Thin disks as sources of stationary

axisymmetric electrovacuum

TOMAS LEDVINKA

Department of Theoretical Physics

Faculty of Mathematics and Physics

Charles University

V Holesovickach Praha Czech Republic

A disertation submitted to the Faculty of Mathematics and Physics Charles University

in accordance with the regulations for admission to the degree of Do ctor of Physics

Branch of the do ctoral study F Theoretical physics astronomy and astrophysics

Praha Octob er

Acknowledgements

I would like to thank professor Jir Bicak for his help He prop osed an inter

esting problem encouraged me in my work and he has always had the time to

discuss my work with me and has made valuable comments on the content of

the thesis

Contents

Intro duction

Relativistic disks as sources of the Kerr metric

Thin layers as sources of the stationary axisymmetric spacetimes

Total angular momentum and charge

Energy conditions and the interpretation of the surface

stressenergy tensor of the disk

Disks formed of counterrotating surface streams

of charged particles

KerrNewman disks

Schwarzschild disks

Extreme Reissner Nordstrom disks

Kerr disks

Charged KerrNewman disks

TomimatsuSato spacetimes

Concluding remarks

App endix A Radial prop erties of Kerr disks

App endix B Radial prop erties of KerrNewman disks

App endix C Radial prop erties of TomimatsuSato disks

Intro duction

Unlike in Newtonian gravity there is a lack of complete exact solutions repre

senting selfgravitating rotating compact ob jects in due to its

essential nonlinearity On the other hand a great progress has b een made within

a class of the stationary axisymmetric vacuum or electrovacuum asymptotically

at spacetimes where p owerful metho ds were applied to obtain analytic solutions

of Einsteins equations Among these solutions the most imp ortant role still

plays the discovered by Kerr One of the reasons consists in the

fact that compact astrophysical ob jects very often p osses large sp ecic angular

momentum and a Kerr black hole is thus a natural candidate for the spinning

remnant of gravitational collapse Indeed assuming that the the cosmic cen

sorship conjecture holds the blackhole uniqueness theorems guarantee that the

Kerr metric represents the unique solution describing all rotating vacuum black

holes cf eg If one takes into account that the prop erties of the interior

Schwarzschild solution describing a static spherically symmetric star surrounded

by its gravitational eld are even now sixty years after its discovery inspiring

astrophysicists in a deep er understanding of the p eculiarities of general relativity

there is no doubt that an exact interior Kerr solution would b e even more

exciting

This of course has b een realized by many workers As the prop erties of Kerrs

spinning particle metric were analyzed the search for such solution started

eg Since that time several authors have b een able to construct interior

solutions that can b e matched with the Kerr vacuum geometry as an external

eld but the material of various balls or shells has very strange prop erties and

allows no reasonable interpretation For example quite recent

work that app eared in gives a toroidal source consisting of a toroidal

shell a disk and an annulus of matter interior to the torus The

of the disk and annulus are negative The review on the Sources for

the Kerr Metric written in contains references and concludes

with Destructive statements denying the existence of a material source for the

Kerr metric should b e rejected until if ever they are reasonably justied To

summarize in Hermann Bondis way the sources suggested so far for the Kerr

metric are not the easiest materials to buy in the shops

In this situation less complicated sources should b e studied the most natu

ral choice seems to b e an axisymmetric thin disk Even among such sources the

most realistic one the thin disk of a nite diameter without a radial pressure was

excluded from a list of p ossible sources of external parts of Kerr spacetime

The situation is somewhat dierent in the sp ecial case of the extreme Kerr metric

where there is a denite relationship b etween mass and angular momentum The

numerical study of uniformly rotating disks indicated how the extreme Kerr

geometry forms around the disk in the ultrarelativistic limit These numerical

results were supp orted recently by the imp ortant analytical work If a re

quirement of the nite diameter is relaxed it can b e shown that such disks can

play a role of the interior Kerr solution with any angular momentum see

also

In this thesis b oth our rst results concerning Kerr disks and their gener

alization to stationary axisymmetric electrovacuum spacetimes are presented

The section is based on letter published in The basic idea which

enables one to construct physically acceptable thin disks as well as completely

unreasonable shells is then presented in section in greater detail Because we

are interested in the interpretation of the stressenergy tensor of the disk using

the usual matter we study only disks without radial pressure Even though the

radius of these disks is innite they have nite integral quantities such as mass

or angular momentum which means that their resp ective densities as dened in

section have reasonable asymptotic b ehavior

In the central region of the disk the situation is quite complicated and the

prop erties of the stressenergy tensor must b e carefully studied using covariant

characteristics the energy conditions which are intro duced in section It is

shown that for a general axisymmetric stationary electrovacuum asymptotically

at spacetime which admits WeylPapap etrou co ordinates the reasonable disk

like source may exist and that the surface stressenergy tensor of such a disk may

consist of two counterrotating streams of charged electro geo desic particles at

each radius if certain conditions studied in section are met Although in last

years several galaxies were found to have the disks built from two nearly identical

counterrotationg stellar comp onents see eg the presence of counter

streaming comp onent apparently makes such disks not so realictic astrophysically

Wee shall see however that the Kerr disks exist with the mass of the corotating

comp onent exceeding of the total mass of the disk

It is clear that physically realistic disks can b e constructed only within certain

limits of the values of total charge angular momentum eective radius of the

disk and other parameters which enter into the metric The survey of the pa

rameter space of the KerrNewman disks is presented in section The analogous

survey of TomimatsuSato disk sources is given in section In these two

sections the prop erties of disks generating considered spacetimes are studied with

an emphasis on p ossible interpretations of the surface stressenergy tensor of the

disk and on various relativistic phenomena such as the existence of ergoregions

or of closed timelike curves in the vicinity of the disks In app endices A B and

C the plots of relevant disk quantities as functions of the circumferential radius

are given

Relativistic disks as sources

of the Kerr metric

This section is based on letter with several notes app ended

Bicak LyndenBell and Katz BLK in the following have shown that

most vacuum static Weyl solutions can arise as the metrics of counterrotating

relativistic disks see also for other references on relativistic disks The

simple idea which inspired the work of BLK is commonly used in Newtonian

galactic dynamics the idea rst app eared in Kuzmins work Imagine

a p oint mass M placed at a distance b b elow the centre of a plane z

see Figure This gives a solution of Laplaces equation ab ove the plane Then

consider the p otential obtained by reecting this z p otential in z so

that a symmetrical solution b oth ab ove and b elow the plane is obtained It is

continuous but has a discontinuous normal derivative on z the jump in which

gives a p ositive surface density on the plane In galactic dynamics one considers

general line distributions of mass along the negative z axis and employing the

metho d describ ed ab ove one nds the p otentialdensity pairs for general axially

symmetric disks In fact Bicak LyndenBell and Pichon found an innite

numb er of new static solutions of Einsteins equations starting from realistic

p otentials used to describ e at galaxies as given recently by Evans and de Zeeuw

z

M

b

b

M

Figure Gravitational eld of a Kuzmin disk illustrated using lines of force

Here we wish to demonstrate that a similar metho d works also for axisymmet

ric reection symmetric stationary spacetimes It is imp ortant to realize that

although now no metric function solves Laplaces equation as in the static case

we may view the pro cedure describ ed ab ove as the identication of the hyp er

surface z b with z b The eld then remains continuous but the jump of

its normal derivative induces a matter distribution in the disk which arises due

to the identication What remains to b e seen is whether the material can b e

b ought in the shops

This idea can b e employed for all known asymptotically at stationary vacuum

spacetimes eg for the TomimatsuSato solutions for the rotating Curzon

solution or for other metrics cf for references Here we shall illustrate the

pro cedure for the simplest but most interesting case the Kerr metric

A stationary axisymmetric vacuum metric can b e written in canonical co or

dinates t z in the form

i h

2 2 2 2 2 2 1

f dt Ad g d dz d ds f

where f g A are functions of z we put c G Spheroidal co ordinates

x y are commonly used and we intro duce b oth prolate and oblate

ones since we wish to include all typ es of Kerr metrics

h i

12

2 2

z xy x y const

For the Kerr solution mass M sp ecic angular momentum a the functions

in are ratios of p olynomials

f LE g LF A B L

2 2 2 2 2 2 2

L p x q y E px q y

2 2 2 2

F p x y B M q y px

2 2

p q q aM M p

Here for a M for a M If a M then M

p q

Now we identify the planes z b const and z b which will lead

to disks with zero radial pressure With the Kerr geometry the matching is more

complicated than in the static cases and therefore we turn to Israels covariant

formalism see for its recent exp osition This enables us using Einsteins

eld equations to link the surface stressenergy tensor S of the disk arising

(a)(b)

h i

from this identication to the jump K K j K j of

z =+b z =b

(a)(b) (a)(b) (a)(b)

normal extrinsic curvature across the timelike hyp ersurface given by z b or

z b The tetrad indices are denoted by a b with the tetrad vectors

b eing chosen so that three vectors tangent to are just e

(a) (a)

p

g is the unit normal fx g ft z g a while n

z z

As a consequence of Einsteins eld equations we nd the nonvanishing com

p onents of the surface stressenergy tensor to b e

Z LF F E L

S

(0)(0)

2

E L F E

F E B ZBF

S

(0)(1)

2

E B F E

2

ZF B L B F E

2

S

(1)(1)

1

2

L L F E B E

2 2 2 2

where L E F B are given by B E

1

12

y E

+

Z y y j

+ z =+b

2 2

F x y

z =b

2 2

and the op erator is dened by the relation f y f x y for

z +

z =b

any f For the Kerr metric

2 2 2 2 2

L xp x q y q

2 2 2

E px px xq y

h i

2 2 2 2

F x p x q y

i h

2 2 2

B M q p y q px

In eqs b oth x and y are evaluated at z b by inverting one nds

2 2

xj xj y j y j Eqs give the stressenergy tensor

z =+b z =b z =+b z =b

of the disks

Let us now show that the disks may b e interpreted as b eing made of two

streams of collisionless particles that circulate in opp osite directions In order to

see this we nd at each radius and for z b the preferred observer for whom

1

the stressenergy tensor acquires a diagonal form Let his velo city the

timelike eigenvector of the tensor read

V N

and his unit spatial basevector in the direction b e

W J

The conditions V V W W V W determine three of the pa

rameters entering in terms of the fourth of say Assume now a tensor

00

T which will b e calculated from has nonvanishing comp onents T

01 11

T T Then by cho osing

h i

12

1 0 1 0 2 0 1 0

T T T T T T T

1 0 1 0 1 0 1

1

The question of existence of such observers was not addressed here In section we show

2

that it is related to the energy conditions and that if S S S V is real

(0)(0) (1)(1)

(0)(1)

vector

T can b e cast into the form

T V V PW W

where

00 2 11

T T

2 2 2

N

11 2 00

T T

P

2 2 2

J

Hence the observers circulating with the velo city will see the diagonal

form of T with

T T V V

(0)(0)

T T V W

(0)(1)

T W W P T

(1)(1)

We call such observers isotropic IOs since the isotropy concerns the

direction only

If P IOs can consider the stressenergy tensor as representing

two equal streams of collisionless particles that circulate in opp osite directions

with the same velo city

12

P

U

If is the surface prop er rest mass density of one stream measured in axes that

p

move with it then the surface density of its rest mass measured by an IO is

12

1

2

U The surface energy density of the pair of streams is

0 p

2

p 0

T

(0)(0)

12

2

2

U

U

2

The tangential pressure caused by the counterrotation is T U The

(1)(1)

sum of the prop er rest mass surface densities of b oth streams is simply

P

p

The condition that the velo city of the streams U do es not exceed the velo city of

light is just the dominant energy condition see eg

What is the relation of IOs to the lo cally nonrotating frames LNRFs

IOs rotate with resp ect to LNRFs with the velo city given by

1

V g

where g g Therefore the streams circulate with dierent velo cities

t

in LNRFs and of course with resp ect to distant stars That is why the disks

pro duce stationary rather than static elds

The physical quantities intro duced in are given in terms of the

metric and the stressenergy tensor Hence al l the physical

quantities describing the disks are given analytical ly The resulting expressions

however are so complicated that it is only reasonable to exhibit them graphically

Yet the central surface density has a simple form

2 2

b M a M

c

32

2 2 2 12 2 2

b a M b M a

In order for the central density to b e p ositive one must make the identication

2 2 2

at b M a This is evident in the blackhole case since the holes interior

2 2 12 2 2 12

is mapp ed onto the ro d M a z M a If a M

one has to cho ose b a M The central density can b ecome arbitrarily large

2 2 2 12

for b M a a region close to the horizon which itself was cut

o is then included can b e large also for a M However in the cases

c

with a M it turns out that the physical condition P cf leads to

the inequality

2 4 2 5 3 4 2

p x x px x x x x

2 2 2

where x bp and p a M cf which restricts b more strongly For

2 2

a M one nds b a M Then we can construct physical disks only

min

with U M a

max

In co ordinates z the ergoregion cf eg has a toroidal character

the center of the generating circle with const is at M q q

0

p

the disks given the radius b eing R M q For aM

2 2 12 2

by M a b M a pro duce the ergoregion Their central density

is p ositive and the graphical results show see b elow that they are physical

everywhere It turns out that physical disks pro ducing an ergoregion exist even

for aM provided that aM

Here we shall conne ourselves to illustrate just one case with aM

The disks corresp onding to dierent choices of b are compared by exhibiting the

circular velo cities etc as functions of the circumferential prop er radius R

p

where g is given by In Figure a we give the velo city curves g

of counterrotating streams as measured by IOs These were calculated starting

from for twelve disks the disks b ecome more relativistic with decreasing b

2

The physical interpretation means in this section counterrotating streams of nongeo detic

particles with equal opp osite velo cities with resp ect to IO which satisfy energy conditions

It is the easiest generalization of the BLK disk mo del If one would like to construct such

disk mo del in a greater detail the mechanism of exchange of radial momentum b etween the

streams should b e invented In section and we distinguish b etween the energy conditions

requirements such as P which is equivalent to V and the existence of b oth co

and counterrotating circular geodesics This is given by radial b ehaviour of metric functions

rather than altitudinal which determines stressenergy tensor of the disk (a) 1 (b) 0.1

0.8 0.08

0.6 0.06 M p 2 2 s s Velocity 0.4 0.04 Stream velocity U * 0.2 0.02 FIOs velocity V 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6

R/M R/M

Figure Velo city and density proles of Kerr disks with aM

In the highly relativistic disks upp er two curves U increases extremely rapidly

as one moves away from the centre approaching the velo city of light at R M

and then start decreasing Also in Figure a velo city curves of the IOs with

resp ect to the LNRFs are plotted by using Although these velo cities achieve

also high values V their maxima o ccur at larger R M In

particular in the two most relativistic disks the IOs the velo city curves of

which cross the other curves from b ottom to the top in Figure b do not move

so rapidly with resp ect to LNRFs close to the centreHere close refers to the

prop er circumferential radius R Just these two disks from the twelve disks

exhibited pro duce an ergoregion

Figure b gives the plots of the sum of the surface prop er rest mass densities

of b oth streams as calculated from For highly relativistic disks rises

p

rapidly towards the centre however it decreases then very rapidly with R and in

the most relativistic case even reaches a lo cal minimum at those R where U

3

The central gravitational redshift is z in this case

c

We are not aware of any exact analytic solutions for the sources of the sta

tionary gravitational elds with such physical prop erties as the disks constructed

here Although extending to innity they have nite mass and exhibit interest

ing relativistic prop erties like high velo cities large redshifts and dragging eects

including ergoregions They may app ear somewhat articial astrophysically but

3

in the Newtonian limit when no dragging arises their features are the same as

for the rotating disks with only one stream of particles in the relativistic regime

the central parts of a realistic highly relativistic rotating attened ob ject should

have some prop erties in common with these disks

3

In fact even in the Newtonian gravity some counterrotation is necessary if the disk should

have prescrib ed values of mass and angular momentum although the latter has no inuence on

the gravitational eld Eg in zero angular momentum case corotating and counterrotating

comp onent of the disk should have the same value of total angular momentum with opp osite

sign

Thin layers as sources of the stationary

axisymmetric spacetimes

In this section we shall investigate the construction of the disk mo dels in a greater

detail and generality

The easiest way to nd a source pro ducing a given stationary solution M of

0

vacuum or electrovacuum Einstein equations It consists in glueing two exactly

+

identical spacetimes M M M together at their b oundary hyp ersurface

0

S They are thus separated by a dimensional layer of innite function typ e

curvature which b ecomes the source of the spacetime rather than various hidden

or naked singularities This metho d avoids the complicated problem of solv

ing the interior eld equations and matching the solution to the electrovacuum

solution outside

+ - S = S

e(2) -n +n

e(1)

Figure A part of a stationary spacetime M depicted as a threedimensional b ox with time

0



co ordinate omitted and hyp ersurfaces S delimiting the emphasized part which represents

+

b oth M and M

Threedimensional hyp ersurfaces S which will b e identied see Figure

may b e characterized by two imp ortant quantities the metric induced on them

by external metrics

e g g e

(a)(b)

(b)

(a)

and a b and their extrinsic curvatures

r e e r n n e K e

(a)(b)

(b) (b)

(a) (a)

a form the basis tangent to S in general not normalized Vectors e

(a)

n Here n and unit vector n is orthogonal to S ie n n e

(a)

+

are normals p ointing from M to M The identication is admissible only if

the induced metrics are equivalent This is guaranteed by the fact that

+

only the orientation of identied b oundaries S and S is dierent whereas basis

+

vectors e are identical We will assume hyp ersurface S to b e prescrib ed by

(a)

+ 12

an equation x which yields the normal vector n

where an appropriate sign must b e chosen

h i

+ +

The jump in the extrinsic curvatures of S and S K K

(a)(b) (a)(b)

K determines the surface density of the stressenergy tensor by the relation

(a)(b)

(c)

K g K S

(a)(b) (a)(b) (a)(b)

(c)

Because whole spacetime is made of two identical copies we have K

(a)(b)

+

K

(a)(b)

Two commuting Killing vector elds of an stationary axisymmetric spacetime

will b e denoted and having closed orbits The co ordinates asso ciated

with the Killing vectors will have their usual names t and

We exclude a case of the dynamic layer generating a stationary spacetime

coincides with and that the and assume that one of the basis vectors e

(0)

stationary basis e is chosen e

(a) (a )

The resulting spacetime should have reasonable physical prop erties One

typical feature of an asymptotically at spacetime is that at spatial innity large

spheres surrounding the center have the ratio of the square of any circumference

to the area of the sphere equal to To meet this requirement we will assume

that the intersection of the b oundary hyp ersurface S and hyp ersurface of con

stant time t is a smo oth surface the layer at given time which at innity

asymptotically approaches a plane we will denote P each part thus delimits the

solid angle

The symmetries of the original spacetime M enable one to cho ose M

+

Q M where Q is some map preserving g and A For example let us assume

that the original spacetime is stationary axi and reectionsymmetric Kerr so

lution with mass M and angular momentum J M a If we intro duce some

altitudinal co ordinate z related to the reection symmetry and omit the time

+

translation the following symmetries can b e used to obtain the M from M

z z

and a a

Note that the z co ordinate entering may even b e the latitudinal co ordi

nate where is one of BoyerLindquists co ordinates as well as z

co ordinate of the original line element by Kerr although we will prefer the

WeylPapap etrou z co ordinate The rst symmetry which enables one

+

to identify M with its reection symmetric copy is illustrated in Figure a

The other p ossible interpretation Figure b is based on comp osition of

and If the b oundary surface is reection symmetric with resp ect to

it can b e rst rotated by angle which means that identication of p oints A

and B should b e made Then the symmetry is plugged in and the natural

identication of A and A can b e done but one has to rememb er that b oth parts

have opp osite angular momentum parameter

z

z

, A B B

z = 0 , A A

A ,

a b

Figure Interpretations of the resulting spacetime based on a reection b axial sym

metry

These two ways of identication are useful as an illustration of the metho d

but they do not alter the main idea that the resulting spacetime consists of two

identical parts and of a thin layer with the surface stressenergy tensor deter

mined by the prop erties of the b oundary hyp ersurface S The new spacetime M

need not necessarily b e axially symmetric as a whole if the asymptotical plane

P is not parallel to the equatorial plane This situation is illustrated in Figure

where the orientation of the angular momentum of b oth M and M is schemat

0

ically presented b efore and after the identication Complicated structure of the

surface stressenergy tensor no symmetry to o many nonvanishing comp onents

yields no obvious interpretation There may b e another reason to prefer P to b e

+

parallel to the equatorial plane If S do es not cross the equatorial plane one

can thing of M as an original spacetime M with removed equatorial region

0

If stationary and axisymmetric electromagnetic p otential of electrovacuum

[ ]

spacetime is a linear combination of Killing vectors and ie A

t

Maxwell tensor implies R R and WeylPapap etrou WP co ordinates

t

i h

2 2 2 2 2 2 2 2 2

e dt Ad e d dz d ds e

Figure Orientation of the angular momentum b efore and after the identication when P

is tilted with resp ect to the equatorial plane

may b e intro duced Even though the resulting spacetime is axially symmetric

its source the surface stressenergy tensor S may not exhibit this symmetry

(a)(b)

It dep ends on the shap e of the hyp ersurface S If S is tangential to the Killing

vector eld ie if the orbits of are contained in this hyp ersurface the basis

vectors may b e chosen such that e which ensures that S is axially

(a)(b)

(a )

symmetric Such sources will b e called disks This means that if n n

we can cho ose e and e Then if e is chosen such that g and

(0)(2)

(0) (1) (2)

g vanish the corresp onding comp onents S and S will cancel out and

(1)(2) (0)(2) (1)(2)

the only nonvanishing comp onents of the surface stressenergy tensor of the disk

are three diagonal comp onents of S energy density and azimuthal and radial

(a)(b)

pressures and one nondiagonal comp onent S angular momentum density

(0)(1)

A similar pro cedure is used to express electric charge and current density of

the disk for a general formalism for charged shells see The pro jection

of an electromagnetic vector p otential into the b oundary surfaces A e A

(a)

(a)

has to b e identical on b oth sides of the surface S the electromagnetic eld F

surrounding the disk is pro duced by the surface electric current density attached

to the layer

i h

J F e n

(a)

(a)

[ ]

If e e and A this relation yields the stationary

(0) (1)

axisymmetric electric current with J

(2)

In our work we will use the hyp ersurface x z b which turns out

to corresp ond to disks without radial pressure for disks with radial pressure see

Then

12

n g

z z

(z )

and the simplest tangent basis is

e

(a) (a)

Since this choice makes g g we will not use brackets to distinguish

(a)(b) ab

pro jected and metric comp onents but one has to rememb er that WP co ordinates

must b e applied to obtain correct results

The fact that S is a surface of constant co ordinate z implies simple formula

12

for the extrinsic curvature K g g The surface stressenergy tensor

ab ab z

and electric current then turns out to b e

p

g

g

ab

S

ab

g

z

A

a

z

J

p

a

g

These formulae yield explicit expressions after any stationary axisymmetric

electrovacuum exact solution g A is plugged in Unfortunately they tend

to b e very long and so they will b e explicitly given only in particular cases eg

Q M when some simplications take place and resulting quantities have clear

physical structure

Total mass angular momentum and charge

The considered metho d of construction of the disk sources guarantees that the

values of integral parameters such as the total mass angular momentum or charge

are equal to those of the original spacetimes The total quantities can also b e

obtained by integration of corresp onding surface densities over the disk If an

electromagnetic eld is present the volume integral of electromagnetic energy

density must also b e considered

In stationary asymptotically at spacetime the mass M dened by asymptotic

b ehavior of metric can b e expressed as an integral of stressenergy tensor com

p onents over spacelike hyp ersurface surrounding the regions of nonvanishing

T The Komar mass integral see eg

Z

r d M



over the dimensional b oundary of spacelike hyp ersurface with b eing

the timelike Killing vector can b e converted using the Stokes theorem to the

volume integral

Z

M T T dS



p

t

g d dz d is a volume element of hyp ersurface t const where dS

Since the surface stressenergy tensor S is equivalent to surface distribution the

ab

total stressenergy tensor can b e written as a sum of the regular electromagnetic

part T and the surface distribution prop ortional to z

p

a b

T T e e S z g

ab z z

where z z b sgn z Thus the Komar mass consists of two comp onents

the mass of electromagnetic eld M and the mass of the uid of the disk M

E D

Z

p

t

M g d dz d T

E

t



Z

q

t a

M S S g g d d

D z z

t a

D

If disks without the radial pressure are considered S and the line element

2

giving g g g is used the last integral reads

z z

Z

p

t

M S S g d

D

t

D

p

g for It is physically reasonable to use the prop er circumferential radius R

lab eling the radial co ordinate when depicting any radially dep endent quantity

in the disks The surface density with resp ect to the circumferential radius

D

may b e dened by the relation

Z Z

p

t

R dR M g R dR S S

D D

t

g

D D

This will enable us to compare mass density curves of various typ es of disks

Using the surface stressenergy tensor the mass of the disk can b e de

termined by appropriate rst derivatives of the metric

Z

t

M g g d d

D tz

This relation can b e expressed in terms of the WeylPapap etrou metric p otentials

and A

Z

AA

z

4

M e d d

D z

2

Here the rst term coincides with the Gauss law of Newtonian gravity M

N ew t

H

1

r dS

The integral expression for M can b e also derived directly from

D

if an appropriate limiting pro cedure is used b ecause after the identication is

made the antisymmetric tensor eld r which dep ends on rst derivatives

of the metric tensor is no longer smo oth As the volume containing the

disk D is shrinking towards the disk the integral approaches M The

D

z =+b

together with the surface element d directly yields so jump r

z =b

the quantity M can b e dened indep endently on the denition of S and its

D ab

interpretation as a distribution

The total angular momentum in axisymmetric stationary spacetime can b e

dened by

Z

J r d



where is the axial Killing vector eld The integral is equivalent with the

volume integral over regions of nonvanishing T

Z

J dS T



Using the metho ds already mentioned these integrals can b e rewritten to the

integral over the surface of the disk

Z

p

t

g J S d d

D

The last integral prop erty of the disks to b e considered here is the total charge

Z

p

t

d g Q J

All integral quantities can b e used as indep endent checks of the co de that

evaluates S and J

ab a

Energy conditions and the interpretation of

the surface stressenergy tensor of the disk

The energy conditions intro duced in the studies of global prop erties of spacetimes

see eg are formulated in an invariant form and can b e used when an

invariant characterization of the surface stressenergy tensor S is needed We

will use the dominant and the weak energy conditions in order to nd physically

realistic sources The weak energy condition requires that any observer with

her velo city W must observe a nonnegative energy density S W W The

dominant energy condition is based on the prop erties of the energymomentum

current S W this should b e future directed timelike vector for classical

matter If this condition is satised the weak energy condition holds

The source which satises the energy conditions may still have an unclear

physical interpretation The next step consists in the diagonalization of S If

the equation

S g X

has two real nonzero eigenvalues and p and p the corresp onding

eigenvectors U and V can b e rescaled into the form U U V V

b ecause U V The equation can b e viewed as a diagonalization of

the matrix dened by the mixedindices tensor S which is no longer symmetric

and has two dierent real eigenvalues corresp onding to the energy density and

azimuthal pressure only if

t 2 t

S S S S

t

t

This inequality also implies that the disks which cannot b e diagonalized must

have det S A B t

AB

The ab ove pro cedure is equivalent to the transformation of S into the frame

of an observer moving with the velo city U so what

S U U pV V

here V is the unit vector in azimuthal direction in observers frame We have

called such an observer the isotropic observer in section If the decomp o

sition of S is p ossible the weak energy condition is equivalent to

and p

and the dominant energy condition reads

and jpj

The eigenvectors U and V can also b e used for the decomp osition of the

electric current inside the disk

J U j V

into the prop er charge density and the current j as measured by IOs

The diagonalization describ ed ab ove also yields a p ossible interpretation of

the surface stressenergy tensor of the disks Imagine that at each radius of the

disk a massive circular ring is placed Let it rotate with the velo city U its

linear energy density is equal to the pro duct of the prop er surface density and

its prop er width The ring is supp orted against the collapse or expansion by its

internal azimuthal pressure p Although usual materials do not provide jpj

they are consistent with energy conditions and used not infrequently in general

relativity The material of such rings must b e sup erconductive if j and

charged when this is consistent with the survey In the following

section however we shall study more physical situations in which the stress

tensor can b e interpreted as counterrotating streams of freely moving particles

Disks formed of counterrotating surface

streams of charged particles

To interpret diagonal surface stressenergy tensor with two nonzero comp onents

S and S placed in the equatorial plane of static axisymmetric spacetime Mor

tt

gan and Morgan intro duced the disk created from counterrotating surface

circular streams of particles

The most noticeable fact is that at a given radius the velo city of counter

rotating particles which generate the azimuthal comp onent of stressenergy ten

sor and the velo city of a particle on circular geo desic on that radius are equal

this is so b ecause the p otentials satisfy the Einstein equations To illustrate this

coincidence we put A in Then the angular velo cities of the circular

geo desic motion

s s

2

e g

tt

1

g

and the angular velo cities which are needed to generate the S comp onent of

the surface stressenergy tensor

v

s

u

2

tt

u

S e

z

t

2

S

z z

are equal b ecause the vacuum Einstein equation

z z

R

z

holds outside the disk

The concept of counterrotating streams can b e extended to general stationary

axisymmetric asymptotically at electrovacuum spacetimes with the reection

symmetric metric and electromagnetic p otentials entering If the radial pres

sure comp onent of the surface stressenergy tensor S attached to the surface of

ab

identication vanishes due to the choice x z b the p ossibility of

construction of the disk from the surface streams of particles on circular orbits

can b e studied in the stationary case as well The nondiagonal metric implies

the nondiagonal surface stressenergy tensor and the counterrotating streams

can no longer have identical surface density and opp osite velo cities If the elec

tromagnetic eld is present it is generated by appropriate charges attached to

the particles of b oth streams

The two streams of charged dust moving in the plane of identication z b

are assumed to have the following energystress tensor and electric current

S U U U U

+

+

+

U J U

+

+

The circular orbits U N of b oth streams have to satisfy the

radial force balance U U F U Due to the symmetries the covariant

(); ()

derivative simplies and the equation of electrogeo desic motion for b oth streams

reads

U U g F U

() ()

Since only the time and angular comp onents of b oth the surface stressenergy

tensor S and the stream particle velo city U are nonzero only the following

seven equations remain to b e satised

tt

S

t

S

+

+

2 2

S

+

t

J

+

J

+

2

F F g g g

t tt t

2

where N and N

Although for given energystress tensor and electric current six quantities

and must b e determined using these seven equations the system is

not overdetermined b ecause the following linear combination of

t tt t

F J F J g S g S g S

t tt t

is equal to the sum of the lefthand sides of b oth equations

So the disk can b e describ ed by the simple dust mo del with electric

current only if the considered combination is zero This is true for any

numb er of streams as long as each consists of electrogeo desic particles

This is guaranteed by virtue of Einstein equations and the contracted Gauss

Co dazzi equations which result in the following relation see eg

F J S

;

where the left hand side of this equation is equal to We can therefore discard

one of the seven equations If the third comp onent of is chosen the

energy and charge densities can b e expressed in a quite symmetric manner

tt t

S S

+

t

J J

+

This can b e immediately substituted into to obtain following system of

two coupled equations for angular velo cities of b oth streams

2 t tt t

F F g g g J J S S

t + tt t +

+

t 2 tt t

F F J J g g g S S

t + tt t +

This system of equations can b e factorized to the pro duct of cubic and

quadratic equations The ro ots of the cubic equation lead to unphysical sym

metric solutions with The quadratic equation

+

2 2

2 tt tt t t tt t t t 2

S g g S g J F g g S S J F g S S g

tt t t

2

2

tt t tt t tt t

S g J F J F S g J F g S S g g J F J

t t t t tt

2

tt t t t tt t

S J F g S g J F J g S F S g g

tt t t t

2

t t t tt 2 t

g J F S J F F S g S

t t t

2 2

2

t t t t t tt 2

J F g S J F J F J F g J F S S g

t t t t t tt

2

tt t t tt t tt

S g S g g S g S J F g J F S g

t tt tt t tt tt

or its vacuum F limit

2

g g g

t tt

which is the radial comp onent of the usual equation of circular geo desic de

termine the stream velo cities Both corotating and counterrotating angular

velo cities must b e checked whether they represent timelike vectors If two

subluminal circular electro geo desics exist their orbital frequencies can b e

substituted into Physically acceptable are only p ositive stream densities

but we do not p ose any restrictions on the sign of the charge density or

the value of the sp ecic charge

The p ositivity of the stream densities note that the proper densities are

AB

is related to the sign of the determinant of the two by two matrix S

A B t b ecause

AB 4 2 t t 2

det S det S pU V U V

AB + +

If is understo o d as the denition of a quadratic form on the twodimensional

vector space of covariant vectors one nds the following statement trivial Both

AB

stream densities are p ositive if S X X for at least one twocomp onent

A B

AB

vector X det S and two subluminal solutions of exist

A

Each disk is characterized by the parameters of the original spacetime M J Q

and the size b of the excluded region The parameter b determines the eective

diameter of the disk this is the radius b elow which eg of the matter

of the disk resides It is clear that for large b elds will b e weak and the only

relativistic eects consist in the fact that the ratio of energy as well as charge

densities of b oth streams is governed by the metric p otentials This

is not the case in Newtonian gravity where the angular momentum is not the

source of dragging and only total angular momentum and charge of the disk are

xed As b b ecomes smaller the relativistic eects play more imp ortant role and

the violation of energy conditions may o ccur For set of spacetimes studied in

the following it was conrmed that unless the energy conditions are violated

closed timelike curves do not exist in the spacetime surrounding the disk Disks

compact enough with parameters close to extreme black holes exhibited the

presence of the ergoregions in the vicinity of the disk but b ecause at least one of

is tachyonic near the ergosphere only disks made of rings are able to generate

ergospheres outside

The z comp onent of electrogeo desic equation

2

d z

z z

g g U U F U

z

(z )

2

d

determines the stability of an individual circular orbit Particles slightly o the

2

d z

equatorial plane are returned back only if Because the issue of

2

d

z =+b

stability was not addressed in our work this inequality was not studied within

the parameter space of the disk sources of dierent electrovacuum solutions

discussed b elow

KerrNewman disks

Although we already discused not only neutral dust disks but also the disks with

electric currents we did not yet study explicitly their external elds In this

section we analyze disks pro ducing KerrNewman metric

To transform the KerrNewman metric from BoyerLindquist co ordinates

tr into WeylPapap etrou WP co ordinates one can put

z r M cos

q

2 2 2 2

r M a Q M sin

Resulting expressions are quite complicated unless the spheroidal co ordinates

z xy

q

p

2 2

y x

M p

are used The parameter discriminates dressed extreme and naked

solutions and the parameter p the dimensionless charge s QM and

the dimensionless angular momentum q aM are related by the equation

2 2 2

s q p The parameter is included in the denition for further

2 2

reference In the extreme case s q this equations do es not

determine p and since in this case p app ears in equations only in the form

of the pro duct px the value of p can b e chosen freely The most natural choice is

p The inverse transform

r r

+

x

r r

+

p

y

q

p

2 2

z is not dened for For extreme black hole where r

metrics transformation has to b e replaced by the usual p olar co ordinate

p p

2 2 2 2

formula y z z while then simplies to x z M Lines

of constant x and y plotted in the plane z form ellipses and hyp erb olas as

showed in Figure

3 3 3

2 2 2

1 1 1

-2 -10 1 2 -3 -2 -10 1 2 3 -3 -2 -10 1 2 3

-1 -1 -1

-2 -2 -2

-3 -3 -3

Figure Prolate spheroidal spherical and oblate spheroidal co ordinate grid

The spheroidal and BoyerLindquist co ordinates have the following simple

relation

r x M

cos y

In Figure the WP co ordinate grid z is plotted in BoyerLindquist co or

dinates In this gure the region covered by WP co ordinates and the shap e of

surfaces z b and also the shap e of region excluded by identication are

illustrated

and using the inverse It is more convenient to express the op erators

z

Jacobi matrix of rather than dierentiating the inverse map

2 2

x y x y

2 2

z x y x y

Figure Planes z const and cylinders const in BoyerLindquist co ordinates

x y

2 2 2

x y x y

The metric p otentials of the KerrNewman spacetime expressed as functions

of the radial and the latitudinal spheroidal co ordinates x and y have the following

form

2 2 2 2 2

p x q y s

2

e

2 2 2

px q y

2 2 2 2 2

p x q y s

2

e

2 2 2

p x y

2

px s

2

A M q y

2 2 2 2 2

p x q y s

The electromagnetic p otential

2

dt mq y d

A dx spx

2 2 2

px q y

yields the comp onents of Maxwell tensor X px Y q y

2

sq y 1y

2 2 2 2 2

F X xX q px X Y

2

z 2 2

2 2

x y

M p(X +Y )

sq

1

2 2 2 2 2 2

p y X Y X y X q F

2

2 2

2 2 2

x y

M p (X +Y )

y

s

2 2 2 2 2

pX Y x X q x y F

2

2 2 z t

2 2

x y

M p(X +Y )

s

1

2 2 2

pxX Y XY F

2

t

2 2

2 2 2 2

x y

M p ( X +Y )

Using these formulae the prop erties of stressenergy tensor distribution can

b e studied Since the most general case includes a numb er of phenomena to

reveal their character several sp ecial cases are discussed separately

Schwarzschild disks

Metric co ecients of the Schwarzschild spacetime simplify to

x

g

tt

x

x

2

g

x

2

x

g g

z z

2 2

x y

The resulting surface stressenergy tensor of the disk

t

p

S

t

g

g

ac

C C B B

b bc

2

S

V

S g

A A

a

2

g V

z

2 3

M b R M M b R

4 2 2 32

R M R M b

2 2

MR R b

2

V

2 3

R M M b R

q q

2 2 2 2

b M b M R

is studied in To interpret the stressenergy tensor authors consider a disk

with two counterrotating streams of particles with prop er density which are

counterrotating with velo cities V If the whole horizon of Schwarzchild black

hole is contained in the removed region ie if b M the velo cities of particles

creating the disk are subluminal although for high central redshifts particles on

certain inner orbits are highly relativistic

Extreme Reissner Nordstrom disks

Metric co ecients of the extreme Reissner Nordstrom spacetime read

2

x

g

tt

2

x

2

x

2

g

2

x

2

x

g g

z z

2

x

2

and the fact that g g implies that only one comp onent of the surface

stressenergy tensor is nonzero

2

x y

S

tt

4

M x

1 2

Normal comp onent of the electric eld F M y x yields

z t

xy

J

t

3

M x

This means that the prop er mass density

b M

t

p p

S

t

2 2 2 2 2

b b M

p

tt

and the prop er charge density g J are identical and p ositive

t

The disk consists of charged dust nonmoving with fourvelo city parallel to

the Killing vector eld In these static disks electric repulsion is just counter

balanced with gravitational attraction as in classical physics

The contribution of the material of the disk to the total mass ie the Komar

integral

M

M b ln

D

b

varies form to M as b changes from to the detailed dep endence M b

D

is depicted in Figure The fact that M as b means that the whole

D

Komar mass of the extreme RN black hole is equal to the energy stored in the

electric eld of the region covered by WP co ordinates ie r M see and

Figure

0.9

0.8

0.7

0.6

0.5

D 0.4 M /M

0.3

0.2

0.1

0

0 1 2 3 4 5

b/M

Figure The dep endence of of the mass of the disk M on the width of the excluded region

D

b The mass M M is of an electromagnetic origin

D

Kerr disks

Because there are no restrictions on the sign or size of the sp ecic charge attached

to particles of b oth counterrotating streams all p eculiarities of the interpreta

tion of the disk concentrate on the prop erties of the the surface stressenergy

tensor and the disks pro ducing the Kerr spacetime exhibit the whole sp ectrum

of b ehavior concerning the p ossibility to interpret a disk as physically realistic

The another reason to study the prop erties of the uncharged case is related to

the astrophysical imp ortance of the Kerr spacetime The detailed description of

the motivation to search for a realistic source of the Kerr spacetime was given in

the rst two sections

Disks without the radial pressure generating Kerr spacetime are characterized

by two dimensionless parameters bM and aM while the scale factor M can

b e omitted

2.0 I. p>0 II.

1.5 50% 60% 70% 80%

1.0 G 90%

b/M H E

0.5 III. III. e

0

0 0.5 1 1.5 2

a/M

Figure The parameter space of Kerr disks I Counterrotating disks I I Rings with

azimuthal pressure I I I Energy conditions violated

Only some regions in the parameter space of Kerr disks corresp ond to accept

able disks which can b e describ ed as a counterrotating material In Figure

such regions are plotted The horizontal dashed curve G which is almost identical

with the line b M delimits the far region where b oth solutions corresp ond

to subluminal velo cities from the near one where the counterrotating streams

decomp osition is not p ossible Below this line the stressenergy tensor can b e

diagonalized but the role of the source play massive rings rotating with the ve

lo city of IOs They must sustain p ositive or negative pressure Because b oth

densities of counterrotating streams must b e p ositive the geo desic particle in

terpretation is p ossible only in the region left to the line lab eled p see

Below the line E b oth energy conditions are violated at some radius of the disk as

well as b elow the line H where the disk hits the horizon of the black hole Using

small circles squares etc several disks with aM and aM are

lab eled Radial prop erties of their relevant physical quantities are summarized in

App endix A Finally b elow the dashed line e toroidal ergoregion is present near

the center of the disk The relation of the ergosphere of the Kerr black hole to

the toroidal ergoregion of the disk spacetime is schematically depicted in Figure

a 1 a 1 a a

p p

M M M M

2 2

Figure The relation of the ergosphere of the black hole or the ergotorus in the naked

case to the toroidal ergoregion of the spacetime of the disk b efore and after the identication

The planes of identication ie the excluded region are represented by the horizontal lines

their p osition represents a typical case when energy conditions considered are fullled If a M

there is no minimal size of the excluded region needed to guarantee the energy conditions 2.0 I. II.

1.5

G p>0 E

1.0 b/M

III. H 0.5

e

0

0 0.5 1 1.5 2

a/M

p

Figure The parameter space of KerrNewman disks with Q M I Counterrotating

disks I I Rings with azimuthal pressure I I I Energy conditions violated

Charged KerrNewman disks

In the general case all expressions b ecome quite long and the only formulae worth

displaying are the central energy and charge density of the disk

s

2 2 2

2 2

M b M a b M a Q bM

c

2 2 2 2 2 2 2

b M a b a Q M

s

2 2

2 2

b M a Q b M a

c

2

2 2 2 2

2 2

b a Q M

b M a

Because b oth the electrostatic and centrifugal forces are acting against the

gravity and the extreme Reissner Nordstrom disks exhibit equilibrium of gravi

tational and electrostatic forces one can conclude that the region in the aM b

plane within the parameter space see Figure where disks can b e made from

counterrotating electrogeo desic particles is shrinking towards the line a as

Q M There are no such disks for Q M although the energy conditions may

b e fullled if b is large enough Then the disk may b e constructed from charged

sup erconductive rings rotating with the velo city of isotropic observers

TomimatsuSato spacetimes

To illustrate the p ossibility of generating disklike sources for arbitrary stationary

axisymmetric vacuum spacetime the prop erties of the disk sources of Tomimatsu

Sato TS spacetimes will b e discussed in this section This is a natural

choice b ecause the Kerr geometry is a memb er of TS family with the parameter

1.0

0.5

0 z /M

-0.5

-1.0

0 0.5 1 1.5 2 2.5 3 3.5

ρ/M

Figure Ergoregion dark grey and regions of closed timelike curves light grey for q

In this gure p oints b elow the plane z corresp ond to p oints with x and y and they

represent TS solution in the region r M

2

TS spacetime is determined by the mass M and angular momentum M q

2 2

The parameter p again satises the relation p q and

2 2

Intro ducing the abbreviations a x and b y the line element is given

by where

4 4 4 4 2 2 2 2

p a q b p q aba b ab

2

e

2 4 2 4 2 2 2 2

p x q y pxa q y pxa px b

4 4 4 4 2 2 2 2

p a q b p q aba b ab

2

e

4 2 2 4

p x y

3 4 2 2 2 2 2 3

p xax x b ax x bap px q b

A M q b

4 4 4 4 2 2 2 2

p a q b p q aba b ab

These metric p otentials are written down in the short form which do es not include

the extreme q case this is identical with the extreme Kerr metric On the

other hand the static limit of TS spacetime is the Darmois solution Weyl metric

with the metric p otential prop ortional with factor to the metric p otential

of the Schwarzschild solution with the mass M and the corresp onding disk

source are thus studied as a sp ecial case of Zip oyVorhees metrics in

One of the most imp ortant dierences b etween Kerr and TS family is that

there remain b oth singular rings and toroidal regions of closed timelike curves in

the region covered by WP co ordinates Their shap e is illustrated in Figure

for TS spacetime with aM

2.0

I. p>0 II.

1.5 E 50% 60% 70% 80%

1.0 G 90%

b/M III. E III. e 0.5

T

0

0 0.5 1 1.5 2

a/M

Figure The parameter space of TS disks I Counterrotating disks I I Rings with

azimuthal pressure I I I Energy conditions violated

In Figure the parameter space of TS disks is presented Its structure is

more complicated than in the Kerr case b ecause of previously mentioned phe

nomena Disks with a M and suciently small b those lying b ellow the curve

T are surrounded by a toroidal region of closed timelike curves and the ring sin

gularity may app ear for even smaller b As shown in Figure such disks violate

energy conditions When a M the dominant energy condition is violated for

disks with nal central redshifts which is not the case for the Kerr disks

Concluding remarks

For a wide class of stationary axisymmetric electrovacuum spacetimes we have

shown a p ossibility to construct disklike interior solutions These sources

provide nice illustration of the well known fact that the Einstein equations govern

not only the b ehavior of the gravitational eld but also yield equations of motion

for the matter Thus if a stressenergy tensor with a nonnegative azimuthal

and vanishing radial pressure is obtained by a removal of the equatorial region

of spacetime considered and two counterrotating circular geo desics exist in the

plane of the disk the disk may b e formed by assigning appropriate energy and

charge densities to b oth counterrotating streams It is generally observed that

such interpretation is allowed as long as the width of the excluded region is large

enough and the gravitational attraction is stronger than electric repulsion If the

eective radius of the disk is small it is not p ossible to construct disks with large

aM In the astrophysically most plausible mo dels around of the total mass

of the disk of dust is rotating in one direction along geo desics and only of

the mass moves along counterrotating geo desics

App endix A Radial prop erties of Kerr disks

0.12

0.10

0.08

0.06 M µ

0.04

0.02

0

0 1 2 3 4 5 6

R/M

Figure Energy density as seen by IOs aM

1.0

0.8

0.6 µ / p 0.4

0.2

0

0 1 2 3 4 5 6

R/M

Figure The ratio of the stress to the energy density as seen by IOs aM 0.25

0.20

0.15 V 0.10

0.05

0

0 1 2 3 4 5 6

R/M

Figure Velo city of IOs with resp ect to LNRFs aM

-2 .10

5

4

3 M µ

2

1

0

0 1 2 3 4 5 6

R/M

Figure Energy density of b oth counterrotating streams aM -2 .10

1.5

1.0 M K µ

0.5

0

0 1 2 3 4 5 6

R/M

Figure Komar mass density aM

-2 .10

4

3 M µ 2

1

0

0 1 2 3 4 5 6

R/M

Figure Energy density as seen by IOs aM 0.2

0

-0.2 µ / p -0.4

-0.6

-0.8

0 1 2 3 4 5 6

R/M

Figure The ratio of the stress to the energy density as seen by IOs aM

0.8

0.7

0.6

0.5

0.4 V

0.3

0.2

0.1

0

0 1 2 3 4 5 6

R/M

Figure Velo city of IOs with resp ect to LNRFs aM -2 .10 3.0

2.5

2.0

1.5 M µ

1.0

0.5

0

0 1 2 3 4 5 6

R/M

Figure Energy density of b oth counterrotating streams aM

-2 .10

2.5

2.0

1.5 M K µ

1.0

0.5

0

0 1 2 3 4 5 6

R/M

Figure Komar mass density aM

App endix B Radial prop erties of KerrNewman disks

0.14

0.12

0.10

0.08 M µ 0.06

0.04

0.02

0

0 1 2 3 4 5 6

R/M

p

Figure Energy density as seen by IOs aM QM

0.9

0.8

0.7

0.6

0.5 µ

/ 0.4 p

0.3

0.2

0.1

0

0 1 2 3 4 5 6

R/M

Figure The ratio of the stress to the energy density as seen by IOs aM 0.20

0.15

V 0.10

0.05

0

0 1 2 3 4 5 6

R/M

p

Figure Velo city of IOs with resp ect to LNRFs aM QM

-2 .10

4

3

2 M µ

1

0

0 1 2 3 4 5 6

R/M

p

Figure Energy density of b oth counterrotating streams aM QM -2 .10

2.5

2.0

1.5

M 1.0 σ

0.5

0

0 1 2 3 4 5 6

R/M

p

Figure Charge density of b oth counterrotating streams aM QM

-2 .10

3

2 M K µ

1

0

0 1 2 3 4 5 6

R/M

p

Figure Komar mass density aM QM -2 .10

5

4

3 M µ

2

1

0

0 1 2 3 4 5 6

R/M

p

Figure Energy density as seen by IOs aM QM

0

-0.2

-0.4 µ / p -0.6

-0.8

-1.0

0 1 2 3 4 5 6

R/M

Figure The ratio of the stress to the energy density as seen by IOs aM 1.0

0.8

0.6 V 0.4

0.2

0

0 1 2 3 4 5 6

R/M

p

Figure Velo city of IOs with resp ect to LNRFs aM QM

-2 .10

2.0

1.5 M µ 1.0

0.5

0

0 1 2 3 4 5 6

R/M

p

Figure Energy density of b oth counterrotating streams aM QM -2 .10

2.0

1.5 M µ 1.0

0.5

0

0 1 2 3 4 5 6

R/M

p

Figure Charge density of b oth counterrotating streams aM QM

-2 .10

5

4

3 M K µ 2

1

0

0 1 2 3 4 5 6

R/M

p

Figure Komar mass density aM QM

App endix C Radial prop erties of Tomimatsu

Sato  = 2 disks

0.10

0.08

0.06 M µ 0.04

0.02

0

0 1 2 3 4 5 6

R/M

Figure Energy density as seen by IOs aM

1.0

0.8

0.6

sµ / p 0.4

0.2

0

0 1 2 3 4 5 6

R/MR

Figure The ratio of the stress to the energy density as seen by IOs aM 0.25

0.20

0.15

V 0.10

0.05

0

0 1 2 3 4 5 6

R/M

Figure Velo city of the isotropic observer with resp ect to LNRFs aM

-2 .10

3.0

2.5

2.0

M 1.5 µ

1.0

0.5

0

0 1 2 3 4 5 6

R/M

Figure Energy density of b oth counterrotating streams aM -2 .10

1.5

1.0 M K µ

0.5

0

0 1 2 3 4 5 6

R/M

Figure Komar mass density aM

-2 .10

4

3 M µ 2

1

0

0 1 2 3 4 5 6

R/M

Figure Energy density as seen by IOs aM 0.2

0

-0.2

µ -0.4 / p

-0.6

-0.8

0 1 2 3 4 5 6

R/M

Figure The ratio of the stress to the energy density as seen by IOs aM

0.9

0.8

0.7

0.6

0.5

V 0.4

0.3

0.2

0.1

0

0 1 2 3 4 5 6

R/M

Figure Velo city of IOs with resp ect to LNRFs aM -2 .10

3

2 M µ

1

0

0 1 2 3 4 5 6

R/M

Figure Energy density of b oth counterrotating streams aM

-2 .10

2.0

1.5 M

K 1.0 µ

0.5

0

0 1 2 3 4 5 6

R/M

Figure Komar mass density aM

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