### Video Transcript

In this video, our topic is
worldlines in Minkowski diagrams. We’re going to learn what Minkowski
diagrams, sometimes also called spacetime diagrams, are. And we’ll also learn how to plot
what are called worldlines on such a diagram.

Now as we mentioned, a Minkowski
diagram is also sometimes called a spacetime diagram. These were developed to help us
visualize the motion of objects with respect to the fastest any object could move,
the speed of light. Keep in mind, though, that we’ve
already seen many of what we could call spacetime diagrams, that is, the plot of the
position of some object versus its time.

For example, if we had some object
moving so that it covered this much distance in this amount of time, then we could
use this graph to get information about that moving object’s speed. Recall that when an object moves at
a constant speed, we can call it 𝑣, that speed is equal to the distance traveled
divided by the time taken to travel that distance. On our graph then, if we divide the
change in this object’s distance by its change in time, then we’ll have figured out
its speed 𝑣 all throughout this motion. Spacetime diagrams then are not
unfamiliar to us. But as we talk about speeds
approaching the speed of light, then a particular type of diagram called a Minkowski
diagram becomes helpful.

These are arranged a bit
differently, though, where, instead of time going on the horizontal axis, the
position 𝑥 does. And it’s on the vertical axis that
we plot time instead of position. And notice that here we have the
time values multiplied by a constant, the speed of light in vacuum. We choose this constant
deliberately so that our whole diagram is oriented around the speed of light. Note, though, that because 𝑐 is a
constant, we really are plotting time on this vertical axis. It’s just time multiplied by some
constant whose value we know.

And there’s an advantage to
plotting 𝑐 times 𝑡 instead of just 𝑡. If we look back at our equation for
the speed of an object, we can see that if we multiply both sides by the time 𝑡,
that cancels out on the right. And we find that the time times the
speed is equal to the distance traveled. This shows us that a speed
multiplied by a time is equal to a distance. And therefore, on our diagram here,
our speed multiplied by a time is equal to a distance 𝑥. This tells us that the units on our
vertical axis are the same as the units on our horizontal axis. They’re both distances. This makes it easy to compare
values on one axis with values on the other. And yet it’s still the case that we
can display the distance against time of some object or objects.

Now, this fact that a Minkowski
diagram plots time versus distance rather than distance versus time, like we saw
earlier, is one of the less intuitive and therefore more important to remember facts
about Minkowski diagrams. Now let’s see what object motion
would look like when it’s plotted on a Minkowski diagram. We can start out by imagining some
object positioned at 𝑥 equals zero and not moving. That is, it has a speed of
zero. On our Minkowski diagram over here,
that would mean that at time 𝑡 equals zero, our object is indeed at the origin, its
position is zero, and then as time passes, that is, as we move up the time axis, its
position stays the same.

This line that we’ve drawn showing
this object’s path through space and time is called its worldline. This word means something like a
trajectory, but it includes not just an object’s motion through space, but also its
motion through time. And that’s why we have a special
term for it. Notice that the slope of this
worldline is infinite. There’s a change in its vertical
extent, with no change in its horizontal extent. If we were working within the old
framework of plotting position versus time, then an infinite slope would be a
problem. That would indicate an object
moving infinitely fast. But here, because we’re plotting
time versus distance instead of the other way around, an infinite slope indicates an
object that’s not moving. And that’s just what we saw
here. Our object started out at 𝑥 equals
zero, and it stays there.

Now let’s imagine that our object
does begin at 𝑥 equals zero, but that it has some nonzero speed with which it moves
through space. If we were to plot the worldline of
that object on our Minkowski diagram, it might look like this. Note that in this case we’re seeing
that our object moves some distance here over some amount of time here. If we call that change in amount of
time 𝑐 times Δ𝑡 and the change in position Δ𝑥, then we can say that the slope of
this worldline here is equal to 𝑐Δ𝑡 divided by Δ𝑥. But then notice this; Δ𝑡 divided
by Δ𝑥, we’re reminded from our equation for speed, is the inverse of speed. In other words, it’s equal to one
over 𝑣, where 𝑣 is the speed of our object.

And this is true in general that
the slope or gradient of any worldline is equal to the speed of light 𝑐 divided by
the object’s speed. Note that this agrees with our
previous worldline, where our object had a speed of zero. Zero in the denominator here would
lead to an infinite slope. And that’s what we found. The fact that the slope of a
worldline is equal to the speed of light in vacuum divided by object’s speed has an
interesting consequence for Minkowski diagrams. Coming back to our sketch over
here, let’s say that now our object once again starts at the origin where 𝑥 is
zero. But it moves off of that spot with
a speed equal to the speed of light, the speed at which a massless particle like a
photon would travel.

To plot this worldline on our
Minkowski diagram, we can use our equation for slope. When 𝑣 is equal to the speed of
light 𝑐, the slope is equal to one, which tells us that the worldline for an object
starting at 𝑥 equals zero and moving with a constant speed equal to 𝑐 would make a
45-degree angle bisecting our two axes. This 45-degree line and in
particular its slope are a reference point for all worldlines that we might plot on
a Minkowski diagram. We’ve seen that for objects that
move at less than the speed of light, their slope is greater than one. So — and this is a bit
counterintuitive, so we must be careful — a slope of one is the lowest possible
slope allowable on a Minkowski diagram.

What we mean by that is if we saw a
worldline with a slope that’s less than one, say this one here, then that would
apply motion at a speed greater than the speed of light in vacuum, not physically
possible. So graphically, if we see a
worldline that starts out at the origin but has a slope below this 45-degree line,
we say that motion is not possible, while if we have a constant sloped worldline
above this 45-degree angle, that is possible. Here again, there’s a bit of a
counterintuitive hurdle to get over. And that hurdle comes from the fact
that we may be used to thinking of position versus time rather than time versus
position. On a Minkowski diagram, it’s slopes
that are greater than or equal to one that are allowed.

So far, we’ve only considered cases
of objects moving with constant speed. These result in worldlines that
have a constant slope all throughout. In general, though, the speed of an
object could change with time. For example, say some object begins
at 𝑥 equals zero and starts out moving quickly, close to the speed of light, but
then slows down over time like this. This worldline represents
physically possible motion because its slope, even though it changes in time, never
gets less than one. This is in contrast to a worldline
that might look like this, where an object starts out moving very slowly and then
speeds up rapidly and then slows back down.

Even though this worldline never
crosses over our 𝑣 equals 𝑐 45-degree angle, it still isn’t physically possible
because its slope, roughly in this region here, does become less than one. This worldline then is claiming
that for this interval of time, the object is moving faster than the speed of
light. For that reason, we say that the
worldline overall represents nonphysical motion. So for a Minkowski diagram, just as
would be the case for a position-versus-time graph, when we have a line that curves,
that means that object’s speed is changing in time. The difference between these cases,
though, comes down to the fact that on a Minkowski diagram at all points the slope
of a line must be greater than or equal to one.

Now, just as one more example of a
worldline on a Minkowski diagram, let’s say that our object, instead of starting out
at 𝑥 equals zero, starts out at some positive displacement. And then from there, it moves
forward in space at some speed 𝑣 less than the speed of light. If we were to draw that worldline
on our Minkowski diagram, it might look like this. At first, this may seem concerning
because we see this world line is in the region between zero and 45 degrees. But we remind ourselves that the
important reality about worldlines is not their values themselves, but rather their
slopes. The slope of this worldline at all
points is greater than one and therefore is allowable motion. It’s just motion that happened to
start at a position advanced of what we’ve called 𝑥 is equal to zero. As we consider worldlines on
Minkowski diagrams then, we’ll keep our eye on the slope and consider whether that
is greater than or equal to one.

Now, taking a step back from our
discussion so far, we can see that we’ve always been observing object motion with
respect to a certain vantage point. That is, we’ve set up a certain
frame of reference, which has a position in space and time that we’ve called
zero. This is completely normal. In any time we observe motion, we
must do it with respect to some reference frame. But it’s worth pointing out that
more than one frame of reference is possible. For example, say that we added a
second reference frame and we called the position here 𝑥 prime. The difference between 𝑥 prime and
𝑥 is that the reference frame we’ve called 𝑥 prime is moving with respect to
𝑥. And specifically, it’s moving with
a speed 𝑣, which is equal to the speed we’ve given our object.

If we were to observe this object
motion from our original frame of reference 𝑥, we would say that it’s moving with a
speed 𝑣 to the right. But now let’s imagine stepping into
this other reference frame, 𝑥 prime, which itself is moving to the right at that
same speed. Well, in that case, in our
particular frame of reference, this object would appear stationary because we’re
moving right along with it. Both of these reference frames
we’ve described here, 𝑥 and 𝑥 prime, are examples of what are called inertial
reference frames, or IRF for short. An inertial reference frame is a
vantage point for observing motion through space and time where that vantage point
is not accelerating. It could be at rest relative to
some moving object, like our 𝑥-coordinate system is, or it could be in motion at a
constant speed, as is 𝑥 prime.

Note, though, that whether we say a
given frame is stationary or in motion is relative. What matters is that it isn’t
accelerating, so long as its speed isn’t changing in time, it’s not accelerating,
and it is an inertial reference frame. Now we bring this up because it’s
possible to show more than one inertial reference frame on a Minkowski diagram. Now, even though here we’ve only
shown two different inertial reference frames with respect to motion through space,
a complete inertial reference frame will include motion through time as well. So, just as one frame of reference
will have 𝑐𝑡 and then 𝑥 on its two axes, we can imagine a second inertial
reference frame with 𝑐𝑡 prime and 𝑥 prime.

Knowing this, let’s say that we
want to analyze the motion of some object, that is, its worldline with respect to
these two separate frames of reference. We can do that by displaying them
both on our Minkowski diagram. We see that our diagram already
displays what we can call our unprimed coordinate frame, 𝑐𝑡 and 𝑥. To display what we can call the
primed coordinate frame on the same diagram, we can recall the fact that our primed
inertial reference frame is moving with a speed 𝑣 with respect to the unprimed
frame. This means that 𝑥 prime equals to
zero, at this point right here, is moving along to the right at a speed 𝑣.

If we were to think of that
location as a particle in motion through space and time, then we could plot its
worldline. And depending on the value of the
speed 𝑣, it would look something like this. But what we’re saying is that this
line actually represents the position of 𝑥 prime is equal to zero over time. And if we look at our unprimed
coordinate system, 𝑥 and 𝑐𝑡, we see what 𝑥 equals zero corresponds to. It corresponds to the 𝑐 times 𝑡
axis. All this to say that this line
we’ve drawn here which we thought of as the worldline of a particle sitting at 𝑥
prime equals zero is actually equal to the time axis of our primed inertial
reference frame. And just like we did before for
various worldlines, we note that the slope of this line is equal to the speed of
light 𝑐 divided by 𝑣, the speed with which this inertial reference frame moves
with respect to the unprimed frame.

Knowing this, we might think that
the 𝑥 prime axis would then be drawn perpendicular to the 𝑐𝑡 prime axis. And so it would look something like
this. But actually, this is not the way
we draw the 𝑥 prime axis because, for inertial reference frames in motion with
respect to ones we’re considering stationary, that motion results in what we could
think of as a squeezing together of the primed axes. The net effect of that is this. If we call the angle between our
𝑐𝑡 prime axis and our line 𝑣 equals 𝑐 𝜃, then our 𝑥 prime axis will actually
be the same angular distance off of a 45-degree line.

At this point then, we’ve
successfully included two inertial reference frames in our Minkowski diagram, one
we’re considering to be stationary, that’s the unprimed frame of reference, while
the primed inertial reference frame is moving with a speed 𝑣 with respect to the
unprimed. Now, the fact that the axes of our
primed inertial reference frame are not at 90 degrees to one another has a very
important consequence. To see what that is, say we
consider the end of the line we’ve drawn 𝑣 is equal to 𝑐, if we wanted to know
what time and position in space this corresponded to in our unprimed frame of
reference, then we could follow these dashed lines and see where they meet the 𝑐𝑡-
and 𝑥-axes. But if we wanted to find the space
and time coordinates of this point in our primed reference frame, we could not look
at similar intersection points along our primed axes. That’s because in our moving
inertial reference frame, space and time are treated differently.

Recall that this line here, our
𝑐𝑡 prime axis, corresponds to a position of 𝑥 prime is equal to zero. That means that this is a line of
constant 𝑥 prime value. And therefore, any other line that
has a constant 𝑥 prime value as well must have the same slope as this one. So if we wanted to find the 𝑥
prime coordinate of the point we’ve marked out right here, that would mean drawing a
line from that point which is parallel to this line here, finding out where it meets
the 𝑥 prime axis and then measuring out this distance to be the 𝑥 prime coordinate
of that point. And then if we want to find the
𝑐𝑡 prime coordinate of this point, we follow the same process. In this case, we can see that the
𝑥 prime axis corresponds to a 𝑡 prime equals zero line. By the way, this is called a line
of constant time.

So to find the 𝑐𝑡 prime
coordinate of this point here, we need to draw a line parallel to the 𝑥 prime axis
until it meets the 𝑐𝑡 prime axis. And then this distance here along
that axis would be the coordinate 𝑐𝑡 prime. And note that this change in time,
𝑐Δ𝑡 prime, and this change in space, Δ𝑥 prime, are not equal to the changes in
space and time in the other inertial reference frame, the unprimed one. This shows us that, depending on
what inertial reference frame we occupy, we’ll measure changes in space and changes
in time differently.

However, and this is a very
important last point, if we were to look at these two ratios, 𝑐Δ𝑡 over Δ𝑥 and
𝑐Δ𝑡 prime over Δ𝑥 prime, it’s actually the case that these ratios in this case
are equal. And the reason they’re equal is
because the point we’re considering is a point along the line 𝑣 is equal to 𝑐, the
speed of light in vacuum. When that is true, then regardless
of the inertial reference frame we’re using the change in time divided by the change
in position is equal to one. This tells us that an object moving
at the speed of light is observed the same way in all inertial reference frames. Note that this isn’t true, though,
for objects moving at other speeds.

Let’s now summarize what we’ve
learned about worldlines in Minkowski diagrams. In this lesson, we saw that a
Minkowski diagram is a way of displaying what are called object worldlines in
different inertial reference frames. A worldline we saw is a path
displaying an object’s trajectory in both space and time. And an inertial reference frame we
saw is a nonaccelerating reference for object motion, again through space as well as
time. We saw further that on a Minkowski
diagram, instead of plotting position versus time, we plot time multiplied by the
speed of light in vacuum versus position and that on such a plot, a worldline
representing motion at the speed of light and starting at the origin forms a
45-degree angle line. This implies that any worldline
must have a slope that’s greater than or equal to one for it to represent physically
possible motion. This is true whether an object’s
speed is constant or varying in time.

Lastly, we studied motion on a
Minkowski diagram in different frames of reference. And we saw that the motion of an
object moving at the speed of light appears the same in all inertial reference
frames. This is a summary of worldlines in
Minkowski diagrams.