# 5.5 The lorentz transformation  (Page 4/12)

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$\text{Δ}r{\prime }^{2}={\left(\text{Δ}x\prime \right)}^{2}+{\left(\text{Δ}y\prime \right)}^{2}+{\left(\text{Δ}z\prime \right)}^{2}.$

That has the same value that $\text{Δ}{r}^{2}$ had. Something similar happens with the Lorentz transformation in space-time.

Define the separation between two events, each given by a set of x , y , and ct along a four-dimensional Cartesian system of axes in space-time, as

$\left(\text{Δ}x,\text{Δ}y,\text{Δ}z,c\text{Δ}t\right)=\left({x}_{2}-{x}_{1},{y}_{2}-{y}_{1},{z}_{2}-{z}_{1},c\left({t}_{2}-{t}_{1}\right)\right).$

Also define the space-time interval $\text{Δ}s$ between the two events as

$\text{Δ}{s}^{2}={\left(\text{Δ}x\right)}^{2}+{\left(\text{Δ}y\right)}^{2}+{\left(\text{Δ}z\right)}^{2}-{\left(c\text{Δ}t\right)}^{2}.$

If the two events have the same value of ct in the frame of reference considered, $\text{Δ}s$ would correspond to the distance $\text{Δ}r$ between points in space.

The path of a particle through space-time consists of the events ( x , y , z¸ ct ) specifying a location at each time of its motion. The path through space-time is called the world line    of the particle. The world line of a particle that remains at rest at the same location is a straight line that is parallel to the time axis. If the particle moves at constant velocity parallel to the x -axis, its world line would be a sloped line $x=vt,$ corresponding to a simple displacement vs. time graph. If the particle accelerates, its world line is curved. The increment of s along the world line of the particle is given in differential form as

$d{s}^{2}={\left(dx\right)}^{2}+{\left(dy\right)}^{2}+{\left(dz\right)}^{2}-{c}^{2}{\left(dt\right)}^{2}.$

Just as the distance $\text{Δ}r$ is invariant under rotation of the space axes, the space-time interval:

$\text{Δ}{s}^{2}={\left(\text{Δ}x\right)}^{2}+{\left(\text{Δ}y\right)}^{2}+{\left(\text{Δ}z\right)}^{2}-{\left(c\text{Δ}t\right)}^{2}.$

is invariant under the Lorentz transformation. This follows from the postulates of relativity, and can be seen also by substitution of the previous Lorentz transformation equations into the expression for the space-time interval:

$\begin{array}{cc}\hfill \text{Δ}{s}^{2}& ={\left(\text{Δ}x\right)}^{2}+{\left(\text{Δ}y\right)}^{2}+{\left(\text{Δ}z\right)}^{2}-{\left(c\text{Δ}t\right)}^{2}\hfill \\ & ={\left(\frac{\text{Δ}x\prime +v\text{Δ}t\prime }{\sqrt{1-{v}^{2}\text{/}{c}^{2}}}\right)}^{2}+{\left(\text{Δ}y\prime \right)}^{2}+{\left(\text{Δ}z\prime \right)}^{2}-{\left(c\frac{\text{Δ}t\prime +\frac{v\text{Δ}x\prime }{{c}^{2}}}{\sqrt{1-{v}^{2}\text{/}{c}^{2}}}\right)}^{2}\hfill \\ & ={\left(\text{Δ}x\prime \right)}^{2}+{\left(\text{Δ}y\prime \right)}^{2}+{\left(\text{Δ}z\prime \right)}^{2}-{\left(c\text{Δ}t\prime \right)}^{2}\hfill \\ & =\text{Δ}s{\prime }^{2}.\hfill \end{array}$

In addition, the Lorentz transformation changes the coordinates of an event in time and space similarly to how a three-dimensional rotation changes old coordinates into new coordinates:

$\begin{array}{ccc}\mathbf{\text{Lorentz transformation}}\hfill & & \mathbf{\text{Axis}}\phantom{\rule{0.2em}{0ex}}–\phantom{\rule{0.2em}{0ex}}\mathbf{\text{rotation around}}\phantom{\rule{0.2em}{0ex}}\mathbit{\text{z}}\mathbf{\text{-axis}}\hfill \\ \left(x,t\phantom{\rule{0.2em}{0ex}}\text{coordinates):}\hfill & & \left(x,y\phantom{\rule{0.2em}{0ex}}\text{coordinates):}\hfill \\ x\prime \phantom{\rule{0.3em}{0ex}}=\left(\gamma \right)x+\left(\text{−}\beta \gamma \right)ct\hfill & & x\prime =\left(\text{cos}\phantom{\rule{0.2em}{0ex}}\theta \right)x+\left(\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \right)y\hfill \\ ct\prime =\left(-\beta \gamma \right)x+\left(\gamma \right)ct\hfill & & y\prime =\left(-\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \right)x+\left(\text{cos}\phantom{\rule{0.2em}{0ex}}\theta \right)y\hfill \end{array}$

where $\gamma =\frac{1}{\sqrt{1-{\beta }^{2}}};\phantom{\rule{0.5em}{0ex}}\beta =v\text{/}c.$

Lorentz transformations can be regarded as generalizations of spatial rotations to space-time. However, there are some differences between a three-dimensional axis rotation and a Lorentz transformation involving the time axis, because of differences in how the metric, or rule for measuring the displacements $\text{Δ}r$ and $\text{Δ}s,$ differ. Although $\text{Δ}r$ is invariant under spatial rotations and $\text{Δ}s$ is invariant also under Lorentz transformation, the Lorentz transformation involving the time axis does not preserve some features, such as the axes remaining perpendicular or the length scale along each axis remaining the same.

Note that the quantity $\text{Δ}{s}^{2}$ can have either sign, depending on the coordinates of the space-time events involved. For pairs of events that give it a negative sign, it is useful to define $\text{Δ}{\tau }^{2}$ as $-\text{Δ}{s}^{2}.$ The significance of $\text{Δ}\tau$ as just defined follows by noting that in a frame of reference where the two events occur at the same location, we have $\text{Δ}x=\text{Δ}y=\text{Δ}z=0$ and therefore (from the equation for $\text{Δ}{s}^{2}=-\text{Δ}{\tau }^{2}\right)\text{:}$

#### Questions & Answers

what is an atom
Aroyameh Reply
All matter is composed of two sets of three dimensions. The first set (1,2,3) decay with a positive charge. The second set (4,5,6) decay with a negative charge. As they decay, they create space (7 8,9) dimensions.
John
Two sets of (1,2,3,4,5,6) dimensions create a proton, a neutron, and an electron. This is the primordial atom.
John
A 10kg mass lift to a height of 24m and release. what is the total energy of the system
ADEPOJU Reply
mechanics is that branch of physical and mathatics that
ADEPOJU
E=Mgh=10*10*24=2400J
Adamu
what is the difference between a molecule and atom
Natanim Reply
Atoms are single neutral particles. Molecules are neutral particles made of two or more atoms bonded together.
Manfred
what I'd dynamic propulsion
Elias Reply
A body quadruples its momentum when its speed doubles.What was the initial speed in units of c.i.e..what was u/c ?
Lekshmi Reply
what is enthalpy?
prabir Reply
a thermodynamic quantity equivalent to the total heat content of a system
RAMLA
proparty of tharmo dainamic
bloch
What is the meaning of Nuclear Fission?
Benita Reply
what do you mean by dynamics single particles
Peacekamei Reply
عند قذف جسم إلى أعلى بسرعة إبتدائية فإنه سيصل إلى ارتفاع معين (أقصى ارتفاع) ثم يعود هابطاً نحو سطح الأرض .   إذا قُذِفَ جسم إلى أعلى ووجد أن سرعته 18 م / ث عندما قطع 1/4 المسافة التي تمثل أقصى ارتفاع سيصله فالمطلوب إيجاد السرعة التي قُذِف بها بالمتر / ث . إن هذه السرعة هي واحدة من الإجابات التالية
Aml Reply
what is light
Ayebanifesunday Reply
light is a kind of radiation That stimulates sight brightness a source of illumination.
kenneth
Electromagnet radiation creates space 7th, 8th, and 9th dimensions at the rate of c.
John
That is the reason that the speed of light is constant.
John
This creation of new space is "Dark Energy".
John
The first two sets of three dimensions, 1 through 6, are "Dark Matter".
John
As matter decays into luminous matter, a proton, a neutron, and an electron creat deuterium.
John
There are three sets of three protons, 9.
John
There are three sets of three neutrons, 9.
John
A free neutron decays into a proton, an electron, and a neutrino.
John
There are three sets of five neutrinoes, 15.
John
Neutrinoes are two dimensional.
John
A positron is composed of the first three dimensions.
John
An electron is composed of the second three dimensions.
John
What is photoelectric
Hsssan Reply
light energy (photons) through semiconduction of N-P junction into electrical via excitation of silicon purified and cristalized into wafers with partially contaminated silicon to allow this N-P function to operate.
Michael
i.e. Solar pannel.
Michael
Photoelectric emission is the emission of electrons on a metal surface due to incident rays reflected on it
Benita
If you lie on a beach looking at the water with your head tipped slightly sideways, your polarized sunglasses do not work very well.Why not?
Rakhi Reply
it has everything to do with the angle the UV sunlight strikes your sunglasses.
Jallal
this is known as optical physics. it describes how visible light, ultraviolet light and infrared light interact when they come into contact with physical matter. usually the photons or light upon interaction result in either reflection refraction diffraction or interference of the light.
Jallal
I hope I'm clear if I'm not please tell me to clarify further or rephrase
Jallal
what is bohrs model for hydrogen atom
Swagatika Reply
hi
Tr
Hello
Youte
Hi
Nwangwu-ike
hi
Siddiquee
hi
Omar
helo
Mcjoi
what is the value of speed of light
Propessor Reply
1.79×10_¹⁹ km per hour
Swagatika
3×10^8
Benita
what r dwarf planet
Sivalakshmi Reply

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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