Time Dilation

Claimed By Frank Hutchison Spring 2022

Note: This article is regarding time dilation due to relative velocity (special relativity). Time dilation is a phenomenon that is exemplified by an apparent disparity in the passage of time within the context of multiple frames of reference.

Background

Before Einstein, the best equations to model the interactions between objects were from Isaac Newton. The concept is known as classical mechanics. For everyday occurrences, such as an apple falling from a tree, Newton’s equations give accurate predictions. The best application of classical mechanics involved the motion of planets. Newton theorized that the Earth was pulling the moon and other objects (like the apple) towards itself. The moon balanced this force of gravity through its inertia. This logic was then applied to planets orbiting the Sun.

Inertial Reference Frames

An important thing to note about observing objects is that a reference point has to made. When a person is sitting, he is usually said to not be moving. However, this is misleading. In reality, the person is moving at the same speed as the Earth is rotating. The person is also moving at the speed the Earth is revolving around the Sun. The reason we don’t feel it is because the rotating and revolving speeds are constant. So, an inertial reference is when we pick something (an example would be the Earth) and assume it is not moving. We then compare other objects to that frame. This then means that velocity is relative or depends on the frame chosen. Example: There is a person standing still (relative to the Earth), a person driving a car at a constant speed, and a person walking. If the reference frame is from the person standing still, the person walking is moving at 5mph and the person in the car is going 20mph. From the reference frame of the person walking, the stationary person is moving at -5mph and the person in the car is going 15mph.

The Main Idea

Time dilation is best understood to be entirely based on reference frames. All effects or observed effects of time dilation are dependent on the observer's particular frame of reference. For example (as the concept of time dilation relates to special relativity), a pilot travelling in a rocket will observe no change in his own situation, despite the fact that, to an outside observer who is "not moving," the pilot and his instruments will be subject to a great deal of time dilation due to the high velocity disparity between the rocket and the observer.

A Mathematical Model

Time dilation in a special relativistic context can be modeled simply with the formula:

[math]\displaystyle{ \Delta t' = \frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}} \, }[/math]

where [math]\displaystyle{ \Delta t }[/math] defines the elapsed time between two events which occur at the same location for one particular observer in his or her frame of reference, [math]\displaystyle{ \Delta t' }[/math] defines a second measured elapsed time between the same two events but by a second particular observer that is moving with a specific velocity of [math]\displaystyle{ v }[/math] with respect to the first observer. In this context, [math]\displaystyle{ c }[/math] refers to the speed of light. For everyday events, the velocity of the observed objects is so small compared to light (300 million meters/second) that the time dilation is negligible. This is not the case for particles moving near the speed of light. Say a particle was moving at 99% the speed of light for one second in our reference frame. In the particles reference frame, it moves:

[math]\displaystyle{ \Delta t' = \frac{\Delta t}{\sqrt{1-\frac{(.99c)^2}{c^2}}} =7.09s \, }[/math]

This gets more extreme the closer the speed is to light. Let’s use the same scenario as above but instead the particle is moving at 99.99% the speed of light.

[math]\displaystyle{ \Delta t' = \frac{\Delta t}{\sqrt{1-\frac{(.9999c)^2}{c^2}}} =70.7s \, }[/math]

These changes in time depending on the reference frame can have startling implications. Below are two famous examples.

Examples

Simple

Imagine a flight from Cleveland to Atlanta. It would seem obvious that the distance covered by a plane that undertakes this journey is identical to a plane that flies from Atlanta to Cleveland. However, it must be considered which plane is actually moving relative to our frame of reference. In order to understand more complicated aspects of time dilation, simple facts about referential velocities are important to learn.

Difficult

Imagine spaceship one is moving at a uniform speed from point A to point B. Spaceship one has an on-board atomic clock that measures time accurately to the nanosecond. Now, imagine a second, identical spaceship (spaceship two) with an identical atomic clock moving at the same speed but this time heading from point B to point A. At the instance that the spaceships pass by one another on their respective routes, the pilot of spaceship one looks into the cockpit of spaceship two and notices that the atomic clock appears to be ticking slower in comparison to his own atomic clock, which seems to operate normally. Compare this to the viewpoint of the pilot of spaceship two, who sees his clock as operating normally whereas the clock in spaceship one appears to be slower.

The reasoning behind the apparent disparity involves the differing frames of reference of the two pilots. But what happens if the two pilots later decide to meet up at the restaurant at the end of the universe? Which one will be older? To answer this question simply, neither will be "older" than one another, but in comparison to a person who had been standing still on earth during the course of the two pilots' travels, the pilots will have aged somewhat slower.

Andromeda Paradox/Simultaneity

In this scenario, Bob is stationary while Alice is running. After they exchange greetings, Alice mentions that an armada from the Andromeda Galaxy has left for Earth. Bob responds by saying they haven’t. Time dilation makes it possible for a circumstance to happen in one frame while the other the frame hasn’t experienced it yet. Put another way, simultaneity has been broken.

Train Simultaneity

Let’s look at another example. There’s a train that can go close to the speed of light. A person decides to board it, while their friend remains at the station. The train then leaves at a constant speed close to light. Suppose the person on the train fires to two lasers, one towards the front of the train, and one towards the back of the train. The person on the train would see the light from the lasers hit the front and back at the same time. It is different than what the person on the station would see. He would see light hit the back of the train before the front of the train. Why does this happen? It has to do with the constant speed of light. The distance an object travels is given by d=vt, and in this case light travels to the person’s eye at speed c, so d1=ct. Remember, the train has a specific length, L, so the distance from the front of the train, d2, will be different (and longer) than d1. Light remains constant, so time has to increase to account for the different distance, or d2=ct'. In this example we are assuming the train is moving towards the right. If it moved to the left, the person at the station would see what was before the front side of the train get struck by lightning first. For more detail on this topic, watch this video:

Twin Paradox

There are two twins, Christian and Alfred. Christian leaves on a rocket ship to Barnard’s star, approximately six light years away. The ship will travel a constant speed at 80% the speed of light. Both brothers start stop watches at the same time Christian leaves. In Alfred’s perspective, he ages 15 years, 7.5 years for the journey to the star and 7.5 years for the journey back to Earth. However, Christian’s perspective is different, and we can calculate the amount of time he aged.

[math]\displaystyle{ \Delta t' = 15 = \frac{\Delta t}{\sqrt{1-\frac{(.8c)^2}{c^2}}} \, }[/math]

[math]\displaystyle{ \Delta t = {15}{\sqrt{1-\frac{(.8c)^2}{c^2}}} = 9yrs \, }[/math]

So, Christian aged 9 years while Alfred aged 15 years. The next question that needs to be answered is, why does Alfred age more than Christian? From Christian’s perspective, Alfred was the moving away and back towards him since Christian was moving at a constant speed. The answer is that Christian was actually in two reference frames, one heading towards the star, and one heading back towards Earth. Alfred was always in one reference frame on Earth.

Length Contraction

There’s another effect that can be witnessed when reference frames travel at speeds close to the speed of light. It has to do with the length of the object. The first thing that goes into the equation is the value of the proper length. Proper length (Lo) is the distance between two objects from the perspective of someone who is not moving towards or away from both objects. It is included in the equation below.

[math]\displaystyle{ \ L = {Lo}{\sqrt{1-\frac{v^2}{c^2}}} \, }[/math]

Here is an example to better understand the concept. Suppose a person observes a particle moving at 95% the speed of light for 3 seconds. What is the length travelled in the particle’s frame? Start by determining the distance in our reference frame.

Next, determine the length in the particle’s frame.

Comparing the two distances, we see that because the particle is moving at a much velocity compared to us, it appears to have travelled a lot farther than it has in its reference frame.

Connectedness

In regards to my particular major, nuclear engineering, this topic does not have a specific connection. However, the concept of time dilation is one that must be taken into special consideration in the field of aerospace engineering, specifically when designing and engineering methods for deep-space flights. Since the effects of time dilation will result in any extremely high-velocity travel, any human subjects will age at a rate much more slowly than scientists and engineers on earth working on deep-space missions. In the future, this phenomenon indeed presents a unique obstacle for exploration efforts.

History

In 1909, Gilbert Lewis used a model of two "light clocks," each of which moved with relative velocities, to describe a theory of time dilation. The clocks operated by bouncing a "signal light" back and forth between two mirrors; within each clock, the mirrors were parallel to each other as well as to the direction of the clock's motion. It was theorized by Lewis that an observer at the reference frame of the first clock would see the second clock as operating "slower."

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