Boyer-Lindquist coordinates

In the mathematical description of the general theory of relativity , the Boyer-Lindquist coordinates represent a generalization of the coordinates for the Schwarzschild metric . They are used in particular in the description of a rotating black hole , i.e. H. when using the Kerr metric (in the unloaded case) or the Kerr-Newman metric (in the loaded case).

The coordinate transformation from Boyer-Lindquist coordinates into Cartesian coordinates is given by: ${\ displaystyle (r, \ theta, \ phi)}$ ${\ displaystyle (x, y, z)}$

{\ displaystyle {\ begin {aligned} x & = {\ sqrt {r ^ {2} + a ^ {2}}} \ sin \ theta \ cos \ phi \\ y & = {\ sqrt {r ^ {2} + a ^ {2}}} \ sin \ theta \ sin \ phi \\ z & = r \ cos \ theta \ end {aligned}}}

Within the Kerr-Newman metric, the line element for a black hole with mass , angular momentum, and charge is given in Boyer-Lindquist coordinates using natural units ( ) by ${\ displaystyle M}$ ${\ displaystyle J}$ ${\ displaystyle Q}$ ${\ displaystyle c = G = k _ {\ mathrm {C}} = 1}$

{\ displaystyle \ mathrm {d} s ^ {2} = - {\ frac {\ Delta} {\ Sigma}} \ left (\ mathrm {d} ta \ sin ^ {2} (\ theta) \ mathrm {d } \ phi \ right) ^ {2} + {\ frac {\ sin ^ {2} \ theta} {\ Sigma}} \ left [\ left (r ^ {2} + a ^ {2} \ right) \ mathrm {d} \ phi - {a} \ mathrm {d} t \ right] ^ {2} + {\ frac {\ Sigma} {\ Delta}} \ mathrm {d} r ^ {2} + \ Sigma \ mathrm {d} \ theta ^ {2},}

the following abbreviations are used:

{\ displaystyle {\ begin {aligned} \ Delta &: = r ^ {2} -2Mr + a ^ {2} + Q ^ {2} \\\ Sigma &: = r ^ {2} + a ^ {2 } \ cos ^ {2} \ theta \\ a &: = J / M \ end {aligned}}}

It should be noted that the sizes , and in natural units all have the unit of measurement of a length. ${\ displaystyle M}$ ${\ displaystyle a}$ ${\ displaystyle Q}$

literature

RH Boyer, RW Lindquist: Maximal Analytic Extension of the Kerr Metric. In: J. Math. Phys. 8, 1967, pp. 265-281.
SL Shapiro , SA Teukolsky : Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects. Wiley, New York 1983, p. 357.

Individual evidence

↑ Brandon Carter: Global Structure of the Kerr Family of Gravitational Fields . In: Physical Review . tape 174 , no. 5 , October 25, 1968, p. 1559–1571 , doi : 10.1103 / PhysRev.174.1559 ( online ).

[1] Brandon Carter: Global Structure of the Kerr Family of Gravitational Fields . In: Physical Review . tape 174 , no. 5 , October 25, 1968, p. 1559–1571 , doi : 10.1103 / PhysRev.174.1559 ( online ).