Physics Lesson 18.5.1 - The Spacetime Lorentz Transformations

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Welcome to our Physics lesson on The Spacetime Lorentz Transformations, this is the first lesson of our suite of physics lessons covering the topic of Lorentz Transformations, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

The Spacetime Lorentz Transformations

Now, let's find in a simple way to obtain the equations which show the transformation of spacetime coordinates of physical events observed in different inertial frames of reference. The need for such a new transformation is evident, especially now that we already are aware for the ultimate speed of light, which must be the same in all systems. Another thing that requires a new approach on relative transformations of spacetime coordinates is the time dilation and length contraction - two elements not prevised in Galilean transformations, in which the time is absolute. Obviously, Galilei could not imagine these things, because they are observed only when objects are moving very fast, at speeds comparable to the speed of light; for V << c such effects are almost zero. Therefore, we must find a set of transformations, which include the Galilean transformations but as a special case, where the time be the same in all inertial systems only in such a case but not in general.

Just like in the other situations discussed so far, we consider a system S' moving due right at velocity V relative to another system S assumed at rest. Obviously, the system S moves at velocity -V relative to the system S' (i.e. the two systems move at the same speed but in opposite direction relative to each other). For simplicity, we will consider any movement only due X and X', despite the two inertial systems S' and S contain all three spatial dimensions, X', Y', Z' and X, Y, Z respectively.

Let's suppose an event appears "somewhere" in S at coordinates x, y, z at a given instant t. The same event appears in S' at coordinates x', y', z' at the instant t'. We choose as time origin the instant in which there is a convergence of origins of the two systems S and S'. In this instant, t₀ = t₀' = 0. Then, the systems separate from each other at velocity V (S' slides away from S in the X-direction at velocity V). Moreover, since now we are dealing with very fast movements, t ≠ t' despite the time origin is the same for both systems.

Let's suppose the event P (for example, the generation of an elementary particle, for which we will discuss in Section 21) does appear in the instant t for S and t' for S' at the corresponding point P shown in the figure. When both systems were at the same position, the particle did not exist yet. Obviously, when the particle is generated, the system S' has moved due right of S by V · t while the system S has not moved. It is evident that the x-coordinate of P at the moment of its generation, is

This is because the origin O' of the system S' is displaced by V · t from the origin O of the system S, while the length has been contracted by the known factor

Rearranging the previous formula, we obtain for the coordinate of the point (event) P in S':

Even if we assume S' at rest and take the system S as moving at -V relative to S', nothing changes except the sign. In this case, we obtain

Combining the last two equations, we obtain

Thus,

Hence, we obtain for the time t' of the event measured in S':

Taking y = y' and z = z' as explained earlier, we obtain the Lorentz Transformation of spacetime coordinates:

These simple but very famous equations are named after Hendrik Lorentz - the famous Dutch scientist, who in 1902 was the first who published them. The only drawback is that Lorentz did not consider the time t' as a real time, still relying on the false concept of cosmic ether.

Example 1

An airplane makes a 6000 km flight at 400 m/s (linear motion at constant speed) when viewed from Earth.

1. How much does the flight last for an observer at rest on the Earth surface?
2. How much does the flight last for the passengers?
3. What conclusion do you draw from the two results obtained at (a) and (b)?

Solution 1

Clues:

x = d = 6000 km = 6 000 000 m = 6 × 10⁶ m
V = 400 m/s = 4 × 10² m/s
(c = 3 × 10⁸ m/s)
t = ?
t' = ?

1. Since the airplane travels a 6000 km distance at 400 m/s, the flight lasts for
   t = d/V
   = 6000 km/400 m/s
   = 6 000 000 m/400 m/s
   = 15 000 s
   = 1.5 × 10⁴ s
2. For the passengers, the flight lasts for
   t' = t - x ∙ V/c²/√1 - V²/c²
   where the distance d given in the clues is denoted by x in the formula. Thus,
   t' = (1.5 × 10⁴ s) - (6 × 10⁶ m) ∙ (4 × 10² m/s)/(3 × 10⁸ m/s)²/√1 - (4 × 10² m/s)²/(3 × 10⁸ m/s)²
   = (15 000 - 0.0000000278) s
3. As you see, the time difference is very small. Therefore, for speeds much smaller than light speed, it is not suitable to use the relativistic approach, as the results are more or less the same.

You have reached the end of Physics lesson 18.5.1 The Spacetime Lorentz Transformations. There are 4 lessons in this physics tutorial covering Lorentz Transformations, you can access all the lessons from this tutorial below.

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6. Continuing learning relativity - read our next physics tutorial: Relativistic Dynamics. Mass, Impulse and Energy in Relativity

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