^{1}

^{*}

^{2}

A certain class K of GR homogeneous spacetimes is considered. For each pair
*E*,
of spacetimes from K,
where conformal transformation
*g* is from
. Each
*E* (being
or its double cover, as a manifold) is interpreted as related to an observer in Segal’s universal cosmos. The definition of separation
*d* between
*E* and
is based on the integration of the conformal factor of the transformation
*g*. The integration is carried out separately over each region where the conformal factor is no less than 1 (or no greater than 1). Certain properties of
are proven; examples are considered; and possible directions of further research are indicated.

The first author has been interested in GR (“GR” is for General Relativity) research for quite a while and he concentrated on a few most symmetric spacetimes ( [

Recall the Lie group

where

where S is the diagonal matrix

where a matrix g from G is determined by its

In

The Lorentzian inner product on

For what follows, it is instrumental to introduce a certain bi-invariant Riemannian inner product on

Notice that, as a group,

Here the variable t is along

It is well-known ( [

Also, it is easily verifiable that for the Riemannian metric

on

It makes sense to mention how a suitable version of the Einstein static universe, _{uc}-action on

More precisely, we deal with two classes of spacetimes:

The separation (or distance)

As mentioned, the totality of all isometries in

G/K. Namely, each element (or coset) x of G/K is specified by an element g from G:_{1}K. For such a pair

there exists such k from K, that_{0}) at z. This inner product has been introduced in our Section 1. To define spacetime E corresponding to a coset x = gK, it is enough to specify the inner product

_{0}. Namely, given vectors _{0}) at z, the inner product

The everywhere positive function

where

Here the right hand side of (2.3L) is calculated in

where the positive definite inner product in the right hand side of (2.3R) has been introduced in our Section 1.

Let us show that, given a coset x in G/K, (2.3L) correctly defines a Lorentzian metric on

Scholium 2.1. The inner product (2.3L) (respectively, the inner product (2.3R)) is independent of the choice of g which represents a coset x.

Proof. If x is represented as g_{1}K, then

where the right hand side of (2.4) is calculated in

where the right hand side is calculated at

However,

Let us notice (see [

Remark 2.2. We have thus defined the class

Given a (1.3)-transformation g of (Lorentzian)

here

where

To further deal with (2.8), we now proceed with more technicalities. Clearly,

Examples of integrals

which follows from (2.8) because in this case

Scholium 2.3. Given the (1.3)-transformation g and isometries

where

Proof. To prove (2.10), we will now show that each of the four numbers (a, b, c, and d) remain the same when we switch from

Similarly,

Let us now use the variable

integrand since

Now, if two cosets are represented as

where

Corollary 2.4. In the above settings,

A word of caution: we use the term distance but we are not sure that the corresponding triangle inequality holds (even locally) for (2.11). However, we prove (below) that (2.11) is symmetric:

for arbitrary f from G (where we have in mind the canonical action of G in G/K).

As regards G-invariance, one can think of a possible relation of our definition (2.11) to the canonical inner product in the symmetric space G/K. This we do not discuss here.

Scholium 2.5. The distance (2.11) is symmetric:

Proof. As justified by our Corollary 2.4, assume that

where

For

Examples of integrals

Alexander Levichev,Andrey Palyanov, (2015) On Separation between Metric Observers in Segal’s Compact Cosmos. Journal of Modern Physics,06,2040-2049. doi: 10.4236/jmp.2015.614210

The following presentation for E^{(2)}, the 2-cover of^{6} = E^{2} Å E^{4} of two Euclidean spaces: E^{2} with rectangular coordinates^{4} with rectangular coordinates^{(2)} is a 6-tuple

and

Clearly, E^{(2)} is S^{1} × S^{3}, topologically. The earlier introduced

The covering map from E^{(2)} onto

Given a matrix z in^{6}, it is helpful to consider a pseudo-Euclidean metric

and an Euclidean metric

It is known (see [^{(2)} = S^{1} ´ S^{3} coincides with metric (1.4) of our Section 1. Similarly, the restriction of (A5R) onto E^{(2)} coincides with metric (1.5).

This group consists of all (1.3)-transformations g of the form:

with

where^{(2)} and a matrix in E_{0}. The statement and the proof of the following theorem presume usage of rectangular coordinates in Euclidean E^{6}: see Appendix A.

Theorem B.1. The image ^{(2)} and the conformal factor

Proof. Notice that due to (A3) and (A4) from Appendix A, the formulas (B.2) correctly define the transformation on the level of E_{0} (when z and

in terms of differentials

Remark B.2. In the case considered, there is an alternative way to determine the conformal factor (B.3). It is as follows [

One can verify that (B.5), when applied in the (B.1)-case, results in (B.3).

It is of interest to determine all fixed points (that is, matrices

Scholium B.3. The totality of all fixed points of (B.1) is a pair of circles. One of the circles is given by equations

Proof. As it follows from (B.1), the totality of all fixed points is the solution set of

equality of two matrices. Comparison of first entries in second rows results in

Since g is not an identity transformation,

Our next goal is to prove that each fixed point (of a given (B.1)―transfor-mation) is an extreme point of the conformal coefficient

Scholium B.4. If

Proof. If

Corollary B.5. If

Corollary B.6. At the point of extremum for the conformal coefficient, either

Proof follows from the expression

Corollary B.7. An extreme value of

We start with the form

on the torus T = S^{1} ´ S^{3}, see our Theorem B.1. Now, T^{+} is for the part of T where^{−} is for the part of T where

where in both cases we have in mind the volume form which has been introduced on T in Section 2.

A word of caution: the function (C.1) is the inverse of the conformal coefficient (B.3). Nevertheless, the findings (which follow) of this Appendix C are relevant to the Appendix A content since k in (C.2), (C.3) can be any integer.

The majority of these Appendix C findings are due to V. V. Ivanov (Sobolev Institute of Mathematics, Novosibirsk, Russia).

Parameterize T as follows:

In terms of these parameters, (C.1) becomes

The integrals (C.2), (C.3) are reduced as follows:

where

Here we consider the rectangle

Our function (C.5) becomes

whereas

r being the distance between

Introduce an (acute) angle

Omitting a few more (straightforward) technicalities, we obtain

The upper limits

Let us conclude in terms of the following statements.

Theorem C.1. For k not equal −1, the integrals (C.2), (C.3) can be evaluated as follows:

For k = −1

Theorem C.2. For every integer k,

Theorem C.3. For a nonnegative k, each of the integrals (C.17) is a finite linear combination of integrals

Remark C.4. Each of the integrals (C.20) is an elementary one and it can be expressed as a polynomial in s and

Recall notations a, b, c, d of Section 2 (see the line prior to Formula (2.9)) for the integrals which are of our utmost interest.

Theorem C.5. The integrals a, b, c, d, are as follows: