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The Derivation of Lorentz Transformations

By Estefania Olaiz

The Lorentz transformations are a set of equations used in relativity to regard the space and time coordinates of two systems moving at the same speed. First introduced by Henrik A. Lorentz in 1904 in the paper “Electromagnetic phenomena in a system moving with any velocity less than that of light,” they formally describe the notion that space and time are dependent on an observer's relative motion.

To derive the transformations, we will examine the Galilean transformation of classical mechanics, relating time, position, velocity, and acceleration in different inertial frames. Then, from the Galilean transformation, which isn’t consistent with special relativity, we will derive the Lorentz transformations.

On another note, it is important to recognize that relativistic phenomena can be explained by the geometrical properties of four-dimensional space-time, in which Lorentz transformations correspond to rotations of axes. Notwithstanding, further explanation was omitted for the sake of brevity.

The Galilean Transformation Equations

An event is any physical occurrence determined by its time and location (x, y, z, t) relative to a primary inertial frame of reference S. A secondary frame of reference S′ moves with velocity v, currently expressed as an occurrence along the x-axis, with respect to the primary frame. It is best depicted by the incoming visual, where S is represented by O, and S′ is is represented by O′.

The relation between the time and coordinates in the two frames of reference should then be

x = x′ + vt, y = y′, z = z′

These equations are known collectively as the Galilean transformation. Therein lies the assumption that time measurements made by observers in S and S′ are the same. Ergo, we can differentiate these equations with respect to time.

ux = u′x+ v, uy + u′y, uz + u′z

ax = a′x, ay + a′y, az + a′z

We denote velocity by u to distinguish it from v, the relative velocity of two reference frames. Per frame, velocities differ whilst acceleration coincides. Additionally, since mass and distance remain constant, observers in both frames see the same forces, F = ma, and Newton’s second and third laws in all inertial frames.

The Lorentz transformations converge to the Galilean transformation when v < c, where velocities are slower than the speed of light.

The Lorentz Transformation Equations

Galilean invariance violates Einstein’s Theory of Relativity. In order to satisfy relativity’s principle in which the laws of physics remain the same in all inertial frames of reference, as well as Maxwell’s equations for electromagnetism, we must replace the Galilean transformation with the Lorentz transformation.

We see the velocity equations state that 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 = ct at time t in the unprimed frame and also has radius r′ = ct′ at time t′ in the primed frame. Denoting these relations in Cartesian coordinates, a pair of numbers in two dimensions that specify distances with respect to the axis, gives

x² + y² +z² - c²t² = 0

x′² + y′² +z′² - c²t′² = 0

Since y = y′, and z = z′ we obtain

x² - c²t² = x′² - c²t′²

To find the correct set of equations, suppose that events that occur at (x, 0, 0, t) in S and at (x′, 0, 0, t′) in S′, coincide. Additionally, assume a scenario in which a spherically spreading pulse of light is emitted the moment the origins of the coordinate systems in S and S′ coincide. At time t, an observer in S finds the origin of S′ to be at x = vt. Via the observations of S, the S′ observer measures the distance from the event to the origin of S′ and finds it to be x′ √ 1 - v²/c². This is plausible because of length contraction, where moving frames perceive a relative shrink. Thus the position of S is

x = vt + x′ √ 1- v²/c²

x′ = x - vt / √ 1-v²/c²

Now, we must employ the spherical wavefront equation, x² - c²t² = x² - c²t², in terms of primed and unprimed coordinates, x² - c²t² = x′² - c²t′². This is necessary since it will allow us to relate x and x′ to obtain the relation between t and t′.

t′ = t - vx / c² ∕ √ 1 - v² / c²

As seen S, the equations relating the time and position of events become

t = t′ +vx/c² / √ 1 - v²/c²

x = x′ + vt′ / √ 1 - v²/c²

y = y′

z = z′

By interchanging the primed and unprimed variables, the inverse transformation expresses the variables in S in terms of those in S′, resulting in

t′ = t +vx/c² / √ 1 - v²/c²

x′ = x + vt / √ 1 - v²/c²

y′ = y

z′ = z′

These are known as the Lorentz transformations. Wherefore, the time and position coordinates of different inertial frames of reference are related. Their application invalidates an absolute frame of reference, and ultimately, endows special relativity with its foundational characteristic, relativity. In studying their derivation, we can begin to comprehensively examine the premise at a conceptual level.



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