Tom Roberts could you please comment the following first try
(SI-units now, and Russian signature convention):

I would like to try following two (possible very rough and I'am not
sure would this be correct at all ???) metrics for H-M's black hole:

ds^2=3D [1-2Gm/(rc^2)] dt^2-[1-2Gm/(rc^2)]^-1 dr^2 +

+ r^2 (dT^2 + sin^2 T dP^2)

(T =3D theta, P =3D phi, this solution valid only for the case 0 < r <
2Gm/c^2
and only for areas where neutrinos can move freely)

ds^2=3D [1+2Gm/(rc^2)] dt^2-[1+2Gm/(rc^2)]^-1 dr^2 +

+ r^2 (dT^2 + sin^2 T dP^2)

(T =3D theta, P =3D phi, this solution valid only for the case 0< r <
2Gm/c^2
and only for areas which absorbs neutrinos, r > 0 now but m has been
replaced
for -m from above metrics, crystal in center has acted like a "mirror"
???)



Compare to copy of my old article below:


"Already, a large number of exact solutions to the Einstein's equations
are known (cf. [3]).



The first of which (the Schwarzschild solution of 1916,

ds^2=3D [1-2Gm/(rc^2)] dt^2-[1-2Gm/(rc^2)]^-1 dr^2 +

+ r^2 (dT^2 + sin^2 T dP^2) (3)

(T =3D theta, P =3D phi, this solution valid only for the case r>2Gm/c^2)
in the commonly-used curvilinear coordinate system) has the meaning
of a sherically-symmetric point mass."

Hannu

Copies of two my articles below:

--COPY 1------

h.poropudas@luukku.com kirjoitti:

I forget to mention the reference. Please take a look.

ftp://ftp.funet.fi/.m/pub/doc/misc/HannuPoropudas/Hanna-Maria-drawing-6=

ftp://ftp.funet.fi/.m/pub/doc/misc/HannuPoropudas/

Hanna-Maria-drawing-6.123.gif
Hanna-Maria-drawing-8.gif

(two H-M's drawings about primordial black hole)

Hanna-Maria-drawing-9.gif
Hanna-Maria-drawing-16.gif

(H-M's drawing of photon in contracting Universe which is area which
absorbs neutrinos)

(remark also that expanding Universe is area where neutrinos can move
freely)

ASCII-text files summaries of my articles (ordinary *.txt files)

Readme.all
Readme.mid
Readme.see

plus all what I have written after these up today (not collected
anymore
to summaries).

Hannu

h.poropudas@luukku.com kirjoitti:

Tom Roberts wrote:

h.poropudas@luukku.com wrote:

About the TRICK in coordinates introduced by Kruskal and Szekeres=

1961
I think that it is questionable to use this mathematical TRICK (s=

below)
for constructing inner metric of Schwarzschild ???

I think your entire argument is flawed. Or rather, I did not see any
argument to support this claim.

First I think that correct form of result of integration should be

ct =3D r + a ln (abs( r - a )) + C (rising photon)

-ct =3D r + a ln (abs( r - a )) + D (falling photon)
(9.1)(correct)

where abs means absolute value and where C and D are constants of
integration.

But anyhow even though we have these new coordinates

UV =3D e^( r/a ) ( r/a - 1 ), V/U =3D e^(ct / a) (9.2)

I would allow to use it only for region r > a, where both U and V must
be positive. I think that we have no physical observation which would
allow
to a passage to negative values (only mathematical reasons are not
enough) ?

Only really considerable physical matters which I know about inside t=

event horizon of black hole are those H-M's drawings which tells us
(as I have understood) something about distribution of areas which
absorbs neutrinos and areas where neutrinos can move freely inside
the event horizon of black hole !!!

We begin in the Schwarzschild coordinate system, which is the onl=

we know.

It may be the only coordinates YOU happen to know, but there are se=

different well-known coordinates for Schwarzschild spacetime.

On any null radial geodesic ( dT and dP both zero, T =3D theta, =

we have [integrating]
ct =3D r + a ln ( r - a ) + C ( rising photon )
- ct =3D r + a ln ( r - a ) + D ( falling photon ),
where C and D are constants of integration.
The TRICK is to use the constants of integration as new coordinat=

That simply is not possible. C and D do indeed label the "photons",=

they are in no way coordinates -- coordinate tuples must be in a 1-=

relationship with points in the manifold, and C and D manifestly ar=

(a given value of C applies to the entire worldline of that particu=

"photon").

I put "photon" in quotes, as that word normally has quantum
implications utterly unrelated to this. Please read "light
pulse" for "photon"....

Indeed, we can
do rather better by using the closely related coordinates U and V

It's not at all clear how U and V are "related" to C and D.

defined by
U V =3D e ^( r / a ) ( r / a - 1 ), V / U =3D e ^( ct / a )

At least those are coordinates (ignoring \theta and \phi, of course=

But it's not clear what their domain of validity is -- it is especi=

not clear whether or not they apply at r=3Da (your a is normally no=

as 2M, the value of r at the event horizon) because r and t are
themselves not well-behaved there....

It is known that Kruskal's U and V are indeed well behaved at r=3Da=

don't have your reference and am not familiar with these particular
coordinates.

Please take a look reference mentioned.

As I said above, I don't see how this relates to your original clai=

Tom Roberts tjroberts@lucent.com

Hannu

--COPY 2--------------------------------------

From: "Hannu Poropudas" <hapor...@koillismaa.fi>
Subject: About ideas behind in mathematics of General Relativity Theory
and Riemann and Ricci tensors
Date: 2000/11/19
Message-ID: <01c05219$76f1ec00$82c89cc3@knet795>
X-Deja-AN: 696873338
Approved:

Originator:

From the physical point of view, solutions to Einstein's equations may

be subdivided into vacuum fields outside local sources, gravitational
fields, Einstein-Maxwell fields, gravitational wave fields,
cosmological solutions, etc.

Various methods for classifying the pseudo-Riemannian spaces that aid
in constructing solutions to Einstein's equations with desired
properties and in interpreting already-known solutions have been
developed.

These are:

1) an algebraic classification by properties of the Weyl
conformal curvature tensor (the three Petrov types - I, II, III, and
two degenerate subtypes D and N, as well as the trivial case, type 0,
corresponding to conformally-flat spaces; it is often taken that N and
III correspond to gravitational wave fields);

2) an algebraic classification by properties of the Ricci tensor (or
energy-momentum tensor); and

3) a classification by groups of motions (isometries: mappings of a
space-time onto itself by Lie displacements without changing the
metric). In 3-dimensional space, supposing it is homogeneous, the last
classification leads to the nine Bianchi types, which play an
important role in the theory of homogeneous cosmological models.

To obtain, and especially, to study solutions of Einstein's equations
one more and more often uses computer; symbolic computations are
successfully employed in this area.

There are reports that the identification problem for metrics given
in differential coordinate systems has been solved with aid of a
computer.

To compare the inferences from Einstein's theory of gravitation and
its various modifications and generalizations there has been an
attempt to develop a phenomenological method of description of the
metric tensor and gravitational effects ('parametrized post-Newtonian
theory of gravitation').

When analyzing concrete solutions of Einstein's equations, an important
role is played by the problem of the completeness of the atlas of
charts of the given manifold (hence the development of extension
methods); the search and investigation of singularities (their
definition is made fundamentally more difficult by the indefineteness
of the metric); and the computation of asymptotic (including
topological) properties of solutions in the case of insular systems.

Problems in Einstein's theory of gravitation lead to the formulation
of a new important concept in pseudo-Riemannian geometry - that of a
horizon.

Although a horizon (one distinguishes between the event, the particles,
the Cauchy, the causality, and the apparent horizon) is not a set of
singular points, it can be invariantly distinguished in space-time.

Thus, the event horizon in an asymptotically-flat world is defined as
the limit of the set of events, i.e. of 4-dimensional points, from
which one still may leave towards infinity while remaining within the
light cone.

If an event horizon exists, then the domain bounded by it in space-time
(from which it is impossible to go to infinity without crossing the
light cone) is called the black hole; the simplest example of it is
described by the extension of the Schwarzschild metric (3).

Inside the black hole there are singularities (in particular, some
invariants of the Riemann curvature of (3) diverge (go to infinity)
as r->0).

Moreover, the Schwarzschild singularity is space-like (ds^2<0),
while for other black holes (the general case is a Kerr-Newmann
metric, [3]) it may also be time-like (ds^2>0).

Astrophysical applications of general relativity theory have shown
that black holes can be formed as result of gravitational collapse
of massive stars; a series of candidates for the role of black hole
have been considered among the observed celestial bodies which could
be disclosed by processes occurring in their outer domains where the
gravitational forces are strong.

It is considered that under so-called energy conditions (in fact:
natural conditions on energy-momentum tensor of matter) a singular
state in the past and future under the evolution of material systems
is inevitable (singularity theorems of R. Penrose, S. Hawking, et al.).

However, it is conjectured that the singularities are hidden by the
horizons (the 'cosmic censorship' hypothesis).

The foundations of the quantum theory of gravitation (both in the
sense of quantizing a gravitational field and of quantizing other
fields on the background of non-flat classical geometries) have been
developed.

One of the consequences is Hawking's effect of particle (photon, etc.)
generation by black holes, leading to their 'evaporation'.

Quantum effects of gravitation are important at the early stages of the
expansion of the Universe.

In the description of the quantum theory of gravitation one uses on the
canonical formalism of field theory, Feynman path integrals, etc.;
related to this one has developed Euclidean field theory (in the sense
of signature), has been investigated instanton solutions to Einstein's
equations, and has developed the Penrose twistor calculus, which is in
its results close to the theory of complexified spaces with a self-dual
or anti-self-dual Weyl conformal curvature tensor (H-spaces of
E. Newman).

Other generalizations of the theory of gravitation include:
the Einstein-Cartan theory with curvature and torsion;
the affine theory of gravitation;
the theory of gravitation with Lagrangian quadratic in the curvature;
the bimetric theories of gravitation, etc. (cf. [6]).

Einstein's theory of gravitation leads to effects that are new as
compared with Newton's theory; however, these effects are difficult to
discover experimentally.

Apart from these, both theories are in mutual agreement. Up till now,
the following effects have been verified:
gravitational red shift;
bending of light rays;
perihelion advance of a planetary orbit;
non-stationary (expansion) of the Universe.

The effects of gravitational radiation have been verified indirectly
(by the loss of energy of a two-body system resolving around a common
centre of mass).

Not a single fact contradicting Einstein's theory of gravitation has
been found.

On the edge of the reach of experiments today are:
gravitational waves coming from cosmic sources, and dragging effects
in gravitational fields on rotating bodies (precession of the axis of
a gyroscope and others).

References:

[1] Einstein, A.: Selected works, 1-2, Moscow, 1966 (in Russian).
[2] Misner, C. W., Thorne, K. S. and Wheeler, J. A.: Gravitation,
Freeman, 1973.
[3] Kramer, D., Stephani, H., MacCallum, M. and Herlt, E.:
Exact solutions of Einstein's field equations,
Cambridge Univ. Press, 1980.
[4] Petrov, A. Z.: Eistein spaces, Pergamon, 1969 (translated
from the Russian).
[5] Hawking, S. W. and Ellis, G. F. R.: The large-scale structure
of space-time, Cambridge Univ. Press, 1973.
[6] Held, A. (ED.): General relativity and gravitation. One hundred
years after the birth of Albert Einstein, 1-2, Plenum, 1980.

N. V. Mitskevich

Editoral comments.

As noted above, a number of features characteristic for soliton
theory (completely-integrable systems theory) also turns up in the
theory of Einstein's equations.

They include B=E5cklund transformations [A1],
dressing transformations
(in this context often referred to as
Kinnersley-Chitre transformations,
Hauser-Ernst transformations,
Hoenselaers-Kinnersley-Xanthopoulos transformations
(HKX-transformations)) [A2],
superposition principles [A3], [A4],
the use of suitable Riemann-Hilbert problems [A2], [A5],
the occurrence of hierarchies, and further considerations
based on symmetry ideas [A6], [A7].

References:

[A1] Kramer, D. and Neugebauer, G.: B=E5cklund transformations
in general relativity, Lect. notes in physics, 205,
Springer, 1984, pp. 1-25.
[A2] Hauser, I.: On the homogeneous Hilbert problem for
effecting Kinnersley-Chitre transformations. Lect. notes
in physics, 205, Spinger, 1984, pp. 128-175.
[A3] Xanthopoulos, B. C.: Superpositions of solutions in general
relativity, Lect. notes in physics, 239, Springer, 1985,
pp. 109-117.
[A4] Chinea, F. J.: Vector B=E5cklund transformations and
associated superposition principle, Lect. notes in physics,
205, Springer, 1984, pp. 55-67.
[A5] Ernst, F. J.: The homogeneous Hilbert problem: practical
application, Lect. notes in physics, 205, Springer,
1984, pp. 176-185.
[A6] Xanthopoulos, B. C.: Symmetries and solutions of the Einstein
equations, Lect. notes in physics, 239, Springer, 1985,
pp. 77-108.
[A7] Schmidt, B. G.: The Geroch group is a Banach Lie group,
Lect. notes in physics, 205, Springer, 1984, pp. 113-127.

AMS 1980 Subject Classification: 70F15, 83CXX, 83C35.

I hope that I could clarify these difficult matters with this little
copy.
=20
Best Regards,
=20
Hannu Poropudas