Relativity Science Calculator - The Heart of Special Relativity: Lorentz Transformation Equations

The Heart of Special Relativity: Lorentz Transformation Equations

"For me personally he [ Lorentz ] meant more than all the others I have met on my life's journey" - The Collected Papers of Albert Einstein ( 1953, Vol. 7 )

Special Relativity was first published in 1905 by Albert Einstein at age 26 working quietly in the Swiss Patent Office, Bern, Switzerland, under the title "On The Electrodynamics Of Moving Bodies", translated from "Zur Elektrodynamik bewegter Körper", Annalen der Physik, volume 17:891, 1905.

"The Einstein Theory of Relativity", by H.A. Lorentz, November 19, 1919, first appearing in The Nieuwe Rotterdamsche Courant, English translation

The Relativity Postulates: 

1). The Principle of Relativity - All the laws of physics in their simplest reduced form are transformable and hence invariant as between

an infinite number of moving reference systems ( inertial systems ), each one of which is moving uniformly

and rectilinearly with respect to any other system and where no one system is privileged or preferred over

any other reference ( inertial ) system when measurements of length or time are taken.

2). The Principle of the Constancy of the Speed of Light - The speed of light in empty ( vacuo ) space is a universal constant as measured

in any reference ( inertial ) system when measured with rods and clocks of the

same kind. This is always true notwithstanding any "relativistic effects" of either

the Lorentz length contraction or time dilation as earlier revealed by the Michelson -

Morley Experiment (1887 ).

The Relativity Corollaries: 

a). The "luminiferous aether" does not exist. [ This entirely comports with the results of the Michelson - Morley Experiment. ]

b). The "Fallacy of Simultaneity at a Distance" - There is no such reality as simultaneously occurring events when measured at great

distances or at velocities approaching the speed of light, .

[ The proof of this will be given later in "Some Results of Lorentz Transformation Equations". ]

The Relativity Assumptions - The Cosmological Principle as regards special relativity: 

i).  Isotropy -  space - time is uniform and symmetric in all directions exhibiting constant values - viz. the velocity of light transmission.

That is, there is no one preferred reference point or direction in space - time.

ii).  Homogeneity -  space - time possesses the quality of uniformity in structure and composition in all directions.

That is, the geometry ( metric ) of space - time is the same from any point to any other point in the universe.

iii).  Systems -

, inertial system at rest relative to

, inertial system moving uniformly and rectilinearly along the - axis with constant

relative velocity with respect to inertial system  , where

- axis is parallel and coincident or common to - axis

and likewise

- axis, - axis are respectively parallel to - axis, - axis. 

Anisotropy versus Isotropy:

While examining the cosmic microwave background ( CMB ) for the large - scale universe, the cosmos appears nearly isotropic although not perfectly. Anisotropy refers to particular regions of cosmic space - time exhibiting different temperature values for the cosmic microwave background ( CMB ) along different measured axes of direction which indicate how tiny perturbations in the distribution of background energies from the earliest times after the Big Bang caused galaxies and other large - scale structures to form ( primordial nucleosynthesis ) from the initial cosmic blast debris. Our cosmic universe on a large scale is approximately isotropic and homogenous ( The Cosmological Principle ) but not precisely so, owing to these tiny anisotropies just described.

The Essential Relativity Problem:

In order to understand an event in relatively moving system for an observer  at the origin in stationary system , we need to understand the rules of transformation for the following coordinates of this event for an observer  at the respective origin in system  :

transformation_rules.png

That is, each space - time point in    is invariably transformable to some other space - time point in  .

The Solution to the above Essential Relativity Problem will give us The Lorentz Transformation Equations:

Therefore the differential form of the above equations becomes

differential_equations.png

for an array of unknown coefficients to whose solution essentially defines the task ahead to the Relativity Problem.

Because of space - time homogeneity all of the coefficients    are independent of event     coordinates    and therefore the equation set (2) is "integrable" and hence must be "linear transformation" equations.

Furthermore, because of space - time homogeneity, all space - time points, in and in , are equivalent under linear transformation.

By calculus integration we get: 

integrating_calculus.png

So far so good. 

Let time at the instant the origin of    in    coincides with    in relatively moving , then

constants.png

Also because there is no relative motion in the or directions,

no_relative_motion.png

A little bit simpler, no?

By The Principle of Relativity and the invariant manner by which parallel lengths of rods in    and   respectively are moving orthogonally ( at right angles ) to the relative direction of motion along the - axis, it follows that they will not experience a Lorentz contraction along their - and - axes.

Hence a rod of length 1 lying along the - axis from  to  in   will also appear to possess a length of 1 in   if this same rod is fixed along the - axis from  to . Likewise for a rod of length 1 along the - axis.


This all implies 

implies.png

And hence,

hence.png

Also because of our basic space - time isotropy assumption (space - time is the same in all directions), and will not be dependent upon the - and - axes since any two ( or more ) - clocks in the - plane placed symmetrically around the - axis will appear to disagree as seen by an observer in    which would otherwise violate our isotropic assumption.

This all implies

implies2.png

which in turn implies the following reduced transformation equations:

reduced_transformation_equations.png

Getting closer, getting closer.

Now event    in moving system   at origin , , must also satisfy  in system    where   is moving rectilinearly and uniformly away from system    with constant velocity    using the following constraints:

Again, our reduced transformation equations become:

reduced_transformation_equations2.png

Applying the 2nd Relativity Principle - The Principle of the Constancy of the Speed of Light - it must be that as   moves past    with a constant velocity at time whose speeding origin coincides exactly with origin of system at the precise moment, time , for an event , flash of light emanating at origin , , there will therefore be an expanding electromagnetic sphere of light propagating with constant speed in all directions in both and systems. Hence the speed of its progress in either system will be equal and can therefore be described by either transformation set of coordinates or as follows: 

light flash.png


And the progress of the light propagation can be described by either equation:

light_propagation.png

equation (6c).png,

since

since.png

Expanding and rearranging,

rearranging.png

But equation for moving system must also satisfy the conditions of for stationary system . We therefore force this condition as follows:

satisfies.png

Whew!

However we must continue ... 

We next solve these three simultaneous equations by first eliminating a11.png as follows:

eliminating a11.png

This entire "elimination process" can be viewed on Page "Solution to Equation ( 7a )" which is rather long and difficult. However the results are:

equation (7a).png

where always the positive (+) sign of the square root is taken.

Have confidence that we are almost at the end! Faith, faith!!

We now substitute equations into  as follows:

substitute.png

Hence,

Lorentz Transformation Equations.png.

The equations   are the famous Lorentz Transformation Equations which are integral to Special Relativity and thereby forms its mathematical basis.

At small values of , where velocities are within the normal range of human experience ( excluding of course experiences of Quantum particle physicists, ha! ), Lorentz Transformation Equations easily reduce to traditional Galilean Transformation Equations as follows:

Galilean Transformation Equations.png

Just to elucidate slightly more, Lorentz Transformation Equations as given above in are those transformation equations where the observer is standing in moving system relative to stationary system and attempting to derive his/her own coordinates relative to system - i.e., as system relatively "moves away".

The inverse of Lorentz Transformation Equations equations are therefore those transformation equations where the observer is standing in stationary system and is attempting to derive his/her coordinates in as system relatively "moves away":

Lorentz Inverse Transformation Equations.png

And,

Galilean Inverse Transformation Equations.png

for small values of .

See Page "Solution to Equation (9)" for this somewhat simpler derivation than that which is shown on Page "Solution to Equation (7a)".

We are thus finished!

These equations are then the necessary tools for Relativity Mathematics and hence for Special Relativity cosmology. It is actually rather simple algebraic equations which form the basis of Special Relativity.

There are also other means and methods for deriving these Lorentz Transformation Equations such as partial differential geometry, etc., nevertheless the final result will always be the same as has already been derived. So why not stay with simple Algebra?

Continuing ...

Because Michelson and Morley [ see Michelson - Morley Experiment ( 1887 ) ]  were able to increase their fringe accuracy to within 1/100th of an interference fringe in their famous experiment, they were able to experimentally demonstrate the inadequacy of the traditional Galilean Transformation Equations to which even Isaac Newton took as ultimate truth.

Notice also that as moves away from , longitudinal velocity transforms from to . This last observation is not trivial as neither element of is directly translatable to .

§ Reading:

  1. "The Einstein Theory of Relativity", by H.A. Lorentz, November 19, 1919, first appearing in The Nieuwe Rotterdamsche Courant, English translation

  2. "On the Relativity Principle and the Conclusions Drawn from It", German version, by A. Einstein, 1907

  3. Funeral Oration for H.A. Lorentz, Feb. 1928 draft document, signed A. Einstein, The Albert Einstein Archives, The Library Authority, The Hebrew University of Jerusalem