Imagine a curved space-time manifold as in orthodox general relativity.
The tangent space fibers are locally flat. The latter is an expression of
the equivalence principle. But can this be generalized?
De Sitter space is a space of constant curvature with scale R so
that the cosmological constant is /\ = 1/R^2.
Can we imagine a more general tangent bundle where the tangent spaces
are De Sitter
spaces of variable R or /\? That would correspond to /\ not as a
constant, but as
a local variable field.
/\ < 0 is De Sitter space (like a sphere), i.e. attractive "dark matter"
and /\ > 0 is anti-Desitter space (like a hyperbolid), i.e. repulsive
Susskind's hologram idea is partly that quantum gravity in D+1
dimensions is dual to a
flat spacetime quantum field or string theory in D dimensions.
Imagine the "strings" have both meanings of a string in computer theory
as well as a string in a space of D dimensions.
Imagine a variable "quantum of area" a in the sense of loop quantum gravity.
Given a bulk region V with boundary A, the thermodynamic entropy in
Shannon c-bits is ~ A/a.
As a increases the thermal entropy in c-bits decreases. This must be
compensated by an increase in qubits.
It works both ways - a tradeoff between string qubits and loop area
c-bits where we have the duality
(Number of Loop c-bits)(Number of string qubits) = A large integer.
What about metric engineering of traversable wormhole star gates and
weightless Alcubierre warp drives for alleged "flying saucers"?
Forget Tuv(Matter). It plays no practical role at all. This is why the
whole Puthoff PV idea is a wrong path to take IMHO.
The basic equation of metric engineering is
Guv + /\zpfguv = 0
The covariant 4 divergence is
Guv^;v + /\zpfguv^;v + /\zpf^,vguv = 0
I include a nonmetricity term ~ guv^;v in case there are extra space
dimensions both bosonic and fermionic. Also there may be torsion fields
and special-conformal and dilaton fields beyond the local gauging of the
4-parameter translation subgroup of special relativity's Poincare group
out of which Einstein's 1915 theory emerges.
/\ = a^-1[a^3/2|Coherence|^2 - 1]
Coherence = Vacuum Coherence + Control Superconductor Coherence
These are MACRO-QUANTUM "Feynman amplitudes".
We metric engineer by a Josephson interference of a control
superconductor with the vacuum coherence wave.
There is an anti-symmetric Cartan 2 form Fuv that also includes the
Berry topological phases as well as the dynamical Bohm-Aharonov phases.
The general phase control parameter is the flux "surface integral" of
Start in the non-exotic vacuum where
a^3/2|Vacuum Coherence| = 1 corresponding to /\ = 0.
Also Superconductor Coherence << Vacuum Coherence
The induced exotic vacuum zero point energy field is then
/\(exotic vacuum) ~ 2(Loop Quantum of Area)^-1/4|Superconductor
Coherence|cos(phase control parameter)
Note that |Coherence| has dimension |area|^-3/4
That is /\ has dimension area^-1
|Superconductor Coherence| = (Condensate density of bound real electron
Where the local De Sitter/Anti-De Sitter fiber parameter for the local
unified dark matter/energy exotic vacuum zero point energy field is R =
Guv^;v is the geometrodynamic current density.
The zero point exotic vacuum Josephson tunnel current density is ~
sin(phase control parameter).
Geometrodynamic current density + Exotic vacuum Josephson current
density = 0
The Josephson current density is controlled by the Berry phase 2-form
Fuv that depends on both EM fields and topological variables.
This theory does not use Tuv(Matter) directly because that is not the
way to control space-time warps because the string tension is too stiff.
On Jan 27, 2004, at 9:45 AM, Tony Smith wrote:
Carlos and Jack, here are some comments on Nottale's work.
Thanks to Carlos for mentioning Nottale's work, as it motivated me
to write up these comments about it, based on the web page at
and some of its linked material.
In summary, I don't fully understand everything Nottale is doing,
and I find his work interesting and worthy of publication etc,
I prefer my way of looking at things (particularly dark matter
and dark energy as I mention in comment 8).
Here are some comments about scale relativity,
based on quotes from the above-cited material (set off by quotation marks):
1 - "... GR seems to be more fundamental .. better founded ...
and thus more complete than QM ...".
I disagree. I think that GR is a large-scale low-energy effective
theory that is derived from a generalized MacDowell-Mansouri mechanism.
2 - "... The fundamental principle of ScR: ... laws of nature must be
valid in every coordinate systems, whatever their state of motion
and of scale. ...".
I like having a scale symmetry, but I get it from the dilation generator,
one of the 15 generators of the conformal group.
I also use the 4 special conformal transformations,
as well as 6 rotations and 4 translations.
3 - "... The fractal Space-Time, explicitely dependent on resolutions,
can be reduced to the definition of a "Space-Time-Zoom"
with 5 dimensions (x,y,z,t,D).
It is the fractal dimension, became a variable,
which plays the role of a 5th dimension for scale laws
(just as relativistic motion laws are implemented
by the interpretation of time as a 4th dimension). ..."
How is the fractal dimension D different from a "second time" dimension,
such as the 2 time dimensions in anti-seSitter Spin(2,3) = Sp(2)
or in the conformal group Spin(2,4) = SU(2,2) ?
Note that the conformal group that I like not only has a "second time"
dimension, it also has a "fourth space" dimension.
4 - "... Equation (40) can be used to theoretically predict
the electron mass from the experimental value of
the fine structure constant. ...".
I prefer my model, because I can calclulate both the electron mass
and the fine structure constant. Although I don't fully understand
everything Nottale is doing here, it seems to me to be primarily
related to renormalization group stuff, which is related to
changes of scale = energy levels.
5 - "... existence of 2 asymptotic, unexceedable and invariant
under dilation scales : lP: Planck length (minimal scale)
and L: cosmological length (maximal scale) ...".
I can understand a physical effective minimal scale such as the
Planck length at which space-time structure breaks down (like a
I don't understand a similarly fixed maximal scale, because it
seems to me that if our universe keeps on expanding (as is indicated
by observation) it will eventually be larger than any such fixed scale.
Also, the dilation generator of the conformal group does not,
as far as I know, have any inherent minimal or maximal scale,
the Planck scale minimum not being due to dilation stuff,
but due to high-energy evaporation of spacetime, which appears
to me to be a condensate related to interactions of ALL gauge boson
generators, not just the dilation one.
6 - "... Elementary fermion mass spectrum: recovered from
a cancellation effect between special scale-relativistic
corrections and radiative corrections. (However, except
concerning the muon mass, this is still a model, not a
totally constrained theory, because an unknown free parameter
remains in this generation mechanism). ...".
I like my model better because fermion mass ratios are in my model
completely fixed with no free paramaters.
7 - "... Top quark mass: predicted by the above mechanism to fall
just beyond the W/Z mass, at 150 ą 50 GeV
(experimental value: 174 ą 17 GeV). ...".
My fundamental 130 GeV t-quark mass calculation has less uncertainty
than the +/- 50 GeV quoted. Also, I consider the experimental results
to be consisent with a ground state of 130 GeV, and excitations
around 173 GeV and 225 GeV, as described in my CERN CDS preprint
8 - "... Value of the cosmological constant:
it is predicted to be L = 1.36 10^-56 cm-2,
under the assumption that the fractal-nonfractal transition
for the vacuum energy density occurs at [about] 70 MeV,
which is the scale of the classical radius of the electron
(or, equivalently, at the 6 flavor QCD scale):
this value could arise from the quark-hadron transition
in the primeval universe. ...".
I don't see how 70 MeV is a fractal-nonfractal transition.
In fact, if the scale relativity has a minimal Planck length
and a maximal universe-size length, then it seems to me that
everything between those lengths/energies (including 70 MeV) would
be in a fractal region.
Also, 70 MeV does not seem to me to be any natural energy level
with respect to particle physics:
The electron mass is 0.5 MeV.
The quark-antiquark pion has mass about 140 MeV.
The 3-quark proton has mass about 938 MeV.
The Higgs vacuum expectation value is about 245 GeV.
I prefer my model's treatment, with dark matter and dark energy,
somewhat similar to Jack's way of looking at it.