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Δ r 2 = ( Δ x ) 2 + ( Δ y ) 2 + ( Δ z ) 2 .

That has the same value that 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

( Δ x , Δ y , Δ z , c Δ t ) = ( x 2 x 1 , y 2 y 1 , z 2 z 1 , c ( t 2 t 1 ) ) .

Also define the space-time interval Δ s between the two events as

Δ s 2 = ( Δ x ) 2 + ( Δ y ) 2 + ( Δ z ) 2 ( c Δ t ) 2 .

If the two events have the same value of ct in the frame of reference considered, Δ s would correspond to the distance 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 = v t , 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 = ( d x ) 2 + ( d y ) 2 + ( d z ) 2 c 2 ( d t ) 2 .

Just as the distance Δ r is invariant under rotation of the space axes, the space-time interval:

Δ s 2 = ( Δ x ) 2 + ( Δ y ) 2 + ( Δ z ) 2 ( c Δ t ) 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:

Δ s 2 = ( Δ x ) 2 + ( Δ y ) 2 + ( Δ z ) 2 ( c Δ t ) 2 = ( Δ x + v Δ t 1 v 2 / c 2 ) 2 + ( Δ y ) 2 + ( Δ z ) 2 ( c Δ t + v Δ x c 2 1 v 2 / c 2 ) 2 = ( Δ x ) 2 + ( Δ y ) 2 + ( Δ z ) 2 ( c Δ t ) 2 = Δ s 2 .

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:

Lorentz transformation Axis rotation around z -axis ( x , t coordinates): ( x , y coordinates): x = ( γ ) x + ( β γ ) c t x = ( cos θ ) x + ( sin θ ) y c t = ( β γ ) x + ( γ ) c t y = ( sin θ ) x + ( cos θ ) y

where γ = 1 1 β 2 ; β = v / 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 Δ r and Δ s , differ. Although Δ r is invariant under spatial rotations and 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 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 Δ τ 2 as Δ s 2 . The significance of Δ τ as just defined follows by noting that in a frame of reference where the two events occur at the same location, we have Δ x = Δ y = Δ z = 0 and therefore (from the equation for Δ s 2 = Δ τ 2 ) :

Questions & Answers

For the question about the scuba instructor's head above the pool, how did you arrive at this answer? What is the process?
Evan Reply
as a free falling object increases speed what is happening to the acceleration
Success Reply
of course g is constant
Alwielland
acceleration also inc
Usman
which paper will be subjective and which one objective
jay
normal distributiin of errors report
Dennis
normal distribution of errors
Dennis
photo electrons doesn't emmit when electrons are free to move on surface of metal why?
Rafi Reply
What would be the minimum work function of a metal have to be for visible light(400-700)nm to ejected photoelectrons?
Mohammed Reply
give any fix value to wave length
Rafi
40 cm into change mm
Arhaan Reply
40cm=40.0×10^-2m =400.0×10^-3m =400mm. that cap(^) I have used above is to the power.
Prema
i.e. 10to the power -2 in the first line and 10 to the power -3 in the the second line.
Prema
there is mistake in my first msg correction is 40cm=40.0×10^-2m =400.0×10^-3m =400mm. sorry for the mistake friends.
Prema
40cm=40.0×10^-2m =400.0×10^-3m =400mm.
Prema
this msg is out of mistake. sorry friends​.
Prema
what is physics?
sisay Reply
why we have physics
Anil Reply
because is the study of mater and natural world
John
because physics is nature. it explains the laws of nature. some laws already discovered. some laws yet to be discovered.
Yoblaze
is this a physics forum
Physics Reply
explain l-s coupling
Depk Reply
how can we say dirac equation is also called a relativistic equation in one word
preeti Reply
what is the electronic configration of Al
usman Reply
what's the signeficance of dirac equetion.?
Sibghat Reply
what is the effect of heat on refractive index
Nepal Reply
As refractive index depend on other factors also but if we supply heat on any system or media its refractive index decrease. i.e. it is inversely proportional to the heat.
ganesh
you are correct
Priyojit
law of multiple
Wahid
if we heated the ice then the refractive index be change from natural water
Nepal
can someone explain normalization condition
Priyojit Reply
please tell
Swati
yes
Chemist
1 millimeter is How many metres
Darling Reply
1millimeter =0.001metre
Gitanjali
Practice Key Terms 4

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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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