Those who are not yet familiar with Relativity may get confused when analyzing two frames of reference in the same space-time diagram. Why are these bars movable? Where do these tilted axes come from?
What does this intermediate fixed line mean? From now on you will realize that many relativistic phenomena can only be understood in terms of space-time. Then you will realize that understanding the diagrams that characterize space-time is a fundamental skill.
Hello everyone, I'm Eudes Fileti. Welcome to Verve Científica. In this video, I will explain in detail how to relate two different frames of reference in a single space-time diagram and I will resolve many difficulties that arise when analyzing space-time.
If this knowledge is important for your understanding of the world, subscribe to the channel now and activate your bell! REVISITING SPACE-TIME When introducing the concept of space-time, we consider the relative movement between two observers: Pedro Parado, who is at rest on the train station platform, and Maria Móvel, who moves to the right in relation to the platform , along with the train car in which it is located. We saw that, if we take prints of the movement at each instant and stack these prints on top of each other, they will form a sequence that will represent the flow of time.
In this sequence, both Pedro and Maria move towards the future, each drawing their own path in time and space. Pedro follows a vertical line towards the future, while Maria, who is in movement, traces an inclined movement. By the principle of relativity, we could have done the opposite as well.
We consider Mary to be still; and in this case, it was Peter who would move, in the opposite direction. And now it is Mary who would follow a vertical world line towards the future, while Peter would move backwards, tracing a world line with the opposite inclination. These graphical schemes illustrate space-time itself.
But this pictorial, illustrative view, although enlightening, is very inconvenient for the general treatment of relative movements. Imagine if for each problem we needed to sketch the entire physical situation, drawing each element of each scene. That would be impractical.
To avoid having to draw all the spatial elements, such as platforms, wagons and people, physicists adopt a concise, simple geometric and mathematical scheme, which contains only the essential elements for analyzing the problem. They use space-time diagrams. These diagrams have already been introduced here in this series, in these videos here, in which I discussed the movement of particles and their world lines through space-time in relation to a single reference frame.
Now I'm going to go further, I'm going to show how the analysis of the relative movement of two different frames of reference is carried out in the same space-time diagram! TWO REFERENCES IN ONE DIAGRAM To compare movements in different references, we first need to construct a space-time diagram that represents the two desired references at once, overlapping the axes of the second reference with the axes of the first. Let us return to the relative movement between Peter and Mary.
Their frames of reference, Peter's platform and Mary's wagon, are in relative motion along the x direction, with constant speed. Now suppose that, at the initial instant, the origins of the two frames of reference coincide. Furthermore, both synchronize their timers in their respective frames of reference, so that, at the initial instant, both are at zero.
As I mentioned, for simplicity, we suppress everything that is not essentially important for the mathematical analysis of the problem. We keep only the relevant information about the frames of reference in their relative motion. Furthermore, we will designate Peter's system by ct - x and Mary's system by ct' - x'.
Here we see that the space-time diagram for Pedro's reference frame, which is standing at the station, is of course the same as what we saw before in the previous video. But how should Mary's reference be represented in this same diagram? Well, to answer this we need to turn to the mathematics behind these diagrams.
The mathematics of spacetime diagrams is that contained in the Lorentz Transformations. That's what we discussed in this video here. These transformations tell us how to translate or convert the position and time coordinates between two different frames of reference into relative motion.
But before you apply the transformations, note two important facts about space-time diagrams. First, the x-axis is nothing more than the set of points in space-time that have zero time coordinates. Second, the time axis is nothing more than the set of points in space-time that have zero position coordinates.
Now, imagine that each of these axes can move, as in fact occurs in the relative movement of Mary and Peter. How do these two axes in the Mary diagram, which move to the right with constant speed, relate to the axes in the Peter diagram, which remain stationary? To know, we just need to imagine separately the movement of both the spatial and temporal origins of Mary's reference.
As the spatial origin of the reference frame moves relative to the platform, it generates a world line that is nothing more than the ct' axis. This axis is the set of all points of the spatial origin of Maria's system, that is, all points with x' = 0. You will be convinced of this when you substitute x' = 0 in the Lorentz transformation for the position.
A little algebra will lead you to a straight line whose inclination is the inverse of the parameter β which is the world line of the movement of the spatial origin, that is, the ct' axis. Now we must also imagine the course of the temporal origin of Mary's reference. As the temporal origin evolves in relation to the time recorded on the platform, it traces another world line that is nothing more than the x' axis.
This axis is the set of all points with t' = 0. Here we can see this when we substitute t' = 0 into the Lorentz transformation for time. A little algebra leads us to a straight line with inclination β in relation to the horizontal axis, which is the world line of the movement of the temporal origin, that is, the x' axis of Maria's reference frame.
Note that the two axes of Maria's mobile system were obtained as a particular case of the Lorentz Transformations and with this we see that Maria's system can be described as two axes superimposed on Pedro's system, mathematically characterizing the relative movement between the two. THE DEFORMATION OF THE MOVING REFERENCE You may question why, over time, the origins of the two systems do not distance themselves from each other, after all Pedro and Maria are moving in relation to each other. The reason for this is mere mathematical convenience!
Since spacetime is invariant under translation from its origin, then we always have the freedom to place origins on top of each other. And here we gave ourselves that luxury; We chose to keep the origins of the coordinates of the two systems always overlapping in the unified diagram, which makes all analyzes much easier. Now I will explain why the moving frame of reference deforms in relation to the fixed frame of reference.
Imagine at first that Maria's wagon was at rest. In this case, both diagrams would be perfectly superimposed; in other words, Mary's frame of reference would be the same as Peter's. Now, consider that the car increases its speed to the right, gradually and slowly.
The increase in relative speed causes Maria's frame of reference to distort, deform, both in time and space. Its axes, of time and space, are projected symmetrically on the line of maximum inclination, imposed by the limiting nature of the speed of light. The greater the speed, the greater the deformation.
In the limit, Maria's wagon would be traveling at the speed of light and its diagram would be reduced to just a world line of a beam of light. This coupled deformation of space and time is a direct consequence of the second postulate of relativity, which states that the speed of light is constant in all frames of reference, which in this example is indicated by the blue straight line of constant inclination. In other words, it is time and space that have to combine so that the speed of light always remains the same.
Of course, according to the principle of relativity, reciprocal movement will also have the same consequences. If Mary's frame of reference is considered to be at rest, then it is the diagram of Pedro that will distort in the opposite direction, traveling backwards, with its axes projecting symmetrically towards the world line of a beam of light moving to the left. EVENTS IN THE DUAL DIAGRAM If we need to describe an event P in this diagram, this event would have a well-defined set in the two coordinate systems, it would be given by the projection of this point on the axes in the usual way.
Thus, in relation to Pedro's system, the point would have x and ct coordinates. And in relation to Maria, x' and ct'. An important thing here!
Don't be confused by the fact that the x and x' axes no longer appear parallel; they are still parallel in space! But remember this is a space-time diagram, which is showing movement in both space and time. Another thing worth highlighting is that even in Mary's frame of reference, the world line for a beam of light still has inclination 1, as is also the case for Peter's frame of reference.
This is a mere mathematical consequence of the fact that the speed of light must be the same in both frames of reference. Thus, in all diagrams, the axes of the moving frame of reference need to adjust so that this line always has the same inclination in all inertial frames of reference. Using a spacetime diagram representing two inertial frames simultaneously, most kinematics problems in special relativity reduce to the task of finding coordinates of relevant events in the two frames of reference.
In the next video, I will introduce the space-time interval, which is another crucial concept in relativity and which is related to phenomena such as time dilation, space contraction and the relativity of simultaneity. This video ends here. If you liked it, give it a like now and subscribe to the channel.
Have Scientific Verve! Hugs and see you next time!
