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 Einstein 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 Einstein 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 spacetime.
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 :
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
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:
So far so good.
Let time at the instant the origin of in coincides with in relatively moving , then
Also because there is no relative motion in the or directions,
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
And hence,
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
which in turn implies the following reduced transformation equations:
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:
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:
And the progress of the light propagation can be described by either equation:
,
since
Expanding and rearranging,
But equation for moving system must also satisfy the conditions of for stationary system . We therefore force this condition as follows:
Whew!
However we must continue ...
We next solve these three simultaneous equations by first eliminating as follows:
This entire "elimination process" can be viewed on Page "Solution to Equation ( 7a )" which is rather long and difficult. However the results are:
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:
Hence,
The equation 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:
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":
And,
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 .
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