Length and Dimension Deformation

Han Erim

May 7, 2012

LENGTH AND DIMENSION DEFORMATION

We never see the objects themselves, but always their image representations. In the Image and Source section, we had seen that an object and its image can be located at different coordinates. The electromagnetic waves — light — deliver the image to us. If there is a velocity difference between reference frames, a deformation occurs on the electromagnetic waves carrying the image. Accordingly, a deformation also forms on the image of the object, and the object may appear shorter, longer, or bent. Length deformation occurs not on the object itself, but always on the image. Length deformation also implies dimensional deformation. For a moving object, the effect is perceived, in general, as a compression or expansion of space. Length deformation is as interesting as it is enjoyable to explore.

In the Image and Source section, the main focus was on where the image of an object appears. Here, we will relate the conclusions from that section to the dimensions of the object, and thus see how length deformation forms.

Like other relativity effects, length deformation is closely related to and defined by the mathematics of (c+v)(c−v). Like all relativity effects, it occurs between frames moving relative to one another.

Principles of Seeing

To define length deformation, we must first address the stages of the act of seeing. Let’s consider our eye as a camera. Suppose it takes a single snapshot. Let’s refer to the moment the photo is taken as the moment of vision, and examine this photo frame. Naturally, we see many objects in a photo. If we had taken a photo of the sky under a tree, we would see both stars and branches of the tree in that photo. Within the same photo, there are traces of light coming from many different places and times. While the electromagnetic waves from the stars began their journey millions of years ago, those from the tree’s branches started only a few nanoseconds ago. This gives us the following insight: electromagnetic waves reach our eyes in groups. Each group contains countless electromagnetic waves that departed from various objects at various times. Naturally, what we perceive depends on the content of the group that reaches our eyes.

Length deformation is closely related to how this grouping of electromagnetic waves occurs. The rules by which the group is formed are the foundational knowledge of dimensional deformation.

Figure 1, How an electromagnetic wave group is formed:

We’ve seen that electromagnetic waves always move toward their targets at the speed “c”. Imagine, in your mind, a large translucent sphere with a shiny surface. Imagine you are inside the sphere, and its center is your eyes. Now imagine the sphere’s radius shrinking at the speed of light, moving toward you. As it shrinks, it passes through many objects around you. Now imagine that these objects "attach" their own electromagnetic waves to the surface of the sphere upon contact. The group I mentioned above is essentially the surface of this sphere. When the sphere fully collapses into your eye, your eye detects the electromagnetic waves that reached it and forms the image brought by the group.

Figure 2 Rule of signal group formation in moving frames.

For simplicity, I will refer to electromagnetic waves as signals from here on.

In this example, there is a car moving relative to the observer. When the car is in motion, the duration of signal group formation for the car changes. The signal group belonging to the car forms in a shorter time when it is moving away from the observer, and in a longer time when approaching the observer. Note that the speed of the sphere’s surface is relative to the observer. The sphere's surface does not move at speed c relative to the car; it moves at (c+v) or (c−v) depending on the direction of motion. Therefore, the time it takes for the signal group to form determines the amount of deformation on the image of the object.

We previously covered where the image (ghost) of an object appears in the Image and Source section. According to the observer’s reference frame, the coordinate where the signal was emitted determines where the object’s image (ghost) will be seen. Here, we apply this rule to every point of the car. As shown, the car will appear shorter or longer depending on its speed and direction of motion relative to the observer.

Since the observer is stationary in this case, the (c+v)(c−v) mathematics takes a back seat (on the car’s side).

Figure 3 In the previous example, the observer was stationary and the car was moving. Now let’s consider the case where the observer is moving and the car is stationary.

As a general rule, two reference systems move relative to one another, and it doesn’t matter which one is in motion. The result observed by the observer in the previous case will be the same here. However, in this case, we need to use the (c+v)(c−v) mathematics and the concept of fields to explain the situation. When conducting such analyses in a moving frame, two points must be paid special attention to:

1) Since the signals arrive at the observer, the center of the signal group’s circle is the observer. Because the observer is moving, the circle will move with them. The signals travel within the observer’s field and reach them at speed c.

2) The coordinates where the signals were emitted are defined relative to the observer’s reference system. These are points within the observer’s field. Therefore, they also move with the observer.

The coordinates at which the signals enter the observer’s field determine where the image of the car will appear. The observer will see the car’s image within these coordinates defined in their reference system. In the end, we obtain the same result as the previous figure. This figure and the previous one are fully equivalent.

Figure 4, THE MATHEMATICS OF LENGTH DEFORMATION

Here we see how to calculate length deformation.

Radio Button 1: Case where the observer is approaching the city: If the observer were stationary, a signal departing from the right would cover the city's length d1 at speed c in time t1. (d1 = c · t1)

However, since the observer is moving toward the city, in order for the signal to cross the entire city, it must now travel a distance d3 instead of d1. (Because the signal travels through the observer’s field.) The distance d3 is covered at speed c in time t2. (d3 = c · t2)

During the signal's travel time t2, the observer moves a distance d2 at speed v. (d2 = v · t2)

From this, we can calculate the ratio of length deformation. For frames approaching each other, length deformation is expressed with the following equation:

Observed length = Original length · c / (c − v)

Radio Button 2: Case where the observer is moving away from the city:
If the observer were stationary, a signal departing from the right would cover the city's length d1 at speed c in time t1. (d1 = c · t1)

But since the observer is moving away from the city, the signal only needs to cover a shorter distance d3 at speed c in time t2. (Because the signal travels within the observer’s field.) (d3 = c · t2)

During the signal's travel time t2, the observer covers a distance d2 at speed v. (d2 = v · t2)

Again, we can compute the length deformation similarly.

General formula for length deformation:
Observed length = Original length · c / (c ± v)
Use (−) if the frames are approaching, and (+) if they are moving apart.

As with all relativity effects, the (c+v)(c−v) mathematics is also determinative for length deformation.

Figure 5, EQUALITY

On this page, we see once more how length deformation occurs depending on the motion direction of the frames. As shown in the animation, whether the observer is moving and the ruler is stationary, or the ruler is moving and the observer is stationary, the result is exactly the same.

Figure 6, Length Deformation Along the X-Axis

The distribution of cross marks shows us how deformation appears on the image.

Figure 7, Length Deformation Along the Y-Axis

Although length deformation occurs in the direction of motion (let's call it the X-axis), it also has an effect along the Y-axis. Here, we observe the deformation on a tall vertical object.

Figure 8, Deformation in Rotational Motion

In rotating objects, we can understand how deformation occurs by observing the distribution of cross marks.

On Length Deformation

Here, we have seen how the speed difference between frames causes length deformation, and we’ve established a foundational understanding. Of course, this topic is not limited to what has been discussed so far. One can identify many subtopics stemming from length deformation and discover very interesting results.

Length deformation is an important component of relativistic effects. Since we perceive the universe the way we see it, dimensional deformation is in fact a factor that defines the kind of environment we live in. Naturally, one would need to move at extremely high speeds to feel this effect clearly. Who knows, maybe one day we will travel at speeds close to the speed of light and see the effect with our own eyes. Here, we only talked about its formation principles. Scientists in their respective fields will decide at which speed ranges this topic becomes important for them.

Perhaps we may soon see this effect accurately depicted in films and computer games. That would certainly be nice.