Time Dilation

Claimed and edited By Alex Ronemus Fall 2024

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. It can be observed when an object is moving close to the speed of light. It is also a crucial concept for understanding much of the uncertainty present in contemporary physics.

Background

Before Einstein, the best equations to model the interactions between objects were those derived by Isaac Newton. His ideas compose what is known today as classical mechanics. Newton's equations give accurate predictions for everyday occurrences, such as an apple falling from a tree or a car moving along a road. The best application of classical mechanics involved the motion of celestial bodies such as planets and Earth's sun. Newton theorized that the Earth was pulling the moon and other objects (like the apple) towards itself and that they were doing the same to the Earth. This force was balanced through inertia, which is the general resistance to movement an object possesses.

One of the most interesting and consequential postulations classical mechanics makes is that time and space are absolute in all inertial reference frames. However, this belief has been challenged by observations made in the 19th and 20th centuries. Modern physics argues that time and space can vary based on how fast an object is moving with effects being noticeable to an observer when an object is moving faster than a tenth of the speed of light.

Inertial Reference Frames

Something important to note about making observations is that they have to be taken with respect to a reference point. For example, take a person who is sitting in a chair. Another person observing the first person would say that the person in the chair is not moving. In the observer's reference frame, this makes sense because from what they can see that person is staying in the same spot. However, according to astrophysics the Earth and by extent, everything on it is rotating. This contradiction can be resolved when taking into account the fact that the observer's reference frame is moving with the Earth so it would appear as though nothing is changing for the observer. However, if the observer's reference frame is not Earth-centric then the person in the chair would be moving at the same speed as the Earth. This means that velocity is relative or, in other words, that it depends on the frame chosen.

For more information, watch this video. [1]

Classical mechanics provides a great framework for analyzing inertial reference frames. However, when objects begin moving at speeds close to that of light; it becomes necessary to use a different model, Special Relativity, to describe their motion.

The Main Idea

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 traveling in a rocket close to the speed of light will observe no change in his own situation, even though, 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. To an observer, the time the pilot was in flight was much longer than what the pilot observed in his frame of reference. This is due to the relativistic effects that occur when an object is moving close to the speed of light.

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 that 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 (3 x 10^8 meters/second) that the time dilation is negligible. This is not the case for particles moving near the speed of light.

GlowScript Simulation

Provided is a link to a GlowScript program that demonstrates how the time dilation functions: https://www.glowscript.org/#/user/aronemus/folder/Alex'sStuff/program/TimeDilationModel

Examples

Simple Examples

Say a particle was moving at 99% the speed of light for one second in our reference frame. In the observer's reference frame of reference, it takes longer for the particle to move a specified distance.

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

Complex Example

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.

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. Here is a simulation of what the person on the train sees. [2] 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. Here is a simulation of what the person on the station would see. [3] 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: [4]

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 length of a object can appear contracted when an observer sees said object moving close to the speed of light. The object has what is known as a proper length that is its length in its rest frame. However, when it is moving at a speed close to the speed of light the object's length can appear shorter to an observer.

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

Rocket Revisited

There is a rocket traveling at a speed, v = 0.25c, and in its rest frame it has a length of 3 meters. What is its length relative to an observer?

[math]\displaystyle{ \ L' = {3}{\sqrt{1-\frac{(0.25c)^2}{c^2}}} \, }[/math]

[math]\displaystyle{ \ L' = {3*}{0.9375} \, }[/math]

[math]\displaystyle{ \ L' = {2.8125} \, }[/math]

The ship appears to be 2.8125 meters long to an observer.

Connectedness

As a physics major with aspirations for research of some form, the time dilation is very important as it has a connection to many concepts in quantum mechanics and statistical mechanics. This page could have been a great resource to me starting out in this class especially given how important it is to what I want to do so I hope by editing it to be more accessible and informative I can help people who are in a similar position to myself.

History

In 1887 the Michelson-Morely Experiment found that the speed of light was constant in all reference frames which gave rise to some of the points made by Einstein in his paper on the theory of relativity eighteen years later. Einstein proposed that the closer an object was moving to the speed of light the slower time seemed to pass for the object. 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."