.28)) there is a term,(1 2=r~), common to thed~t2 andd~r2 elements, which o¤ers an insight into some of the important properties
of the Schwarzschild geometry. As r~! 1, (i.e. the properties of the line element are considered at locations far distant from the gravitating body), the Schwarzschild line element asymptotically approaches that of Minkowski space (equation (1.16)) . For de- creasing values ofr~, the spacetime curvature becomes more pronounced; and a coordinate singularity exists atr~= 2. The quantity, t~=t=M, which appears in the Schwarzschild metric through the term, d~t2, corresponds to the Schwarzschild or coordinate time (nor- malised by dividing by the mass of the black hole, M) as it would be measured in a stationary frame at an in…nite distance away (see Susskind and Lindesay [47]).
At the radius r~= 2; the coe¢ cient ofd~r2 is at a coordinate singularity. This radial
position, the Schwarzschild radius, de…nes the event horizon of a Schwarzschild black hole (SBH). As an exercise in visualising the scale of this coordinate singularity, one may neglect any rotational angular momentum and calculate the Schwarzschild radius of a given body, whether it is massive or not.
Table 1.1: Schwarzschild radii for spherically symmetrical bodies of various masses. The data is presented in MKS units; but geometrised units, for which c= 1 and G= 1 will be used henceforth, see table 1.2.
Category Object Mass [kg] True Radius [m]
Schwarzschild Radius [m] lepton electron 9:11 10 31 <10 22 [48] 1:4 10 57
hadron proton 1:67 10 27 5:0 10 16 2:5 10 54
sports bocce ball 1:00 5:5 10 2 1:5 10 27
large satellite Moon 7:35 1022 1:7 106 1:1 10 4
planet Earth 5:97 1024 6:4 106 8:9 10 3
white dwarf Sirius B 1:95 1030 5:7 106 2:9 103
star Sun 1:99 1030 7:0 108 3:0 103
neutron star PSR 1913+16 2:87 1030 1:0 104 4:3 103
M supergiant Orionis 3:78 1031 8:2 1011 5:6 104
massive black hole 107 M 1:99 1037 3:0 1010 3:0 1010
The values tabulated in table 1.16 demonstrate that the Schwarzschild radius is so
small as to be beyond the realm of our everyday experience; MKS units, which are familiar to the general reader, are used at this point. The Schwarzschild radius of a bocce ball is …ve orders of magnitude smaller than the estimated upper limit of the electron radius. The Schwarzschild radius of the Earth is less than1 cmin size. Consequently, virtually all the of mass of these bodies lies outside the Schwarzschild radius. This circumstance is even true for an object as massive as the Sun, which has a Schwarzschild radius of
3 km. But this work focuses on black holes and extreme mass-ratio binary black hole systems, for which the event horizon is no longer an abstraction.
Henceforth in this work, geometrised units will be used. By setting the speed of light, c = 1, distance can be expressed in units of seconds. In addition, by setting the gravitational constant, G = 1, it is also possible to express mass and energy, as well as momentum, in units of seconds. An important result can be calculated for the Sun: one solar mass = 5 s.
6The calculations of the Schwarzschild radius given here are for illustrative purposes; they are based
Table 1.2: The representation of metres and kg in units of time.
Parameter Symbol Geometrised Units speed of light c [26] 299;795;458 m=s = 1 )1 m = 1=299795458 s gravitational constant G [49] 6:67428: : : 10 11 m3kg 1 s 2 = 1 )1 kg = 2:47702 10 36 s
1.3.5.3 Kerr Black Hole - an axisymmetric solution
Because an SBH does not rotate, it possesses spherical symmetry and serves as a useful idealisation. But one expects most black holes to possess some spin angular momentum (J), therefore, the spherical symmetry associated with an SBH is broken, becoming an axisymmetric spacetime, orientated parallel to the axis of rotation. In 1963, Roy Kerr derived the analytical solution to the Einstein …eld equations for a spinning black hole [50]. This result was the culmination of an e¤ort made by many researchers over a period of several years; hence a black hole with spin angular momentum is called a Kerr black hole (KBH).
The form of the Kerr spacetime line element used today [50]:
d~2 = ~ S~2sin2( ) ~ d~t 2+ ~ ~d~r 2+ ~d 2 4 ~ Sr~sin2( ) ~ d~td + ~ r2+ ~S2 2 S~2~ sin2( ) ~ sin 2 ( )d 2; (1.29) where ~ S = jJj M2 =M2 ~r2 2~r+ ~S2 ) ~ = ~r2 2~r+ ~S2 and = 2 =M2 ~r2+ ~S2cos2( ) ) ~ = ~r2+ ~S2cos2( );
the parameters in the equation are normalised with respect to the mass,M, of the Kerr black hole (KBH). The use of Boyer-Lindquist coordinates simpli…es the metric to a form with a single o¤-diagonal element7 for d~td (see Appendix 3.B.1). Further, in the limit
as S~ ! 0, equation (1.29) approaches the Schwarzschild geometry. As in the case of the Schwarzschild geometry the Kerr metric is also stationary, and R = 0 throughout the region of free space outside the rotating body [51, 52]. The terms are autonomous. But under time reversal, the o¤-diagonal elements (those that containd dt) change sign, therefore, the Kerr spacetime geometry is stationary, but not static.
The free space surrounding the Kerr black hole contains some important regions. The event horizon for a KBH corresponds to the singularity of the coe¢ cient for dr~2, hence
~
rH = 1
p
1 S~2; (1.30)
where the positive root corresponds to the event horizon of an SBH (S~= 0). Interest- ingly, the value of r~H is independent of the polar angle . But this does not mean the
event horizon of a KBH is spherically symmetrical; it only appears so when represented in Boyer-Lindquist coordinates.
By solving for the roots of the coe¢ cient of d~t2 one …nds the de…nition of the ergo-
sphere of the KBH:
~
rErgo= 1
q
1 S~2cos2( ); (1.31)
which describes another null surface (positive root) that extends farther into space than the event horizon.
7There are two, symmetrically placed o¤-diagonal elements, both of which correspond to d~td .