<< Chapter < Page Chapter >> Page >

The lorentz transformation equations

The Galilean transformation nevertheless violates Einstein’s postulates, because the velocity equations state that a pulse of light moving with speed c along the x -axis would travel at speed c v in the other inertial frame. Specifically, the spherical pulse has radius r = c t at time t in the unprimed frame, and also has radius r = c t at time t in the primed frame. Expressing these relations in Cartesian coordinates gives

x 2 + y 2 + z 2 c 2 t 2 = 0 x 2 + y 2 + z 2 c 2 t 2 = 0 .

The left-hand sides of the two expressions can be set equal because both are zero. Because y = y and z = z , we obtain

x 2 c 2 t 2 = x 2 c 2 t 2 .

This cannot be satisfied for nonzero relative velocity v of the two frames if we assume the Galilean transformation results in t = t with x = x + v t .

To find the correct set of transformation equations, assume the two coordinate systems S and S in [link] . First suppose that an event occurs at ( x , 0 , 0 , t ) in S and at ( x , 0 , 0 , t ) in S , as depicted in the figure.

The axes of frames S and S prime are shown. S has axes x, y, and z. S prime is moving to the right with velocity v and has axes x prime, y prime and z prime. S and S prime are aligned along the horizontal x and x prime axes and are separated by a distance v t. An event on the horizontal x and x prime axes is indicated by a point which is a distance x from the y z plane of the S frame and a distance x prime from the y prime, z prime plane of the S prime frame.
An event occurs at ( x , 0, 0, t ) in S and at ( x , 0 , 0 , t ) in S . The Lorentz transformation equations relate events in the two systems.

Suppose that at the instant that the origins of the coordinate systems in S and S coincide, a flash bulb emits a spherically spreading pulse of light starting from the origin. At time t , an observer in S finds the origin of S to be at x = v t . With the help of a friend in S , the S observer also measures the distance from the event to the origin of S and finds it to be x 1 v 2 / c 2 . This follows because we have already shown the postulates of relativity to imply length contraction. Thus the position of the event in S is

x = v t + x 1 v 2 / c 2

and

x = x v t 1 v 2 / c 2 .

The postulates of relativity imply that the equation relating distance and time of the spherical wave front:

x 2 + y 2 + z 2 c 2 t 2 = 0

must apply both in terms of primed and unprimed coordinates, which was shown above to lead to [link] :

x 2 c 2 t 2 = x 2 c 2 t 2 .

We combine this with the equation relating x and x to obtain the relation between t and t :

t = t v x / c 2 1 v 2 / c 2 .

The equations relating the time and position of the events as seen in S are then

t = t + v x / c 2 1 v 2 / c 2 x = x + v t 1 v 2 / c 2 y = y z = z .

This set of equations, relating the position and time in the two inertial frames, is known as the Lorentz transformation    . They are named in honor of H.A. Lorentz (1853–1928), who first proposed them. Interestingly, he justified the transformation on what was eventually discovered to be a fallacious hypothesis. The correct theoretical basis is Einstein’s special theory of relativity.

The reverse transformation expresses the variables in S in terms of those in S . Simply interchanging the primed and unprimed variables and substituting gives:

t = t v x / c 2 1 v 2 / c 2 x = x v t 1 v 2 / c 2 y = y z = z .

Using the lorentz transformation for time

Spacecraft S is on its way to Alpha Centauri when Spacecraft S passes it at relative speed c /2. The captain of S sends a radio signal that lasts 1.2 s according to that ship’s clock. Use the Lorentz transformation to find the time interval of the signal measured by the communications officer of spaceship S .

Solution

  1. Identify the known: Δ t = t 2 t 1 = 1.2 s ; Δ x = x 2 x 1 = 0 .
  2. Identify the unknown: Δ t = t 2 t 1 .
  3. Express the answer as an equation. The time signal starts as ( x , t 1 ) and stops at ( x , t 2 ) . Note that the x coordinate of both events is the same because the clock is at rest in S . Write the first Lorentz transformation equation in terms of Δ t = t 2 t 1 , Δ x = x 2 x 1 , and similarly for the primed coordinates, as:
    Δ t = Δ t + v Δ x / c 2 1 v 2 c 2 .

    Because the position of the clock in S is fixed, Δ x = 0 , and the time interval Δ t becomes:
    Δ t = Δ t 1 v 2 c 2 .
  4. Do the calculation.
    With Δ t = 1.2 s this gives:
    Δ t = 1.2 s 1 ( 1 2 ) 2 = 1.6 s.

    Note that the Lorentz transformation reproduces the time dilation equation.
Got questions? Get instant answers now!

Questions & Answers

what is force
Afework Reply
The different examples for collision
Afework
What is polarization and there are type
Muhammed Reply
Polarization is the process of transforming unpolarized light into polarized light. types of polarization 1. linear polarization. 2. circular polarization. 3. elliptical polarization.
Eze
Describe what you would see when looking at a body whose temperature is increased from 1000 K to 1,000,000 K
Aishwarya Reply
how is tan ninety minus an angle equals to cot an angle?
Niicommey Reply
please I don't understand all about this things going on here
Jeremiah Reply
What is torque?
Matthew Reply
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment, moment of force, rotational force or turning effect, depending on the field of study.
Teka
Torque refers to the rotational force. i.e Torque = Force × radius.
Arun
Torque is the rotational equivalent of force . Specifically, it is a force exerted at a distance from an object's axis of rotation. In the same way that a force applied to an object will cause it to move linearly, a torque applied to an object will cause it to rotate around a pivot point.
Teka
Torque is the rotational equivalence of force . So, a net torque will cause an object to rotate with an angular acceleration. Because all rotational motions have an axis of rotation, a torque must be defined about a rotational axis. A torque is a force applied to a point on an object about the axis
Teka
When a missle is shot from one spaceship towards another, it leaves the first at 0.950c and approaches the other at 0.750c. what is the relative velocity of the two shipd
Marifel Reply
how to convert:m^3/s^2 all divided by kg to cm^3/s^2
Thibaza Reply
Is there any proof of existence of luminiferious aether ?
Zero Reply
mass conversion of 58.73kg =mg
Proactive Reply
is Space time fabric real
Godawari Reply
What's the relationship between the work function and the cut off frequency in the diagram above?
frankline Reply
due to the upthrust weight of the object varise with force in which the body fall into the water pendincular with the reflection of light with it
Gift
n=I/r
Gift
can someone explain what is going on here
falanga
so some pretty easy physics questions bring em
falanga
what is meant by fluctuated
Olasukanmi Reply
If n=cv then how v=cn? and if n=c/v then how v=cn?
Natanim
convert feet to metre
Mbah Reply
what is electrolysis
Mbah
Electrolysis is the chemical decomposition of electrolyte either in molten state or solution to conduct electricity
Ayomide
class ninekasindhtextbookurdusave
Ayesha Reply
can someone help explain why v2/c2 is =1/2 Using The Lorentz Transformation For Time Spacecraft S′ is on its way to Alpha Centauri when Spacecraft S passes it at relative speed c /2. The captain of S′ sends a radio signal that lasts 1.2 s according to that ship’s clock. Use the Lorentz transformati
Jennifer
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 3' conversation and receive update notifications?

Ask