Friday, April 22, 2011

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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