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Albert Michelson - Edward Morley  1887 Experiment


The Heart of Special Relativity Physics: 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 )

Albert Michelson - Edward Morley  1887 Experiment

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 Albert Michelson - Edward Morley  1887 Experiment "On The Electrodynamics Of Moving Bodies", translated from "Zur Elektrodynamik bewegter Körper", Annalen der Physik, volume 17:891, 1905.

Albert Michelson - Edward Morley  1887 Experiment "The Einstein Theory of Relativity", by H.A. Lorentz, November 19, 1919, first appearing in The Nieuwe Rotterdamsche Courant, English translation

The Einstein Relativity Physics 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 Einstein Relativity Physics 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, Albert Michelson - Edward Morley  1887 Experiment. [ 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 physics: 

i).  Isotropy -  space - time is uniform and symmetric in all Albert Michelson - Edward Morley  1887 Experiment 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 Albert Michelson - Edward Morley  1887 Experiment directions.

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

iii).  Systems -

Albert Michelson - Edward Morley  1887 Experiment, inertial system at rest relative to Albert Michelson - Edward Morley  1887 Experiment

Albert Michelson - Edward Morley  1887 Experiment, inertial system moving uniformly and rectilinearly along the Albert Michelson - Edward Morley  1887 Experiment - axis with constant relative velocity Albert Michelson - Edward Morley  1887 Experiment with respect to inertial system  Albert Michelson - Edward Morley  1887 Experiment, where

Albert Michelson - Edward Morley  1887 Experiment - axis is parallel and coincident or common to Albert Michelson - Edward Morley  1887 Experiment - axis

and likewise

Albert Michelson - Edward Morley  1887 Experiment - axis, Albert Michelson - Edward Morley  1887 Experiment - axis are respectively parallel to Albert Michelson - Edward Morley  1887 Experiment - axis, Albert Michelson - Edward Morley  1887 Experiment - 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 Physics Problem:

In order to understand an event Albert Michelson - Edward Morley  1887 Experiment in relatively moving system Albert Michelson - Edward Morley  1887 Experiment for an observer Albert Michelson - Edward Morley  1887 Experiment  at the origin in stationary system Albert Michelson - Edward Morley  1887 Experiment, we need to understand the rules of transformation for the following coordinates of this event Albert Michelson - Edward Morley  1887 Experiment for an observer Albert Michelson - Edward Morley  1887 Experiment at the respective origin in system  Albert Michelson - Edward Morley  1887 Experiment:

lorentz transformation rules

That is, each Albert Michelson - Edward Morley  1887 Experiment space - time point in  Albert Michelson - Edward Morley  1887 Experiment  is invariably transformable to some other Albert Michelson - Edward Morley  1887 Experiment space - time point in  Albert Michelson - Edward Morley  1887 Experiment.

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

Therefore the differential form of the above equations becomes

differential calculus equations

for an array of unknown coefficients Albert Michelson - Edward Morley  1887 Experiment to whose solution essentially defines the task ahead to the Relativity Problem.

Because of space - time homogeneity all of the coefficients  Albert Michelson - Edward Morley  1887 Experiment  are independent of event  Albert Michelson - Edward Morley  1887 Experiment   coordinates  Albert Michelson - Edward Morley  1887 Experiment  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, Albert Michelson - Edward Morley  1887 Experiment in Albert Michelson - Edward Morley  1887 Experiment and Albert Michelson - Edward Morley  1887 Experiment in Albert Michelson - Edward Morley  1887 Experiment, are equivalent under linear transformation.

By calculus integration we get: 

integrating_calculus.png

So far so good. 

Let time Albert Michelson - Edward Morley  1887 Experiment at the instant the origin of  Albert Michelson - Edward Morley  1887 Experiment  in  Albert Michelson - Edward Morley  1887 Experiment  coincides with  Albert Michelson - Edward Morley  1887 Experiment  in relatively moving Albert Michelson - Edward Morley  1887 Experiment, then

calculus integration constant

Also because there is no relative motion in the Albert Michelson - Edward Morley  1887 Experiment or Albert Michelson - Edward Morley  1887 Experiment directions,

relative motion

A little bit simpler, no?

By The Principle of Relativity and the invariant manner by which parallel lengths of rods in  Albert Michelson - Edward Morley  1887 Experiment  and  Albert Michelson - Edward Morley  1887 Experiment respectively are moving orthogonally ( at right angles ) to the relative direction of motion along the Albert Michelson - Edward Morley  1887 Experiment - axis, it follows that they will not experience a Lorentz contraction along their Albert Michelson - Edward Morley  1887 Experiment - and Albert Michelson - Edward Morley  1887 Experiment - axes.

Hence a rod of length 1 lying along the Albert Michelson - Edward Morley  1887 Experiment - axis from Albert Michelson - Edward Morley  1887 Experiment  to Albert Michelson - Edward Morley  1887 Experiment  in  Albert Michelson - Edward Morley  1887 Experiment  will also appear to possess a length of 1 in  Albert Michelson - Edward Morley  1887 Experiment if this same rod is fixed along the Albert Michelson - Edward Morley  1887 Experiment - axis from Albert Michelson - Edward Morley  1887 Experiment  to Albert Michelson - Edward Morley  1887 Experiment. Likewise for a rod of length 1 along the Albert Michelson - Edward Morley  1887 Experiment - axis.


This all implies 

Albert Michelson - Edward Morley  1887 Experiment

And hence,

Albert Michelson - Edward Morley  1887 Experiment

Also because of our basic space - time isotropy assumption ( space - time is the same in all directions ), Albert Michelson - Edward Morley  1887 Experiment and Albert Michelson - Edward Morley  1887 Experiment will not be dependent upon the Albert Michelson - Edward Morley  1887 Experiment - and Albert Michelson - Edward Morley  1887 Experiment - axes since any two ( or more ) Albert Michelson - Edward Morley  1887 Experiment- clocks in the Albert Michelson - Edward Morley  1887 Experiment - plane placed symmetrically around the Albert Michelson - Edward Morley  1887 Experiment - axis will appear to disagree as seen by an observer in  Albert Michelson - Edward Morley  1887 Experiment  which would otherwise violate our isotropic assumption.

This all implies

implies2.png

which in turn implies the following reduced transformation equations:

reduced transformation equations

Getting closer, getting closer.

Now event  Albert Michelson - Edward Morley  1887 Experiment  in moving system  Albert Michelson - Edward Morley  1887 Experiment at origin Albert Michelson - Edward Morley  1887 Experiment, Albert Michelson - Edward Morley  1887 Experiment, must also satisfy Albert Michelson - Edward Morley  1887 Experiment  in system  Albert Michelson - Edward Morley  1887 Experiment  where  Albert Michelson - Edward Morley  1887 Experiment is moving rectilinearly and uniformly away from system  Albert Michelson - Edward Morley  1887 Experiment  with constant velocity  Albert Michelson - Edward Morley  1887 Experiment  using the following constraints:

Albert Michelson - Edward Morley  1887 Experiment

Again, our reduced transformation equations become:

reduced transformation equations

Applying the 2nd Relativity Principle - The Principle of the Constancy of the Speed of Light - it must be that as  Albert Michelson - Edward Morley  1887 Experiment moves past  Albert Michelson - Edward Morley  1887 Experiment  with a constant velocity Albert Michelson - Edward Morley  1887 Experiment at time Albert Michelson - Edward Morley  1887 Experiment whose speeding origin Albert Michelson - Edward Morley  1887 Experimentcoincides exactly with origin Albert Michelson - Edward Morley  1887 Experiment of system Albert Michelson - Edward Morley  1887 Experiment at the precise moment, time Albert Michelson - Edward Morley  1887 Experiment, for an event Albert Michelson - Edward Morley  1887 Experiment, flash of light emanating at origin Albert Michelson - Edward Morley  1887 Experiment, Albert Michelson - Edward Morley  1887 Experiment, there will therefore be an expanding electromagnetic sphere of light propagating with constant speed Albert Michelson - Edward Morley  1887 Experiment in all Albert Michelson - Edward Morley  1887 Experiment directions in both Albert Michelson - Edward Morley  1887 Experiment and Albert Michelson - Edward Morley  1887 Experiment systems. Hence the speed of its progress in either system will be equal and can therefore be described by either transformation set of coordinates Albert Michelson - Edward Morley  1887 Experiment or Albert Michelson - Edward Morley  1887 Experiment as follows: 

light flash


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

light propagation

Albert Michelson - Edward Morley  1887 Experiment,

since

Albert Michelson - Edward Morley  1887 Experiment

Expanding and rearranging,

rearranging calculus terms

But equation Albert Michelson - Edward Morley  1887 Experiment for moving system Albert Michelson - Edward Morley  1887 Experiment must also satisfy the conditions of Albert Michelson - Edward Morley  1887 Experiment for stationary system Albert Michelson - Edward Morley  1887 Experiment. We therefore force this condition as follows:

integral calculus

Whew!

However we must continue ... 

We next solve these three simultaneous equations Albert Michelson - Edward Morley  1887 Experiment by first eliminating differential calculus as follows:

light flash propagating

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

relatively moving frame system,

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 Albert Michelson - Edward Morley  1887 Experiment equations into Albert Michelson - Edward Morley  1887 Experiment as follows:

Planck's quantum constant

Hence,

Lorentz Transformation Equations

The equation Albert Michelson - Edward Morley  1887 Experiment  are the famous Lorentz Transformation Equations which are integral to Special Relativity and thereby forms its mathematical basis.

At small values of Albert Michelson - Edward Morley  1887 Experiment, 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

Just to elucidate slightly more, Lorentz Transformation Equations as given above in Albert Michelson - Edward Morley  1887 Experiment are those transformation equations where the observer Albert Michelson - Edward Morley  1887 Experiment is standing in moving system Albert Michelson - Edward Morley  1887 Experiment relative to stationary system Albert Michelson - Edward Morley  1887 Experiment and attempting to derive his/her own coordinates Albert Michelson - Edward Morley  1887 Experiment relative to system Albert Michelson - Edward Morley  1887 Experiment - i.e., as system Albert Michelson - Edward Morley  1887 Experiment relatively "moves away".

The inverse of Lorentz Transformation Equations equations are therefore those transformation equations where the observer Albert Michelson - Edward Morley  1887 Experiment is standing in stationary system Albert Michelson - Edward Morley  1887 Experiment and is attempting to derive his/her coordinates Albert Michelson - Edward Morley  1887 Experiment in Albert Michelson - Edward Morley  1887 Experiment as system Albert Michelson - Edward Morley  1887 Experiment relatively "moves away":

Lorentz Inverse Transformation Equations

And,

Galilean Inverse Transformation Equations

for small values of Albert Michelson - Edward Morley  1887 Experiment.

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 Albert Michelson - Edward Morley  1887 Experiment 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 Albert Michelson - Edward Morley  1887 Experiment moves away from Albert Michelson - Edward Morley  1887 Experiment, longitudinal velocity transforms from Albert Michelson - Edward Morley  1887 Experiment to Albert Michelson - Edward Morley  1887 Experiment. This last observation is not trivial as neither element of Albert Michelson - Edward Morley  1887 Experiment is directly translatable to Albert Michelson - Edward Morley  1887 Experiment.

§ Reading:

  1. Albert Michelson - Edward Morley  1887 Experiment "The Einstein Theory of Relativity", by H.A. Lorentz, November 19, 1919, first appearing in The Nieuwe Rotterdamsche Courant, English translation

  2. Albert Michelson - Edward Morley  1887 Experiment "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

  4. Albert Michelson - Edward Morley  1887 Experiment "A Planck-scale limit on spacetime fuzziness and stochastic Lorentz invariance violation", published March 14, 2015. This research was supported by an ERC advanced grant; the I-CORE Israeli Centers Of Research Excellence; the Israel Space Agency; the joint Israel Science Foundation (ISF) and the National Natural Science Foundation of China (NSFC) program; and the Templeton Foundation. Conclusion: that the speed of light is an immutable constant of nature at all energy - frequency levels. See: "Is Spacetime Foam a Physical Reality?"

Albert Michelson - Edward Morley  1887 Experiment

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