Lorentz Transformation: UNIZOR.COM - Relativity 4 All - Einstein View

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Lorentz Transformation

As in the previous lecture about time dilation, consider two inertial reference frames:
frame α with Cartesian coordinates X, Y, Z and time T
and frame β with Cartesian coordinates x, y, z and time t.

Assume that at time T=t=0 both frames coincide, that is their origins are at the same point in space and their corresponding axes overlap, but frame β is moving relative to α along the X-axis with constant speed v, so the velocity vector of β-frame in α-coordinates has components {v,0,0}.

Since the movement is relative, we can say at the same time that frame α is moving relative to β along its x-axis with speed −v, so the velocity vector of α-frame in β-coordinates has components {−v,0,0}.

We still rely on two main principles:
(a) Principle of Relativity that states that all the Physics laws must be the same, if expressed quantitatively, using corresponding coordinates and time, in all inertial reference frames.
(b) Speed of light is constant and the same in all inertial reference frames in empty space, we will use a symbol c for this universal constant.

Our current task is purely mathematical - express the transformation of space and time coordinates from one inertial reference frame to another that moves relative to the first, preserving the integrity of the above principles.

Since the direction of β-frame movement is along X-axis, coordinates y and z will be the same as, correspondingly, Y and Z. This simplifies our task, reducing it to transformation of only two coordinates - X in space and T in time into x and t.

We will look for a simple linear transformation from {X,T} system to {x,t} system of the form
x = p·X + q·T
t = r·X + s·T
where p, q, r and s are four unknown coefficients of transformation, which we are going to determine.

We should not add any constants into above transformations since coordinates (X=0,T=0) should transform into (x=0,t=0).

Equation 1
Since β-frame moves along X-axis of α-frame with speed v, its origin of space coordinate (point x=0) must at any moment of α-time T be on a distance v·T from the origin of coordinates of a stationary α-frame.
Hence, if X=v·T then x=0 for any T.
Therefore, 0 = p·v·T + q·T for any T. Hence, p·v + q = 0

Equation 2
Consider a ray of light issued at time T=t=0 from the point of coinciding origins of both reference frames in the direction of positive coordinates X and x.
Since the speed of light c is the same in both systems {X,T} and {x,t}, according to Principle of Relativity, an equation of the motion of the front of the light wave in the α-frame must be X=c·T and in the β-frame it is x=c·t.
Therefore, if X=c·T, then x=c·t.
Hence, p·c·T + q·T = r·c²·T + s·c·T for any T.
Hence, p·c + q = r·c² + s·c

Equation 3
Consider a ray of light issued at time T=t=0 from the point of coinciding origins of both reference frames in the direction of negative coordinates X and x.
Repeat the logic of a previous paragraph, getting
−p·c + q = r·c² − s·c

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