Special relativity is almost always presented through "time dilation" and "space contraction", which has already been covered in this series. While these are perfectly correct concepts, they reflect the antiquated tendency to treat space and time as separate entities. But now you will realize that reality is much deeper: for the theory of relativity, the only thing that makes sense is the interdependent relationship between space and time, the two come together in a physical entity: space-time.
This interdependence is graphically described by space-time diagrams, which not only facilitate the analysis of relative movements, but also allow us to have a better physical idea of what the past, present and future are. Hi, I'm Eudes Fileti. Welcome to the Verve Científica channel!
In this video, you will have your first contact with the space-time diagrams, which are a stage for the theory of relativity to take place. But before continuing, subscribe to the channel, because here you will find content like this, capable of improving your understanding of how the world works. Also activate the bell to never miss any of our videos.
SPACE-TIME Imagine that Pedro Parado is standing on the station platform, while Maria Móvel moves to the right in relation to the platform, along with the train carriage in which she is. If we took a print of the movement at each instant and stacked these prints on top of each other, they would form a sequence that would 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.
Peter follows a vertical line towards the future, while Mary, who is in motion, traces an inclined path. Each slice of this scheme indicates its location at a given time. Pedro remains in the same place in space, while Maria moves to the right.
This construction, which allows us to simultaneously describe the movement of both observers, is what we call space-time. As we have already seen in this series, both space and time are both distorted and even merged into one another, due to relative motion. In relativity, it is no longer true that space and time have separate, objective meanings.
What actually exists is spacetime, and our division of it into space and time is just a useful convention. In fact, one of the main reasons relativity has a reputation for being difficult to understand is that all of our intuitions train us to think of space and time as separate things. Considering this, it becomes clear the importance of space-time geometry and why it must be explored in detail; just as we will do in this and the next videos in this series.
RELATIVE MOVEMENT IN SPACE-TIME Note that Mary's movement, analyzed here, is described from the point of view of Pedro, who is standing on the station platform. But what would space-time be like when we analyze movement from Mary's point of view? To obtain a symmetrical scheme, simply slide all the prints, one after the other, to place it in the center of the scene.
Thus, in Maria's frame of reference, she and the carriage are stationary and Pedro is the one who appears to be moving backwards. By analyzing it this way, all we have done is perform a Galilean transformation. The sliding of the prints represents the change from one reference point to another.
Note that these notions of spacetime are not exclusive to Special Relativity; They are also valid for Newtonian mechanics and could even have been used since the 17th century, when physicists already understood well what relative motion was. For Peter, it is Mary who is moving, but for Mary, it is Peter who moves. Neither of these two statements is more valid than the other.
Movement is relative. An idea that becomes even clearer when we examine it in the light of space-time. SPACE-TIME DIAGRAMS Much of the intuition we have about distances, times and speeds, for example, simply does not work in relativity in the same way as it does in Classical Physics.
And that's why space- Time is such a useful mathematical tool to help us understand this new scenario. Spacetime consists of a four-dimensional continuum with three spatial dimensions and one temporal dimension. A point in space-time represents an event, an event.
A physical occurrence, which can happen in a defined place in space and at a defined instant in time. Such events could be, for example, the flashing of flashbulbs, the collision of two rockets or the explosion of a star. Just as space is the collection of all "places" or positions, spacetime is the collection of all "happenings" or events.
When considering space-time as a continuous set of four unified dimensions, questions arise about its nature. Is it flat or is it curved? Is it static or dynamic?
Is it finite or infinite? How to model or map spacetime? This is what the space-time diagram was designed for.
The Minkowski model, for example, describes a flat, static and infinite space-time. Albert Einstein, over the course of a decade, investigated how to incorporate gravity into his theory, and came to realize that space-time can also be curved and dynamic. And it is precisely this curvature, what we call "gravity" today.
This revolutionary idea resulted in the theory of general relativity. Thus, special relativity deals with a static and flat space-time, while general relativity deals with a dynamic space-time, where curvature generates gravity. EVENTS IN SPACETIME As I mentioned, in order to study events and processes of interest that occur in spacetime, we need to know how to map it using a mathematical map, which we project onto real spacetime, that is, using a "diagram of space-time".
In practice, a space-time diagram is a graph that records the entire history of an event or process. In it, we can graphically represent the spatial and temporal coordinates of many events in one or more inertial frames. For simplicity and clarity, frequency diagrams often present only the temporal component and a single spatial component, instead of three.
This makes its appearance two-dimensional and much simpler to analyze. The limitation of a two-dimensional space-time diagram is that it is restricted to events or movements that, although they can occur at any instant, are restricted only along a single direction in space. This simplification, however, does not in any way affect the understanding of the physical phenomena involved.
Spacetime diagrams highlight that we cannot separate space from time, since they are inherently intertwined. This intertwining, as we saw before, is expressed in mathematical terms by the Lorentz Transformations, which, as I will show below, play a crucial role in the construction of these diagrams. MINKOWSKI DIAGRAM All of the relativistic phenomena we have discussed so far in this series involve relative motions limited to a single direction.
Furthermore, all movements between the frames of reference were uniform, without the presence of gravitational effects. This physical condition is a simpler case and can be described by a special space-time diagram: the Minkowski diagram. Its name is given in honor of Hermann Minkowski, who was a Polish mathematician and former teacher of Einstein, and who gave an elegant interpretation to Relativity, in terms of a four-dimensional space-time.
Minkowski was the first to realize that Lorentz transformations could be better understood in the context of a non-Euclidean space, he was the first to propose the union of space and time into a single unified space-time. In doing so, he launched much of the mathematical work that makes up the formal basis of special and general relativity. He conceived of what is now known as Minkowski spacetime, which is the arena where special relativity takes place.
Unlike a trajectory graph, where each axis measures distance and each point is nothing more than a location in space, a Minkowski diagram presents a spatial axis and a temporal axis, and each of its points is no longer a mere position in space, but rather an event, a physical event. Naturally, the spatial location of each event is marked along the spatial axis, while its temporal location, that is, the instant at which the event occurred, is marked along the vertical axis. In other words, the x-axis is the set of events that exist simultaneously, at the same instant.
This axis is what we could call NOW. On the other hand, the t axis is the set of events that happen in the same place, an axis that we could call HERE. CHARACTERISTICS OF THE MINKOWSKI DIAGRAM In special relativity, we deal with very high speeds, which implies that the time it takes for the object to travel everyday distances is very short.
To give you an idea, the time it takes for light to travel 30 cm is just 1 nanosecond. Therefore, when we try to sketch a 'time x space' coordinate pair for a given event in the diagram, using conventional units such as seconds for time and meters for length, the line representing the movement of that body will have such a low slope that we would be unable to distinguish it from the x-axis itself. It is for this reason that in space-time diagrams, the vertical axis does not describe time itself, but rather time multiplied by a constant: the speed of light.
The speed of light converts units of time into units of space, so we can use the same scale and the same units for both axes. From then on, the new time coordinate also has distance dimensions, and we interpret it as the distance that the light travels in the specified time. Thus, when a particle travels for a value ct = 3m, we say that it traveled for 3m of time; that is, for a time equivalent to the time it takes for light to travel 3m.
This diagram shows four different events. Events A and D, for example, occur in the same location, at a position of x = 2 m of space, but at different times: while A occurs at t = 2 m of time, event D occurs at an instant of half a meter of space. time.
Note that the time interval could be measured with either clock, but it is clock 2 that determines the proper time interval, as this clock is in the same position where both events occur. In this framework, events A and B occur in different locations, but at the same time. Finally, event C occurred at some point in the past, since ct = -1 m time.
In this video you had contact with the basic characteristics of an important mathematical instrument, crucial for relativity: The space-time diagram. In particular I discussed the Minkowski diagram, which specifically describes special relativity. But know that there is an even more important aspect of the diagrams that I will only present in the next video: the world lines of the objects!
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