Kerr - Newman Decision

The Kerr – Newman solution is the exact solution of the Einstein equations describing a non-perturbed electrically charged rotating black hole without a cosmological term. The astrophysical significance of the solution is unclear, since it is assumed that naturally occurring collapsars cannot be substantially electrically charged.

Content

Solution form and its properties

The three-parameter Kerr-Newman family is the most general solution corresponding to the final equilibrium state not perturbed by the external fields of a black hole (according to the “no-hair” theorems for known physical fields ). In the Boyer – Lindquist coordinates, the Kerr – Newman metric is given by the expression: ^[1]

ds^{2}=-\left(1-{2\,Mr-Q^{2} \over \Sigma }\right)\,dt^{2}-2(2\,Mr-Q^{2})a{\sin ^{2}\theta  \over \Sigma }\,dt\,d\varphi \,+

{\ displaystyle ds ^ {2} = - \ left (1- {2 \, Mr-Q ^ {2} \ over \ Sigma} \ right) \, dt ^ {2} -2 (2 \, Mr-Q ^ {2}) a {\ sin ^ {2} \ theta \ over \ Sigma} \, dt \, d \ varphi \, +}

+\left(r^{2}+a^{2}+{(2\,Mr-Q^{2})a^{2}\sin ^{2}\theta  \over \Sigma }\right)\sin ^{2}\theta \,{d\varphi ^{2}}+{\Sigma  \over \Delta }\,dr^{2}+{\Sigma \,{d\theta ^{2}}},

{\ displaystyle + \ left (r ^ {2} + a ^ {2} + {(2 \, Mr-Q ^ {2}) a ^ {2} \ sin ^ {2} \ theta \ over \ Sigma} \ right) \ sin ^ {2} \ theta \, {d \ varphi ^ {2}} + {\ Sigma \ over \ Delta} \, dr ^ {2} + {\ Sigma \, {d \ theta ^ { 2}}},}

Where $\Sigma \equiv r^{2}+a^{2}\cos ^{2}\theta$ ${\ displaystyle \ Sigma \ equiv r ^ {2} + a ^ {2} \ cos ^ {2} \ theta}$ ; $\Delta \equiv r^{2}-2Mr+a^{2}+Q^{2}$ ${\ displaystyle \ Delta \ equiv r ^ {2} -2Mr + a ^ {2} + Q ^ {2}}$ and $a\equiv L/M$ ${\ displaystyle a \ equiv L / M}$ where $L$ ${\ displaystyle L}$ Is the angular momentum normalized to the speed of light, and $Q$ ${\ displaystyle Q}$ - similarly normalized charge.

From this simple formula it easily follows that the event horizon is on a radius: $r_{+}=M+{\sqrt {M^{2}-Q^{2}-a^{2}}}$ ${\ displaystyle r _ {+} = M + {\ sqrt {M ^ {2} -Q ^ {2} -a ^ {2}}}$ , and therefore the parameters of a black hole cannot be arbitrary: the electric charge and angular momentum cannot be greater than the values corresponding to the disappearance of the event horizon. The following restrictions must be met:

a^{2}+Q^{2}\leqslant M^{2}

{\ displaystyle a ^ {2} + Q ^ {2} \ leqslant M ^ {2}}

- this restriction for the BH Kerr - Newman .

If these restrictions are violated, the event horizon will disappear, and the solution instead of a black hole will describe the so-called “bare” singularity , but such objects, according to popular belief, should not exist in the real Universe (according to the not yet proven, but plausible principle of cosmic censorship ). Alternatively, under the horizon there can be a source of collapsed matter, which closes the singularity, and therefore the Kerr or Kerr-Newman external solution should be continuously coupled with the internal solution of the Einstein equations with the energy-momentum tensor of this matter. The singularity disappears along with the restriction on the Kerr-Newman BH parameters.

Back in 1970, V.Israel considered the source of the Kerr-Newman solution in the form of a rotating disk covering this move. This direction was developed by C. Lóopez, who showed that the Kerr singularity can be closed by a rotating shell (bubble), and in this case the restriction on the parameters of the Kerr – Newman solution does not apply. Moreover, as B. Carter (1968) noted, the Kerr-Newman solution has the same gyromagnetic ratio as the electron has according to the Dirac equation. The history of this direction for the Kerr-Newman solution is presented in arXiv: 0910.5388 [hep-th] .

The Kerr – Newman metric (and just Kerr, but not Schwarzschild) can be analytically continued across the horizon in such a way as to connect in the black hole infinitely many “independent” spaces. These can be either “other” universes or remote parts of our Universe. In the spaces thus obtained there are closed timelike curves: the traveler can, in principle, get into his past, that is, he can meet with himself. Around the event horizon of a rotating black hole there also exists a region called the ergosphere , which is practically equivalent to the ergosphere from the Kerr solution; the stationary observer located there must rotate with a positive angular velocity (in the direction of rotation of the black hole).

Coordinates of Kerr - Schild

The most simple expression of the Kerr and Kerr – Newman solutions is in the Kerr – Schild form (KSh) ^[2] , in which the metric has the form

g_{\mu \nu }=\eta _{\mu \nu }+2Hk_{\mu }k_{\nu }

{\ displaystyle g _ {\ mu \ nu} = \ eta _ {\ mu \ nu} + 2Hk _ {\ mu} k _ {\ nu}}

,

Where $\eta _{\mu \nu }$ ${\ displaystyle \ eta _ {\ mu \ nu}}$ is a metric of auxiliary Minkowski space with Cartesian coordinates $x=x^{\mu }(x)=(t,x,y,z)$ ${\ displaystyle x = x ^ {\ mu} (x) = (t, x, y, z)}$ .

In this form $k^{\mu }(x)$ ${\ displaystyle k ^ {\ mu} (x)}$ is a vector field of light-like directions. Often they say "zero" directions, because $k_{\mu }k^{\mu }=g_{\mu \nu }k^{\mu }k^{\nu }=0$ ${\ displaystyle k _ {\ mu} k ^ {\ mu} = g _ {\ mu \ nu} k ^ {\ mu} k ^ {\ nu} = 0}$ . Note that the specific structure of the KS metric form ensures that the field $k^{\mu }(x)$ ${\ displaystyle k ^ {\ mu} (x)}$ is also zero with respect to auxiliary flat space, i.e. $\eta _{\mu \nu }k^{\mu }k^{\nu }=0$ ${\ displaystyle \ eta _ {\ mu \ nu} k ^ {\ mu} k ^ {\ nu} = 0}$ .

The function H has the form

H={\frac {Mr-|Q|^{2}/2}{r^{2}+a^{2}\cos ^{2}\theta }},

{\ displaystyle H = {\ frac {Mr- | Q | ^ {2} / 2} {r ^ {2} + a ^ {2} \ cos ^ {2} \ theta}},}

Where $r,\theta$ ${\ displaystyle r, \ theta}$ - these are oblate spheroidal Kerr coordinates, which are determined by the ratio

x+iy=(r+ia)e^{i\phi }\sin \theta ,\ z=r\cos \theta .

{\ displaystyle x + iy = (r + ia) e ^ {i \ phi} \ sin \ theta, \ z = r \ cos \ theta.}

and move away from the BH in the usual spherical coordinates. In these coordinates, the components of the vector $k_{\alpha }$ ${\ displaystyle k _ {\ alpha}}$ are determined from the differential form

k_{\alpha }dx^{\alpha }=dr-dt-a\sin ^{2}\theta d\phi

{\ displaystyle k _ {\ alpha} dx ^ {\ alpha} = dr-dt-a \ sin ^ {2} \ theta d \ phi}

by comparing coefficients before differentials. This is one example of calculation using a very convenient apparatus of external forms, which was used by Kerr to obtain a solution in the first and subsequent works.

In fact, the Kerr’s angular coordinate $\phi$ ${\ displaystyle \ phi}$ very unusual, and the simple form of KSH is due to the fact that all the complexity of the solution is hidden in the form of a vector field $k^{\mu }(x)$ ${\ displaystyle k ^ {\ mu} (x)}$ , which is a vortex light-like flow, forming the so-called Main Zero Congruence (GOC). In Cartesian coordinates, the components of the vector field $k_{\mu }$ ${\ displaystyle k _ {\ mu}}$ determined by the form

k_{\mu }dx^{\mu }=-dt+{\frac {z}{r}}dz+{\frac {r}{r^{2}+a^{2}}}(xdx+ydy)-{\frac {a}{r^{2}+a^{2}}}(xdy-ydx)

{\ displaystyle k _ {\ mu} dx ^ {\ mu} = - dt + {\ frac {z} {r}} dz + {\ frac {r} {r ^ {2} + a ^ {2}}} (xdx + ydy) - {\ frac {a} {r ^ {2} + a ^ {2}}} (xdy-ydx)}

.

In the theory of CS to determine this field are also used "zero" (light) Cartesian coordinates

$u=(z-t)/{\sqrt {2}},\quad v=(z+t)/{\sqrt {2}},\quad \zeta =(x+iy)/{\sqrt {2}},\quad {\bar {\zeta }}=(x-iy)/{\sqrt {2}}$ ${\ displaystyle u = (zt) / {\ sqrt {2}}, \ quad v = (z + t) / {\ sqrt {2}}, \ quad \ zeta = (x + iy) / {\ sqrt { 2}}, \ quad {\ bar {\ zeta}} = (x-iy) / {\ sqrt {2}}}$ ,

in which the congruence has components defined by the differential form

k_{\mu }^{(\pm )}dx^{\mu }=P^{-1}(du+{\bar {Y}}^{\pm }d\zeta +Y^{\pm }d{\bar {\zeta }}-Y^{\pm }{\bar {Y}}^{(\pm )}dv)

{\ displaystyle k _ {\ mu} ^ {(\ pm)} dx ^ {\ mu} = P ^ {- 1} (du + {\ bar {Y}} ^ {\ pm} d \ zeta + Y ^ {\ pm} d {\ bar {\ zeta}} - Y ^ {\ pm} {\ bar {Y}} ^ {(\ pm)} dv)}

.

This expression is defined by a complex function. $Y(x)$ ${\ displaystyle Y (x)}$ which has two solutions $Y^{\pm }(x)$ ${\ displaystyle Y ^ {\ pm} (x)}$ that gives for a vector field $k^{\mu }(x)$ ${\ displaystyle k ^ {\ mu} (x)}$ two different congruences (GOC). Thus, the solution for rotating BHs can be written in two different forms, which are based on the BH “congruent” or “outgoing” BH congruences, which corresponds to the so-called algebraically special solutions of type D (according to Petrov’s classification ).

Representation in the form of KS has several advantages, since the congruence, all coordinates and the form of solutions for the electromagnetic (EM) field and metrics are rigidly related to the coordinates of the auxiliary flat space and do not depend on the position of the horizon and the ergosphere boundary. Moreover, the KS solutions unambiguously continue analytically across the horizon inside the BH and further to the “negative” sheet - the area of negative values of the oblate radial coordinate $r$ ${\ displaystyle r}$ .

In the coordinates of Kerr $\theta ,\phi$ ${\ displaystyle \ theta, \ phi}$ function $Y(x)$ ${\ displaystyle Y (x)}$ has the appearance

Y(x)=e^{i\phi }\tan {\frac {\theta }{2}}

{\ displaystyle Y (x) = e ^ {i \ phi} \ tan {\ frac {\ theta} {2}}}

.

Geometrically, it is a projection of the celestial sphere with coordinates $\theta ,\phi$ ${\ displaystyle \ theta, \ phi}$ on the complex plane $Y$ ${\ displaystyle Y}$ however addiction $x\to Y(x)$ ${\ displaystyle x \ to Y (x)}$ is very nontrivial and is given by the Kerr theorem, which is closely related to the twistors . In fact, the GNC forms the backbone of the Kerr solution as a whirlwind of twistor rays. Function $P$ ${\ displaystyle P}$ for a resting solution it looks like

$P={\frac {1}{\sqrt {2}}}(1+Y{\bar {Y}})$ ${\ displaystyle P = {\ frac {1} {\ sqrt {2}}} (1 + Y {\ bar {Y}})}$ .

Like the form of the KS metric, all tensor characteristics of the solution must be consistent with the vector field of the GNC, and in particular, the vector potential of the EM field of the Kerr – Newman solution is expressed

A^{\mu }=\Re e{\frac {Qr}{r^{2}+a^{2}\cos ^{2}\theta }}k^{\mu }

{\ displaystyle A ^ {\ mu} = \ Re e {\ frac {Qr} {r ^ {2} + a ^ {2} \ cos ^ {2} \ theta}} k ^ {\ mu}}

.

The Kerr singularity is under the horizon. It is associated with the singularity of the function H and corresponds to the values $r=0$ ${\ displaystyle r = 0}$ and at the same time $\theta =0$ ${\ displaystyle \ theta = 0}$ . It is a ring that opens a passage to the negative sheet of the Kerr geometry, $r<0$ ${\ displaystyle r <0}$ where the values of mass and charge, as well as the direction of the fields are reversed (Do not confuse with the maximum analytical expansion of solutions across the horizon of the BH, described somewhat below.) This second sheet (“Alisovo mirror-glass”) has long been a mystery of the Kerr decision.

Literature

C. Minzer, C. Thorne, J. Wheeler. Gravity. - Mir, 1977. - T. 3. - 512 p.
Subramanian Chandrasekhar . Mathematical theory of black holes. In 2 volumes = Mathematical theory of black holes. n V. A. Berezina. Ed. D.F.-M. n D.A. Galtsova. - M .: Mir, 1986.
I.D. Novikov, V.P. Frolov. Black hole physics - M .: Science, 1986. - 328 p.

Notes

↑ C. Minzer, K. Thorne, J. Wheeler. Gravity, Vol. 3, 1977 , Supplement 33.2. KERRA - NEWMAN GEOMETRY AND ELECTROMAGNETIC FIELD, c. 88
↑ Debney GC, Kerr RP and Schild A. Solutions of the Einstein and Einstein-Maxwell Equations (Eng.) // Journal of Mathematical Physics . - 1969. - Vol. 10 - P. 1842-1854 . - DOI : 10.1063 / 1.1664769 .

[_70e89ac4dfae2863-1] C. Minzer, K. Thorne, J. Wheeler. Gravity, Vol. 3, 1977 , Supplement 33.2. KERRA - NEWMAN GEOMETRY AND ELECTROMAGNETIC FIELD, c. 88

[ksch-2] Debney GC, Kerr RP and Schild A. Solutions of the Einstein and Einstein-Maxwell Equations (Eng.) // Journal of Mathematical Physics . - 1969. - Vol. 10 - P. 1842-1854 . - DOI : 10.1063 / 1.1664769 .