Post date: Jun 12, 2013 4:08:54 AM
MATHEMATICAL PHILOSOPHY OF TIME IN MINKOWSKIAN SPACE
Keywords
Algebra on event field, proper time, real number field, time loop, proper world line, proper observer.
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.
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 an 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 spcetime 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 two events
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 space
time. 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 Mspace,
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 restmass has a co-ordinates description
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