**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Fri, 3 Sep 1999 08:07:05 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**WDEshleman@aol.com: "[time 661] Re: [time 653] Change of State"**Previous message:**Stephen P. King: "[time 659] Cylindrical But Locally Lorentzian Universes"**Next in thread:**Stephen P. King: "[time 666] Re: [time 660] Re: [time 659] Cylindrical But Locally Lorentzian Universes"

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

**Next message:**WDEshleman@aol.com: "[time 661] Re: [time 653] Change of State"**Previous message:**Stephen P. King: "[time 659] Cylindrical But Locally Lorentzian Universes"**Next in thread:**Stephen P. King: "[time 666] Re: [time 660] Re: [time 659] Cylindrical But Locally Lorentzian Universes"

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