Euler's `e', sheer relativistic ecstasy

IT WAS a revelation to me after sixty years of dalliance with special relativity that Euler's e unlocks space-time unity enabling us to understand all the concepts and their interconnections with clarity and rigour.

We start with Euler's definition of e as

e = Limit n<108,SYM,103><108,SYM,165> of ( 1+[1/n]){+n} and

e<108,SYM,97> = Limit n<108,SYM,103><108,SYM,165> (1+[<108,SYM,97>/n]){+n}

If <108,SYM,97> is the Newtonian velocity of a particle, it can be considered the sum of n Newtonian velocities <108,SYM,97>/n. (we are dealing with only one dimensional motion).

But according to Albert Einstein, velocities are compounded by the velocity transformation formula

V = (v{-1}+v{-2})/(1+v{-1}v{-2}) =

(1+v{-1})(1+v{-2})... - (1-v{-1})(1-v{-2})...

/ (1+v{-1})(1+v{-2})... +(1-v{-1})(1-v{-2}).

Where v{-1} is the velocity of particle 2 with respect to particle 1, v{-2} is the velocity of particle 3 with respect to 2 and V is the velocity of particle 3 with respect to particle 1.

If we have n such velocities in sequence, then the corresponding velocity of the (n+1)th particle with respect to the first is

V = (P{++}{-n} - P{+-}{-n}) / (P{++}{-n} + P{+-}{-n})

Where P{++} {-n} = (1+v{-1})(1+v{-2})... (1+v{-n}),

P{+-}{-n} = (1-v{-1})(1-v{-2})... ..(1-v{-n}).

If we assume v{-1} = v{-2}... .= v{-n} = <108,SYM,97> / n, then as n<108,SYM,103><108,SYM,165>, the Einstein velocity

V = (e<108,SYM,97> - e-<108,SYM,97> ) / (e<108,SYM,97> + e-<108,SYM,97> )

= tanh<108,SYM,97>, v{lt}1

<108,SYM,97> is the Newtonian velocity, the sum of n velocities each equal to<108,SYM,97> /n as n tends to infinity.

If x and t are the Einstein distances and time of travel of the particle, then x = t{-o} sinh<108,SYM,97>, t = t{-o} cosh<108,SYM,97>.

The corresponding Newtonian velocities are

X{-(}{-n}{-e}{-w}{-t}{-o}{-n}{-)} = <108,SYM,97>t{-o} , t{-(}{-N}{-e}{-w}{-t}{-o}{-n}{-)} = t{-o}

In Einstein's approach, space-time obey the `principle of same behaviour' (enunciated by the author) when expressed in terms of the unit of light velocity (c = 1).

Why should time interval change as we move from one observer to another? This is the direct consequence of Lorentz contraction of a moving `Rod'. A `Rod', according to the author is defined as two points moving with the same velocity but separated by a space interval. Also, `Events' are defined as the crossing of the end points of `Rods' while a point is a `Rod' with space interval (length) zero. When a point travels with velocity v across a stationary `Rod' of length x, it takes time t to cross the `Rod' and the `Events' are the crossing of the point with the end points of the `Rod'.

It was shown by the author that the Lorentz transformation is the direct consequence of the Lorentz contraction. A Lorentz matrix is just a circulant matrix with determinant unity.

[ a b ] with a{+2} - b{+2} = 1

[ b a ]

Hence its elements can be expressed in terms of one parameter.

If the parameter is

<108,SYM,97>, then a = cosh<108,SYM,97>, b = sinh<108,SYM,97>,

V = tanh<108,SYM,97>

<108,SYM,113>, then a = sec <108,SYM,113>, b = tan <108,SYM,113>, V = sin <108,SYM,113> with t and x as hypotenuse and side.

V, then a = 1 /(1-v{+2}){+1}{+/}{+2},

b = V/(1-v{+2}){+1}{+/}{+2}

The product of Lorentz matrices is a Lorentz matrix

L<108,SYM,97> = L<108,SYM,97>{-1}L<108,SYM,97>{-2}... .L<108,SYM,97>{-n}

L<108,SYM,113> = L<108,SYM,113>{-1}L <108,SYM,113>{-2}... L <108,SYM,113>{-n}

LV = Lv{-1}Lv{-2}... ..Lv{-n}

While <108,SYM,97>= <108,SYM,97>{-1}+<108,SYM,97>{-2}... .+<108,SYM,97>n

But, <108,SYM,113> {Dagger} <108,SYM,113>{-1}+<108,SYM,113>{-2}... +<108,SYM,113>n

V {Dagger} v{-1}+v{-2}... ..+v{-n}

Now, v = tanh<108,SYM,97> implies

<108,SYM,97> = log{lcub}(1+V)/(1-V){rcub}{+1}{+/}{+2}

{lcub}(1+V)/(1-V){rcub}{+1}{+/}{+2} = B is called BIAS (called by sir Herman Bondi) and its importance is due to its multiplicative property

B = B{-1} B{-2}... .B{-n}.

B{-k} = {lcub}(1+V)/(1-V){rcub}{+1}{+/}{+2}

<108,SYM,97> is consequently additive. B and 1/B are the eigenvalues of the Lorentz matrix with parameter V representing the equal fractions in which the distance travelled and the time taken by the light with velocity �1 (light velocity c = 1). The corresponding eigenvectors are

[1] and [ 1 ]

[1] [ -1]

If a `Rod' of light of length l moves with velocity �1, its length is altered by fractions with reversed values 1/B and B.

B = e<108,SYM,97>

Where <108,SYM,97> is the Newtonian velocity.

Does it not convince us that Euler's e is sheer mathematical ecstasy?

Of incredible beauty is the correspondence between the functions and matrices.

e<108,SYM,97> and [ cosh<108,SYM,181> sinh<108,SYM,181> ]

<15,,,0><11,13m,0m>[ sinh<108,SYM,181> cosh<108,SYM,181> ] ,

<11,0m,0m>cosh<108,SYM,181> and [ cosh<108,SYM,181> 0 ]

<11,16m,0m>[ 0 cosh<108,SYM,181> ]

<11,0m,0m>but, sinh<108,SYM,181>and [ 0 sinh<108,SYM,181> ]

<11,20m,0m>[ sinh<108,SYM,181> 0 ]

and tanh<108,SYM,181> and [ 0 tanh<108,SYM,181> ]

<11,22m,0m>[ tanh<108,SYM,181> 0 ]

Professor Krishnaswami Alladi has noticed that such correspondence applies to any function of a single variable and a circulant matrix with its elements as the even and odd part of the function. If the determinant is normalised to unity it is a Lorentz matrix..

Alladi Ramakrishnan

Founder- director


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