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BLACKHOLES:


In 1939, J. Robert Oppenheimer and Hartland Snyder "... studiedthe collapse of a spherically symmetric pressureless star of uniformdensity, and noted that the star's implosion, as seen by a stationaryobserver who remains outside, would slow and ultimately freeze as thestar's surface approached the critical Schwarzschild radius. Yet theyalso clearly explained that no such freezing of the implosion wouldbe seen by observers riding in with the collapsing matter - theseobservers would cross inside the critical circumference in a finiteproper time, and thereafter would be unable to send a light signal toobservers on the outside. This extreme difference between thedescriptions in the two reference frames proved exceptionallydifficult to grasp. The two descriptions were not clearly reconcileduntil

1958, when David Finkelstein... indicated ... that ...

[see Past-Future Asymmetry of the Gravitational Field of aPoint Particle (Phys. Rev. 112 (May 15, 1958) 965-967) - (279kpdf file)] ... once the star fell through its criticalcircumference, its

compression to form a...[Black Hole]...wasinevitable...."

In 1958, "... In Moscow, Landau ... while believing Oppenheimerand Snyder's calculations, had severe trouble reconciling these twoviewpoints ... [until he] read the article [in Phys. Rev.by David Finkelstein]. It was a revelation. Suddenly everythingwas clear. Finkelstein ... lectured at Kings College in London. RogerPenrose ... [heard] Finkelstein's lecture, and returned toCambridge enthusiastic. In Princeton, Wheeler was intrigued at first,but was not fully convinced. ... He would become convinced onlygradually, over the next several years. ...".

First above quoted material edited from the foreword of John Preskill and Kip S. Thorne to the Feynman Lectures on Gravitation, edited by Brian Hatfield (Addison-Wesley 1995)). Second above quoted material from Thorne, Black Holes and Time Warps (Norton 1994). Lev Landau (1908-1968) won the 1962 Nobel Prize in Physics.

In his 1958 paper, David Finkelstein states:

"... The Schwarzchild surface r = 2m is not a singularitybut acts as a perfect unidirectional membrane: causal influences cancross it but only in one direction. ...".

In units such that the Schwarzschild radius and the speed of lightare unity, David Finkelstein's line element is:

ds^2 = (1 - r^(-1)) (dx0)^2 + 2 r^(-1) dx0 dr - (1 + r^(-1))dr^2 - (dxi dxi - dr^2)

where r = (xi xi)^(1/2) and dr = r^(-1) xi dxi.

For r greater than 1, David Finkelstein transforms by X0 = x0 +ln( r - 1 ) and Xi = xi to get the Schwarzchld line element

ds^2 = ( 1 - 1/R ) (dX0)^2 - ( 1 - 1/R )^(-1) dR^2 - ( dXidXi - dR^2 )


Note that for a number P exp(i W) that is not a positive realnumber, its logarithm is a Complex Number,

such as ln(P exp(i W) = ln(P) + i W or ln (-1) = ln (exp i pi) = ipi

so that for r less than 1, the transformation X0 = x0 + ln( r - 1) produces a Complex Number

consistent with Complex SpaceTimeinside a Compton Radius Vortex.

In the D4-D5-E6-E7 physics model,Complex SpaceTime is based onthe Symmetric Space

Spin(4,2) / Spin(3,1)xU(1)

where Spin(4,2) = SU(2,2) is the15-Real-dimensional conformal group.The Symmetric Space has realdimension 15 - 6 - 1 = 8. It has local Spin(3,1) Lorentz symmetry,for special relativity. It has a local U(1) symmetry, so it is a4-Complex-dimensional Hermitian SymmetricSpace that is the physical SpaceTime inside a ComptonRadius Vortex.

What about the Real SpaceTime outside the Compton Vortex? It is aShilov Boundary.

The Type IV HermitianSymmetric Space has a corresponding compact version that isisomorphic to a Bounded Complex Homogeneous Domain, the ShilovBoundary of which is RP1 x S3 (topologically, S1 x S3). If you letthe S1 correspond to time and the S3 to space, the Shilov Boundary isthe 4-Real-dimensional SpaceTime outside a ComptonRadius Vortex.

 

The relevant Complex Structure can be seen in such physicalconcepts as Momentum Space,Position-MomentumComplementarity, Type IV(2)Domains, Hyperspace,Wavelets, and ConformalSpaceTime.

 

 


The following edited illustration from Gravitation, by Misner,Thorne, and Wheeler (Freeman 1973) shows David Finkelstein'scoordinates based on photons going into the Black Hole. Their bookuses units such that the Schwarzschild radius is 2M and the speed oflight is unity, so that the Schwarzschild line element is

ds^2 = 0 = - ( 1 - 2 M / r ) dt^2 + ( 1 - 2 M / r )^(-1) dr^2

Note in particular that, after the Black Hole forms, its EventHorizon (the cylinder r = 2M) is made up of the world lines ofphotons and other light-cone particles that are going out from theBlack Hole. Since gravitons, like photons, are massless light-coneparticles, the Event Horizon can be considered to be constructed fromgravitons that go out from the Black Hole, but cannot escape beyondthe Event Horizon.

A Black Hole of the mass of the Sun would have a radius(proportional to its mass) of about 10^5 cm, or 1 km.

In the D4-D5-E6 model, physicalspin-2 gravitons are made fromConformal Group (with de Sitter subgroup)gravitons by the MacDowell-Mansourimechanism.

Since outside observers cannot interact directly with gravitonsfrom the Black Hole itself, but can interact with the Event Horizon,the Event Horizon acts as a intermediary to answer the question,asked of me by Paul A. J. Wagg: If gravity is the exchange ofgravitons, and gravitons from the Black Hole cannot get outside theEvent Horizon to an outside observer, how can an outside observer beattracted by gravity to the Black Hole?

About 1972, Jacob Beckensteinidentified theArea of the Event Horizon of a Black Hole withEntropy.

The Entropy of a Black Hole, and hence its Area, can also beidentified with the amount of Information itcarries. Jack Sarfatti, indiscussing Wheeler's book Geons, Black Holes and Quantum Foam (W. W.Norton 1998), says:

"... Wheeler points out that Bekenstein showed that this Planck area of 10^-66 cm^2 [of a Planck Mass Black Hole] corresponds exactly to one c-bit of classical Shannon on the surface (event horizon) of a classical black hole. Note that a classical c-bit is not the same as a quantum q-bit of information. Wheeler also discusses Christodoulou's marvelously simple profound new application of the ancient Pythagorean theorem: "If two balls of putty collide and stick together, the mass of the new, larger ball is the sum of the masses of the balls that collided. Not so for black holes. If two spinless, uncharged black holes collide and coalesce - and if they get rid of as much energy as they possibly can in the form of gravitational waves as they combine - the square of the mass of the new, heavier black hole is the sum of the squares of the combining masses. That means that a right triangle with sides scaled to measure the masses of two black holes has a hypotenuse that measures the mass of the single black hole they form when they join. Try to picture the incredible tumult of two black holes locked in each other's embrace, each swallowing the other, both churning space and time with gravitational radiation. Then marvel that the simple rule of Pythagoras imposes its order on this ultimate cosmic maelstrom." p.p. 300-301 ... However, oddly enough, Wheeler does not, it appears, make the obvious, to me at least, connection between Christodoulo's Pythagorean theorem and Bekenstein's black hole information theory that the black hole horizon's surface area is proportional to the c-bits it has swallowed up. Wheeler's discussion of Bekenstein is later on p. 314. Clearly, Christodoulo's theorem simply means that the information of each black hole add linearly when they fuse together and attain dynamical equilibrium. ..."

The Pythagorean Property of the Mass of a Black Hole formed bymerger of two Black Holes supports the ComptonRadius Vortex picture of Gaja/Ganesha Physical ElementaryParticles in that it is consistent with the many Mass-squared,Charge-squared, and Square-Root of the Sum of Mass-squared propertiesof Particle Masses and Force Strengths in Particle Physics, suchas:

The Higgs scalar particle H can be considered to be a merger of 3 Compton Radius Kerr-Newman Black Holes corresponding to W+, W-, and Z0.

What about Higgs mass? Then the mass mH = sqrt( mW+^2 + mW-^2 + mZ0^2 ) = sqrt( 80^2 + 80^2 + 90^2 ) = sqrt( 20,900 ) = 145 GeV. Although the Higgs H has not yet been observed, mH = 145 GeV is consistent with indirect Standard Model limits.

What about Higgs spin? The Higgs H is a scalar with spin = 0. Each of the Weak Bosons W+, W-, and Z0 is a vector with spin = 1. Look at the process of merger of the 3 Compton Radius Kerr Black Holes corresponding to W+, W-, and Z0. Since it is a 3-body merger, and I don't know a reference for 3-body Black Hole mergers, I will approximate it by using the textbook treatment of a 2-body merger of W+ and W- with the Z0 being added in a way that I guess (but have not yet fully calculated) is correct. When the W= and W- merge, their spins should be antiparallel because of a gravitational spin-spin force which is attractive for antiparallel spins (see Wald, General Relativity (Chicago 1984) page 338, problem 12.4). Since angular momentum cannot be radiated away in an axisymmetric spacetime, the resulting angular momentum from the W+ and W- should be +1 -1 = 0. What about the Z0? The Z0 violated axisymmetric symmetry, so its angular momentum should be radiated away by gravitational radiation. Therefore the final spin state of the Black Hole merged from W+, W-, and Z0 should be spin = 0, a scalar, consistent with its identification with the Higgs scalar.

Some other Particle Physics Phenomena supported by the PythagoreanProperty are:

In contrast, Quantum Many Worlds Sum-Over Histories phenomena arenot square-root-of sum-of-squares Black Hole merger phenomena, butare linearly additive. Examples include:

If the Event Horizon of a Black Hole has Entropy, it should alsohave Temperature. The temperature of a Black Hole increases as itevaporates, with a temperature that is inversely proportional to itsmass, and, equivalently, inversely proportional to its radius. ABlack Hole of the mass of the Sun would have a temperature of 10^(-6)degrees K.

If a Black Hole has Temperature, it should emit Radiation with aspectrum corresponding to that Temperature. If a Black Hole hasgreater than the stable minimum Planck mass, it will decay with alifetime that is proportional to the cube of its mass. For a BlackHole to survive about 20 billion years, it must have a mass of atleast 2 x 10^14 grams. For comparison, the Earth has mass of 6 x10^27 grams and the Sun has a mass of 2 x 10^33 grams.

In his book Black Holes and Time Warps (Norton 1994) Kip Thornesays, on pages 428-430 and 432-433:

"... in June 1971 ... Zeldovich announced ... A spinning black hole must radiate ... a spinning metal sphere emits electromagnetic radiation ... The radiation is so weak ... that nobody has ever observed it, nor predicted it before. However, it must occur. The metal sphere will radiate when electromagnetic vacuum fluctuations tickle it. ... Zeldovich's mechanism by which vacuum fluctuations cause a spinning body to radiate. ... His sketch ...

... showed a wave flowing toward a spinning object, skimming around its surface for a while, and then flowing away. The wave might be electromagnetic and the spinning body a metal sphere ... or the wave might be gravitational and the body a black hole ... The incoming wave is not a "real" wave ... but rather a vacuum fluctuation. ... the wave's outer parts ... are in the "radiation zone" while the inner parts are in the "near zone" ... the wave's outer parts move at ... the speed of light ... its inner parts move more slowly than the body's surface is spinning ... the rapidly spinning body will ... accelerate ...[the inner parts of the incoming wave] ... The acceleration feeds some of the body's spin energy into the wave, amplifying it. The new, amplified portion of the wave is a "real wave" with positive total energy, while the original, unamplified portion remains a vacuum fluctuation with zero total energy. ... Zeldovich ... proved that a spinning metal sphere radiates in this way; his proof was based on the laws of quantum electrodynamics ...

.... In September 1975 ... Zeldovich and Starobinsky ... described how their version of the laws of quantum fields in a black hole's curved spacetime ... was really equivalent ...[to]... Hawking's ...[ theory of black hole radiation ]...[ with spacetime curvature tides having a similar effect to the spinning/dragging of the metal sphere in Zeldovich's electromagnetic model ]... They had concluded that ...[ non-spinning ]... black holes cannot evaporate because of an error in their calculations ... With the error corrected ... black holes evaporate ...".

 

In 1974, StephenHawking showed, as illustrated by this figure from his book ABrief History of Time (Bantam 1988), how the Event Horizon emitsRadiation:

The Zeldovich-Hawking Process, in which a Virtual Pair ofMusaka/Ganesha Particles near the EventHorizon can be separated with

one of the Virtual Pair going into the Black Hole

and

the other going into External Spacetime,

can be applied to Quark-AntiQuark VirtualPairs

showing that

a Black Hole can carry Color Charge of theSU(3) color force.

What about Entropy in the Zeldovich-Hawking Process?

Cerf andAdami have written a paper "Prolegomenato a Non-Equilibrium Quantum Statistical Mechanics" in whichthey

"... suggest that the framework of quantum information theory ... is a reasonable starting point to study non-equilibrium quantum statistical phenomena. As an application, [they] discuss the non-equilibrium quantum thermodynamics of black hole formation and evaporation. ...

For the brief transitory periods, ... the time during which a system approaches equilibrium, our bag of tricks containing the tools of statistical mechanics is of little use. The canonical phenomena of this type are relaxation or transport processes, phenomena which are usually termed "irreversible", and phase transitions for which the entropy is not a constant.

The standard approach to deal with such situations is to study the N-body dynamics of the system, with a Hamiltonian that includes an interaction term (in equilibrium statistical mechanics the Hamiltonian is a sum of non-interacting one-body terms) and the construction of equations that follow the N-particle distribution function through time: the Boltzmann equation. This approach suffers from the drawback that it can only be solved in perturbation theory, which obscures the relation to the "exact" formalism of thermodynamics. ...

[A] formalism in which the second law [of thermodynamics] is replaced by a conservation law for entropy (... in which case the second law ... [appears] as a corollary) exists in the form of the classical theory of information, introduced by Shannon. Its extension to the quantum regime is particularly interesting as it consistently describes quantum unitary dynamics which dictates that the von Neumann entropy - the quantum extension of the Shannon entropy - is a constant. ... [This] formalism ... can quantitatively describe even the approach to equilibrium and other non-equilibrium statistical phenomena. ...

The discovery of Hawking radiation appears to have plunged quantum mechanics into a deep crisis, as it seems to imply that the evaporation of black holes violates unitarity ... Hawking pointed out that the process of thermal evaporation of a black hole leads to an "information paradox". If we assume that the black hole [of mass M] is formed from a quantum mechanically pure state S = 0, the entropy of the purely thermal blackbody radiation left behind after evaporation should be of the order M^2, i.e., a pure state evolved to a mixed one. This contradicts the unitary evolution of quantum states, according to which ... the entropy of a closed system is a constant, in this particular case the constant zero. ...

... beyond the information paradox pointed out by Hawking, as observed by Zurek we also need to match the black hole entropy ... with the entropy of approximately thermal radiation ... with black hole temperature ... .

we will focus here on the most conservative explanation, namely that Hawking radiation is effectively non-thermal (in the sense that quantum correlations between the radiation and the state of the black hole exist in principle) ...

While the fields do live in a product Hilbert space, the wavefunction of an EPR pair created at the event horizon of the black hole indirectly becomes entangled with the hole the moment one of the particles crosses the horizon (even though the quantum fields are separated by space-like distances) and the combined quantum state becomes inseparable. This situation is not unlike the scenario we noted in the formation of the black hole, where the accreted particle and the radiation it emits when tumbling into the black hole can be considered an entangled, EPR-type state (albeit with real rather than virtual energy). Just as in that case the radiation ... shared no entropy with the black hole, neither does the Hawking radiation, while still being entangled with it. Thus, the Hawking radiation carries "information" about the inside of the hole in the same manner as the measurement of EPR partners separated by space-like distances reveals correlations in measurement devices that are at space-like distances.

Yet, a fundamental problem remains that is unlikely to be solved within the present formalism. The Hawking radiation - while emitted in a unitary manner and while information loss certainly does not take place - remains causally uncorrelated to the black hole as long as the horizon separates the black hole entropy from the radiation field. In a sense, we have to wait until the last moment - the disappearance of the black hole - for the entropy balance to be restored. This appears to put a severe strain on current black hole models, as it is hard to imagine that this much entropy can be stored in an ever-shrinking black hole. This problem is likely due to our incomplete understanding of late-stage black holes, rather than a problem intrinsic to quantum mechanics.

... An alternative solution would present itself if the Bekenstein-Hawking entropy could be understood in terms of a conditional entropy. In that case, entropy flow from the black hole to the outside via the formation of virtual pairs is understood easily, as the member of the pair that crosses the horizon not only has negative energy but also negative conditional entropy. As a conditional entropy can become as negative as the marginal entropy of the system it is a part of, we can circumvent the argument that "the black hole cannot store the information until the end because it runs out of quantum states", because the radiation could "borrow" as much entropy as necessary from the black hole until the horizon has disappeared. ... It is not inconceivable ... that a quantum statistical information theory extended to curved space-time would reveal ... that the Bekenstein-Hawking entropy is in fact conditional ...

... We emphasize that great care is needed in using the concepts of entropy and information consistently: information, for example, can never be "stored" in one system (e.g., a black hole). Rather, information is a measure of correlation between two systems, which implies that information is always stored in correlations. ... radiation and black hole matter are unstable at any time, and transitions must occur as long as matter of either kind is present. Yet, a consistent formulation of the correlations between radiation and matter shows that entropy is not created during the process, and consequently that information is conserved. ....".

 

As Kip Thorne says in his book Black Holes and Time Warps (Norton1994, pages 491-492),

"... in 1974 ... Hawking inferred as a byproduct of his discovery of black hole evaporation ... that vacuum fluctuations near a [Black Hole]'s horizon are exotic: They have negative average energy density as seen by outgoing light beams near the hole's horizon. ... it is this exotic property of the vacuum fluctuations that permits the hole's horizon to shrink as the hole evaporates ... The horizon distorts the vacuum fluctuations away from the shapes they would have on Earth, and by this distortion it makes their average energy density negative,that is, that is, it makes the fluctuations exotic. ... Gunar Klinkhammer ... has proved that in flat spacetime ... vacuum fluctuations can never be exotic ... Robert Wald and Ulvi Yurtsever have proved that in curved spacetime ... the curvature distorts the vacuum fluctuations and thereby makes them exotic. ...".

 

Some relationships among ExoticSpaceTime, Ordinary SpaceTime, Compton Radius Vortices, ClassialBlack Holes, and Planck Pivot Vortices are discussed HERE.

 

 


Marcus Chown, in an article in the NewScientist, 21/28 December 2002, pages 55-56, says: "... In 1989,Paul Davies ... discovered that a rotating black hole flips from anegative to a positive specific heat when the square of its massdivided by the square of its spin parameter is equal to thegolden ratio....".

 


Finkelstein's Black Hole, which shows how Mass curvesSpaceTime by Gravity, can be generalized to deal with Spin andElectric Charge.

The generalization, called a Kerr-Newman BlackHole,

was developed by Kerr (who generalized to add angular momentum Jto mass M in 1963) and by Newman (who generalized to add charge e in1965), according to the book General Relativity, by Robert Wald(Chicago 1984). Let a = J / M. Then the Kerr-Newman SpaceTime metricin spatially spherical coordinates t, r, T, P (where the angularmomentum axis is T = 0) is

ds^2 = - ( ( D - a^2 sin^2(T) ) / S ) dt^2 -

- ( 2a sin^2(T) (r^2 + a^2 - D ) / S ) dt dP +

+ ( ( ( r^2 + a^2 )^2 - D a^2 sin^2(T) ) / S ) sin^2(T) dP^2 +

+ S / D dr^2 +

+ S dT^2

and the Electromagnetic Vector Potential is

A = - ( e r / S ) ( dt - a sin^2(T) dP )

where

S = r^2 + a^2 cos^2(T)

and

D = r^2 + a^2 + e^2 - 2Mr

If e = M = 0 and a is nonzero, then the Kerr-Newman metric is justthe metric of Minkowski space in spatially spherical coordinates, andthe singularity at S = 0 is a coordinate singularity located at aRing of radius a in the plane z = 0 in cartesian coordinates t, x, y,z, that is,

the Ring Singularity is at z = 0 and x^2 + y^2 =a^2


I would like to thank GeraldKaiser very much for his comment at this point(by e-mail 10 June 2002):

"... I'd like to point out that the linearized Einstein-Maxwell fields are actually double-valued and can be made single-valued by choosing a branch cut which spans the ring and makes it into a DISK. In fact, there is a footnote to that effect in one of Ted Newman's papers (Maxwell's equations and complex Minkowski space, J. Math. Phys. 14, 102, 1973) which he told me was in response to a comment by the anonymous referee who turned out to be Roger Penrose! A careful analysis of the EM case in flat R^4 appears in my paper http://arXiv.org/pdf/gr-qc/0108041 (Of course, you could instead go to the covering space, but people like Newman don't think that has physical significance.)

PS: I've been playing with complex spacetime since 1966. For example, see my book Quantum Physics, Relativity, and Complex Spacetime (now out of print, to be updated some day). ...".


If M and a are nonzero, then there are ClosedTimelike Loops in a neighborhood of the Ring.

 

There are 3 cases:

e^2 + a^2 is less thanM^2

e^2 + a^2 = M^2

e^2 + a^2 is greater thanM^2

In all 3 cases,

Kerr Black Holes have natural CliffordAlgebra Structure:

 

In astro-ph/9707165,Lasenby, Doran, Dobrowski, and Challinor say:

"... Gauge-theory gravity,expressed in the language of Geometric Algebra [CliffordAlgebra],

allows very efficient numerical calculation of photon paths. ...We discuss ... applications of a gauge theory of gravity ... Thetheory employs [two] gauge fields in a flat Minkowskibackground spacetime to describe gravitational interactions. ... Thefirst of these, h(a), is a position-dependent linear function mappingthe vector argument a to vectors. The position dependence is usuallyleft implicit. Its gauge-theoretic purpose is to ensure covariance ofthe equations under arbitrary local displacements of the matterfields in the background spacetime. The second gauge field, W(a), isa position-dependent linear function which maps the vector a tobivectors. Its introduction ensures covariance of the equations underlocal rotations of vector and tensor fields, at a point, in thebackground spacetime. Once this gauging has been carried out, and asuitable Lagrangian for the matter fields and gauge fields has beenconstructed, we find that gravity has been introduced. ... the theoryis formally similar in its equations (hence local behaviour) to theEinstein-Cartan-Kibble-Sciama spin-torsiontheory, but it restricts the Lagrangian type and the torsion type(... torsion that is not trivector type leads to minimally coupledLagrangians giving non-minimally coupled equations for quantum fieldswith non-zero spin). ...

If we restrict attention to situations where the gravitatingmatter has no spin, then there are still differences between generalrelativity and the theory presented here. These differences arisewhen time reversal effects are important (e.g. horizons), whenquantum effects are important, and when topological issues areaddressed. ...

... within the framework of gauge-theory gravity, the Kerrsingularity is composed of a ring of matter, moving at the speed oflight, which surrounds a disk of pure isotropic tension. ...

... As an interesting aside, we note that self-consistenthomogeneous cosmologies, based on a classical Dirac field, requirethat k = 0 (the universe is spatially flat). ... ".

 

In the paper Gravity, gauge theories and geometric algebra,downloadable from the webpage of The Geometric Algebra Research Group at Cavendish Laboratory,University of Cambridge, Lasenby, Doran, and Gull say: that"...fermionic matter would be able to detect the center of theuniverse if k=/= 0 [if the univese were not spatiallyflat] ...".

 

In the paper Effects of Spin-Torsionin Gauge Theory Gravity, downloadable from the webpage of The Geometric Algebra Research Group at Cavendish Laboratory,University of Cambridge, Doran, Lasenby, Challinor, and Gull say:that "... Within [Gauge-TheoryGravity], torsionis viewed as a physical field derived from the gravitational gaugefields. This viewpoint has some conceptual advantages over thatused in differential geometry, where torsion is regarded as aproperty of a non-riemannian manifold. ... for a massive spinningpoint-particle, moving in a gravitational background with torsion...the motion is not generally geodesic, the spin vector is notFermi-transported, and the particle couples to the torsion. ...spinning point particles see a preferred direction in space due tothe spin of the matter field. ... with spin there are extraphysical fields present which have observable consequences. ...".

 

In gr-qc/9910099,Chris Doran says: "...

A new form of the Kerr solution

... is global and involves a time coordinate which represents thelocal proper time for free-falling observers on a set of simpletrajectories. ... The Kerr solution ... is global, making it suitablefor studying processes near the horizon. ... the time coordinatemeasured by a family of free-falling observers brings the Diracequation into Hamiltonian form ... This form of the equations alsopermits many techniques from quantum field theory to be carried overto a gravitational backgroundwith little modification. ... ".

 

 

 


If e^2 + a^2 is less than M^2, then D = 0 and

the Outer Event Horizon is at r = r+ = M + sqrt( M^2 -a^2 - e^2)

and

the Inner Event Horizon is at r = r- = M - sqrt( M^2 - a^2- e^2)

Outside the Outer Event Horizon you can escape the Black Hole, butyou must rotate in the same direction as the Black Hole if you arewithin the Static Limit outer boundary of

the Ergosphere at r = M + sqrt( M^2 - a^2 cos^2(T) -e^2)

This effect can be said to be due to extreme frame dragging, or totimelike translations becoming spacelike as though they had been Wickrotated in ComplexSpaceTime.

Ergosphere (white), Outer Event Horizon (red), Inner Event Horizon(green), and Ring Singularity (purple) from Black Holes - ATraveller's Guide, by CliffordPickover (Wiley 1996).

 

In his paper "Generationand Evolution of Magnetic Fields in the Gravitomagnetic Field of aKerr Black Hole", Ramon Khanna says: "... a rotating black holecan generate magnetic fields in an initially un-magnetized plasma. Inaxisymmetry a plasma battery can only generate a toroidal magneticfield, but then the coupling of the gravitomagnetic potential withtoroidal magnetic fields generates poloidal magnetic fields. Even anaxisymmetric self-excited dynamo is theoretically possible, i.e.Cowling's theorem does not hold close to a Kerr black hole. Due tothe joint action of gravitomagnetic battery and gravitomagneticdynamo source term, a rotating black hole will always be surroundedby poloidal and toroidal magnetic fields (probably of low fieldstrength, though). The gravitomagnetic dynamo source may generateclosed poloidal magnetic field structures around the hole, which willinfluence the efficiency of the Blandford-Znajek mechanism[whereby coupling of the gravitomagnetic potential with amagnetic field results in an electromotive force that drives currentsthat may extract rotational energy from a black hole].The"shear-of-space" driven fraction of a global current can be assessedby a kinematic simulation of a zero-angular-momentum flow. The majorpart of the resulting current system is generated and closed in thecorona near the hole. In a realistic accretion-ejection flow, theplasma shear in the ergosphere of a rapidly rotating black hole willbe similar to the shear of space. The current system should thereforehave a similar structure ... which means that the energy extractedfrom the hole is likely to be deposited in the disk corona.

The Dyadosphere is defined by Remo Ruffini in astro-ph/9811232as being the region, outside the horizon of a black hole with anelectromagnetic field (EMBH), where the electromagnetic field exceedsthe Heisenberg-Euler critical value E = m^2 c^3 / hbar e (where m ande are the electron mass and charge) for e+ e- pair production. TheDyadosphere will dissipate in about 10^7 sec, or less than ayear. In a very short time on the order of hbar / m c^2 a verylarge number of pairs is created there and reaches thermodynamicequilibrium with a photon gas. In the ensuing enormousPair-ElectroMagnetic pulse (PEM pulse), a large fraction of theextractable energy of the EMBH in the sense of will be carried away.The PEM-pulse will interact with some of the baryonic matter ofthe uncollapsed material and the associated emission is closelyrelated to the observed properties of GRBsources. The electromagnetic field of the remnant willfurther dissipate in the acceleration of cosmicrays or in the propulsion of jets on much longer time scales. Upto 50% of the mass energy of an extreme EMBH, whose charge is Qmax =R c^2 / sqrt(G) (where R is the outer horizon radius), can be storedin its electromagnetic field. Even in the case of an extreme EMBH thecharge to mass ratio is one quantum of charge per 10^18 nucleons. Alarge fraction of the energy of an EMBH can be extracted byHeisenberg-Euler electromagnetic field pair production. This energyextraction process only works for EMBH black holes whose irreduciblemass is less than 10^6 solar masses. The Dyadosphere radius ismaximized for the extreme case. The Dyadosphere exists for EMBH'swith mass larger than the upper limit for neutron stars, about 3.2solar masses, all the way up to a maximum mass of 6 x 10^5 solarmasses. Correspondingly smaller values of the maximum mass areobtained for EMBH's with less than the maximal charge of an extremeEMBH. In the following figure, the redand blue lines are theDyadosphere radius andOuter Event Horizon radius for anextreme EMBH, while the gold andpurple lines areDyadosphere radius andOuter Event Horizon radius for an EMBHwith charge Q = 0.01 Qmax.

For EMBH's with mass larger than the maximum value, theelectromagnetic field (whose strength decreases inversely with themass) never becomes critical. For an EMBH of 10 solar masses, theenergy of pairs near the horizon can reach 10 GeV. For an EMBH of10^5 solar masses, the energy of pairs never goes over a few MeV.These two values of the mass are representative of objects typical ofthe galactic population or for the nuclei of galaxies compatible withour upper limit of the maximum mass of 6 x 10^5 solar masses. Thetime scale of the dyadosphere discharge is of the order of 10^19 sec.Such a time scale is much shorter than the characteristicmagnetohydrodynamical time scales. We expect that the formation ofthe dyadosphere should only occur during the gravitational collapseitself and in the process of formation of the EMBH, with theformation of a charge depleted region with an electric fieldsufficient to polarize the vacuum. As Ruffini says, "... From afundamental point of view, the process occuring in the Dyadosphererepresents the first mechanism capable of extracting large amounts ofenergy from a Black Hole with an extremely high efficiency. ... Basicenergy requirements for gamma ray bursts(GRB), including GRB971214 recentlyobserved at z = 3.4, can be accounted for by processes occurringin the Dyadosphere.". Ruffini ( along with, in some papers,co-authors ) discusses The Dyadosphere of Black Holes and Gamma-RayBursts in astro-ph/9905071,astro-ph/9905072.,and astro-ph/0106532.

 

The Maximal Extension of SpaceTime for a Black Hole with e^2 +a^2 is less than M^2 is shown in Figure 12.4 from General Relativityby Robert M. Wald (Chicago 1984):

Wald says: "... Region I ... is the asymptotically flat regioncovered in a nonsingular fashion by the original coordinates with rgreater than r+. By extending through the Coordinate Singularity at r= r+, we obtain Region II representing a Black Hole, Region IIIrepresenting a White Hole, and Region IV representing anotherasymptotically flat region ... However, ... instead of encountering atrue singularity at the Top Boundary of Region II and the BottomBoundary of Region III, we encounter merely another CoordinateSingularity at r = r-. Thus, we can extend Region II through r = r-to obtain Regions V and VI. These regions contain the RingSingularity at SIGMA = 0 and ... [ as in the case in whiche^2 + a^2 is greater than M^2] ... onecan pass through the ring singularity to obtain anotherasymptotically flat Region VI' with r going to negative infinity. (Inthis asymptotically flat Region VI' the Ring Singularity is a nakedsingularity of negative mass. ... ) One may then continue to extendthe ... SpaceTime upward ad infinitum to obtain a Region VII,identical in structure to Region III, and obtain Regions VIII and IX,identical in structure to Regions IV and I, etc. Similarly, one mayextend ... downward ad infinitum. ... Thus, an observer starting inRegion I ... may cross the Event Horizon at r = r+ and enter theBlack Hole Region II. ... the observer may pass through the InnerHorizon r = r- (which is a Cauchy Horizon for the hypersurface S ...), thereby entering Region V or VI. From there, he may ... passthrough the Ring Singularity to a new asymptotically flat Region V'or VI', or he may enter the White Hole Region VII and from thereenter the new asymptotically flat Region VIII or IX. From there hemay enter the new Black Hole associated with these asymptoticregions, and continue his journey. ..."

Is such a journey physically reasonable?

As Wald says, "... we would, in general, expect a complicateddynamical evolution which only settles down to a stationary geometryat late times ... Thus, we are not in a position to follow thedynamical evolution of the gravitational collapse of a body whichforms a Kerr Black Hole and thereby determine the detailed SpaceTimegeometry inside the Black Hole. ...", so we really don't know theanswer.

Wald is pessimistic, saying: "... there is good reason to believethat in a physically realistic case ... the Cauchy Horizon r = r- ...will become a true, physical singularity ...".

Wald's pessimism is supported by Hodand Piran in gr-qc/9803004, who "... study the inner-structure ofa charged black-hole which is formed from the gravitational collapseof a self-gravitating charged scalar-field ...". According toPhysicsNews Update 386-1, Hod and Piran "... have now supported previousindications showing that these Cauchy horizons are unstable; smalldisturbances in the black hole instantly transform them intosingularity regions. In fact, their calculations suggest that genericblack holes contain two singularities that are connected to eachother so that all infalling matter reaches one or the other. ..."

However, I do not agree that the results of such calculations,which due to computational difficulty deal with very simplifiedstructures, are applicable in all physically reasonable cases.Therefore, I am optimistic that such a journey might be physicallyreasonable.

 

Such a journey goes through 3 Regions inside EventHorizons,

such as for example, Regions II, VI, and VII.

Also, there is

the possible journey through the RingSingularity

such as from Region VI to Region VI' and back to RegionVI.

 

 


If e^2 + a^2 = M^2, then D = 0 at r = M the Outer andInner Event Horizons coincide.

 


If e^2 + a^2 is greater than M^2, then D = 0 impliesthat r at the Event Horizon is Complex

and the only singularity is the Ring Singularity at z = 0 and x^2+ y^2 = a^2, which is then a Naked Singularity in that theKerr-Newman metrics are not strongly asymptotically predictable.

Hawking and Ellis, in The LargeScale Structure of Space-Time(Cambridge 1973), show the Maximal Extension of SpaceTime at theBlack Hole Ring Singularity (whether e^2 + a^2is greater than, equal to, or less than M^2).

and say that:

The metric in the (x,z) SpaceTime is extended through the Ring Singularity to the (x',z') SpaceTime, with the entire maximally extended SpaceTime having coordinates (r, Theta, Phi, t) (abbreviated to (r,T,P,t). The metric in the (x,z) region has Positive r, while the metric in the (x',z') region has Negative r.

At large Negative values of r, the SpaceTime is asymptotically flat but with Negative Mass.

For small Negative values of r near the Ring Singularity, the circles (r = constant, T = constant, t = constant) are Closed Timelike Curves that can be deformed to pass through any point of the extended SpaceTime.

Robert Forward, in Indistinguishable from Magic (Baen 1995),describes SpaceTime travel through the Ring Singularity. Since ClosedTimelike Curves can be deformed to pass through any point of theextended SpaceTime. As Forward says, "... by making and opening[Ring Singularities] in orbit around distant stars we cantravel from one star system to another by merely popping into the[Ring Singularity] in our solar system and popping out againin another [Ring Singularity] around some distant starsystem.

As Kip Thorne says, in Black Holes and Time Warps (Norton 1994),quoting Tom Roman, "... If ... one [can] travel overinterstellar distances far faster than light, ... one can also ...travel backward in time ...", so Forward's Ring Singularity SpaceTimetravel is travel in both Space and Time.

Forward also describes SpaceTime travel by SpaceTime Tunnel asproposed by Mike Morris and his teacher Kip Thorne, who call aSpaceTime Tunnel a Wormhole. As Thorne says, " ... Einstein's fieldequation predict[s] that ... left to [its] owndevices ... [a SpaceTime Tunnel] ... opens up briefly, andthen pinches off and disappears - and its total life span fromcreation to pinch-off is so short that nothing can travel through it,from one mouth to the other. ... the only way to hold the[SpaceTime Tunnel] open is to thread [it] with withsome sort of material that pushes [its] walls apart. ... suchmaterial ... must have a Negative average energy density, as seen bya light beam travelling through it. ..."

As Don Page remarked to Morris and Thorne, Proposition 9.2.8 ofthe book of Hawking and Ellis shows that any SpaceTime Tunnelrequires such Negative Material to hold it open.

Since Ring Singularity SpaceTime travel goes from our ordinaryPositive Mass (x,z) region of SpaceTime through the Negative Mass(x'z') region and then back to our ordinary region, RingSingularity SpaceTime travel is consistent with Morris-ThorneSpaceTime Tunnel travel with the tunnel being held open by NegativeMaterial.

Thorne discusses other possibilities for getting Negative Materialto hold open a SpaceTime Tunnel, such as ZPFvacuum fluctuations, which Hawking has shown to have Negativeaverage energy density near the horizon of a Black Hole (thusallowing the Black Hole to decay). However, Gunnar Klinkhammer, astudent of Thorne, has shown that vacuum fluctuations in flatSpaceTime can never have Negative average energy density. Such vacuumfluctuations must occur in suitably curved SpaceTime.

Therefore, I think that SpaceTimetravel is most likely to be done by Ring Singularity, as wasshown in the movie StarGate

that linked the StarGate to the GizaPyramids (image from the CD-ROM Secrets of StarGate (Le StudioCanal 1994)). ( Compare a GraviPhoton2-Torus Ring Ship. )

As Jack Sarfatti says:

"... This may be the real meaning of The Whale thatswallowed up Jonah and spat him out again ..."

 

Shinkai and Hayward, in gr-qc/0205041,say: "... We study numerically the stability of Morris & Thorne'sfirst traversiblewormhole, shown previously by Ellis to be a solution for amassless ghost Klein-Gordon field. Our code uses a dual-nullformulation for spherically symmetric space-time integration, and thenumerical range covers both universes connected by the wormhole. Weobserve that the wormhole is unstable against Gaussian pulses ineither exotic or normal massless Klein-Gordon fields. Thewormhole throat suffers a bifurcation of horizons and either explodesto form an inflationary universe or collapses to a black hole, if thetotal input energy is respectively negative or positive. As theperturbations become small in total energy, there is evidence forcritical solutions with a certain black-hole mass or Hubble constant.The collapse time is related to the initial energy with an apparentlyuniversal critical exponent. For normal matter, such as atraveller traversing the wormhole, collapse to a black hole alwaysresults. However, carefully balanced additional ghostradiation can maintain the wormhole for a limited time. Theblack-hole formation from a traversible wormhole confirms therecently proposed duality between them. The inflationary caseprovides a mechanism for inflating, to macroscopic size, aPlanck-sized wormhole formed in space-time foam. ...".

 

Maybe StarGates for Informationcarried by Gamma Rays could be built fromSuperHeavy Ring-Shaped Nuclei, as described by Robert Forward inIndistinguishable from Magic (Baen Books 1995):

 


As Wald says in General Relativity (Chicago 1984),

if e^2 + a^2 is greater than M^2, the Kerr-Newman metrics do notdescribe the usual Gravitational Black Holes resulting fromGravitational curvature of SpaceTime.

Such Kerr-Newman metrics do describe ComptonRadius Vortices

and

the Gyromagnetic Ratio of a Kerr-Newman Black Hole is 2,the same as the Gyromagnetic Ratioof a spin 1/2 Electron.

For a spin 1 particle, with Mass m, Electric Charge Q, MagneticMoment M, and Angular Momentum L,

M / L = Q / 2m

so that its Gyromagnetic Ratio is 1. (see, for example, Leighton,Principles of Modern Physics, page 186, McGraw-Hill 1959).

According to pages 883 and 891-892 of Misner, Thorne, and Wheeler(Gravitation, Freeman 1973), a Kerr-Newman Black Hole Mass m,Electric Charge Q, Magnetic Moment M, and Angular Momentum L,

M / L = Q / m = 2 (Q / 2m)

so that a Kerr-Newman Black Hole has Gyromagnetic Ratio 2, or,equivalently, spin 1/2.

 

 


Since the charge-to-mass ratio e / M of the Electron is about10^21

the Electron can be considered to be a ComptonRadius Vortex.

 


Since the charge-to-mass ratio (in geometric units) e / M of theProton is about 10^18

the Proton can be also considered to be a ComptonRadius Vortex, although its internal structure, with 3valence Quarks, is more complicated than that of theElectron.

 


The Pion can also be described in terms of ComptonRadius Vortex.

 


 

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In 1997, DavidFinkelstein and his wife Shlomit visited India, where Shlomittook this picture:

 

In 1962, David wrote (with Jauch,Schiminovich, and Speiser) the firstQuaternion-SU(2) paper about Massive Charged WeakBosons.

 

 

I learned about Clifford Algebras fromstudying under DavidFinkelstein at Georgia Tech since around 1981. Both DavidFinkelstein and I use the the Periodicity-8tensor product Cl(N) = Cl(8) x...N/8 times... x Cl(8)) in our physics models. With respect tohis model, DavidFinkelstein says "... An assembly with Clifford statistics we call asquad. The Spinorial Chessboard shows how the dynamics, a large squadof chronons, can spontaneously break down into a Maxwellian assemblyof squads of 8 chronons each. Squads of 8 are special this way....". Also, see, for example, thepaper hep-th/0009086, Clifford Algebra as Quantum Language, by JamesBaugh, David Ritz Finkelstein, Andrei Galiautdinov, and HeinrichSaller, and the paper by D. Finkelstein and E. Rodriguez, Thequantum pentacle, International Journal of Theoretical Physics 23,887-894 (1984).

 


 

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