Video Transcript
In this video, our topic is events in Minkowski diagrams. We’ll learn what events are, how they appear on these diagrams, and how to compare them from one inertial reference frame to another. This will lead us to an understanding of the phenomena of time dilation and length contraction.
As we get started, we can define an event as an occurrence at a specific time and a specific place. So, if we display the time and position axes of some inertial reference frame on a Minkowski diagram, then an event could look like this. This is an occurrence that happens at a specific time, this time, and a specific place, here. An example of an event might be, say, a light bulb flashing briefly on and then off from a certain location. Any number of events could be occurring in this reference frame. Say we have an event here as well as one here. Each event has its own particular combination of space and time coordinates.
If we give these events names, say, event 𝐴, 𝐵, and 𝐶, we see though that in some cases there is agreement on these coordinates. For example, events 𝐵 and 𝐶 both take place at the same moment in time. We say then that these are simultaneous events. Notice also that events 𝐴 and 𝐵 occur at the same location. They happen at different time values separated by this amount, but we could say they are colocated. When multiple events happen at the same location, we say that they lie along the same line of constant position. Similarly, simultaneous events lie along the same line of constant time. Later on, we’ll see how lines of constant position and time in a given reference frame can be used to measure separations in time as well as space between different events.
For now though, let’s consider instead that for these three events in our inertial reference frame, the times at which they actually occur may be different from the times at which they are perceived to occur. We’ve seen, for example, that event 𝐵 actually takes place at this time value and at this location. But imagine now that we position ourselves at 𝑋 equals zero in this reference frame and we look to observe this event taking place. The only way we could perceive an event taking place is if light travels from the event to our eye. So say that event 𝐵 as it happens at this actual time value does give off a flash of light.
We know that in a Minkowski diagram, a particle moving at the speed of light follows a 45-degree path. So light given off from event 𝐵 if it were moving forward in position would follow this 45-degree line. But since we’re standing back at 𝑋 equals zero, we need light to move backward in space while still moving forward in time. The world line followed by this light would look this way. Notice that it’s moving forward in time in the positive direction on the 𝑐𝑡-axis but backward in position towards negative position values.
We’ll observe this ray of light when it reaches the position 𝑋 equals zero. We see that happens at this time value, meaning that there’s a time difference between when event 𝐵 actually takes place and when we perceive it to. We see something similar going on if we consider making an observation of event 𝐶. Once more to perceive this event, we need light from it to reach our position at 𝑋 equals zero. If event 𝐶 consisted of a brief flash of light, then this light would follow this world line in moving forward in position and this one in moving back toward us. The light from event 𝐶 then would reach us at this time value here, which we see is very different from the actual time at which it takes place. In reality, events 𝐵 and 𝐶 are simultaneous. Yet, if we rely on our perceptions to tell us when they occur, we would see a time interval — we’ve called it 𝑡 sub 𝐵𝐶 — between them.
Taking this one step further, let’s say we had a fourth event taking place and that it happened along this world line of the light from event 𝐶. In this case, the light from event 𝐶 and the light from this new event — we’re calling it event 𝐷 — would reach the position 𝑋 equals zero at the same time value. We would therefore perceive them to be simultaneous. Though by comparing the actual time values at which these two events take place, we can see that really they’re not. We can summarize this by saying that perceived event times and actual event times may not agree and that this disagreement would be due to the travel time of light.
Now, if we go back to just having events 𝐴, 𝐵, and 𝐶, we can say that so far we’ve analyzed these events from a reference frame that’s at rest with respect to them. This means that the locations of these three events in this frame of reference don’t change over time. But we know that Minkowski diagrams in general allow us to display multiple inertial reference frames. Let’s add then an inertial reference frame, which we’ll say is moving at half the speed of light with respect to our rest frame. The orange 𝑐𝑡 prime and 𝑋 prime axes will represent this frame. And let’s now compare the order in which our events 𝐴, 𝐵, and 𝐶 occur in these two different reference frames.
Before we do that though, it’s worth pointing out that the notion of lines of constant time and constant position apply to any inertial reference frame. We’ve seen what those look like in what we can call our unprimed reference frame represented by the blue axes. And to draw similar such lines in the primed frame represented by the orange axes, we can recall first that any line of constant time must be parallel to the 𝑋 prime axis which represents a time of 𝑡 prime equals zero. Therefore, one line of constant time in our primed reference frame would look like this.
And second, we recall that a line of constant position must be parallel to the 𝑐𝑡 prime axis. We know this because this axis corresponds to a constant position in the primed reference frame of 𝑋 prime equals zero. An example line of constant position then in the primed frame would look this way. Knowing this, let’s return to our goal of determining the order of events first in the unprimed frame and then in the primed frame.
Since we’re looking to understand the relationship of events 𝐴, 𝐵, and 𝐶 in time, we can start by drawing a line of constant time at a position of 𝑡 equals zero. In our unprimed reference frame, none of the three events have yet occurred. So, we’ll slide this line of constant time up, keeping the same slope. We do this until the line reaches an event. In this reference frame, we see that incidence with one event corresponds to incidence with two of them, events 𝐵 and 𝐶. And that’s because in this frame, they’re simultaneous.
We then continue to slide our line of constant time up, advancing it in time until it reaches our next event, event 𝐴. That done, we can then apply a similar sweeping technique in the primed reference frame. Notice though that if we put a line of constant time at 𝑡 prime equals zero, we’ve already passed by one event. The fact that in this frame event 𝐶 occurs at a negative time is not a problem. It just means that from where we’ve defined 𝑡 prime equals zero, this event occurs before that. In our primed reference frame then, event 𝐶 by itself occurs first. Knowing this, we then sweep our line of constant time up, being careful to keep its slope constant. Using this method, the next event we encounter is event 𝐵 and then lastly event 𝐴.
What we’ve learned then is a technique for determining the order in which events occur in any inertial reference frame. Notice also that for events 𝐴, 𝐵, and 𝐶, the order of events is not the same in our two frames and that this difference is the difference in actual measured values for these events. That is, it doesn’t have to do with event perception, which depends on the travel time of light. The way that space and time are mapped out in these two different inertial reference frames is itself different. This leads to the fact that in the unprimed reference frame, events 𝐵 and 𝐶 are simultaneous, while in the primed frame, they are not.
To better understand this difference, we’re going to draw grids on each one of our inertial reference frames. These grids will consist of lines of constant time and position in those frames, and they’ll show us how space and time are evenly divided in each one. We’ll start with a grid in our unprimed reference frame corresponding to our blue axes. First, we’ll sketch in evenly spaced lines of constant position and then add in identically spaced lines of constant time. We can say that on either axis, the distance between adjacent grid lines corresponds to movement through one unit of either time or space. This means that we’ve defined the basic unit of travel in our unprimed reference frame.
If we then move into the primed reference frame represented by the orange axes, once more, we sketch in evenly spaced lines of constant position and then lines of constant time. Our grid spaces reflect the fact that our primed axes are not at 90 degrees to one another. We can see then how moving from our unprimed into our primed frame changes the layout of space and time.
And notice something else. If we start at the origin of both sets of axes, we can see that, for example, one unit of space in our unprimed coordinate system corresponds to this distance, while moving into the prime frame, one unit of distance corresponds to this length. These lengths are clearly different, and yet this accurately reflects how space is measured in these two different reference frames.
The same thing is true for time. Here’s one unit of time in our unprimed reference frame, and here’s one unit in the primed frame, again corresponding to different lengths. These differences in space and time distances, which again are not due to perception but are actually embedded in the two inertial reference frames, lead to the very interesting phenomena of time dilation and length contraction.
These effects can be calculated mathematically using equations, but to see them on a Minkowski diagram requires a very carefully drawn set of grids. To make these effects clearly visible, we’re going to use grids that are drawn with machine precision. Here, we once again have two inertial reference frames: the unprimed frame in black and the primed frame in blue. We can say that relative to the two events 𝑄 and 𝑃, the black frame is at rest and the blue frame is in motion. In each frame, a series of evenly spaced lines of constant time have been drawn. Adjacent lines then represent a difference in one unit along the time axis.
Considering once more the origin of these two pairs of axes, notice that once again one unit of time in the unprimed reference frame is not the same as one unit of time in the primed frame. This will have an effect, we’ll see, on how far apart in time these two events 𝑃 and 𝑄 are measured to occur in the two different frames. First, in the unprimed reference frame, we see that these two events are separated by exactly one unit of time. But then, looking in the blue primed frame, we see event 𝑃 occurring along this line of constant time but that event 𝑄 occurs farther away than one unit of time from this point.
In other words, in the primed reference frame, these two events are separated by more than one time unit. This means that if we took a stopwatch into the unprimed reference frame and measured the time difference between these two events, if we got a time of, say, one second and that this time interval was based on the actual times of events 𝑃 and 𝑄, then if we next stepped into the primed reference frame and used the same stopwatch to measure the time interval between the events, we would measure a time of more than one second.
So here’s a way that we can summarize time dilation. If we call the difference in time between events in our unprimed reference frame Δ𝑡 and the difference in time between those same two events in our primed frame Δ𝑡 prime, then we’re seeing that time dilation means that Δ𝑡 is less than Δ𝑡 prime. Note that this is true when our unprimed reference frame is at rest with respect to the events under consideration. So, time dilation means that events seem to be stretched out in time as we move faster and faster relative to them in an inertial reference frame.
Next, let’s consider the related phenomenon of length contraction. Say we have the same two frames of reference, the unprimed and the primed, but now our two events 𝑃 and 𝑄 are at these times and locations. And now, instead of having lines of constant time drawn in for each reference frame, we have lines of constant position. Once more, these lines show us even intervals in these two reference frames. If we look at the separation in position between events 𝑃 and 𝑄, we can call that separation Δ𝑋. We see that it’s equal to exactly two units in the unprimed frame.
Then, looking at the primed blue frame, we see that event 𝑃 occurs at this location, while event 𝑄 occurs at this one. In this frame of reference, this distance interval is less than two units. So if we call this interval Δ𝑋 prime, we can see that Δ𝑋 is greater than Δ𝑋 prime, meaning that as we move into our primed frame from our unprimed, the actual distance in space separating these two events gets smaller. That’s length contraction. The greater the relative speed between our primed and unprimed frames, the more clear these effects of time dilation and length contraction are. In our everyday life, we don’t notice them because our frames of reference have such low relative speeds. Nonetheless, both phenomena do still occur.
Let’s summarize now what we’ve learned about events in Minkowski diagrams. In this lesson, we learned that an event is an occurrence at a specific time and location. We also saw that events lying along the same line of constant time are called simultaneous and events on the same line of constant position are colocated. We learned further that perceived and actual event order may differ due to the travel time of light but also that actual event order may differ in different inertial reference frames. Moreover, evenly spaced grids defining intervals of space and time can be drawn for any inertial reference frame. And events compared to cross reference frames in relative motion to one another show evidence of time dilation and length contraction. This is a summary of events in Minkowski diagrams.
