Lesson Video: Lorentz Transformation

In this video we learn how to map one moving frame of reference onto another stationary one in a classical setting, and use a Lorentz transformation to do this in a relativistic setting where the moving frame of reference travels at a significant proportion of the speed of light.


Video Transcript

In this video, we’re going to talk about the Lorentz transformation. What it is? Why it matters? And how to use it to solve problems. To start off, let’s recall a little bit about coordinate reference frames and transformations between them. Imagine that you are standing in a particular location. And a friend of yours is standing in another spot some distance away. From your perspective, from where you’re standing, you have a particular vantage point on the world. We could call it your frame of reference. Everything that you see happening happens with respect to this particular frame. Your friend also has a frame of reference, their own particular way of seeing things. And their frame is not the same as yours. In order to clarify and to keep track of the different frames involved, we can call your reference frame the unprimed reference frame. And your friend’s frame we’ll call the primed frame.

Let’s imagine that, in this setup, you take a stone and place it one meter in front of you in the positive 𝑥-direction. Now if somebody asked you “What’s the position of that stone?,” you would give its position in reference probably to your own frame. You might say the position of the stone is one, zero, zero. Now if we asked your friend the exact same question, “What’s the position of the stone?” He or she would give a different answer. That’s because the origins of these two different reference frames are not colocated. They’re not in the same spot. Given this, you may wonder, “Well then, how do we ever talk about the position of anything?” Often what we do is establish a third independent reference frame that all observers can agree on. Then we might then say that the true position of the stone is in reference to that independent frame.

Now the point here is that different frames of reference give rise to different values that we assign to physical phenomena. But we’d like to have a common language in which to speak about these phenomena, such as the position of an object or an object’s speed. This desire led to the creation of coordinate transformations. To get an idea for what a transformation might look like, imagine that your friend rather than being stationary is in motion with a speed 𝑣 in the positive 𝑥-prime direction. Let’s imagine further that you place the stone at the one-meter mark in the 𝑥-direction in your reference frame at a time value of 𝑡 equals one second. Let’s further imagine that your friend, starting initially at the same location as you, has been moving since 𝑡 equals zero at a speed of 𝑣 equals two meters per second in their positive 𝑥-prime direction.

So initially, the reference frames are on top of one another, colocated. But your friend is moving at a speed of two meters per second in the positive 𝑥-prime direction, while you stay still. In your reference frame, the 𝑥-, 𝑦-, 𝑧-, and 𝑡-coordinates of this event of putting the stone down are one, zero, zero, one. But what about in your friend’s reference frame. That is, what are 𝑥 prime, 𝑦 prime, 𝑧 prime, and 𝑡 prime for the same event? For starters, we know that since the motion is entirely in the 𝑥-prime direction, 𝑦 prime and 𝑧 prime won’t change from 𝑦 and 𝑧 in the unprimed frame. Those values are both zero, similarly, with 𝑡 and 𝑡 prime. If we make a classical assumption that time passes at the same rate in different reference frames, then there’s no difference between 𝑡 prime and 𝑡. They’re both one second.

Finally, we get to 𝑥 prime. This value is different from 𝑥 because of the speed of the primed reference frame. We can write that 𝑥 prime is equal to 𝑥 minus 𝑣 times 𝑡. And when we plug in the values for 𝑥, 𝑣, and 𝑡, we see that 𝑥 prime is negative one meter. We’ve just performed a classical or a Galilean transformation of coordinates. This transformation for motion that’s exclusively in the 𝑥-direction says that, across coordinate frames, the 𝑦-coordinate will not change. Neither will the 𝑧-coordinate, again assuming motion is only in the 𝑥-direction. That 𝑥 prime, the position of events in the primed coordinate frame, equals the speed of that frame minus 𝑥 times 𝑡. And that 𝑡 prime equals 𝑡. That time passes the same in both initial reference frames.

This series of coordinate transformations is what was used for hundreds and hundreds of years. And we never saw any issue with it until much more recently in the 20th century. It was around this time frame that we began to notice experimentally the subtle differences implied by relativistic effects. So the Galilean transformation, that it worked perfectly well and been used for so long, was still perfectly useful for classical cases where speeds are relatively low compared to the speed of light. But this transformation does have the flaw of not being able to account for relativistic effects. And that is where our topic comes in, the Lorentz transformation.

When no motion is involved in the 𝑦-prime and 𝑧-prime directions, the Lorentz transformation looks just like the classical version. But when it comes to time and directions in which motion does occur, we can see there’s a marked difference. It has to do with this Greek letter 𝛾. As well as in the time transformation this extra factor of 𝑣𝑥 over 𝑐 squared. Another name for 𝛾 is the Lorentz factor. And as an equation, 𝛾 is equal to one divided by the square root of one minus 𝑣 squared over 𝑐 squared. Where 𝑣 is the speed of one of the inertial reference frames relative to the other. And 𝑐 is the speed of light in vacuum. Now you may wonder, “What’s the reason for the particular form that 𝛾 takes, why this mathematical equation?” When Lorentz was deriving these transformations, he had one constraint. And that is that the speed of light is constant in all inertial reference frames. An inertial reference frame, remember, is a nonaccelerating frame. It’s this condition that leads to all the relativistic effects we observe: time dilation, length contraction, and so on. And it’s this condition that governs the mathematical form for 𝛾.

Speaking of that mathematical form, let’s take a look at a plot of the function 𝛾, the Lorentz factor. What we’ll do is plot 𝛾 versus speed 𝑣. And in particular, we’ll mark out where 𝛾 is one and where 𝑣 is the speed of light 𝑐. If we look back at the equation for 𝛾, we can see that when 𝑣 is zero, that is, when the speed of our reference frame is zero. The entire denominator of this fraction reduces to one and 𝛾 is one. Now what about when 𝑣 goes in the other direction at its upper limit of the speed of light? In that case, our denominator in this fraction approaches zero and 𝛾 approaches infinity. If we were to plot 𝛾 as a function of 𝑣 as 𝑣 gets closer to the speed of light, it would look something like this. With 𝛾 approaching infinity as 𝑣 approaches 𝑐, but never quite reaching it for a massive object.

As we look at this graph, it helps us understand better why the Galilean transformation has worked so well for so long. When speeds are not significantly close to 𝑐, then 𝛾 is effectively one. Which, for example, would reduce our Lorentz transformation for motion in the 𝑥-direction to a mathematically identical form as our classical transformation. So we really only start to see relativistic effects in the Lorentz transformation as our speed 𝑣 approaches the speed of light. With this background on the Lorentz transformation, let’s work a practice problem involving these concepts.

What is the Lorentz factor 𝛾 produced by a relative velocity of 0.250𝑐?

We can call that relative velocity, 0.250𝑐, 𝑣. And we want to solve in this exercise for the Lorentz factor, 𝛾. To do that, we can recall the mathematical form for this term. 𝛾 is defined as one divided by the square root of one minus 𝑣 squared over 𝑐 squared. In our case, we’ve been told the velocity 𝑣. So we’re ready to plug in at this equation to solve for 𝛾. When we do, we see under the square root sign that the factors of the speed of light 𝑐 cancel out, simplifying our denominator. When we enter this value on our calculator to solve for 𝛾, we find that, to three significant figures, it’s 1.03. So even when our speed is a quarter of the speed of light, 𝛾 is only slightly above one.

So in summary of the Lorentz transformation, this transformation accounts for relativistic effects. And therefore is an advancement beyond the Galilean transformation. It uses the Lorentz factor 𝛾, which is equal to one divided by the square root of one minus 𝑣 squared over 𝑐 squared. To account for the fact that the speed of light is constant in all nonaccelerating reference frames. And finally, this transformation is most useful at high speeds, that is, speeds that approach the speed of light. Ultimately, using the Lorentz transformation lets us move from one coordinate frame to another while accurately accounting for the effects of relativity.

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