Mathematical Philosophy Of Time In Minkowskian Space

Mathematical Philosophy Of Time In Minkowskian Space

Sudhakaran Gopalan

(7/M.Sc/Physics/1964,IIT Madras) Author of gr-qc/0106029 in xxx.lanl.gov

Former Professor Of Physics, Sree Narayana College, Kollam,Kerala,India Pin:691001

Abstract

Time is a monotonic strictly increasing single valued real parameter that exists in spacetime. Here we consider an observer in his rest frame belonging to the Minkowskian spacetime. The order of the sequence of events on his World line is strictly preserved in the sense that the order of the sequence of events remains invariant under Lorentz transformations in Minkowskian spacetime: because the world line of the observer is always time-like.

Keywords – Algebra on event field, proper time, real number field, time loop, proper world line, proper observer.

1. Introduction

   Time is awake when all things sleep. Time stands straight when all things fall. Time shuts in all and will not be shut. Is,was,and shall be are Time`s children. O Reasoning, be witness, be stable. [1]

   VYASA,the Mahabharata [ca.A.D 400]

   This universe has basic temporal structure. The fundamental nature of TIME in relation to human consciousness is evident as soon as we think that our judgements related to time and events in time appear themselves to be IN TIME.

   Our analysis concerning SPACE do not appear in any obvious sense to be IN SPACE. But SPACE seems to be appeared to us all of a piece, whereas TIME comes to us only BIT by BIT. The Past exists only in our memory and the Future is hidden from us. Only the Present is the physical reality

   experienced by us. Thus TIME is always a ONE-WAY membrane. We cannot go from Present to the Past; while one can perform backward and forward motion in SPACE.

   The free mobility in SPACE leads to the idea of transportable rigid rods. The absence of free mobility in TIME leads to the concept ONE-WAY membrane

   TIME is a monotonic strictly increasing single valued real parameter corresponding to a non – spatial dimension represented by a straight line in Minkowskian spacetime and the SPACE is three dimensional. Minkowski unified space and time to a single entity called spacetime which is absolute. Einstein used the concept of spacetime for constructing spacetime geometry so that physics becomes part and parcel of geometry in Minkowskian spacetime. Einstein introduced the concept square of the distance between twoevents ds2 = -dx2-dy2-dz2+dT2 [2]

   here ds is distance between two events P(x,y,z,T) and Q(x+dx,y+dy,z+dz,T+dT).If ds2 is greater than zero the separation between events is called time- like;if ds2= 0, the separation between events is called null-like leading to the concept of Light Cone Structure in Special Theory of Relativity([3] &[4]) and if ds2 is less than zero, the separation between events is called space-like. Time-like events are causally connected and also null-like events are causally connected; there is no causal connection between events separated by space-like interval. All real particles trace curves in spacetime. These curves are called time-like curves. Light rays travel along null curve in spacetime.Here we are concerned only with time-like curve so that the order of sequence of occurrence of events shall

   be the same for every observer under admissible co- ordinates transformations.

   The world view proposed by Minkowski is often termed as Minkowskian spacetime [5] or M-space. It is said to have a (3+1) description of spacetime. Here 3 represents the Three Dimensional Euclidean space and 1 the One Dimensional time.

   We introduce spacetime co-ordinates to order events. In M-space, the co-ordinates of an event can be represented by an ordered set of four real numbers, <x1*,x2*,x3*,x4*>. Here the numbers x1*, x2*, x3* and x4* are taken to be PURE real numbers. 1,2,3,4 are superscripts used to specify the co- ordinates. It is always convenient to consider a Lorentz frame with orthonormal basis vectors e1,e2,,e3,and e4 [1]. Relative to the origin of this frame the time-like worldline of a particle with real non-zero rest mass has a co-ordinates description

   S=x1*() e1+ x2*()e2+x3*() e3+x4*()e4 ………..(1)

   Here S is the position 4-vector of the particle on its time-like wordline and is the proper time parameter defined along the time-like worldline of the particle.

   M-Space = M={<x1*,x2*,x3*,x4*>xi* belongs to real number field}

   Here we shall have inner product of e ,e , e , and

   Here c is the speed of light in free space.c=3 lakhs km/sec.Then

   S2= -4 + c2t2=-4+(9×4)=32;

   so S = (32) lakhs km which is the distance between two events. If S2 is +ve, the spatio-temporal separation of events is time-like. If S2 is equal to zero, the separation is null-like and if S2 is less than zero, the separation between events is space-like. Since we consider only (3+1) description of space time co-ordinates in M-space we assume that (x1*,x2*,x3*) represents the rectangular Cartesian co- ordinate system to specify the spatial position vector of an event in M-space and x4* represents the temporal position of the event. Thus the spatio- temporal position is represented by

   < x1*, x2*, x3*,x4*>.

   We write the event =<x1*, x2*, x3*,x4*> and we may put x1*=x,x2*=y, x3*,=z,x4*=T

   The totality of events M={< x1*, x2*, x3*,x4*>} constitutes the Minkowskian spacetime (M-space). The three dimensional Euclidean space(contained in Minkowskian spacetime) is homogeneous and isotropic.

2. Homogeneity of Space

   Because of the homogeneity of space a physical phenomenon is always the same, if physical conditions are the same,wherever it is observed: the same physical experiment performed in California gives the same result if performed in New Delhi

   e4as follows:

   1 2 3

   under the same physical conditions. The physical equivalence of different points of space- the

   homogeneity of space- demands certain restrictions

   e1.e1 = e2.e2 = e3.e3 = -1 ande4.e4 = +1. ei.ej=0 if ij

   ei.ejare the covariant components of the metric tensor of spacetime. This tensor determines the nature of geometry of spacetime.Here the spacetime is taken to have signature -2.

   Now taking the inner product of S in equation (1) we write

   S.S=S2= -1{[x1*]2 + [x2*]2 + [x3*]2} + [x4*]2= -D2+[x4*]2

   D is the spatial separation between the two events and [x4*] is temporal separation between the same two events. The minus sign associated with D2 and the plus sign associated with (X4*)2 show that the space and time are distinct physical entities even though they are interdependent to form SPACETIME which is absolute. For illustration let D=2 lakhs km and time (t) separation between the two events be 2 sec.Then in conventional units we may put [x4*]=ct.

   on physical phenomena leading to the expression of law of conservation of linear momentum.

3. Isotropy of Space

   Isotropy of space says that the physical properties of space at any particular point of space are the same in every direction. Because of this property of space, we do have what is called law of conservation of angular momentum.

4. Homogeneity of Time

   Because of time homogeneity a physical phenomenon is always the same if physical conditions are the same, whenever it is observed. This physical equivalence of different instants of time- the homogeneity of time-imposes restrictions

   on physical phenomena. It leads to the expression of law of conservation of energy (Energy can neither be created nor destroyed).But the time is not isotroic; because we cannot go from present to the past.

   These are two of the basic fundamental symmetric properties of space and time. [6]

   Three dimensional Euclidean space is taken to be a space where Euclidean geometrical notions are true. In this space distinct straight lines parallel to one another do meet only at infinity, sum of the angles of a triangle is always 180 degree etc. Often this space is called a flat space.

   Any physical body does have a spatio-temporal existence, in the sense that it has both spatial extension and temporal existence. And it can be represented by world band in spacetime.

   But energy only has temporal existence. Energy is purely temporal being. It does not have any spatial extension. In that sense photon does not have any spatial dimension and we may think that photon is quantized action per unit time.

5. Event Field in Minkowskian Spacetime (M-Space)

   We define the event field in M-space F=[{<x1,x2,x3,x4>},,]

   =[{P0,P1,P2,.P.}= {P},,] ……………….. (2) Here, {P}= (M-{Q0Q0 P0})

   Q0 is the event that does not have multiplicative inverse.

   P0 = < 0, 0, 0,0 > is the additive identity.

   P=<x1, x2, x3, x4> is the th event .. (3) Now introduce two binary operations &in event field of M- space.

   : {P} x {P} {P}

   : ({P} {P0}) x ({P} {P0}) ({P} {P0})

   is the event Addition operator andis the event Multiplication operator. We consider only the event field in M-space which obeys the following algebra.

   Event field Algebra

6. Postulates A

   (A1) F is an abelian group under , P0 =<0,0,0,0>is the additive identity is called the addition of events in M-space

   P P= <x1 , x2, x3, x4> <x1 , x2, x3, x4>=

   <x1+x1, x2+x2, x3+x3, x4+x4> belongs to M- space ………………………………………………………….. (4)

   Now P P= P P (thus commutative property and closure property under are obeyed)

   (A2)P (PP)= (PP) P (Associative Law under holds good)

   (A3) P(-P)=(-P)P=P0 i.e. (5)

   <x1,x2,x3,x4><-x1,-x2,-x3,-x4>=<0,0,00>= P0 which is the additive identity and event (-P) is the additive inverse of the event P.

   (A4)P P0 = P0P=P ……………………………… (6)

7. Postulates B

   B1) P P= P P=<x1, x2, x3, x4>

   <x1 , x2, x3, x4>=<x1 x1 ,x2 x2,x3 x3, x4 x4>,

   ……………………………………………………………………………….. (7)

   (Thus Closure Property under Multiplication is obeyed)

   B2) P (PP) = (PP) P

   (Associative Property under Multiplication is

   obeyed) ………………………………………………….(8)

   B3) PPI = PIP= Phere PI=<1,1,1,1> is

   Multiplicative Identity…….(9)

   B4) PP -1= P-1P=< x1,x2, x3 ,x4>< (x1 )-1 ,( x2)-

   1,(x3)-1,(x4)-1>=<1,1,1,1> = PI=P-1P.. (10)

   Here P is not equal to P0: that isinverse does not exist for P0.Event (P-1) is the multiplicative inverse of the event P.

   B5) P (PP) = (PP) (PP) (P P) P = (PP) (PP)

   (Distributive law holds good)……………………………..(11)

   B6) PP= PP(commutative) thus we have an event field which is commutative in M-space.

   …………………………………………………………………………………..(12)

8. Proper Time, Proper World Line, Proper Observer in M-Space

   In Minkowskianspacetime,time is homogenous but not isotropic. Even though space and time are interdependent entities in M-space,but not identical in all respects, we may denote event P in Minkowskian space time (when the observer is spatially at rest) P=<x1,x2,x3x4>=<spatial position of occurrence of Ptemporal position of P>

   We consider an observer in his time-like world line. He is always in his own rest frame. That is he is always spatially at rest with respect to his own rest frame. This observer can be represented by

   <x1=c1,x2=c2,x3=c3x4=>; is a real number.

   where c1,c2,c3areconstants. For sake of convenience we may put the spatial co-ordinates (c1,c2,c3)

   =(0,0,0)=0=spatial zero vector (zero spatial vector) which remains constant vector throughout the history of the observer in his rest frame. It is his HERE. We may put 0() to show the HERE of the observer on his world line at the proper time. We may call this observer, the proper observer possessing the proper worldline

   W= }.

   ={<O()>};is the proper time measured by the observer.

   Consider the ordered sequence of events on his proper worldline W along which only varies. The set of events on his worldline is represented by

   {P0, P1..}. P0 is the identity = < 0(0)0>.

   P1 = < 0(1)1 >is themultiplicative identity. Event(-P) = <0(-)->is the additive inverse of the event P.

   Event (P-1) = < 0 (-1)-1>is themultiplicative inverse of event P.

   a1) P1P2 = P2 P1 .. (13)

   {<o(1)1>}{<o(2)2>}(<o(1+2)1+2>: closureproperty obeyed

   a2) (P1P2)P3 =P1(P2P3) associative law obeyed under ………………………….. (14)

   a3)<O(0)o><O()>=<O()> ; identity

   ………………………………………………………………….. (15)

   a4) {<O()><O(-)->=<O(o)o> Thus inverse…………… (16)

   hence we have an abelian group under event addition .

   b1) <O(1)1><O(2)2>=O(12)12>…(17)

   closure property obeyed

   b2)[P1P2] P3= P1[P2P3] …………………..(18)

   associative property obeyed

   b3) <O()><O(1)1>=<O()>……………… (19)

   <O(1)1> is multiplicative identity

   b4 <O()><O(-1)-1>=<O(1)1>…………(20)

   Thus inverse exists except for <O(0)0> b5)

   <O(1)1>[<O(2)2>O(3)3>]=<O(12)12

   ><O(13)13>

   =<O(12+13)12+13>

   Similarly

   [P2 P3] P1=<O<21+13)21+13>……..(21)

   Distributive law holds

   b6) P1 P2 =P2 P1 …………………………………….. (22)

   Thus we have an event field being commutative on proper world line of the observer in M-space.

9. Total Order of Event Field constituting the world line of a proper observer [7] & [8]

The event field of proper observer on his world line,

F=[{<O()>

such thatbelongs to real number field},,]together with a TOTAL ORDER on the event field if the order satisfies the following properties. Here stands for precedes

This relation satisfies the following properties.

1. P P for all P belonging to W the proper worldline of the observer (reflexivity).

2. If P Q and Q P, then P=Q ( antisymmetry)

3. If P Q and Q R, then P R ( transitivity)

4. Either P Q or Q P ( totality)

   Here P,Q and R are events on the WORLDLINE W

   of the observer.

   Also we may have the following properties.

   • If event P event Q then

(P R) (QR) here P = <O(p)p (23) Q = <O(q)q> …………………………. (24)

R = <O(r)r>………………………..(25)

If P0 P and P0 Q then P0 (P Q) where P0 is <O(o)o> which is the additive identity of event field in M-space

i.e., <O(o)o><O( ) ><O(o)o><O( ) >

10. Acknowledgements

The author is grateful to Dr.C. Sivaram, Professor of Theoretical Astrophyics,Indian Institute of Astrophysics, Bangalore, Dr. A. V GopalaRao,former Professor of Theoretical Physics,University of

p p

then

q q Mysore, Prof.G.Aravindakshan,Former Professor of

Mathematics, SreeNarayana College, Kollam for

<O(o)o><O(p)p><O(q)q>

i.e., <O(o)o><O(pq)pq>iff proper time of occurrence of P and that of Q are NOT LESS THAN ZERO.

Here p,q and rare proper times of occurrence of events P,Q and R respectively.

The world line

W = {<O()>; belongs to }.

Every event belonging to W does have a number associated with that. This number is the proper time associated with the event. And every number representing proper time is associated with an event on W.There does exists one to one correspondence

between W and { belongs }

Thus the totality of events on world line W of the proper observer is a real mathematical field. Consider the inequality i j.

Here i<j must imply that the event Pi occurs prior to the event Pj on the world line W of the observer and if i= j, Pi becomes identical to Pj: thus the order is also preserved for the event field W in M-space. Since this field of events is ISOMORPHIC to real number we have shown that W is an ordered mathematical field. Here W does have one to one correspondence with {} and that {} is also an ordered mathematical field being ISOMORPHIC

.So closed time loop cannot exist in Minkowskianspacetime, for can never be a closed curve.

Thus we do have the theorem

Theorem :

If proper time is an ordered mathematical field, no closed time loop can exist in Minkowskianspacetime.

discussions and to anonymous referees for accepting this paper for publication in the International Journal of Engineering Research and Technology.

The author is deeply indebted to Dr.M.RBiju(Editor, South Asian Journal of Socio Political Studies),Professor,Department of Political Science, SreeNarayana College Kollam for his valuable suggestions, Smt.N.Letha,Mr. Akshayabodth and Mr.BinduNatesan for the constant support they have given to the author.

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