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.
