This (meaning: the whole affair that this posting was a small part of)
has come to my attention, only now, because the twins involved in it
have died together on the same day yesterday.
On Wednesday, October 23, 2002 at 11:43:50 PM UTC-5, John Baez wrote:
... I hear that two brothers have managed to publish 3 meaningless
papers in physics journals as a hoax - and even get Ph.D. degrees
in physics from Bourgogne University in the process!
on the "reverse Sokal" affair (as Baez elsewhere called it)
that may or may not have been a hoax or may have been for real,
because apparently nobody could make any sense out of what they
were trying to say.
So, I'm going to take a closer look at this here and provide some
perspective on the issues underlying the matters raised by them,
since it concerns matters that are of interest to me.
It is also an issue that is coming to a head with the recent
launch of the Webb telescope.
The theses are available in PDF format online, at least for now:
Igor Bogdanov
ETAT TOPOLOGIQUE DE L'ESPACE TEMPS A ECHELLE 0
currently here:
https://tel.archives-ouvertes.fr/tel-00001503v1/document
FLUCTUATIONS QUANTIQUES DE LA SIGNATURE DE LA METRIQUE
A L'ECHELLE DE PLANCK
currently here:
https://tel.archives-ouvertes.fr/tel-00001502/document
Other publications:
Topological field theory of the initial singularity of spacetime,
Classical and Quantum Gravity 18 (2001), 4341-4372.
Grichka Bogdanov and Igor Bogdanov,
Spacetime Metric and the KMS Condition at the Planck Scale,
Annals of Physics, 295 (2002), 90-97.
Grichka Bogdanov and Igor Bogdanov,
KMS space-time at the Planck scale,
Nuovo Cimento, 117B (2002) 417-424.
Igor Bogdanov,
Topological origin of inertia,
Czechoslovak Journal of Physics, 51 (2001), 1153-1236.
1. Igor
Igor's thesis asserts that the Big Bang is a zero-size singularity
involving a signature change from (++++) to (+++-) or (---+), depending
on which convention is used.
The FRW[L] (or "Big Bang") metrics are generally of the form
ds^2 = dt^2 - A(t) (dx^2 + dy^2 + dz^2)
for the proper time (s), where A(t) > 0 for t > 0 and A(t) -> 0 as t -> 0, where the time (t) is set to 0 at the Big Bang. Variations on this allow
for the spatial geometry to be curved rather than 3D Euclidean (and even
more general versions may even allow for it to be a manifold for a 3D
Lie algebra - i.e. the root of the Bianchi classification.)
1A. What Is "Zero Size"?
There is actually an ambiguity here when one says "zero size". Does that
mean that the entire geometry is 0-dimensional at t = 0 or 3-dimensional
but with metric distances of 0 along the hypersurface?
This is an ambiguity that leads to misrepresentations and false
visualizations, that unfortunately pop science presentations are replete
with - that many professional physicists have grown up on and which have
been internalized in their own language and visualizations!
In fact, it is not consistent to take the geometry as 0D, unless the
function A(t) satisfies certain conditions on its growth. In particular,
if A(t) does not grow fast enough (e.g. A(t) ~ t^n for powers n less
than 2), then the light cones different spacetime points land on the
t = 0 surface at different 3D subsurfaces.
The limiting case n = 2 is also the threshold for the weak energy
condition, and it is only for n >= 2 that the light cones splay out to
spatial infinity as t -> 0. You can't have a 0D geometry at t = 0,
unless every single point at the t = 0 hypersurface is contained in the
causal past, as they are all being identified as one and the same point.
Otherwise, it is a 3D surface. The most interesting feature of it is
that it yields a one-to-one correspondence with the spacetime points on
the t > 0 sector of the geometry - each sphere on the 3D surface is the intersection of a past light cone coming from a unique spacetime point.
1B. What Is The "Singularity"?
The nature of the "singularity" is that the metric goes from signature
(+---) to (+000). At each point, (x,y,z,t), where t > 0, the light cone
is given by the equation
(dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2 = 1/A(t)
so that the local light speed is root(1/A(t)). As long as A(t) > 0,
this is locally Minkowski and the kinematic group in the region of the
point (x,y,z,t) is Poincare'.
As t -> 0, if A(t) -> 0, then the metric reduces to rank 1 ... and the
inverse metric (up to conformal factor) to rank 3. This is actually the characteristic property of a Newton-Cartan geometry. In effect, light
speed goes to infinity at time 0; the light cones become flattened out
with the t = 0 surface being an envelope of them all.
This puts proper perspective on what's going on with "scale". Since
distances in Minkowski geometry are measured in light speed units,
if light speed goes to infinity, then you have an extreme case of the
King's Thumb Problem - distances measured in light speed units will
go to 0.
But that is not the same thing as saying that space, itself, goes to
a 0D point! It is *still* a 3D surface at t = 0, for virtually any
function A(t) that can be considered (again: the exceptional case being
if A(t) ~ t^2).
In other words: the spatial metric and temporal metric become
decoupled as t -> 0, because their conversion factor has become
infinte.
I parodied this situation, here, a long time ago. If you look back in
the s.p.r. archives, you'll find "The Untold Story Of Genesis", where
I alluded to this.
So, it is at this point that we can put the following in perspective:
Here's the abstract of Igor Bogdanov's thesis:
We propose in this research a new solution regarding the existence
and the content of the initial spacetime singularity. In the context
of topological field theory we consider that the initial singularity
of space-time corresponds to a zero size singular gravitational instanton characterized by a Riemannian metric configuration (++++) in dimension
D = 4. Connected with some unexpected topological data corresponding
to the zero scale of space-time, the initial singularity is thus not considered in terms of divergences of physical fields but can be resolved
in the frame of topological field theory.
It all hinges on the question: what is "zero size" supposed to be here?
single-point 0D?
or
3D with zero-metric but distinct points?
They didn't make this crystal clear. But to their credit: neither did any
of you. So, it's not really their problem, but all of your problem, collectively. And it is because that ambiguity was left unnoticed and unchecked, that their discussion passed through the literature. It's not
a flaw in the review process, but a flaw in the reviewers' frame of mind
by virtue of their having been imbued with the widely-propagated, but misleading (and false) "Big Bang Means All Of Space Contracts To A
Single Point" image that pervades the folklore.
So, in all the places where you see this (as per Baez' description):
Then it is suggested that the "zero scale singularity" can be understood
in terms of topological invariants,
you have to ask the same thing: what is "zero scale"?
1C. The Transition And Geodesic Completeness
The nature of the t = 0 envelope depends on the function A(t), itself.
For metrics that are asymptotically radiation dominant as t -> 0, it goes
as A(t) ~ t^1. For such metrics, there is (indeed) a smooth transition
over from signature (+---) to (++++) at time 0, with the boundary
case at t = 0 (+000) being the rank 1 metric typical of a Newton-Cartan geometry.
For t < 0, one has a locally Euclidean geometry, the kinmetic group
being that for the 4D Euclidean geometry; while at t = 0, it reduces to
the Galilei group - and its central extension, the Bargmann group.
The case A(t) ~ t^1 for t ~ 0 is of special interest, because of a
property it has - widely neglected and rarely (if ever) mentioned -
that totally blows open folklore on pseudo-Riemannian geometries.
It's not a Riemannian or pseudo-Riemannian geometry at all, because of
the signature change, so one should not expect familiar theorems
learned on those contexts to be true.
The light cones don't just flatten out at t = 0, they reverse! Past
light cones have the shape of a pointed Mexican hat. They splay out,
back into the future direction. The generators of the cones are
parabolic along the t = 0 surface.
Past-directed timelike geodesics, almost all, reflect off the t = 0
surface as catenary curves and go back toward the future. This
includes geodesics that can intersect the worldline of a given object
at a given time, reflect back off the t = 0 surface and intersect the
future of that object at a future point.
The sole exceptions are the timelike geodesics associated with the
co-moving frame. They pass through the t = 0 surface over into the
t < 0 sector.
Overall, the geometry is GEODESICALLY COMPLETE - notwithstanding the
metric singularity! The irregularity of the metric shows up, not by
a rupture of the geodesics, but by the emergence of a distinctly
non-Riemannian property on the t = 0 surface:
Uniqueness Of Geodesics breaks down:
Along the t = 0 surface: a given point and given direction, laying
within the surface, does not uniquely determine a geodesic.
1D. Signature-Changing Geometries As Lie Bundles
Now we can look at another part of the description:
We propose hereafter that the signature of the Space-Time metric
(+++-) is not anymore frozen at the Planck scale and presents quantum fluctuations (++++/-) until 0 scale where it becomes Euclidean (++++).
(i) At the albraic level we suggest an oscillation path (3,1) (4,0)
excluding (2,2).
The type of geometry I alluded to above (and which is also alluded to here)
has not been much studied. Yes, there is literature on signature-changing geometries, but generally they don't focus on what happens to the local kinematic group. What you're talking about, here, is a type of geometry
in which different points may have different Lie algebras associated with
them - which is what we call a Lie Bundle.
You need a framework, like this, to be able to talk about continuous transitions between different signatures.
Lie Bundles have been considered in the literature, but I'm not aware of
who works with them for signature-changing cosmologies. To model the Big
Bang, if you take the FRW metric seriously all the way down to time 0,
requires such a geometry, where the t -> 0 threshold provides some kind
of physical realization for the passage of the Poincare' group (and
actually, for an extension of it, as I'm about to describe below) to the Bargmann group, and then on over at t < 0 to the Euclid 4D group.
1E. "Relativistic" Bargmann Geometries and 5D Cosmologies
To see exactly what this is, consider the Minkowski metric, which we
write as a metric for proper time (s):
ds^2 = dt^2 - (1/c)^2 (dx^2 + dy^2 + dz^2).
Literally throw in the proper time as an extra coordinate and make this
a 5-dimensional geometry with a 5D metric and constraint:
ds^2 - dt^2 + (1/c)^2 (dx^2 + dy^2 + dz^2) = 0.
Now, rewrite it in terms of time dilation, but tack it up an order of
c^2:
u = c^2 (s - t).
Why the extra factor? Because when you substitute into the line element
you get a metric that looks like this ... after removing a constant
conformal factor:
dx^2 + dy^2 + dz^2 + 2 dt du + (1/c)^2 du^2 = 0.
That is, in fact, the relativistic version of what is already known as
the Bargmann Geometry, whose metric has the form:
dx^2 + dy^2 + dz^2 + 2 dt du = 0
which, in turn, is the limiting case as (1/c)^2 -> 0 of the above
geometry, which (for lack of a better term) we could just call the "relativistic" Bargmann geometry.
When the light cone constaint is applied to the metric, then the
proper time can be expressed in terms of u as
s = t + (1/c)^2 u
and, as expected, in the non-relativistic limit s and t coincide.
The Bargmann group is the symmetry group that possesses the following
two invariants:
dx^2 + dy^2 + dz^2 + 2 dt du
dt
By comparison, the relativistic version of this would be a symmetry
group that leaves the corresponding deformations of these two invariants
dx^2 + dy^2 + dz^2 + 2 dt du + (1/c)^2 du^2
dt + (1/c)^2 du
fixed. It is, in fact, just the Poincare' group ... but with a twist:
it has an additional, 11th generator. It's a (trivial) central extension
of Poincare'.
For the corresponding representation theory, the extra generator gives
you an additional degree of freedom that shows up as an additive term
for the mass and energy. Among other things, the extension provides
extra room to be able to talk about zero point energy.
The Bargmann geometry (both relativistic and non-relativitic) has
a 4+1 signature. In the relativistic case, it reduces to 3+1 when you
impose the light cone condition on the metric
dx^2 + dy^2 + dz^2 + 2 dt du + (1/c)^2 du^2 = 0.
Doing so removes a degree of freedom that, in effect, eliminates the
11th generator.
But to have a consistent passage from the t > 0 sector of a space-time
geometry with signature change to the t < 0 sector, where the t = 0
boundary is that for a (rank 1 metric + rank 3 dual metric up to
conformal equivalence) requires keeping in the 11th generator, if you
also want to make the transition for the Lie Bundle continuous.
There are actually quite a few publications on 5D cosmologies, and in
some cases you may recognize the underlying 5D geometry as being that
for a curved space-time version of the relativistic Bargmann geometry.
The passage to the Euclid 4D group may be obtained by generalizing
the extra factor (0 versus (1/c)^2) into a deformation parameters
and rewriting the invariants as:
dx^2 + dy^2 + dz^2 + 2 dt du + alpha du^2
dt + alpha du
The case alpha = 0 yields the Bargmann group. The other cases are
deformations of it, with non-zero alpha. If alpha > 0, it is the
(trivially) central extended Poincare' group, and if alpha < 0, it
is a trivial central extension of the Euclid 4D group.
I don't know if anyone working with 5D cosmology uses this as a way
to also model or account for signature change. But one of the more
interesting properties is that it has two senses of time in it:
the coordinate time (t)
and
the proper time (s), which is an invariant.
They can be combined into a single "complex" time t + is, and then the
metrics - except for the alpha = 0 case - could be rewritten as the
real part of a metric that has complex time.
It might, then, be possible to associate s with thermal time.
In light of what I just described, it's interesting to see this passage:
We built the quotient topological space describing
the superposition of the Lorentzian and the Riemanian metrics. In
terms of quantum groups we evidence a relation between q-deformation
and deformation of the signature. We have obtained a new algebraic construction (a new cocycle bicrossproducts by twisting) which allowed
us to unify the Lorentzian and the Euclidean signatures within a
unique quantum group structure.
But I strongly disagree with the notion of any singularity being "zero
scale" or "0-size" as here:
The initial singularity must be considered as
a singular 0-size gravitational instanton.
1F. The Initial Hypersurface: Hawking vs. Mansouri & Junction Conditions
There's actually a lot more I wanted to talk about here, both for Igor
and his twin's thesis, but I may have to defer the rest for a later time,
apart from a few brief closing observations.
Hawking use of complex time and signature change - as best I understand
it - was meant to be a technical fix devoid of physical content, used
to just "get the job done". But one of the most important properties of
his "fix" is that he had a discontinuous transition at t = 0. The
geometry did NOT pass through a Newton-Cartan intermediate phase, but
rather: the t = 0 surface was spacelike. So, it lay firmly on the
(+---) side of the geometry.
Others, like Mansouri, have adopted a different starting point, asserting
that the t = 0 hypersurface is (indeed) a null surface ... i.e. an
envelope of light cones; thereby effectively asserting c -> infinity as
t -> 0. Though they don't call the t = 0 slice an equal-time slice of Newton-Cartann geometry, that's what it is.
In important property of this (or any other interface, like Hawking's)
is that it entains the kind of consistency conditions known in the
literature as Junction Conditions. These are the extra conditions
required to be able to pair off the (+---) Lorentzian geometry in
the t > 0 sector with the (++++) locally-Euclidean-4D geometry in the
t < 0 sector.
One of the most important ones that comes out of Mansouri's work is
the prediction of inflation, as a consequence of having the initial
t = 0 hypersurface being null.
If I have more time, I'll get to the "Topological Origin of Inertia"
thesis and discuss the issues surrounding this.
--- SoupGate-Win32 v1.05
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