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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


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 .


  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.
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Questions & Answers

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torque is actually the rotational equivalent of linear force....as this equation tells..... however in general terms it is the turning effect of applied force
Torque is moment of Force if a force is applied at some finite distance from COM of a body it produce rotation. For pure rotation we need to apply Couple not Torque...
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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.
Two sets of (1,2,3,4,5,6) dimensions create a proton, a neutron, and an electron. This is the primordial atom.
A 10kg mass lift to a height of 24m and release. what is the total energy of the system
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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.
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a thermodynamic quantity equivalent to the total heat content of a system
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What is the meaning of Nuclear Fission?
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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
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Ayebanifesunday Reply
light is a kind of radiation That stimulates sight brightness a source of illumination.
Electromagnet radiation creates space 7th, 8th, and 9th dimensions at the rate of c.
That is the reason that the speed of light is constant.
This creation of new space is "Dark Energy".
The first two sets of three dimensions, 1 through 6, are "Dark Matter".
As matter decays into luminous matter, a proton, a neutron, and an electron creat deuterium.
There are three sets of three protons, 9.
There are three sets of three neutrons, 9.
A free neutron decays into a proton, an electron, and a neutrino.
There are three sets of five neutrinoes, 15.
Neutrinoes are two dimensional.
A positron is composed of the first three dimensions.
An electron is composed of the second three dimensions.
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What is photoelectric
Hsssan Reply
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i.e. Solar pannel.
Photoelectric emission is the emission of electrons on a metal surface due to incident rays reflected on it
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?
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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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