Time Dilation

Han Erim

May 7, 2012

TIME DILATION

With the inclusion of the concept of Image and Source into the Alice Law in 2009, the Relativity Theory of the Alice Law gained significant momentum and reached its final form. Image and Source was indeed a very important topic and brought with it many new insights. Thanks to this, topics such as Time Dilation and Length Deformation became understandable.

Time Dilation is one of the foremost topics that occupy physics the most. Because it appears in many areas. Therefore, it is necessary to understand this topic well and to know the mechanism of its formation. In this section, we will see in full detail how and why time dilation occurs.

Time Dilation arises as a result of the electromagnetic interaction between relatively moving frames. Like all other relativistic effects, it is defined by the mathematics of (c+v)(c-v).

Figure 1, Let's imagine two clocks moving towards each other. Suppose a rod is placed on the protrusions on the minute hands of the clocks, and Alice is sitting on this rod. We assume that Alice's position always remains at the midpoint of both clocks.

Will these two clocks always work synchronously according to Alice? This is not a difficult question at all. Since both clocks are approaching and moving away at the same speed according to Alice, and they always remain at equal distances, the clocks will always work synchronously from Alice’s point of view.

Figure 2, The stopping of one clock and the movement of the other does not change the situation for Alice. Again, since both clocks approach and move away from Alice equally, and the distances of the clocks to Alice remain the same, both clocks will work equally according to Alice.

Actually, Alice observes a time dilation on the clocks. However, since this time dilation occurs equally on both clocks, they still operate equally according to Alice. But let’s proceed a bit slowly. Time dilation is a topic that needs careful attention and is somewhat difficult.

Figure 3, Now let's place a second observer on the clock on the left. Here, another character from Alice in Wonderland, Gryphon, will accompany us.

Now there is an interesting situation: When Gryphon compares the tick-tock intervals of the clock on the right, which is moving relative to him, with the intervals of the clock he is sitting on, he clearly detects a time dilation on the moving clock. That is, he sees that the tick-tock intervals of the two clocks are different. However, this is not the case for Alice. According to Alice, both clocks are working synchronously, and their tick-tock intervals are equal. How and why can such a thing happen? This will be the question we seek to answer.

To be able to answer this question, we first need to understand whether the time dilation observed – detected – measured by Gryphon is a real effect or merely a perceptual effect.

If the time dilation seen on the moving clocks emerges as a real effect (i.e., if it causes a change in the operating speed of the clock), then according to Gryphon, the two clocks must run at different speeds. In that case, the hour and minute hands of both clocks must rotate at different speeds. And in such a case, the inclination of the rod on which Alice is sitting must change.

However, we see that Alice is sitting at a point that does not allow us to make such a logical conclusion. Because, from Alice’s perspective, the fact that both clocks operate synchronously is a primary, definitive piece of information. According to Alice, the rod must always remain parallel to the ground.

Figure 4, Let’s definitively determine whether the inclination of the rod changes according to Gryphon. If we clarify this topic, we will have largely solved the issue.

Let’s mount a pen to the right end of the rod. While the rod is moving, let the pen mark its current position. The line that the pen draws will be definitive for both Alice's and Gryphon's reference frames.

It is clear that if the inclination of the rod is changing, the pen will draw an irregular sine curve; if not, it will draw a regular sine curve. We already know that, in Alice's reference system, the inclination of the rod does not change. Therefore, the pen must draw a regular sine wave for Alice's reference system. While the pen is drawing the line for Alice, it cannot simultaneously draw a different line for Gryphon. A pen cannot draw two different lines at once. The sine wave drawn must also be valid for Gryphon’s reference system. From this, we conclude: According to Gryphon as well, the inclination of the rod does not change.

The fact that the inclination of the rod does not change means that both clocks are always operating synchronously for both Alice and Gryphon. From this, we reach an important conclusion: Time dilation is not a real effect. Moving clocks do not mechanically slow down.

Figure 5, Now let’s address how Gryphon sees the rod. Since both ends of the rod are at different distances from Gryphon and the rod moves vertically up and down, Gryphon will never see the rod as straight.

Due to the motion of the rod, a deformation will occur in the rod's image, and as a result, Gryphon will perceive the rod as bent. What he sees is not the actual rod but its ghost. The clock's ghost will also be in a different position. We had already covered these in the Length Deformation section.

Of course, we are discussing principles here. The speed of light being 300,000 km/s does not easily allow us to perceive such effects. But since we are presenting the speed of light with very small velocity values here, we can see what kind of effects occur in all their detail. No matter how fast light is, such effects always occur, even if they are extremely small.

Figure 6, Now let’s examine how Gryphon sees the clock. Gryphon will not see the actual clock but the image of the clock. A transmitted signal will reach Gryphon according to the rules we previously discussed. (I use the term signal to refer to electromagnetic waves.)

In this animation, only one clock signal is considered. After the clock emits the signal, it continues moving. When the signal reaches Gryphon, he sees the image of the clock at the coordinate where the signal was emitted. The hands on the dial of the observed clock will point to the moment when the signal was transmitted.

Figure 7, Clearly, to determine whether time dilation occurs on a clock, we must consider successive signals. Time dilation can only be detected by measuring the interval between consecutive arriving signals. As you might guess, Gryphon's observation of time dilation is related to a change in the duration between two consecutive signals that reach him.

I created this animation to give a preliminary idea. Suppose the clock emits a new signal every second. When one signal reaches him, Gryphon sees the clock in a new position with a new value on its dial. But let’s be careful — Gryphon does not see the actual clock, he sees its ghost.

PRINCIPLE

The animation we will base everything on contains all the knowledge related to Time Dilation. From now on, we will examine the details of this animation step by step. First, I would like to explain the principle on which the animation is based.

Principle:

Let’s start the animation.

The clock that Alice is sitting on is approaching or moving away from Gryphon. During this time, it emits signals at equal time intervals. The red hour and minute hands show the location and time of each signal’s emission. The signals are numbered. We can see where and when each numbered signal was emitted, which ones are en route, and which one has reached Gryphon.

The clock showing Alice’s Ghost follows the signals left by Alice’s actual clock. The ghost's motion follows the same rule we saw on the previous page. For example, when signal number 4 reaches Gryphon, the Ghost will be at the coordinate where signal number 4 was emitted, and the dial will display the moment it was sent. I repeat this often, but let’s note again: Gryphon sees the clock of the ghost, not the actual clock.

To prevent visual clutter in the animation, only one out of every 25 signals is visible. You can change the visible signal interval using the Numeric Stepper below. Invisible signals continue working in the background, ensuring smooth animation.

The transparent gray rod oscillating up and down indicates that the clocks at Alice's and Gryphon's positions are working synchronously. (This is the rod that Alice is sitting on, as mentioned at the beginning.)

You can adjust Alice’s speed using the sliding button on the left side. A value of 5 pixels/frame is used for the speed of light in the animation. Alice’s speed can be varied between 0–5 pixels/frame. Therefore, the animation can illustrate what happens even at or near light-speed movement.

Of course, here we are focusing only on the topic of time dilation. Although length deformation also occurs alongside time dilation, it has not been included in these animations.

INTERVAL

The signals emitted by Alice’s clock travel across Gryphon’s field at speed c and reach him. For Gryphon, the duration between two successive arriving signals determines the time dilation. In the table second from the left at the bottom, we can see the time intervals between signals.

Since the clocks at the original sources operate synchronously, both sources have equal intervals. However, Alice’s movement alters the distance between signals traveling through Gryphon’s field. As a result, Gryphon observes Time Dilation. When Gryphon measures the intervals of the signals arriving to him, we again see the result in the table.

If Alice is moving away from Gryphon, the signal intervals increase.
If Alice is approaching Gryphon, the signal intervals shorten.
Only if Alice is stationary will the signal intervals be equal.

THE MATHEMATICS OF TIME DILATION

Let’s click the Math Button. As we will see, Time Dilation is expressed and calculated using the mathematics of (c+v)(c-v). In the pop-up window, you can see how the time difference between two signals changes.

SPEED OF THE GHOST

Not sure if you’ve noticed, but now pay attention to the speed of the ghost. When Alice is moving away from Gryphon, the ghost moves more slowly; when Alice is approaching Gryphon, the ghost moves faster.

To see this more clearly, set Alice’s speed to a value close to the speed of light and follow the value in the text box that shows the ghost’s speed. You’ll see that the ghost can move much faster than the speed of light. The image is not a material object, nor is it an electromagnetic wave. The ghost is a virtual reality created by electromagnetic waves. Therefore, there is no limit like the speed of light for it. The ghost’s speed can easily exceed the speed of light.

If you click the Speed Button, you can observe how the ghost’s speed changes in the pop-up window. Again, the ghost’s speed is expressed by the mathematics of (c+v)(c-v).

COMPARISON

On this page, we will compare the clocks. After starting the animation, click the Compare button. In the pop-up window, we will see the clocks side by side. In the background, the animation will continue running. When Gryphon compares his own clock with Alice’s clock (the Ghost), we will be able to more easily see what situation emerges in the opened window.

Gryphon sees that when Alice is approaching him, her clock runs faster than normal, and when she is moving away, her clock runs slower. Of course, Gryphon sees the clock in the ghost, not the original one. By now, we already know that the clocks at the sources (originals) operate synchronously.

As a result, time dilation occurs in both directions — as speeding up and slowing down. Time dilation is not a real effect; it is a perception.

Even though time dilation is a perceptual effect, it carries real meaning for us. Because we experience life within the frameworks we perceive. Time dilation is our virtual reality.

GRYPHON

On previous pages, we discussed topics of importance to Time Dilation one by one. Here, you can examine Time Dilation as a whole from Gryphon’s perspective.

PRINCIPLE FOR ALICE

Until now, we have considered an event from both reference frames. We will do the same here. We saw how time dilation occurs for Gryphon. Now, we will examine how time dilation occurs for Alice. Because Alice also observes — detects — measures time dilation on Gryphon’s clock. We will apply the same principles we used for Gryphon, but now for Alice. Since Alice and Gryphon are two frames moving relative to each other, any event that happens in one frame must occur simultaneously in the other.

Gryphon’s clock sends signals to Alice at equal time intervals. The coordinates and time of the signals are shown by red hour and minute hands.

Since Alice’s frame is moving and the signals are reaching Alice, pay attention that the coordinates of the signals are defined according to Alice’s reference system (Alice’s field). Therefore, these coordinates move along with Alice’s reference system. We had examined this in detail in the Length Deformation section. Here, we apply the same principles in a similar way.

The movement of Gryphon’s ghost will occur in accordance with the signals. Again, for example: when signal number five reaches Alice, the ghost of Gryphon’s clock will be at the location where signal number five was emitted, and the dial will show the emission time of that signal. Gryphon’s ghost will follow the signals one by one and move in its own specific way according to this rule.

If you have available space on the sides of your program window, try expanding it horizontally as much as possible. That way, you can better observe Gryphon’s ghost.

FINAL

Finally, here we see the event as it occurs for both frames together. All controls are active here. You can use them to navigate the animation as you wish.

Let’s imagine a symmetry axis that lies between Alice and Gryphon (between their origins) and always remains equidistant to both. Events occurring on both sides of the symmetry axis must be equal. This symmetry axis serves as a verification tool to confirm the accuracy of the animation.

About Time Dilation

To be honest, this was the section I had the hardest time writing. I actually knew what I needed to write, but it was still challenging. I hope I was able to explain what Time Dilation is.

Of course, this section will cause a lot of noise in the field of physics. Because it changes many things in physics. It will affect all theories about time, require already published studies to be rewritten or corrected, and necessitate the reevaluation of practical applications. What I explained here will also make its way into school curriculums. I don’t know when that will happen, but it was a joy to explain it.

Ah, Alice, you sat in such a beautiful place that the whole world turned to look at you.