# Space – Time as a Complete Measure Manifold of Dimension-4

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#### Space – Time as a Complete Measure Manifold of Dimension-4

S. C. P. Halakatti* Department of Mathematics, Karnatak University Dharwad,

Karnataka India,

Research scholar, Department of Mathematics, Karnatak University Dharwad,

Karnataka India,

Abstract: – In this paper we introduce the structure of space- time as a complete measure manifold of dimension-4 endowed with a partial order relation defined by causal connections. This new approach to space-time structure gives us insight into the dark reason of a space-time that is still measurable.

Key words: Causal connectedness, complete measure manifold, measure manifold, sequential connectedness.

Subject Classification: 57N13, 58C35.

1. INTRODUCTION

Spacetime is the 4-dimensional manifold in which all physical events take place. An event is a point in space- time specified by its space and time co-ordinates. In physics space- time is a mathematical model that combines space and time into a single manifold called Minkowski space time. In cosmology the concept of space-time combines space and time to a single abstract universe. The common practice to study Minkowski spacetime is by selecting Lorentz metric or Minkowski metric that measures the interval between two events in space-time that is, ds2=dx2+dy2+dz2+(icdt)2. The interval ds2 may be classified into three different types, time-like (ds2<0), light like (ds2=0) and space like (ds2>0).

For some physical applications, a space-time continuum is mathematically a 4-dimensional smooth connected Lorentzian manifold (M, g), g is a Lorentz metric with signature (3, 1). The causal structure of Lorentzian manifold describes the causal relation between the events in the manifold. Hawking, S. W. Ellis, G.F.R(1973)[4], Penrose R. (1972)[7] have studied causal structure in terms of causal future, causal past, chronological future , chronological past.

The major distinction in the study of physics is the difference between local and global structures, measurement in physics are done in the local neighborhood of the events in the space- time, leading to the study of local structure of space-time in general relativity. The study of global space- time structure is vital in cosmological problems.

Generally, over lapping coordinate charts covers a manifold. The intersection of two charts represents a region of spacetime in which two observers can measure physical quantities and compare their results. The concepts of co- ordinate charts as local observer who perform their measurements to collect and compare their results in the

non empty intersection of charts locally, is vital to our mathematical model. The concept of connectedness serves this purpose. Without connectedness this would not be possible.

In this paper we study the concept of causal connected 1-a.e., sequentially connected 1-a.e. and maximally connected 1-a.e., on complete measure manifold of dimension-4 introduced by S. C. P. Halakatti ([9], [10], [11], [12], [13],[14]). We have proved some results on space-time manifold (M4,1,1,1). The significance of these results are, sequentially connected 1-

a.e. is an invariant property under measurable homeomorphism and measure invariant function F. Also, we have developed the concept of maximal connectedness on (M4,1,1,1) using sequential connected 1-a.e. property. It is interesting to see that if i is sequentially connected 1-a.e. to j and j is sequentially connected 1-

a.e. to l, under the composition of two measurable homeomorphic and measure invariant function GF on

i j l Ak(M4) then (M4,1,1,1) is maximally connected.

The following concepts are introduced and developed by S. C. P. Halakatti to generate a causally connected network manifold, whose applications are in the field of neural network, brain structure, in the study of large scale structures and engineering science.

2. PRELIMINARIES Some basic definitions referred are as follows

Definition 2.1: Convergence point wise almost everywhere on (M, 1, 1, 1)

Let fn f point wise almost everywhere in (n, , , ) and if any measure manifold (M, 1, 1, 1) is measurable homeomorphic to (n, , , ) then for every x An (n, , , ) 1 x = p 1 (An ) denoted by,

S = 1(An) (U, ) (M, 1, 1, 1) and (fn ) f

• point wise a.e. in S (U, ) (M, 1, 1, 1) such that,

S = 1 (An ) ={p=1 x (M, 1, 1, 1): |(fn )(p) – (f

• )(p) | < , n N} on the chart (U, ) satisfying the following conditions:

i 1(S) > 0, if | (fn )(p) – (f )(p) | < , n N,

ii 1(S) = 0, if | (fn )(p) – (f )(p) | , n N, that is, 1(S) = 0 as n .

Definition 2.2: Convergence 1-a.e. on (M, 1, 1, 1)

1 n

a.e

Let f

real valued measurable functions converging to f and g pointwise 1 a.e. on (U, ) and (V, ) belonging to the atlas respectively. The ordered pairs ({fn }, f )

and ({gn }, g ) induce two Borel subsets S U,

n f in An ( , , , ) and if any measure

manifold

(M, 1, 1, 1) is measurable homeomorphic to (n, , ,

) then for every An (n, , , ) S (U,)

A Ak(M) and R (V, ) Ak(M) satisfying the following condition:

S = {p M, 1, 1, 1): | (fn ) (p) – (f ) (p) | <

(M, , , ) such that, (f

) 1a.e

(f ), x on S

, n N} on the chart (U, ) for which 1(S) > 0.

1 1 1 n

(U,) (M, 1, 1, 1) and

S = { p (M, 1, 1, 1) : | (fn )(p) – (f )(p) | < , n

N} with,

i 1(S) > 0, if | (fn )(p) – (f )(p) | < , n N,

ii 1 (S) = 0, if | (fn )(p) – (f )(p) | , n N, that is, 1(S) = 0.

Definition 2.3: Complete Measure Manifold

If (M, 1, 1, 1) is a measure manifold of dimension n and suppose that for every measure chart (U, ) (M, 1, 1,

), ( U )=0 and every V (U, ), if ( V ) = 0, then

R = {q M, 1, 1, 1): | (gn ) (q) – (g ) (q) | <

, n N} on the chart (V, ) for which 1(R) > 0.

Note: We denote the Borel subsets 1(An ) = S U, M,1,1,1) and 1(Bn ) = R (V, ) M,

1, 1, 1).

Definition 2.6: Interconnected 1 a.e on complete measure manifold

The Borel subset S U, A M, 1, 1, 1) is interconnected to the Borel subset R (V, ) A M, 1,

1, 1) 1 a.e. if a C- map

: [0, 1] S R Ak(M) such that

1 1 1

(M, 1, 1, 1) is called as a complete measure manifold.

Let (M, 1, 1, 1) be a complete measure manifold of dimension n which is measurable homeomorphic to a measure space (n, , , ). Let

{f }, {g } be measurable real valued functions converging

(0) = p S ,

(1) = q S , such that 1(S) > 0 and 1(R) > 0. That is, p is interconnected 1 a.e. to q in S R A Ak(M).

That is, interconnectedness 1 a.e. is between two charts in the same atlas Ak(M).

n n

to f and g respectively in (n, , , ).

Since is measurable homeomorphism from (M,

1, 1, 1) to (n, , , ) for every {fn} and {gn} on (n,

, , ) there exist corresponding measurable real valued functions {fn } and {gn } converging to f and g on (M, 1, 1, 1).

The ordered pair ({fn }, f ) induces a Borel subset S (U, ) M, 1, 1, 1) satisfying the follwing condition:

S = {p M, 1, 1, 1): | (fn ) (p) – (f ) (p) | < , n

N} on the chart (U, ) for which 1(S) > 0.

Definition 2.4: Locally path connected 1 a.e. on complete measure manifold

The Borel subset S is locally path connected 1 a.e. if a

C- map

: [0, 1] S (U, ) such that

(0) = p S ,

(1) = q S, such that 1(S) > 0.

That is, p is locally path connected 1 a.e. to q in S

(U, A Ak(M).

If 1(S) =0 and 1(R) =0 then a path between p and q.

Definition 2.7:

If 1(S)=0 where {fn } f in (U,) and 1(R)=0 where {gn } g in (V, ), then S is called as dark region in the chart (U, ) and R is called as dark region in the chart (V, ) belonging to the same atlas A in Ak(M).

Let (n, , , ) be a measure space and {fn},

{gn}and {hn}are sequences of measurable functions on (n,,,) converging to f, g and h point wise almost everywhere on (n, , , ). The ordered pairs ({fn},f), ({gn}, g) and ({hn}, h) induce the following Borel subsets An , Bn and Cn .

We define Borel subsets

An = { x (n, , , ) : | fn(x) – f(x) | < } , n N, where

(i) (An ) > 0, if |fn(x) – f(x) | < , n N,

(ii) (An ) = 0, if |fn(x) – f(x) | , n N, that is, (An )

= 0 as n.

Similarly,

That is, locally path connectedness 1

two points in the same chart A Ak(M).

a.e. is between

(i)Bn = { y (n, , , ) : | gn(y) – g(y) | < }, n N, where

If 1(S) = 0, then there does not exist a path between p and q.

Definition 2.5:

If 1(S) = 0 where (fn ) f , then S (U, ) (M,

1, 1, 1) is called as a dark region in the chart (U, ).

Let (M, 1, 1, 1) be a complete measure manifold on which {fn } and {gn } are sequence of

(Bn ) > 0, if | gn(y) – g(y) | < , n N,

(ii) (Bn ) = 0, if | gn(y) – g(y) | , n N, that is, (Bn )

= 0 as n and

Cn = { z (n, , , ) : | hn(z) – h(z) | < }, n N, where

(i) (Cn ) > 0, if | hn(z) – h(z) | < , n N,

(ii) (Cn ) = 0, if | hn(z) – h(z) | , n N, that is,

(Cn ) = 0 as n.

If (M, 1, 1, 1) is a complete measure manifold that is measurable homeomorphic and measure invariant to (n, , , ). Then a measurable homeomorphism and measure invariant transformation

: (M, 1, 1, 1) (n, , , ), such that, {fn }, {gn } and {hn } are sequences of real valued measurable functions converging to f , g and h point wise 1-

a.e. on (U, ), (V, ) and (W, ) belonging to the atlases

i, j, l respectively. Also, for every induced Borel subsets An , Bn and Cn in (n, , , ), the corresponding induced Borel subsets namely

S (U, i Ak(M), R (V, ) j Ak(M) and

Q (W, ) l Ak(M) on (M, 1, 1, 1). Now, we define S, R and Q as follows:

S = {p M, 1, 1, 1): | (fn ) (p) – (f )(p) | < , n

N} on the chart (U, i Ak(M), for which

1(S) > 0,

R = {q M, 1, 1, 1): | (gn ) (q) – (g ) (q) | <

, n N} on the chart (V, ) j Ak(M), for which

1(R) > 0 and

Q = {r M, 1, 1, 1): | (hn ) (r) – (h )(r) | <

, n N} on the chart (W, ) l Ak(M), for which

1(Q) > 0.

Note: We denote the Borel subsets 1(An ) = S U, i Ak(M), 1(Bn )= R(V, ) jAk(M) and 1 (Cn )=Q (W, ) l Ak(M).

Definition 2.8: Maximal connected 1-a.e on complete measure manifold

Let (M, 1, 1, 1) be a complete measure manifold and let Ai, j and l Ak(M) be atlases on (M, 1, 1, 1). Let S, R and Q be Borel subsets of i, j and l. Then, we say that Ak(M) M, 1, 1, 1) is maximally connected if a map :[0,1] S R Q i j l Ak(M) such that, (0) = p S (U, ) i Ak(M) for which 1(S) > 0,

(1) = q R (V, ) j Ak(M) for which 1(R) > 0 and

3. DIFFERENT VERSIONS OF CONNECTEDNESS ON SPACE-TIME MANIFOLD OF DIMENSION – 4

Let (M4, 1, 1, 1) be a complete measure manifold of dimension- 4.

Let pi = (xi, yi, zi, ti) and pj = (xj, yj, zj, tj) are events in (M4,1,1,1), where (x, y, z) are space coordinates and t is the time co-ordinate.

Let fn and f be measurable functions on the chart (U, ) of a complete measure manifold (M4, 1, 1, 1) such that

fn : U converges to f: U 1-a.e., pi (U, ) (M4,1,1,1) then, the ordered pair ({fn}, f) induces a set S of all events pi, that is S ={pi (Ui, i) (M4,1,1,1) :

|fn(pi) – f(pi)| < , i=1,2.,n}with the following conditions: (i) 1(S) > 0 if |fn(pi) – f(pi) | < n N , i=1,2

(ii) 1(S) = 0 if |fn(pi) – f(pi)| > n N ,i=1,2..

We introduce Causal Connectedness 1-a.e., sequentially connected 1-a.e. and maximally connectedness 1-a.e. on (M4,1,1,1).

Definition 3.1: Causally connected 1-a.e.

Let (U, ) (M4, 1,1,1) and let pi and pj , i < j, i, j=1,2,,n are events in (U, ) then we say that (U, ) is causally connected 1-a.e. if there exists a C map : [0,1]S U M4 between pi and pj :

(0) = pi S U (M4, 1,1,1), 1(S) > 0,

(1) = pj S U (M4,1,1,1), 1(S) > 0, i, j I, where pi < pj for ti < tj , ij I. Then a relation < is called as a causal connection 1-a.e between pi and pj. pi is called causally connected 1-a.e. to pj in S (U,) (M4,1,1,1), if 1(U) > 0, 1(S) > 0.

Note:

1. If 1(S) > 0 then the events are separated by time-like interval.

2. If 1(U) = 0, then the events are not causally connected in (U, ) and they are separated by space-like interval.

2 k Then S can be recognized as dark region of (U, ).

(1) = r Q (W, ) l A (M) for which 1(Q) > 0. That is, for each p (U, ) i is path connected to each q

(V, ) j, for i j Ak(M), 1(i j) > 0 for each q (V, ) is path connected to each r (W, )

Definition 3.2: Sequentially connected 1-a.e.

Let (M4, 1,1,1) be a complete measure manifold and let

k j k l be a atlas in Ak(M4), where (U , ) then we say

A (M) and for , j l A (M), 1(j l) > 0. Then, if

for each p (U, ) i is path connected to each r (W,

) Ak(M) and for Ak(M) , ( ) > 0

i i

is sequentially connected

1-a.e. if a C-map : [0, 1] Ak(M) such that,

l

then (

i l 1 i l

k

(0) = p1 S (U1, 1) (M4, 1, 1, 1), 1(S) > 0,

i j l) A (M) M, 1, 1, 1) is maximally

path connected if ( ) > 0 on complete

( k ) = p S (U , ) (M4, , , ), k

(0,

measure manifold.

1 i j l

2n i i i

1), k < 2n

1 1 1 2n

If 1(S) = 0 and 1(R) = 0 then there does not exist a path between p S and q R.

Definition 2.9:

If 1(S) = 0 where {fn } f in (U, ) and 1 (R) = 0 where

{gn } g in (V, ) and 1 (Q) = 0 where {hn }

h in (W, ), then S is called as dark region in the chart

i I, 1(Ui) > 0 and 1(S) > 0.

(1) = pn S (Un, n) (M4, , 1, 1), 1(S) > 0 satisfying a causal relation < such that p1 < < pi <.< pn for t1 < < ti <.< tn , i < j.

Then the relation < is called a sequentially connected 1-

a.e. on S

=

(U, ) (M4,1,1,1), if 1(S)> 0, 1(U)> 0, where U

(U, ) i, R is called as dark region in the chart (V, )

and Q is called as dark region in the chart (W, )

n i=1

(Ui, i).

j l

in Ak(M).

Let (M4,1,1,1) be a complete measure manifold and let i, j and l Ak(M4) and {fn}, {gn}, {hn} are measurable functions converging to real valued functions f, g , h on i, j, l. Then the ordered pairs ({fn}, f ), ({gn},g), ({hn},h ) induces the following sets S1, S2, S3 on

i, j, l respectively, where i is sequentially connected to j and j is sequentially connected to l by the functions F and G respectively.

S1 = {pi (Ui, i) i Ak(M4): |fn(pi) – f(pi) | < n N

, i=1,n},

S2 = {qi (Vi, i) j Ak(M4): |gn(qi) – g(qi) | < n

N , i=1,n},

where gn = fn F-1 on j and

S3 = {ri (Wi, i) l Ak(M4): |hn(ri) – h(ri) | < n N

, i=1,n},

where hn = (gn fn ) F-1 on l .

homeomorphism and measure invariant map. If i is sequentially connected 1-a.e., then j is also sequentially connected 1-a.e.in Ak(M4).

Proof: Let (M4, 1, 1, 1) be a complete measure manifold and i and j Ak(M4). Let F: ij be a C measurable homeomorphism and measure invariant map. We show that if i is sequentially connected

1-a.e. then j is also sequentially connected 1-a.e.

Let i be sequentially connected 1-a.e.on S (Ui, i) i

Ak(M4).

By definition of sequentially connectedness, the induced set S is defined as, S = { pi (Ui, i) i Ak(M4) : |fn (pi)-f(pi)| < , i=1,.,n} satisfying, 1(S) > 0 in (Ui, i)

i and C-map :[0 ,1]i Ak(M4), such that,

(0) = p1 (U1, 1) i Ak(M4) , 1(S) > 0 , 1(U1) > 0,

( ) = p (U , ) Ak(M4), ( ) (0,1), k < 2n,

Since

and

are sequentially connected by

2 i i i i

2

i j 1< i < n, 1(S) > 0 and 1(Ui) > 0,

measurable homeomorphism and measure invariant map F and j and l are sequentially connected by measurable

(1) = pn S (Un, n) i Ak(M4) , 1(S) > 0 and

(Un) > 0 satisfying a Causal relation < in S (U , )

homeomorphism and measure invariant map G, if i is 1 k 4 1 1

sequentially connected to l by measurable

i A (M ) such that, p1 << pi << pn for t1 < < ti <

4 < tn , i < j , where 1(S) > 0, 1(Ui)>0 where i= (

homeomorphism and measure invariant map GF then (M ,

U , ).

=1

1, 1 , 1) is maximally connected. Therefore, one can i i

develop the following definition.

Definition 3.3: Maximal Connectedness -a.e. on (M4,

Since F: ij is a measurable homeomorphism and measure invariant map, for every (Ui, i) i Ak(M4) the corresponding F(Uj) = Vj j Ak(M4), where (Vj, j)

1,1,1)

1 is a chart in the atlas j Ak(M4).

Also, for every S in i a corresponding set F(S) in j

Let (M4, 1,1 ,1) be a complete measure manifold and i,

j, l Ak(M4) are mutually sequentially connected by measurable homeomorphism and measure invariant maps F and G and G F respectively, then (M4, 1,1 ,1) is maximally connected 1-a.e.

4. SOME RESULTS ON SPACE-TIME MANIFOLD

It is necessary to note that if any atlas is

defined as,

F(S) = {qj (Vj, j) j Ak(M4) 😐 (qj)- (qj) | < , j=1,.,n} satisfying, 1(F(S)) > 0 in (Vj, j) j and C-map F :[0 ,1]j Ak(M4), such that,

F(0) = q1 F S (V1, 1) j Ak(M4) , 1(F(S)) > 0 ,

1(V1) > 0,

F( ) = q F S (V , ) Ak(M4) , (F(S)) > 0 ,

i

sequentially connected 1-a.e. then we show that any other

2 i

1(Vi) > 0,

i i j 1

atlas j, ij N is also sequentially connected 1-a.e. that is, sequentially connected 1-a.e. is an invariant property under measurable homeomorphism and measure invariant function F.

F(1) = qn F S (Vn, n) j Ak(M4) , 1(F(S)) > 0,

1(Vn) > 0

such that, q1 < ….< qi < .. < qn for t1 < .< ti < . < tn ,

i < j,

=

where 1(F1(S)) > 0, 1(Vi) > 0 and 1(j) > 0 where j

Theorem 4.1:

Let (M4, , , ) be a complete measure-manifold, where

=1

(Vi,i).

1 1 1

i, j Ak(M4). Let F: i j be a C measurable

i j

(Ui, i)

. pi

F

Fo

(Vj, j)

. qj

0 1

Therefore, the events in j are sequentially connected 1-

a.e. in

(M4, 1, 1, 1) if 1(S) > 0, 1(Vi) > 0 and 1(j) > 0. Therefore, if i is sequentially connected 1-a.e. in S (Ui, i) i Ak(M4) then j is also sequentially connected

#### Fig 1

1-a.e. in F(S) (V, ) j Ak(M4) under measurable homeomorphism and measure invariant transformation.

In other words, sequentially connectedness 1-a.e. on (M4, 1, 1, 1) is invariant under measurable homeomorphism and measure invariant function F.

Theorem 4.2:

Let (M4, 1, 1, 1) be a complete measure manifold that is maximally path connected 1-a.e. where i, j, l Ak(M4). If i is sequentially connected 1-a.e to j by measurable homeomorphism and measure invariant map F and if j is sequentially connected 1-a.e to l by measurable homeomorphism and measure invariant map G then i is sequentially connected 1-a.e. to l by measurable homeomorphism and measure invariant map G F, i, j, l N, ti < tj < tl.

Proof: Let (M4, 1, 1, 1) be a complete measure manifold which is maximally path connected 1-a.e. and i, j Ak(M4). Let F: i j and G: j l be C measurable homeomorphism and measure invariant maps such that i is sequentially connected 1-a.e. to j and j is sequentially connected 1-a.e. to l. Now, to show that i is sequentially connected 1-a.e. to l a C measurable homeomorphism and measure invariant map G F: i

l, since F and G are measurable homeomorphism and measure invariant maps therefore composite function G F is measurable homeomorphism and measure invariant.

According to definition 3.2, i is sequentially path connected 1-a.e to j and j is sequentially path connected

1-a.e to l in Ak(M4), then we show that i is sequentially path connected 1-a.e to l in Ak(M4):

Let S1 = {pi (Ui, i) i Ak(M4): |fn(pi) – f(pi) | < n

N, i=1,n},

S2 = {qi (Vi, i) j Ak(M4): |gn(qi) – g(qi) | < n

N , i=1,n},

where gn = fn F-1 on j and

S3 = {ri (Wi, i) l Ak(M4): |hn(ri) – h(ri) | < n N

, i=1,n}, where hn = (gnfn) F-1 on l be the induced sets on i, j and l respectively. Since i is sequentially connected 1-a.e. to j, according to definition 3.2 and theorem 4.1, sequentially connectedness is invariant under measurable homeomorphism and measure invariant function. Since, for every C map : [0,1] i Ak(M4) that connects all the events in sequential way p1 <.< pi

<.pn for t1 <. ti….tn, i < j where 1(S1) > 0, 1(Ui) > 0,

F : [0,1] j such that, it connects all the events in

j in sequential way q1 <.< qi <.qn for t1 <. ti….tn ,

i< j where 1(F(S1)) > 0, 1(Vi) > 0, 1(j) > 0.

i

(Ui, i)

. pi

F

GoF

j

(Vi, i)

. qj

GoFo

(Wi, i)

.ri

G

l

0 1

Fig. 2

Similarly, if for every F : [0,1] j G F :

[0,1] l where

G F connects all the events sequentially in l, such that,

G F (0) = r1 S (W1, 1) l Ak(M4), 1(S3) > 0 ,

1(W1) > 0,

G F ( k ) = r S (W , ) Ak(M4), ( k ) (0,1)

invariant function G F. This means if sequentially connectedness is invariant under measurable homeomorphism and measure invariant transformation on Ak(M4), i, j and l, i, j, l I then M4 is maximally connected.

A maximally connectedness property on (M4, 1, 1, 1)

2n i

i i l 2n

defines a causa structure on space-time of dimension 4.

, k < 2n,1< i <n, 1(S3) > 0 and 1(Wi) > 0,

G F (1) = rn S (Wn, n) l Ak(M4), 1(S3) > 0 and 1(Wn) > 0 satisfying a Causal relation < in S3 (W1, 1) l Ak(M4), such that

r1 <..< ri <. < rn for t1 < t2<. < ti < < tn , i < j ,

Definition 4.3:

A complete measure manifold (M4, 1, 1, 1) endowed with a causal structure induces a partial ordered relation

that generates a network manifold of dimension 4.

i=1

where 1(S3) > 0, 1(Wi) > 0 where l= n ( Wi, i).

Hence, i is sequentially connected 1-a.e to l , i, j, l I, ti <tj < tl.

Therefore, if i is sequentially connected 1-a.e to j and if j is sequentially connected 1-a.e to l, we have shown that i is sequentially connected 1-a.e. to l under the composition of measurable homeomorphism and measure

5 CONCLUSION

A measure manifold (M4, 1, 1, 1) admitting a partial ordered relation on it denoted by (M4, 1, 1, 1, ) generates a network manifold of dimension-4. This approach provides a new vision to the space-time as a 4- dimensional complete measure manifold. The advantage of

such approach is to generate a causally connected network manifold, whose applications are in the field of neural network, brain structure, in the study of large scale structures and engineering science.

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