<< 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

where are the mcq
Fred Reply
ok
Giorgi
acids and bases
Navya
How does unpolarized light have electric vector randomly oriented in all directions.
Tanishq Reply
unpolarized light refers to a wave collection which has an equal distribution of electric field orientations for all directions
pro
In a grating, the angle of diffraction for second order maximum is 30°.When light of wavelength 5*10^-10cm is used. Calculate the number of lines per cm of the grating.
Micheal Reply
state the law of gravity 6
cletus Reply
what is cathodic protection
Ebe Reply
its just a technique used for the protection of a metal from corrosion by making it cathode of an electrochemical cell.
akif
what is interferometer
Sonu Reply
Show that n1Sino1=n2Sino2
javan Reply
what's propagation
Vikas Reply
is it in context of waves?
Edgar
It is the manner of motion of the energy whether mechanical(requiring elastic medium)or electromagnetic(non interference with medium)
Edgar
determine displacement cat any time t for a body of mass 2kg under a time varrying force ft=bt³+csinkt
Felix Reply
A round diaphragm S with diameter of d = 0.05 is used as light source in Michelson interferometer shown on the picture. The diaphragm is illuminated by parallel beam of monochromatic light with wavelength of λ = 0.6 μm. The distances are A B = 30, A C = 10 . The interference picture is in the form of concentric circles and is observed on the screen placed in the focal plane of the lens. Estimate the number of interference rings m observed near the main diffractive maximum.
Jyoti Reply
A Pb wire wound in a tight solenoid of diameter of 4.0 mm is cooled to a temperature of 5.0 K. The wire is connected in series with a 50-Ωresistor and a variable source of emf. As the emf is increased, what value does it have when the superconductivity of the wire is destroyed?
Rupal Reply
how does colour appear in thin films
Nwjwr Reply
hii
Sonu
hao
Naorem
hello
Naorem
hiiiiii
ram
🎓📖
Deepika
yaaa ☺
Deepika
ok
Naorem
hii
PALAK
in the wave equation y=Asin(kx-wt+¢) what does k and w stand for.
Kimani Reply
derivation of lateral shieft
James Reply
hi
Imran
total binding energy of ionic crystal at equilibrium is
All Reply
Practice Key Terms 4

Get the best University physics vol... course in your pocket!





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