# [time 660] Re: [time 659] Cylindrical But Locally Lorentzian Universes

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Fri, 3 Sep 1999 08:07:05 +0300 (EET DST)

Hi All, My friend Paul Hanna pointed me to this page. It is very similar
to what he is working on! http://www.seanet.com/~ksbrown/kmath422.htm
Onward, Stephen Cylindrical But Locally Lorentzian Universes

Cylindrical But Locally Lorentzian Universes

A three-dimensional space that is everywhere locally Eulcidean and
yet cylindrical in all directions can be constructed by embedding the
three spatial dimensions in a six-dimensional space according to the
parameterization

x1 = R1 cos x/R1 x2 = R1 sin x/R1
x3 = R2 cos y/R2 x4 = R2 sin y/R2
x5 = R3 cos z/R3 x6 = R3 sin z/R3

so the spatial Euclidean line element is

dx1^2 + dx2^2 + dx3^2 + dx4^2 + dx5^2 + dx6^2 = dx^2 + dy^2 + dz^2

giving a Euclidean spatial metric in a closed three-space with total
volume (2*pi)^3*R1*R2*R3. It's been suggested (by Klaus Kassner,
among others) that we can subtract this from an ordinary temporal
component to give an everywhere-locally-Lorentzian spacetime that
is cylindrical in the three spatial directions, i.e.,

ds^2 = c^2 dt^2 - (dx^2 + dy^2 + dz^2)

However, this "subtraction" doesn't seem well motivated. One way
of providing a motivation, and in the process making the universe
cylindrical in ALL directions, temporal as well as spatial, would
be to embed the entire 4D spacetime into a space of 8 dimensions,
2 of which are purely imaginary, like this

x1 = R1 cos x/R1 x2 = R1 sin x/R1
x3 = R2 cos y/R2 x4 = R2 sin y/R2
x5 = R3 cos z/R3 x6 = R3 sin z/R3
x7 = i R4 cos t/R4 x8 = i R4 sin t/R4

leading (again) to a locally Lorentzian 4D metric

(ds)^2 = (dx)^2 + (dy)^2 + (dz)^2 - (dt)^2

but now *all four* of the dimensions x,y,z,t are periodic. So
here we have an everywhere-locally-Lorentzian manifold that is
closed and unbounded in every spatial and temporal direction.
(Obviously this manifold contains closed time-like worldlines.)

This reminds me a bit of Stephen Hawking's recent attempts to
describe a universe that is finite but unbounded in time as well
as space. He too invokes an imaginary time dimension, but aside
from that I don't think his model is related to the locally-flat
cosmology described here. Anyway, the causal structure of such
a universe is interesting.
*********

[MP] In TGD closed spacetime surfaces are possible absolute minima
of Kaehler action. They have necessarily finite extension in both
time and spatial directions. One can imagine of constructing them
by taking two finite pieces of Minkowski space in M^4_+xCP_2, slightly
deforming them in CP_2 directions, gluing along their boundaries,
smoothening resulting edges, and requiring that absolute minimum
of Kaehler action is in question. These surfaces decompose
to regions with Minkowskian signature and Euclidian signature
or have global Euclidian signature. They are not locally vacua:
globally they are: classical energies, momenta, etc are
of opposite sign on the two sheets. These surfaces are analogous
to vacuum bubbles appearing in perturbative quantum field
theories and contributing nothing to S-matrix elements.

Physical considerations pose no restrictions on the density of these
purely 'mindlike' objects (using the terminology of TGD inspired
theory of consciousness): they represent pure thought.
I have told about amusing structural isomorphism between the properties of
asymptotic selves (selves for which all subsystems have vanishing p-adic
entanglement) and between the closed Euclidian 4-surfaces.
I would identify them as liberated Buddhas! (Ontogeny recapitules
phylogeny.)
*****************

We might imagine that a flat, closed, unbounded universe of this
type would tend to collapse if it contained any matter, unless a
non-zero cosmological constant is assumed. On the other hand, I'm
not sure what "collapse" would mean in this context. It might
mean that the R parameters would shrink, but R is not a dynamical
parameter of the model. The 4D field equations operate only on
x,y,z,t. Also, any "change" in R would imply some meta-time
parameter T, so that all the R coefficients in the embedding
formulas would actually be functions R(T).

It seems that the flatness of the 4-space is independent of the
value of R(T), and if the field equations are satisfied for one
value of R they would be satisfied for any value of R.

But I'm not sure how the meta-time T would relate to the internal time
t for a given observer. It might require some "meta field
equations" to relate T to the internal parameters x,y,z,t.
Possibly these meta-equations would allow (require?) the value
of R to be "increasing" versus T, and therefore indirectly
versus our internal time t = f(T), in order to achieve stability.

[MP] In critical Robertson-Walker cosmology one has somewhat
similar situation. 3-space is Euclidian and could be compactified to
3-torus. Allowing R to on t one obtains curved 4-space as is clear from
the fact that mass density (G^tt component of Einstein tensor) is
nonvanishing. In special case one would have the

ds^2= t^2(dt^2 -dx^2-dy^2-dz^2)

I checked from general formula for mass density in RW cosmology
that mass density goes like 1/t^4, that is scales. This is to be expected
since line element is Weyl equivalent to flat metric. It seems
that this is nothing but the radiation dominated critical cosmology which
is scale invariant (massless particles dominate).
In this case R indeed increases with t linearly.

**********************

On the general method of embedding a locally flat n-dimensional
space in a flat 2n-dimensional space, I wonder if every orthogonal
basis in the n-space maps to an orthogonal basis in the 2n-space
according to a set of formulas formally the same as those shown
above. If not, is there a more general mapping that applies to
all bases?

By the way, the above totally-cylindrical spacetime has a natural
expression in terms of "octonion space", i.e., the Cayley algebra
whose elements are two ordered quaterions

x1 = i R1 cos x/R1 x2 = i R1 sin x/R1
x3 = j R2 cos y/R2 x4 = j R2 sin y/R2
x5 = k R3 cos z/R3 x6 = k R3 sin z/R3
x7 = R4 cos t/R4 x8 = R4 sin t/R4

Thus each point (x,y,z,t) in 4D spacetime represents two quaterions

q1 = x1 + x3 + x5 + x7

q2 = x2 + x4 + x6 + x8

To determine the absolute distances in this 8D space we again consider
the eight coordinate differentials, exemplified by

d x1 = i R1 (-sin(x/R1)) (1/R1) (dx)

(using the rule for total differentials) so the squared differentials
are exemplified by

(d x1)^2 = - sin^2(x/R1) (dx)^2

Adding up the eight squared differentials to give the square of the
absolute differential interval leads again to the locally Lorentzian
4D metric

(ds)^2 = (dt)^2 - (dx)^2 - (dy)^2 - (dz)^2

[MP] It is often believed that quaternions and octonions are inherently
Euclidian objects. This is actually not true as I discovered for year or
two ago. The point is that one can define inner product as
real part Re(z1z2) of z1z2: the resulting inner product is standard
Minkowskian inner product. If one defines inner product as Re(z1^*z2)
one obtains the usual Euclidian inner product.

Best,
MP

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