Time Dilation: First Possible Step Towards Time Travel

Introduction

The concept of time travel has been one of dreams, fantasies, and the imagination. What would you do if you traveled to the past? Would you make corrections or changes to your past, maybe fix what you regret, what if we had more time to do what we can not: these are some of the questions that people ask themselves when thinking of what life would be like if time travel was possible. Thousands of books and movies have been created centralizing on the infinite amount of possibilities that could occur if mankind was able to travel to any point or occurrence in the history of the universe. Although scientists and mathematicians have not found a possible method of time travel, there comes questions: is it still possible to manipulate time? Is time relative or constant?

During an IB Physics trip to a university, I discovered the concept of time dilation, a principle which sparked my interest and curiosity. As we had recently studied optics and light in physics, I was able to see certain variables within the equation for velocity time dilation that were associated with that of optics, such as Snell’s Law. With this discovery of time dilation, I couldn’t help but ask questions about the implications of the equation, replacing variables with equations that calculate them. Though this replacing only went as far as optics, I found myself wanting to spend more time with the subject. As a result, with the informing of the IB Mathematics Internal Assessment, I decided to make my topic time dilation, learning more about the concept without implications of time outside of class.

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Before going into the mathematics behind time dilation, it is important to define and discuss the history behind time dilation. Time dilation is a concept predicted by Albert Einstein’s theory of special relativity. This theory, published in 1905, relates to phenomena that are detectable, or, not negligible, as an object travels near the speed of light (“Special Relativity”). In regards to time dilation, to measured length of time between two events depends upon the frame of reference of the observer (“Frame of Reference”). To further explain, time appears to occur slower for objects moving with respect to the observer, meaning that time appears to have lengthened, hence the name “time dilation.”

To further explain the meaning of this concept, it would be better to apply other concepts which relate to time dilation. The first is simultaneity. Before Einstein’s theory of special relativity, there was a perspective that time was a single quantity, an absolute quantity. This means that time would be the same for everyone in every location; there would then be such a thing such as a universal time, where the universe functions on the same time. However, with Einstein’s theory of special relativity, the idea of an absolute time was abandoned. The first part of simultaneity is defining what an event is, and how events can be simultaneous. An event is something that occurs at a specific place and particular time, and in addition, two events can be described as occurring simultaneously if they occur at exactly the same time. After several thought experiments, it was determined that two events that may appear simultaneously to one person, or one observer, may not appear simultaneous to another, implying that simultaneity is not an absolute concept, but is instead relative with respect to the observer. As a result, because simultaneity deals with time, there comes the question as to whether or not time occurs differently from observers in different reference frames, or a “set of coordinate axes that help describe the position or movement of an object” (“Reference frames”). This means that two people may measure the same object with different results (“Frame of reference”).

This concept applies to that of time dilation. To explain this scenario, I will first present the question for velocity time dilation, and a thought experiment which results in the same equation. The equation for velocity time dilation is the following: t = t01- V2C2. Now that the equation for velocity time dilation has been provided, we can now continue with a thought experiment which will result in the given equation.

Thought Experiment: Spaceship (Case #1)

Here we have a spaceship that is moving at a constant velocity (v) in the direction of the arrow shown below. Inside the spaceship, there is a mirror attached to one end of the spaceship, and on the other end of the ship, there is a stationary observer with respect to the spaceship (the spaceship, in this scenario, is the spaceship). As seen in the picture, the distance between the observer and the mirror is “d.”

In this scenario, the observer, holding a flashlight in the direction of the mirror, flashes the flashlight, turning the flashlight on and off. This allows the observer to send a beam of light directly across to the mirror. This action can be defined as event #1. Because the object is a mirror, the light will then travel across to the mirror, reflects, and travels back to the observer. The beam of light returning to the owner can be defined as event #2. With this in mind, we now have two events which are occurring in the same reference frame. Now, we have the following question: how long will it take the light to travel from the observer directly across and reflects off the mirror (event #1), and travels directly back to the observer (event #2), or to rephrase the question, what is the time between event #1 and event #2?

To answer this question, we can use the following equation: t0=2dc. In this equation, t represents t- nought or proper time, 2d representing the two times that the light travels the distance (d) , and c representing the speed of light, which is constant according to Einstein's theory of special relativity (“Special Relativity”).

Case #2

Now that we have covered the first case of this experiment, we will now change reference frames. How we have a stationary observer on Earth, observing the same ray of light. However, now the ray of light will travel a different pathway. This is because the spaceship is traveling with respect to the observer on Earth with velocity (v). As a result, instead of the light traveling straight across, as seen previously, the light will travel the pathway as seen in orange. Relative to the stationary observer on Earth, the spaceship is moving, and as a result, the light travels the diagonal shown in the diagram.

As a result, since light is traveling with identical speed in both cases, but it is traveling a long distance in case #2, the time between event one and event two is longer for the stationary observer on Earth. So, as we did previously, we will not calculate the time interval, t, as measured by the observer on Earth between events 1 and 2.

c = total distance total time = 2l2+ d2t

(Using Pythagorean Theorem. In addition, we have two triangles, and so the square root is multiplied by 2)

V is = 2lt, so l = vt2

As a result, c = 2v2(t)24+ d2t (square both sides)

c2 = 4(v2(t)24) + d2(t)2

c2 = v2+4d2(t)2

t = 2dC1-v2c2

t = t01- V2C2

So what exactly does this equation mean? This equation means that the time for a moving observer passes more slowly. Because v is always less than c, the quantity of v2c2 is always less than 1, so 1 minus a quantity less than 1, and a square root of a number less than 1 is also a number less than 1. As a result, the following relationship occurs: t will always be >t0.

The time that has passed for the person inside the spaceship is always less than the time lapsed for the person that is stationary on Earth that is observing that spaceship. As a result, time for the moving observer always passes more slowly with respect for the person that is stationary, and this is a result of time dilation.

Graphical Representation of the Velocity Time Dilation Equation

Here is a graphical representation of the velocity time dilation equation. As you can see, the relationship between velocity as a fraction of the speed of light and time dilation is exponential. One of the most significant pieces of information is how rapidly increasing the rate of time dilation is towards the farther end of the graph.

Examples of the Velocity Time Dilation Equation

There are several examples of situations where time dilation applies. One of the most prominent examples, however is that which explains why Muons can be detected at the surface of Earth. Muons are unstable particles that are similar and have 207 times more mass electrons (“Time Dilation”). The most important information, however, is that they have a mean life of “0.0000022 seconds before decaying into smaller particles,” so how are we able to detect these particles with such an extremely low mean life (“Time Dilation”)? This is because of the velocities which Muons are produced with. 'Muons are produced with velocities close to 99.8% the speed of light from cosmic ray collisions with atoms in Earth's upper atmosphere, roughly 10 km (6 mi) above sea level” (“Time dilation”). Without the concept of time-dilation, “the Muons would travel a distance equal to only about 660 m (2200 ft) before decaying, and therefore they should not be detectable at the surface of Earth” (“Time dilation”). However, since the Muons is traveling at 99.8% the speed of light with respect to the surface of Earth, the lifetime of the Muon appears to be longer, due to the applying of this value into the equation: t = 11 - (0.998)2, which is approximately 15.82, which means that the Muon appears to live for 16 times longer, or for about 0.000035 seconds. With the speeds which Muons travel, traveling at 99.8% the speed of light for the duration of the Muon’s dilated life allows the Muon to travel about 10.4 km, or 6.5 miles before decaying. Because of the great amount of distance traveled as a result of a dilated mean life, time dilation explains why we are able to detect Muons at sea level.

Another example of time-dilation can be seen in the twin paradox: twin brothers Scott and Mark Kelly. “In a famous Einsteinian thought experiment called the twin paradox, a twin who embarked on a whirling flight through space would age more slowly compared to the twin left back home on Earth, a result of time dilation when traveling near light speed” (Wanjek). However, NASA was able to do a real-life study on the two twin astronauts, an opportunity to truly test the twin paradox. “Mark flew on four NASA space missions, each one lasting under two weeks, and retired in 2011” (Wanjek). On the other hand, Scott also flew on four NASA space missions, and the first three were short” (Wanjek). For the fourth mission, however, Scott spent 342 days in 2015 in the International Space Station (Wanjek). The goal of this study was to “study the health effects of a long space flight, similar in length to the nine months it would take to journey to Mars”. However, in the results of the study, instead of finding that the twin who embarked in space would age more slowly when traveling at near the speed of light, scientists studying the NASA twin astronauts found a somewhat opposite result: “Astronaut Scott Kelly may have aged just a little faster as a result of his yearlong stint on the International Space Station (ISS) compared with his Earthbound brother, Mark” (Wanjek). As a result, the NASA twin paradox study is another example of time dilation at work in a real life scenario.

Reflection

In this math exploration, I learned about the velocity time dilation, and its implications in real world situations. Through thought experiments, Albert Einstein established his theory on special and general relativity, establishing principles such as reference frames, simultaneity, and time dilation. Within the concept of time dilation, the equation t = t01- V2C2 shows that the time for the moving observer always passes more slowly with respect for the person that is stationary, and this is a result of time dilation. As a result, through this exploration, I was able to further explore a topic and concept related to my planned college major: physics, learning more about the relativity of physics in the world and in the universe.

Works Cited

1. “Frame of reference”. World of Physics, Gale, 2001. Gale In Context: Science, https://link.gale.com/apps/doc/CV2434500188/SCIC?u=j199901004&sid=SCIC&xid=f9e5f35d. Accessed 30 Jan. 2020.
2. Mars, Kelli. “NASA's Twins Study Results Published in Science Journal.” NASA, NASA, 28 Sept. 2018, https://nasa.gov/feature/nasa-s-twins-study-results-published-in-science
3. “Reference frames”. World of Physics, Gale, 2001. Gale In Context: Science, https://link.gale.com/apps/doc/CV2434500435/SCIC?u=j199901004&sid=SCIC&xid=a84dd8c6. Accessed 30 Jan. 2020.
4. “Special Relativity”. World of Physics, Gale, 2001. Gale In Context: Science, https://link.gale.com/apps/doc/CV2434500442/SCIC?u=j199901004&sid=SCIC&xid=e9121f55. Accessed 30 Jan. 2020.
5. “Time Dilation.” Aklectures.com, , https://aklectures.com/lecture/simultaneity-time-dilation-and-length-contraction/time-dilation.
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7. Wanjek, Christopher. “Scott Kelly's Year in Space May Have Aged Him - But He's Mostly Fine.” LiveScience, Purch, 11 Apr. 2019, , https://livescience.com/65214-nasa-twin-study-mark-scott-kelly.html.