# Kerr metric

Black Hole Metrics
static ${\ displaystyle (J = 0)}$ rotating ${\ displaystyle (J \ neq 0)}$
uncharged ${\ displaystyle (Q = 0)}$ Schwarzschild metric Kerr metric
loaded ${\ displaystyle (Q \ neq 0)}$ Reissner-Nordström metric Kerr-Newman metric
${\ displaystyle Q}$: Electric charge ; : Angular momentum${\ displaystyle \ J}$

The Kerr metric is a stationary and axially symmetric vacuum solution of Einstein's field equations . It describes uncharged, rotating black holes and is named after Roy Kerr , who published it in 1963. Shortly after the discovery of the Schwarzschild or Kerr metric, the corresponding generalizations for the case of electrically charged black holes were found. In contrast to the Schwarzschild metric, which also applies to the outer area of ​​a non-rotating and spherically symmetrical body of any size, the Kerr metric exclusively describes the field of a black hole, because rapidly rotating stars often have a non-negligible multipole moment and different density gradients, so whose spacetime geometry only approaches the Kerr metric at a certain distance from the surface of the star.

## Line element

### Boyer-Lindquist coordinates

With the covariants

${\ displaystyle g_ {tt} = \ zeta -1, \ g_ {rr} = {\ frac {\ Sigma} {\ Delta}}, \ g _ {\ theta \ theta} = \ Sigma, \ g _ {\ phi \ phi} = {\ frac {\ chi \ sin ^ {2} \ theta} {\ Sigma}}, \ g_ {t \ phi} = - a \ zeta \ sin ^ {2} \ theta}$

and the contravariants obtained by matrix inversion

${\ displaystyle g ^ {tt} = - {\ frac {\ chi} {\ Delta \ Sigma}}, g ^ {rr} = {\ frac {\ Delta} {\ Sigma}}, g ^ {\ theta \ theta} = {\ frac {1} {\ Sigma}}, g ^ {\ phi \ phi} = {\ frac {\ Delta -a ^ {2} \ sin ^ {2} \ theta} {\ Delta \ Sigma \ sin ^ {2} \ theta}}, \ g ^ {t \ phi} = - {\ frac {a \ zeta} {\ Delta}}}$

metric coefficients is the line element of Kerr space-time in Boyer-Lindquist coordinates and geometrical units, i.e. H. : ${\ displaystyle G = c = 1}$

${\ displaystyle {\ mathrm {d} s} ^ {2} = g _ {\ mu \ nu} \ mathrm {d} x ^ {\ mu} \ mathrm {d} x ^ {\ nu} = g_ {tt} \ mathrm {d} t ^ {2} + g_ {rr} \ mathrm {d} r ^ {2} + g _ {\ theta \ theta} \ mathrm {d} \ theta ^ {2} + g _ {\ phi \ phi} \ mathrm {d} \ phi ^ {2} + 2g_ {t \ phi} \ mathrm {d} t \ mathrm {d} \ phi,}$

${\ displaystyle \ mathrm {d} s ^ {2} = (\ zeta -1) \ mathrm {d} t ^ {2} + {\ frac {\ Sigma} {\ Delta}} \ mathrm {d} r ^ {2} + \ Sigma \ mathrm {d} \ theta ^ {2} + {\ frac {\ chi \ sin ^ {2} \ theta} {\ Sigma}} \ mathrm {d} \ phi ^ {2} - 2a \ zeta \ sin ^ {2} \ theta \ mathrm {d} t \ mathrm {d} \ phi}$

The D'Alembert operator is:

${\ displaystyle \ partial ^ {\ mu} \ partial _ {\ mu} = g ^ {\ mu \ nu} \ left ({\ frac {\ partial} {\ partial x ^ {\ mu}}} \ right) {\ frac {\ partial} {\ partial x ^ {\ nu}}} = g ^ {tt} \ left ({\ frac {\ partial} {\ partial t}} \ right) ^ {2} + g ^ {rr} \ left ({\ frac {\ partial} {\ partial r}} \ right) ^ {2} + g ^ {\ theta \ theta} \ left ({\ frac {\ partial} {\ partial \ theta }} \ right) ^ {2} + g ^ {\ phi \ phi} \ left ({\ frac {\ partial} {\ partial \ phi}} \ right) ^ {2} + 2g ^ {t \ phi} \, {\ frac {\ partial} {\ partial \ phi}} {\ frac {\ partial} {\ partial t}}}$

The following applies:

${\ displaystyle r _ {\ mathrm {s}} = 2M, \ quad a = J / M, \ quad \ Sigma = r ^ {2} + a ^ {2} \ cos ^ {2} \ theta, \ quad \ Delta = r ^ {2} -r _ {\ mathrm {s}} \ r + a ^ {2}, \ quad \ chi = \ left (a ^ {2} + r ^ {2} \ right) ^ {2 } -a ^ {2} \ sin ^ {2} \ theta \ Delta, \ quad \ zeta = r _ {\ mathrm {s}} r / \ Sigma}$

${\ displaystyle M}$is the field-generating, gravitational mass including the rotational energy. If all of its rotational energy is withdrawn from a black hole, for example with the help of a Penrose process , its gravitational mass is reduced to the irreducible mass . For these applies: ${\ displaystyle M}$${\ displaystyle M _ {\ mathrm {irr}}}$

${\ displaystyle M _ {\ mathrm {irr}} = {\ sqrt {\, {\ frac {M ^ {2} + {\ sqrt {M ^ {4} -a ^ {2} M ^ {2}}} } {2}}}}}$

After resolved, the following also applies: ${\ displaystyle M}$

${\ displaystyle M = 2 {\ sqrt {\ frac {M _ {\ mathrm {irr}} ^ {4}} {4M _ {\ mathrm {irr}} ^ {2} -a ^ {2}}}}}$

The rotational energy corresponds to a mass in accordance with the equivalence of mass and energy . In the event that the body also rotates, the result is a mass equivalent that is a factor higher than for a static body with the same irreducible mass. ${\ displaystyle E _ {\ mathrm {red}} = M-M _ {\ mathrm {irr}}}$${\ displaystyle a = M}$${\ displaystyle {\ sqrt {2}}}$

${\ displaystyle r _ {\ mathrm {s}}}$is the Schwarzschild radius . The parameter is also called the Kerr parameter. It is proportional to the angular momentum of the black hole. A positive angular momentum describes a counterclockwise rotation when viewed from the North Pole. A negative angular momentum describes the opposite direction. ${\ displaystyle a}$ ${\ displaystyle J}$

In Cartesian coordinates with

${\ displaystyle x = {\ sqrt {r ^ {2} + a ^ {2}}} \ sin \ theta \ cos \ phi, \ quad y = {\ sqrt {r ^ {2} + a ^ {2} }} \; \ sin \ theta \ sin \ phi, \ quad z = r \ cos \ theta}$

results from the line element above:

${\ displaystyle \ mathrm {d} x = {\ frac {r \ cos \ phi \ sin \ theta} {\ sqrt {a ^ {2} + r ^ {2}}}} \ mathrm {d} r + {\ sqrt {a ^ {2} + r ^ {2}}} \ cos \ phi \ cos \ theta \ mathrm {d} \ Theta - {\ sqrt {a ^ {2} + r ^ {2}}} \ sin \ phi \ sin \ theta \ mathrm {d} \ phi}$
${\ displaystyle \ mathrm {d} y = {\ frac {r \ sin \ phi \ sin \ theta} {\ sqrt {a ^ {2} + r ^ {2}}}} \ mathrm {d} r + {\ sqrt {a ^ {2} + r ^ {2}}} \ sin \ phi \ cos \ theta \ mathrm {d} \ Theta - {\ sqrt {a ^ {2} + r ^ {2}}} \ cos \ phi \ sin \ theta \ mathrm {d} \ phi}$
${\ displaystyle \ mathrm {d} z = \ cos \ theta \ mathrm {d} rr \ sin \ theta \ mathrm {d} \ theta}$

In the case of vanishing rotation , the line element is reduced to the Schwarzschild line element in Schwarzschild coordinates, and in the case of vanishing mass to the line element of Minkowski space-time in spherical coordinates . ${\ displaystyle a = 0}$${\ displaystyle M = 0}$

### Kerr shield coordinates

In order to avoid the coordinate singularity at the event horizon, it can be transformed into Kerr-Schild coordinates. In these the line element is

${\ displaystyle \ mathrm {d} s ^ {2} = \ mathrm {d} x ^ {2} + \ mathrm {d} y ^ {2} + \ mathrm {d} z ^ {2} - \ mathrm { d} {\ hat {t}} ^ {2} + {\ frac {r _ {\ mathrm {s}} \ r ^ {3}} {r ^ {4} + a ^ {2} \ z ^ {2 }}} \ \ left (\ mathrm {d} {\ hat {t}} + {\ frac {r \ (x \ \ mathrm {d} x + y \ \ mathrm {d} y)} {r ^ { 2} + a ^ {2}}} + {\ frac {a \ (y \ \ mathrm {d} xx \ \ mathrm {d} y)} {r ^ {2} + a ^ {2}}} + {\ frac {z \ \ mathrm {d} z} {r}} \ right) ^ {2}}$.

${\ displaystyle r}$ is determined by the following equation:

${\ displaystyle x ^ {2} + y ^ {2} + z ^ {2} = r ^ {2} + a ^ {2} \ left (1 - {\ frac {z ^ {2}} {r ^ {2}}} \ right).}$

With the coordinate time

${\ displaystyle {\ hat {t}} = t + r _ {\ mathrm {s}} \ int {\ frac {r \ \ mathrm {d} r} {\ Delta}} \, \ \ \ mathrm {d} {\ hat {t}} = \ mathrm {d} t + r _ {\ mathrm {s}} \ \ mathrm {d} r \ r / \ Delta,}$

the azimuthal angle

${\ displaystyle {\ hat {\ varphi}} = \ phi + a \ \ int {\ frac {\ mathrm {d} r} {\ Delta}} \, \ \ \ mathrm {d} {\ hat {\ varphi }} = \ mathrm {d} \ phi + \ mathrm {d} r \ a / \ Delta}$

and the transformation between spherical and Cartesian background coordinates

${\ displaystyle x = (r \ \ cos {\ hat {\ varphi}} + a \ \ sin \ {\ hat {\ varphi}}) \ \ sin \ theta \, \ \ y = (r \ \ sin { \ hat {\ varphi}} - a \ \ cos \ {\ hat {\ varphi}}) \ \ sin \ theta \, \ \ z = r \ cos \ theta}$

the non-vanishing covariant metric components are in spherical coordinates

${\ displaystyle g _ {{\ hat {t}} {\ hat {t}}} = \ zeta -1, \ g _ {{{\ hat {t}} r} = \ zeta, \ g _ {{\ hat {t }} {\ hat {\ varphi}}} = - \ zeta a \ sin ^ {2} \ theta, \ g_ {rr} = \ zeta +1, \ g_ {r {\ hat {\ varphi}}} = - (\ zeta +1) a \ sin ^ {2} \ theta, \ g _ {\ theta \ theta} = \ Sigma, \ g _ {{\ hat {\ varphi}} {\ hat {\ varphi}}} = {\ frac {\ chi \ sin ^ {2} \ theta} {\ Sigma}}}$

and the contravariant components

{\ displaystyle {\ begin {aligned} g ^ {{\ hat {t}} {\ hat {t}}} & = - \ zeta -1, \\ g ^ {{\ hat {t}} r} & = \ zeta, \\ g ^ {rr} & = - {\ frac {\ zeta ^ {2} a ^ {2} \ Sigma \ sin ^ {2} \ theta - \ zeta \ chi + \ chi} {( \ zeta +1) a ^ {2} \ Sigma \ sin ^ {2} \ theta - \ chi}}, \\ g ^ {r {\ hat {\ varphi}}} & = - {\ frac {a \ Sigma} {(\ zeta +1) a ^ {2} \ Sigma \ sin ^ {2} \ theta - \ chi}}, \\ g ^ {\ theta \ theta} & = {\ frac {1} {\ Sigma}}, \\ g ^ {{\ hat {\ varphi}} {\ hat {\ varphi}}} & = - {\ frac {\ Sigma \ csc ^ {4} \ theta} {(\ zeta +1 ) a ^ {2} \ Sigma - \ chi \ csc ^ {2} \ theta}}. \ end {aligned}}}

The radial coordinate and the polar angle are identical to their Boyer-Lindquist counterparts. However, the local observer is not on a fixed radial coordinate, but coincides radially with it ${\ displaystyle r}$${\ displaystyle \ theta}$

${\ displaystyle \ mathrm {d} r / \ mathrm {d} t = -r _ {\ mathrm {s}} \ r \ \ Delta / \ chi}$

towards the central mass, while like the local Boyer-Lindquist observer (also called ZAMO for “zero angular momentum observer” in literature) with the angular velocity

${\ displaystyle \ mathrm {d} \ phi / \ mathrm {d} t = r _ {\ mathrm {s}} \ r \ a / \ chi}$

rotates around the axis of symmetry.

These coordinates were used by Roy Kerr in his original 1963 work. With , the line element is reduced to the Schwarzschild line element in extended Eddington-Finkelstein coordinates. ${\ displaystyle a = 0}$

## Special areas

### Horizons

Geometric representation of the event horizons and ergospheres of the Kerr space-time in Cartesian background coordinates. The ring singularity is due to the equatorial bulge of the inner ergosphere${\ displaystyle R = a.}$
Size comparison of the shadow (black) and the special areas (white) of a black hole. The spin parameter runs from 0 to with the left side of the black hole rotating towards the observer.${\ displaystyle a}$${\ displaystyle M,}$

In Boyer-Lindquist coordinates, the Kerr metric degenerates over several surfaces. With the designations from above, for example, the denominator of the purely radial component of the Kerr metric can become zero if it is set and resolved to. The event horizons are thus on ${\ displaystyle g_ {rr}}$${\ displaystyle \ Delta = 0}$${\ displaystyle r}$

${\ displaystyle r _ {\ text {H}} ^ {\ pm} \ = \ M \ \ pm \ {\ sqrt {M ^ {2} \ - \ a ^ {2}}}.}$

At maximum rotation with both values ​​coincide with the gravitational radius. With minimal rotation with the positive value coincides with the Schwarzschild radius and the negative value falls on the center. Therefore, these two surfaces are also referred to as the inner and outer event horizons . Although the radial coordinate has a constant value for both event horizons, the curvature behavior of the event horizons shows that they have more the geometric properties of an ellipsoid of revolution. The inner event horizon, which is a Cauchy horizon, eludes direct observation as long as the spin parameter applies. Since the space-time inside is extremely unstable, it is considered rather unlikely that such a space will actually form in the event of a real collapse of a star. ${\ displaystyle a = M}$${\ displaystyle r_ {G} = M}$${\ displaystyle a = 0}$${\ displaystyle r_ {s} = 2r_ {G}}$${\ displaystyle r}$${\ displaystyle a \ leq M}$

### Ergospheres

Two further areas result in Boyer-Lindquist coordinates due to a change in sign of the time-like component . The condition here leads again to a quadratic equation with the solutions ${\ displaystyle g_ {tt}}$${\ displaystyle g_ {tt} = 0}$

${\ displaystyle r _ {\ text {E}} ^ {\ pm} = M \ pm {\ sqrt {M ^ {2} -a ^ {2} \ cos ^ {2} \ theta}}.}$

Because of the term under the root, these two surfaces can be represented as flattened spheres or ellipsoids of revolution if the spin parameter is low. The outer surface touches the outer event horizon at the two poles that are defined by the axis of rotation. The two poles correspond to an angle of and . With a higher spin parameter, the ergosphere bulges away from the poles in a pumpkin-like manner on the z-axis, while the inner event horizon converges towards the outer one and coincides with it. ${\ displaystyle \ cos ^ {2} \ theta}$${\ displaystyle \ theta}$${\ displaystyle 0}$${\ displaystyle \ pi}$${\ displaystyle a = M}$

The space between the two outer surfaces with and is called the ergosphere . For a particle with mass, is negative along its world line . Since the component of the metric is positive within the ergosphere , this is only possible if the particle rotates with the internal mass with a certain minimum angular velocity . It can therefore not be any particles within the ergosphere which rest or rotating in the opposite direction to the mass on the ring singularity, because the local transverse velocity of the space-time swirl (the frame dragging effect) equal to the speed of light from the outer edge of ergosphere greater is . ${\ displaystyle r = r _ {\ text {H}} ^ {\ text {+}}}$${\ displaystyle r = r _ {\ text {E}} ^ {\ text {+}}}$${\ displaystyle \ mathrm {d} s ^ {2}}$${\ displaystyle g_ {tt}}$${\ displaystyle \ Omega}$${\ displaystyle M}$${\ displaystyle v _ {\ mathrm {zamo}} = \ Omega \ {\ bar {R}} \ \ varsigma}$${\ displaystyle c}$

The shadow of a black hole is the black area that an observer sees where the black hole is located. So it is the apparent expansion of the black hole, which is always larger than the outer event horizon due to the strong curvature of space-time in the vicinity of the black hole.

The outline of the shadow can either be calculated with numerical integration of the light-like geodesics or with Fourier-transformed limaçons .

In the following, the observer is assumed to be at a great distance from the black hole and stationary. denotes the polar angle of the observer's position. and thus corresponds to a position on the symmetry axis of the space-time considered. on the other hand corresponds to a position in the equatorial plane. The wavelength of the light is considered to be negligibly small compared to the gravitational radius. The contour lines are given by ${\ displaystyle \ theta}$${\ displaystyle \ theta = 0}$${\ displaystyle \ theta = \ pi}$${\ displaystyle \ theta = \ pi / 2}$

${\ displaystyle 0 = (x ^ {2} + z ^ {2} -x \ A) ^ {2} -B ^ {2} \ (x ^ {2} + z ^ {2})}$

with the two parameters

${\ displaystyle A = \ alpha \ sin \ theta + {\ bar {a}} \ sin ^ {3} \ theta \ cos ^ {2} \ theta / 5}$
${\ displaystyle B = \ beta +0 {,} 23 \ cos ^ {4} \ theta \ (1 - {\ sqrt {1 - {\ bar {a}} ^ {4}}}),}$

which still depend on the Kerr parameter and the position of the observer. The following series expansion also applies

{\ displaystyle {\ begin {aligned} \ alpha = & - 8892 {,} 68 {\ bar {a}} ^ {10} +30413 {,} 2 {\ bar {a}} ^ {9} -46107 { ,} 4 {\ bar {a}} ^ {8} + \\ & + 37064 {,} 7 {\ bar {a}} ^ {7} -18685 {,} 4 {\ bar {a}} ^ { 6} +4666 {,} 5 {\ bar {a}} ^ {5} -3894 {,} 54 {\ bar {a}} ^ {4} + \\ & + 49 {,} 5645 {\ bar { a}} ^ {3} -9672 {,} 25 {\ bar {a}} ^ {2} +2 {,} 27392 {\ bar {a}} + 9669 {,} 01 {\ bar {a}} \ \ tan ({\ bar {a}}) \ end {aligned}}}
${\ displaystyle \ beta = 5 {,} 19058-0 {,} 343743 {\ bar {a}} \ \ tan ({\ bar {a}}) + 0 {,} 0284803 {\ bar {a}} - 0 {,} 0470795 {\ bar {a}} ^ {\ 27 {,} 5224} \ tan ({\ bar {a}})}$

with , whereby the observed length scales are considered here in units of . The observed radius of the shadow in polar coordinates is thus . From the polar view at , the black hole rotates counterclockwise from the observer's point of view and clockwise from the perspective . The observed radius of the shadow of a non-rotating black hole is thus just above it . This also applies to rotating black holes when viewed from the polar perspective. The further the position of the observer lies in the equatorial plane, the stronger the asymmetrical distortion becomes. The shadow is indented on the side rotating towards the observer and bulged on the side rotating away from him. ${\ displaystyle {\ bar {a}} = a / M}$${\ displaystyle GM / c ^ {2}}$${\ displaystyle r _ {\ mathrm {obs}} = A \ cos \ vartheta + B}$${\ displaystyle \ theta = 0}$${\ displaystyle \ theta = \ pi}$${\ displaystyle 5GM / c ^ {2}}$

### Circumference and area formulas

The non-Euclidean geometry does not result in the circumference , but in the axial direction ${\ displaystyle U = 2 \ pi \ r}$

${\ displaystyle U _ {\ phi} = \ int _ {0} ^ {2 \ pi} {\ sqrt {| g _ {\ phi \ phi} |}} \ \ mathrm {d} \ phi = 2 \ pi {\ bar {R}}}$

and in the polodial direction

${\ displaystyle U _ {\ theta} = \ int _ {0} ^ {2 \ pi} {\ sqrt {| g _ {\ theta \ theta} |}} \ \ mathrm {d} \ theta = 4 {\ sqrt { a ^ {2} + r ^ {2}}} \ \ xi \ left ({\ frac {a ^ {2}} {a ^ {2} + r ^ {2}}} \ right)}$.

where the function denotes the elliptic integral of the 2nd kind . Therefore the surface of the event horizon is not , but ${\ displaystyle \ xi}$${\ displaystyle 4 \ pi r _ {\ text {H}} ^ {2}}$

${\ displaystyle A _ {\ mathrm {H}} = \ int _ {0} ^ {\ pi} 2 \ pi {\ bar {R}} \ {\ sqrt {\ Sigma}} \, \ mathrm {d} \ theta = 8 \ pi Mr _ {\ text {H}}}$

with the axial radius of the gyration

${\ displaystyle {\ bar {R}} = {\ sqrt {| g _ {\ phi \ phi} |}} = {\ sqrt {\ frac {\ chi} {\ Sigma}}} \ \ sin \ theta,}$

which coincides for all with the Schwarzschild radius at the outer event horizon on the equatorial plane . ${\ displaystyle a}$${\ displaystyle r _ {\ mathrm {s}}}$

## Spin

In a'd naked singularity occur because at such high angular momentum values no event horizon can exist. As early as 1974, Kip Thorne concluded from computer simulations of the growth of black holes from accretion disks that black holes would not reach this limit (his simulations at that time indicated a maximum spin of ). Simulations of the collision of two black holes at high energies from 2009 by E. Berti and colleagues also showed that the limit value is indeed very close ( ), but it is not exceeded, since the energy and angular momentum are radiated by gravitational waves. ${\ displaystyle a> M, \ r = r _ {\ text {H}} ^ {+}}$${\ displaystyle a \ approx 0 {,} 998M}$${\ displaystyle a = 0 {,} 95M}$

It is generally assumed that the limit value cannot be exceeded in principle (as part of the Cosmic Censorship Hypothesis). However, this limitation for black holes does not apply to stars and other objects with an extent that is significantly larger than their outer event horizon. Before they collapse into a black hole, they have to throw off part of their excess angular momentum to the outside, so that the spin of the resulting black hole is ultimately at . ${\ displaystyle a

With a spin parameter of , the event horizon would also rotate at the speed of light. This limit value is not reached in nature, but some black holes such as e. B. that in the core of the spiral galaxy NGC 1365 or Markarian 335 comes very close to this limit. ${\ displaystyle a = M}$

As with the Schwarzschild metric in Schwarzschild coordinates , the poles of the Kerr metric, which describe the position of the event horizons, are also only coordinate singularities in Boyer-Lindquist coordinates. By choosing a different coordinate, the spacetime of the Kerr metric can also be described continuously and without poles in the metric up to the interior of the event horizons.

## Path of test bodies

Prograde orbit of a test body around a rotating black hole with ${\ displaystyle a = 0 {,} 9 \ M}$
Retrograde orbit with a spin parameter of ${\ displaystyle a = 0 {,} 95 \ M}$

The equation for the movement of a test body in Kerr space-time can be obtained from suitable Hamilton-Jacobi equations . A form of these equations in Boyer-Lindquist coordinates that is suitable for numerical integration is in natural units , with lengths in , times in and the spin parameter in : ${\ displaystyle G = M = c = 1}$${\ displaystyle GM / c ^ {2}}$${\ displaystyle GM / c ^ {3}}$${\ displaystyle a = Jc / (GM ^ {2})}$

${\ displaystyle {\ dot {t}} = {\ frac {2 \ E \ r \ \ left (a ^ {2} + r ^ {2} \ right) -2 \ a \ L_ {z} \ r} {\ Delta \ \ Sigma}} + E = {\ frac {\ varsigma} {\ sqrt {1-v ^ {2}}}}}$
${\ displaystyle {\ dot {r}} = {\ frac {\ Delta \ p_ {r}} {\ Sigma}}}$
${\ displaystyle {\ dot {p}} _ {r} = {\ frac {(r-1) \ left (\ mu \ \ left (a ^ {2} + r ^ {2} \ right) -k \ right) +2 \ E ^ {2} \ r \ left (a ^ {2} + r ^ {2} \ right) -2 \ a \ E \ L_ {z} + \ Delta \ \ mu \ r} { \ Delta \ \ Sigma}} - {\ frac {2 \ p_ {r} ^ {2} \ (r-1)} {\ Sigma}}}$
${\ displaystyle p_ {r} = {\ frac {v ^ {r}} {\ sqrt {1+ \ mu \ v ^ {2}}}} {\ sqrt {\ frac {\ Sigma} {\ Delta}} }}$
${\ displaystyle {\ dot {\ theta}} = {\ frac {p _ {\ theta}} {\ Sigma}}}$
${\ displaystyle {\ dot {p}} _ {\ theta} = {\ frac {\ sin \ theta \ \ cos \ theta \ left (L_ {z} ^ {2} / \ sin ^ {4} \ theta - a ^ {2} \ left (E ^ {2} + \ mu \ right) \ right)} {\ Sigma}}}$
${\ displaystyle p _ {\ theta} = {\ frac {v ^ {\ theta} \ {\ sqrt {\ Sigma}}} {\ sqrt {1+ \ mu \ v ^ {2}}}}}$
${\ displaystyle {\ dot {\ phi}} = {\ frac {2 \ a \ E \ r + L_ {z} \ \ csc ^ {2} \ theta \ (\ Sigma -2r)} {\ Delta \ \ Sigma}}}$

In the case of a mass test body, the point above the variables stands for the differentiation according to the proper time and in the case of a massless test particle according to the affine parameter, which instead of the proper time denotes the distance of the photon locally integrated in the system of the ZAMOs. In the case of a mass-laden test particle, the total physical distance covered is obtained with the integral of the proper time over the local speed of 3 ${\ displaystyle \ tau}$

${\ displaystyle \ mathrm {d} {\ bar {s}} = \ mathrm {d} \ tau \ \ mathrm {d} v \ \ gamma \, \ \ {\ bar {s}} = \ int _ {0 } ^ {\ tau} v (\ tau ') \ \ gamma \, \ mathrm {d} \ tau'}$with the Lorentz factor .${\ displaystyle \ gamma = 1 / {\ sqrt {(1-v ^ {2})}} = {\ dot {t}} / \ varsigma}$

There are , and the components of the local 3 speed ${\ displaystyle v ^ {r}}$${\ displaystyle v ^ {\ theta}}$${\ displaystyle v ^ {\ phi}}$

${\ displaystyle v = {\ sqrt {(v ^ {r}) ^ {2} + (v ^ {\ theta}) ^ {2} + (v ^ {\ phi}) ^ {2}}} = { \ sqrt {(v ^ {x}) ^ {2} + (v ^ {y}) ^ {2} + (v ^ {z}) ^ {2}}}}$

along the respective axes, and it results

${\ displaystyle v = {\ sqrt {\ frac {\ chi (E-L_ {z} \ \ Omega) ^ {2} - \ Delta \ Sigma} {\ chi (E-L_ {z} \ \ Omega) ^ {2}}}} = {\ frac {\ sqrt {{\ dot {t}} ^ {2} - \ varsigma ^ {2}}} {\ dot {t}}}}$.

${\ displaystyle E}$and are the obtained specific energy and the component of the specific angular momentum along the axis of symmetry of the space-time. is the Carter constant named after its discoverer Brandon Carter : ${\ displaystyle L_ {z}}$ ${\ displaystyle C}$

${\ displaystyle C = p _ {\ theta} ^ {2} + \ cos ^ {2} \ theta \ left (a ^ {2} (\ mu ^ {2} -E ^ {2}) + {\ frac { L_ {z} ^ {2}} {\ sin ^ {2} \ theta}} \ right) = a ^ {2} \ (\ mu ^ {2} -E ^ {2}) \ \ sin ^ {2 } I + L_ {z} ^ {2} \ \ tan ^ {2} I}$

These go with

${\ displaystyle k = a ^ {2} \ left (E ^ {2} + \ mu \ right) + L_ {z} ^ {2} + C}$

into the equations of motion. is the polar -, the radial - and the constant the azimuthal component of the orbital angular momentum, which results from the canonical specific momentum components${\ displaystyle p _ {\ theta}}$${\ displaystyle \ theta}$${\ displaystyle p_ {r}}$${\ displaystyle r}$${\ displaystyle p _ {\ phi} = L_ {z}}$${\ displaystyle \ phi}$

${\ displaystyle p _ {\ mu} = g _ {\ mu \ nu} {\ dot {x}} ^ {\ nu}}$

surrender. The time pulse is proportional to the energy: . is the orbit angle of the test particle. For mass-afflicted test particles, the following applies , while for massless particles such as photons . ${\ displaystyle p_ {t} = - E}$${\ displaystyle I}$${\ displaystyle \ mu = -1}$${\ displaystyle \ mu = 0}$

The 4 constants of movement are therefore and . Energy and angular momentum can be obtained from the proper time derivatives of the coordinates or the local velocity: ${\ displaystyle E, L_ {z}, C}$${\ displaystyle \ mu}$

${\ displaystyle E = \ - \ g_ {tt} \ {\ dot {t}} \ - \ g_ {t \ phi} \ {\ dot {\ phi}} = {\ sqrt {{\ frac {(\ Sigma -2 \ r) \ left ({\ dot {\ theta}} ^ {2} \ \ Delta \ \ Sigma + {\ dot {r}} ^ {2} \ \ Sigma - \ Delta \ \ mu \ right) } {\ Delta \ \ Sigma}} + {\ dot {\ phi}} ^ {2} \ \ Delta \ \ sin ^ {2} \ theta}} \}$${\ displaystyle = {\ sqrt {\ frac {\ Delta \ \ Sigma} {(1+ \ mu \ v ^ {2}) \ \ chi}}} + \ Omega \ L_ {z}}$
${\ displaystyle L_ {z} = g _ {\ phi \ phi} \ {\ dot {\ phi}} \ + \ g_ {t \ phi} \ {\ dot {t}} = {\ frac {\ sin ^ { 2} \ theta \ ({\ dot {\ phi}} \ \ Delta \ \ Sigma -2 \ a \ E \ r)} {\ Sigma -2 \ r}} = {\ frac {v ^ {\ phi} \ {\ bar {R}}} {\ sqrt {1+ \ mu \ v ^ {2}}}}}$

## Moving inertial systems

Corotation of locally stationary measuring buoys due to the inertial frame dragging effect

Due to the frame-dragging effect, even the inertial system of an observer that is free of angular momentum and locally at rest corotates with the angular velocity

${\ displaystyle \ Omega = - {\ frac {g_ {t \ phi}} {g _ {\ phi \ phi}}} = {\ frac {r _ {\ mathrm {s}} \ a \ r} {\ chi} }}$

with the rotating central mass, the angular velocity being described according to the coordinate time of the observer, who is stationary relative to the fixed stars and is at a sufficiently large distance from the mass. ${\ displaystyle t}$

Since the ZAMO is at rest relative to the space surrounding it locally, the description of the local physical processes in its reference system takes on the simplest form. So is z. B. only in his reference system the speed of a light beam passing him is equal to 1, while in the system of an observer stationary relative to the fixed stars it would be slowed down due to the gravitational time dilation and shifted in amount and direction due to the frame dragging. The ZAMO can therefore be used as a local measuring buoy, relative to which the speed is determined on site . ${\ displaystyle (v)}$

The gravitational time dilation between an observer who is moving along and sitting on a fixed position and an observer who is far away is ${\ displaystyle \ Omega}$${\ displaystyle r}$

${\ displaystyle \ varsigma = {\ frac {\ mathrm {d} t} {\ mathrm {d} \ tau}} = {\ sqrt {g ^ {tt}}}}$.

The radial local escape speed is thus obtained from ${\ displaystyle v _ {\ mathrm {esc}}}$

${\ displaystyle \ varsigma = {\ frac {1} {\ sqrt {1-v _ {\ mathrm {esc}} ^ {2}}}} \ \ to \ v _ {\ mathrm {esc}} = {\ frac { \ sqrt {\ varsigma ^ {2} -1}} {\ varsigma}}}$.

For a test body with results , d. i.e., he escapes the crowd with the exact escape speed. ${\ displaystyle E = 1 \, \ L = 0}$${\ displaystyle v ^ {r} = v _ {\ mathrm {esc}}}$

## Circular paths

Pro- and retrograde orbit velocity as a function of and${\ displaystyle a}$${\ displaystyle r}$
Photon
orbit on r ° = (1 + √2) GM / c² at a local inclination angle of 90 ° Because of the twisting of space-time, the photon moves along the -axis despite vanishing axial angular momentum . This leads to a track inclination of 61 ° being measured from a distance.${\ displaystyle (L_ {z} = 0).}$${\ displaystyle \ phi}$
A rotating black hole has 2 radii, between which photon orbits of all conceivable angles of inclination are possible. In this animation all photon orbits are shown for.${\ displaystyle a = M}$

The pro- and retrograde orbit speed (relative to the ZAMO) results from

${\ displaystyle {\ dot {p}} _ {r} = v ^ {r} = v ^ {\ theta} = 0 \, \ \ \ theta = \ pi / 2 \, \ \ v = v ^ {\ phi} \, \ \ {\ bar {a}} = a / M \, \ \ {\ bar {r}} = r / M}$

set and after is resolved. This results in the solution ${\ displaystyle v ^ {\ phi}}$

${\ displaystyle v _ {\ pm} ^ {\ circ} = {\ frac {{\ bar {a}} ^ {2} \ mp 2 {\ bar {a}} {\ sqrt {\ bar {r}}} + {\ bar {r}} ^ {2}} {{\ sqrt {{\ bar {a}} ^ {2} + ({\ bar {r}} - 2) r}} \ left ({\ bar {a}} \ pm {\ bar {r}} ^ {3/2} \ right)}}}$

for the prograde (+) and retrograde (-) orbital velocity. For photons with , therefore, results ${\ displaystyle v = 1, \ \ mu = 0}$

${\ displaystyle r _ {\ pm} ^ {\ circ} = r _ {\ mathrm {s}} \ \ left (\ cos \ left ({\ frac {2} {3}} \ cos ^ {- 1} (\ mp {\ bar {a}}) \ right) +1 \ right)}$

for the pro- and retrograde photon circle radius in Boyer-Lindquist coordinates. For a photon with vanishing axial angular momentum, i.e. a local inclination angle of 90 °, a closed orbit results

${\ displaystyle r _ {\ perp} ^ {\ circ} = r _ {\ mathrm {s}} \ {\ sqrt {1 - {\ frac {{\ bar {a}} ^ {3}} {3}}} } \ cos \ left ({\ frac {1} {3}} \ cos ^ {- 1} \ left ({\ frac {1 - {\ bar {a}} ^ {2}} {\ left (1- {\ frac {{\ bar {a}} ^ {2}} {3}} \ right) ^ {3/2}}} \ right) \ right) + {\ frac {r _ {\ mathrm {s}} } {2}}.}$

Between and photons are orbits of all possible path angle between ± 180 ° (retrograde) and 0 ° (prograde) possible. Since all photons orbits have a constant radius Boyer-Lindquist, can of the respective and found suitable inclination angle by the radial pulse derivation as above at 0, the initial latitude set to the equator and after is dissolved. ${\ displaystyle r _ {+} ^ {\ circ}}$${\ displaystyle r _ {-} ^ {\ circ}}$${\ displaystyle a}$${\ displaystyle r}$${\ displaystyle {\ dot {p}} _ {r}}$${\ displaystyle \ theta _ {0}}$${\ displaystyle v ^ {\ phi}}$

For photon orbits on there is also an equatorial inclination angle of 90 ° observed from a distance. The local angle of inclination relative to a co-rotating observer on site (ZAMO) is higher (the axial angular momentum is then negative), but is compensated due to the frame dragging effect. In the Schwarzschild limit with the photon bits of all orbital inclination angles are noticed and form the spherical shell-shaped photon sphere. ${\ displaystyle r = 3M}$${\ displaystyle a}$${\ displaystyle a = 0}$${\ displaystyle r ^ {\ circ} = 3M}$

In the extreme case of , both equatorial photon orbits with and at the same time particle circle orbits would result. The reason for this is that the circles starting from the center can have the same value on the radial coordinate, while in the Euclidean embedding they can also have an infinite distance from one another if they have the same local circumference as in the case of . ${\ displaystyle a = M}$${\ displaystyle r = M}$${\ displaystyle v _ {+} ^ {\ circ} = 1}$${\ displaystyle v _ {+} ^ {\ circ} = 1/2}$${\ displaystyle a = M}$${\ displaystyle 2 \ pi {\ bar {R}}}$

## literature

• Robert H. Boyer, Richard W. Lindquist: Maximal Analytic Extension of the Kerr Metric. In: Journal of Mathematical Physics. Vol. 8, Issue 2, 1967, pp. 265-281. doi: 10.1063 / 1.1705193 .
• Barrett O'Neill: The geometry of Kerr black holes. Peters, Wellesley 1995, ISBN 1-56881-019-9 .
• David L. Wiltshire, Matt Visser, Susan M. Scott (Eds.): The Kerr spacetime: Rotating Black Holes in General Relativity. Cambridge University Press, Cambridge 2009, ISBN 978-0-521-88512-6 .
• Roy P. Kerr: The Kerr and Kerr-Schild Metrics. In: Wiltshire, Visser, Scott: The Kerr Spacetime. Cambridge UP, 2009, pp. 38-72 (First publication: Discovering the Kerr and Kerr-Schild metrics. Arxiv : 0706.1109 ).

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43. ^
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