Can anybody point me to the reference for the PROOF, by Penrose (Oxford
mathematician) & Hawking (Cambridge physicist), that singularities are
an inevitable consequence of the mathematical apparatus of Einstein's
Field Equation?

You need to read Hawking and Ellis, The large scale structure of Space-

Can anybody point me to the reference for the PROOF, by Penrose (Oxford
mathematician) & Hawking (Cambridge physicist), that singularities are
an inevitable consequence of the mathematical apparatus of Einstein's
Field Equation?

(I already have Einstein's comments on this subject, prior to death,
and I do know that this proof postdated that death)

Can anybody point me to the reference for the PROOF, by Penrose (Oxford
mathematician) & Hawking (Cambridge physicist), that singularities are
an inevitable consequence of the mathematical apparatus of Einstein's
Field Equation?

Can anybody point me to the reference for the PROOF, by Penrose (Oxford
mathematician) & Hawking (Cambridge physicist), that singularities are
an inevitable consequence of the mathematical apparatus of Einstein's
Field Equation?

(I already have Einstein's comments on this subject, prior to death,
and I do know that this proof postdated that death)

John (Liberty) Bell
(Change John to Liberty to respond by email)

Can anybody point me to the reference for the PROOF, by Penrose (Oxford
mathematician) & Hawking (Cambridge physicist), that singularities are
an inevitable consequence of the mathematical apparatus of Einstein's
Field Equation?

Can anybody point me to the reference for the PROOF, by Penrose (Oxford
mathematician) & Hawking (Cambridge physicist), that singularities are
an inevitable consequence of the mathematical apparatus of Einstein's
Field Equation?
++++

(I already have Einstein's comments on this subject, prior to death,
and I do know that this proof postdated that death)
=20
John (Liberty) Bell
(Change John to Liberty to respond by email)

"John (Liberty) Bell" <john.bell@accelerators.co.uk> wrote:
Can anybody point me to the reference for the PROOF, by Penrose (Oxford
mathematician) & Hawking (Cambridge physicist), that singularities are
an inevitable consequence of the mathematical apparatus of Einstein's
Field Equation?

I think the proof you're looking for is the one that, given positive
energy conditions (and maybe some other technical conditions which I
forget), the existence of any trapped surfaces implies that there exists
an inextensible geodesic (which is the mathematical property usually
used for the intuitive notion of a singularity).

This proof is given in proposition 9.2.8 of
@Book{Hawking73a,
author = {Stephen W. Hawking and George F. R. Ellis},
title = {The large scale structure of spacetime},
publisher = {Cambridge University Press},
year = 1973,
address = {Cambridge, England},
isbn = {0-521-09906-4},
}

I'm not sure if this gives references to the original papers. If not,
you might try
@Article{Penrose65,
key = {Penrose65},
author = {Roger Penrose},
title = {Gravitational Collapse and Space-Time Singularities},
journal = {Phys. Rev. Lett.},
year = 1965,
volume = 14,
pages = 57
}
and/or
@Article{Penrose70a,
author = {Roger Penrose and Stephen W. Hawking},
title = {The singularities of gravitational collapse and
cosmology},
journal = {Proc. Roy. Soc. Lond. A},
year = 1970,
volume = 314,
pages = 529
}

ciao,

Interesting that, as confirmed by jacques.f...@neuf.fr, the originator
was Roger Penrose (which kind of ties in with my own prior
understanding of their relative merits). Equally interesting that many
respondents merely quoted Hawking and Ellis, thus apparently leaving
Roger out of the equation entirely.
+++

John

I additionally found http://www.hawking.org.uk/pdf/time.pdf (reproduced
at http://www.arxiv.org/abs/hep-th/9409195 )
This starts with the sentence "In these lectures Roger Penrose and I
will put forward our related but rather different viewpoints on the
nature of space and time."
However, in inimicable Hawking style, no further mention was made of
the responses of Roger Penrose, and no reference to those responses was
given.

Jonathan Thornburg -- remove -animal to reply wrote:



"John (Liberty) Bell" <john.b...@accelerators.co.uk> wrote:
Can anybody point me to the reference for the PROOF, by Penrose (Oxford
mathematician) & Hawking (Cambridge physicist), that singularities are
an inevitable consequence of the mathematical apparatus of Einstein's
Field Equation?

I think the proof you're looking for is the one that, given positive
energy conditions (and maybe some other technical conditions which I
forget), the existence of any trapped surfaces implies that there exists
an inextensible geodesic (which is the mathematical property usually
used for the intuitive notion of a singularity).

This proof is given in proposition 9.2.8 of
@Book{Hawking73a,
author = {Stephen W. Hawking and George F. R. Ellis},
title = {The large scale structure of spacetime},
publisher = {Cambridge University Press},
year = 1973,
address = {Cambridge, England},
isbn = {0-521-09906-4},
}

I'm not sure if this gives references to the original papers. If not,
you might try
@Article{Penrose65,
key = {Penrose65},
author = {Roger Penrose},
title = {Gravitational Collapse and Space-Time Singularities},
journal = {Phys. Rev. Lett.},
year = 1965,
volume = 14,
pages = 57
}
and/or
@Article{Penrose70a,
author = {Roger Penrose and Stephen W. Hawking},
title = {The singularities of gravitational collapse and
cosmology},
journal = {Proc. Roy. Soc. Lond. A},
year = 1970,
volume = 314,
pages = 529
}

ciao,Thanks

Interesting that, as confirmed by jacques.f...@neuf.fr, the originator
was Roger Penrose (which kind of ties in with my own prior
understanding of their relative merits). Equally interesting that many
respondents merely quoted Hawking and Ellis, thus apparently leaving
Roger out of the equation entirely.
+++

I also foundhttp://en.wikipedia.org/wiki/Penrose-Hawking_singularity_theoremswhich
has the advantage of (a) being online for free, and (b) giving a
concise and simple explanation of what it is all about.

I additionally foundhttp://www.hawking.org.uk/pdf/time.pdf(reproduced
athttp://www.arxiv.org/abs/hep-th/9409195)

This starts with the sentence "In these lectures Roger Penrose and I
will put forward our related but rather different viewpoints on the
nature of space and time."

However, in inimicable Hawking style, no further mention was made of
the responses of Roger Penrose, and no reference to those responses was
given.

I did say in my OP "I already have Einstein's comments on this subject,
prior to death". However, it turns out that I probably don't. What I
remembered was comments made by Einstein on unified field theory and
singularities, in the 1956 paperback version of "The Meaning Of
Relativity", which had been referenced by an author in class. quantum
grav. in 1988. However, I have just got the (ordered) 1950 hardback
version from my local library, and can find no mention of this subject.
At least, there is nothing in the index.

John

Interesting that, as confirmed by jacques.f...@neuf.fr, the originator
was Roger Penrose (which kind of ties in with my own prior
understanding of their relative merits). Equally interesting that many
respondents merely quoted Hawking and Ellis, thus apparently leaving
Roger out of the equation entirely.

On 24 jan, 16:48, "John (Liberty) Bell" <john.b...@accelerators.co.uk
wrote:> Interesting that, as confirmed by jacques.f...@neuf.fr, the originator
was Roger Penrose (which kind of ties in with my own prior
understanding of their relative merits). Equally interesting that many
respondents merely quoted Hawking and Ellis, thus apparently leaving
Roger out of the equation entirely.+++

Around 1965, Hawking was completing his PHD at Cambridge under
supervision of D.Sciama who used to direct his students ( S.Hawking, B.
Carter) to Penrose (in London) for dealing with all tricky
mathematical problems involved with BH .
Penrose was the first to use successfully "the global methods"
involving topology in GR for dealing with "singularity" problems.
As reported by K. Thorne in his book (Black Holes,..), in 1965 there
was a fierce debate with the Russian
relativists (Kalatnikov, Lifchitz,..) who thought having demonstrated
that the central singularity of a BH is just an artefact of the perfect
spherical symetry (instabilities due to asymetry would prevent
collapse) and the western ones who thought that singularities are
inevitable, even when symetry is not perfect.
In a conference in July 1965 in London , at end of Kalatnikov speech
relative to their supposed "demo", C. Misner protested vehemently
arguing the brand new theorem of R. Penrose just published in Jan 65.
But as this theorem was based on topological arguments, and that the
relativist community was not awre of such methods, they conclude that
Kalatnitov should be right. Several later Kalatnikov will regognise his
error, as meanwhile,they have found the BKL (Belinski, Kalatnikov,
Lifchitz) oscillatory singularity showing that even in case of large
perturbations the instabilities cannot prevent the system to finally
collapse into a singularity.
Jacques

On 1/29/2007 9:44 PM, John (Liberty) Bell wrote:
.. Einstein then
stated that singularities were indicative of a weakness of the
mathematical apparatus used to derive solutions.

In sci.math.research (path 'Symmetries reflect unilaterality and vice
versa', my reply, message ID <ern4qj$587$1@news.ks.uiuc.edu>) I gave an
example in order to justify my suggestion to accept |sign(0)|=1 instead
of |sign(0)|=0. Would this have implications for physical theories?

It's just a convention. I don't think there would be any real