Special
theory of relativity: space and time quantization
An abstract of my theory:
Special relativity is the theory of measurement in inertial
frames of reference proposed in 1905 by Einstein.
In his initial presentation in 1905 he expressed these
postulates:
The Principle of
Relativity – The laws by which the states of physical systems undergo change
are not affected, whether these changes of state be referred to the one or the
other of two systems in uniform translatory motion relative to each other.
The Principle of
Invariant Light Speed – "... light is always propagated in empty space
with a definite velocity [speed] c which is independent of the state of motion
of the emitting body." (from the preface). That is, light in vacuum
propagates with the speed c (a fixed constant, independent of direction) in at
least one system of inertial coordinates (the "stationary system"),
regardless of the state of motion of the light source.
The special
theory of relativity is contained in the postulate:
The laws of physics are invariant with respect to Lorentz
transformations (for the transition from one inertial system to any other
arbitrarily chosen inertial system).
The special
theory of relativity has a wide range of consequences which have been
experimentally verified such as length contraction and time dilatation.
For simplicity, we will restrict
consideration motion in one direction.
Observers are simply people or
instruments capable of making and recording measurements.
Length contraction
The length of an object in a moving frame will appear
contracted in the direction of motion. The amount of contraction is calculated
from the Lorentz transformation; the length is maximum in the frame in which
the object is at rest.
l = l0( 1 – v2 / c2 )1/2
where: l = the length
measured by the "other" observer
l0 = the length measured
by the observers on reference frame
v = the
speed of the object
c = the
speed of light in a vacuum ( c = 2.99792458 × 108 m s−1 )
If the object is moving horizontally, then it is the
horizontal dimension which is contracted; there would be no contraction of the
height of the object.
For example if a spaceship in motion has the speed v = 0.95c
and l0 = 17.6 m (for you inside the spaceship), for an observer on
earth the spaceship has l = 5.5 m.
Time dilatation
The time lapse between two events is not invariant from one
observer to another but is dependent on the relative speeds of the observers'
reference frames.
Consider a clock consisting of two mirrors A and B, between
which a light pulse is bouncing. The distance between the mirrors is L and the
clock ticks once each time it hits a given mirror.
In the frame where the clock is at rest the period of the
clock:
Δt = 2L / c
From the frame of reference of a moving observer traveling
at the speed v the light pulse traces out a longer, angled path, the period of
the clock:
Δt’ = Δt / ( 1 – v2 / c2 )1/2
This means for the moving observer the period of the clock
is longer than in the frame of the clock itself.
Planes travel about a million times more slowly than c but
atomic clocks are very precise and so this tiny effect can actually be
measured.
The twin paradox: there are two twin brothers. On their
thirtieth birthday, one of the brothers goes on a space journey in a rocket
that travels at 99% of the speed of light. The space traveler stays on his
journey for precisely one year, whereupon he returns to Earth on his 31st
birthday. On Earth seven years have elapsed, so his twin brother is 37 years
old at the time of his arrival.
Space and time quantization
I formulated a postulate in Special theory of relativity: in
any frame of reference the Planck constants are the same.
This means for the frame of reference at rest and for the
frame of reference moving with the speed v, the Planck length and the Planck
time are the same.
This means in any frame of reference a spaceship can not
travel in time less than the Planck time and in space can not move less than
the Planck length.
Length contraction
Because any system can not attain the speed of light:
l = l0( 1 – v2 / c2 )1/2 and l > lP
Planck length lP = 1.6162 × 10−35 m
This means l0( 1 – v2 / c2 )1/2 > lP
Time dilatation
Because the period of the clock
Δt’ = Δt / ( 1 – v2 / c2 )1/2
This means the time measured by the clock:
t’ = t( 1 – v2 / c2 )1/2 and t’ > tP
Planck time tP = 5.39124 × 10−44 s
This means t( 1 – v2 / c2 )1/2 > tP
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