About

Worldline and special relativity theory

Activities A in the real world happen in four dimensions, three of space (x, y ,z) and one of time (t):

A = A (x, y, z, t)

For their description we need a four dimensional coordinate system. As our imagination is only capable of grasping three dimensional objects, we must use two or three dimensional projections to visualize them. Most often we restrict graphic illustrations to events of a single point object moving along one coordinate axis (x). Then we can represent it by a two dimensional graph of position over time:

x = x(t)

In contrast to the familiar way-over-time scheme of allocating position to the ordinate and time to the abscissa of a plane coordinate system, in special relativity is has become habitual to allocate abscissa to time, and ordinate to position (Minkowski Diagram).

t = t(x)

This kind of presentation is of special interest when objects move at a speed not small compared to the speed of light (c  = 2.99 792 458 *10⁸ m/s), measured relative to a resting observer at the origin of the system. Using ct instead of t for the ordinate, both axes get the same dimension of length.

ct = ct(x)

A trajectory (curve) in this scheme is called a wordline. For t < 0 it shows the entire past of the event, for t > 0 its future. Any point on the world line is called an event.

To achieve reasonable scaling for fast objects 2.99 792 458 *10⁸ m/s * (1 unit of time) is used as unit for the x axis. If time is measured in seconds the x- unit will be 2.99 792 458 *10⁸ m ≈ 300 000 km =  1 lightsecond.

With this scaling of space−time geometry a light signal passing the origin will appear as a straight line at 45 degrees to the axes (a light cone if one includes two spatial directions).

The simulation will demonstrate the movement of a particle under constant acceleration at its worldline.

Under the laws of classical mechanics there would be no limit to the speed that an object can achieve under constant acceleration, relative to an observer resting at its starting place. It would follow a parabola in space−time.

Special relativity theory tells us that this is not possible. A real, accelerated object can approach the speed of light at most. As it approaches this range, from the standpoint of the resting observer its mass increases while the speed increment decreases.

In the view of an observer moving with the object, speed continues to increase. The resting observer interprets this impression as caused by dilatation of the time scale in the moving object.

The base of special relativity is the experimentally proven fact that light (a photon, which has no rest mass) travels with constant velocity c in any system. Its world line is a diagonal both for the resting and the moving observer. A consequence is that no observed object can travel faster than light. Therefore any events that have a causal connection lie above the light cone. The "classical" path becomes unreal when it reaches the gradient of the cone, independent of how great the acceleration would be.

Classical and relativistic worldline

With Play an object (black: wheel classical, magenta point: relativistic) starts at the origin in x direction with constant acceleration b. Its "classical" worldline is shown in light gray. It would cross the red light cone at x = 2 and achieve speed of light at x = 1 (gradient equals that of the light cone).

At the beginning classical and relativistic objects seem to coincide and to travel along the paraboloid classical worldline, while a red photon runs ahead on its light cone with the speed of light. When speed is no longer small compared to that of light, classical and relativistic worldlines separate. While the "classical", black wheel pursues the gray parabola, the real, magenta colored object draws its relativistic blue worldline (a hyperbola), which finally runs parallel to the light cone when the object is approaching the speed of light.

The difference between "classical" and relativistic case becomes obvious only when the speed is of the order of magnitude of the speed of light. With t scaled in seconds (x = 1 corresponds to 300 000 km), all happenings in "normal life" will be restricted to the immediate neighborhood of the origin.

The gray line is the classical solution for constant acceleration b:

x = 1/2 bt²

derived from the differential equation

d ²x / d t ²= b

The relativistiv movement is numerically calculated with the differential equation

d ²x / d ²t = b sqrt(1-((dx /dt)/c)²)  = b sqrt(1-(v(t)/c)²) 

The red line is the light cone with x = ct

Play/Pause starts and stops the animation. Reset leads back to the starting condition.

Acceleration can be changed with the slider b. Default value is b = 1 which results in the classical object achieving the speed of light after 1 unit of ct at a distance of 1/2 unit of x km. After 2 ct units the object would surpass the simultaneously started photon.

In scaling it is open which is the unit of time. If time is scaled in seconds, the unit of the x scale is light seconds; if it is scaled in years, it is light years.

The real relativistic path (blue) is a hyperbola, which for small speed is not visibly discernible from the classical parabola. At high speed the hyperbola becomes nearly parallel to the light cone. The object can approach the speed of light, but not achieve it.

E1: Start the simulation and check when relativistic effect become noticable. Try this for different accelerations.

E2: Observe the light cone originating from the moving object. How will the observer at rest appraise light signals emitted forward or backward?

E3: Reflect about the scaling of "earthly" happenings. Take as an example a rocket with constant acceleration of 10 times that of earthly gravitation g  = 9,8 m/s². How long will it take till relativistic effects become noticeable? How far away will the object be at that time?

E4: Hint for E3: The graphic shows that relativistic effects are manifest above 1/2 c. Calculate t classically for that limit with v = bt.  Then s  = b/2 t.²

E5: The time resulting will not be extreme. What is the real limiting resource? Do some calculation!

 

Translations

Code	Language		Translator	Run

Credits

Dieter Roess - WEH-Foundation; Fremont Teng; Loo Kang Wee

Overview

This document summarizes the key concepts and functionalities presented in the "Worldline and Special Relativity Theory JavaScript Simulation Applet HTML5" available on the Open Educational Resources / Open Source Physics @ Singapore website. The resource aims to visually demonstrate the differences between classical mechanics and special relativity in the context of a constantly accelerating object, utilizing the concept of a "worldline" in a Minkowski diagram.

Main Themes and Important Ideas

1. Four-Dimensional Spacetime: The resource introduces the idea that real-world events occur in four dimensions: three of space (x, y, z) and one of time (t). Describing these events requires a four-dimensional coordinate system. However, due to our limitations in visualizing four dimensions, we often use two or three-dimensional projections.
2. "Activities A in the real world happen in four dimensions, three of space (x, y ,z) and one of time (t ): A = A (x, y, z, t) For their description we need a four dimensional coordinate system."
3. Worldlines and Minkowski Diagrams: The simulation focuses on a single point object moving along one spatial axis (x), allowing its trajectory to be represented in two dimensions. In special relativity, it is customary to use a Minkowski diagram where time is plotted on the abscissa and position on the ordinate, a reversal of the familiar position-over-time graph. To give both axes the same dimension of length, time is often scaled by the speed of light (ct). A trajectory in this spacetime diagram is called a worldline.
4. "In contrast to the familiar way-over-time scheme of allocating position to the ordinate and time to the abscissa of a plane coordinate system, in special relativity is has become habitual to allocate abscissa to time, and ordinate to position (Minkowski Diagram)."
5. "ct = ct(x) A trajectory (curve) in this scheme is called a wordline."
6. Scaling in Spacetime: To effectively visualize fast-moving objects, the unit for the spatial axis (x) is often chosen such that 1 unit of time corresponds to the distance light travels in that unit of time (e.g., 1 lightsecond if time is measured in seconds). With this scaling, the path of light passing through the origin appears as a straight line at a 45-degree angle, forming a light cone when two spatial dimensions are considered.
7. "To achieve reasonable scaling for fast objects 2.99 792 458 *108 m/s * (1 unit of time) is used as unit for the x axis. If time is measured in seconds the x- unit will be 2.99 792 458 *108 m ≈ 300 000 km = 1 lightsecond . With this scaling of space−time geometry a light signal passing the origin will appear as a straight line at 45 degrees to the axes (a light cone if one includes two spatial directions)."
8. Classical vs. Relativistic Motion: The simulation demonstrates the worldline of a particle under constant acceleration according to both classical mechanics and special relativity.

• Classical Mechanics: Predicts no limit to the speed an object can achieve, resulting in a parabolic worldline in spacetime.
• Special Relativity: Dictates that no object with mass can reach the speed of light. As an object accelerates and approaches the speed of light, its mass appears to increase to a resting observer, and the rate of speed increase diminishes. From the moving object's perspective, speed continues to increase, which the resting observer interprets as time dilation in the moving frame. The relativistic worldline is a hyperbola that asymptotically approaches the light cone.

1. "Under the laws of classical mechanics there would be no limit to the speed that an object can achieve under constant acceleration, relative to an observer resting at its starting place. It would follow a parabola in space−time."
2. "Special relativity theory tells us that this is not possible. A real, accelerated object can approach the speed of light at most."
3. "The real relativistic path (blue) is a hyperbola, which for small speed is not visibly discernible from the classical parabola. At high speed the hyperbola becomes nearly parallel to the light cone. The object can approach the speed of light, but not achieve it."
4. The Speed of Light as a Limit: A fundamental postulate of special relativity is the constant speed of light (c) in all inertial frames of reference. The worldline of light is always a diagonal line at 45 degrees in the scaled Minkowski diagram, for both resting and moving observers. This constancy implies that no observed object can travel faster than light, and causally connected events lie within the light cone.
5. "The base of special relativity is the experimentally proven fact that light (a photon, which has no rest mass) travels with constant velocity c in any system. Its world line is a diagonal both for the resting and the moving observer. A consequence is that no observed object can travel faster than light."
6. Simulation Functionality: The JavaScript applet allows users to:

• Observe the "classical" (black wheel) and relativistic (magenta point) worldlines of an object starting at the origin with constant acceleration.
• Visualize the light cone (red line).
• Use Play/Pause and Reset buttons to control the animation.
• Adjust the acceleration (b) using a slider. The default value (b=1) is set such that the classical object reaches the speed of light after 1 unit of ct.
• Double-click to toggle full screen (when the simulation is not playing).

1. "With Play an object (black: wheel classical, magenta point: relativistic) starts at the origin in x direction with constant acceleration b . Its 'classical' worldline is shown in light gray. It would cross the red light cone at x = 2 and achieve speed of light at x = 1 (gradient equals that of the light cone)."
2. "Play/Pause starts and stops the animation. Reset leads back to the starting condition. Acceleration can be changed with the slider b. Default value is b = 1 which results in theclassical object achieving the speed of light after 1 unit of ct at a distance of 1/2 unit of x km."
3. Experimental Activities: The resource suggests several experiments for users to explore:

• Observing when relativistic effects become noticeable for different accelerations.
• Analyzing the light cone originating from the moving object as perceived by a resting observer.
• Considering the scaling of everyday events and estimating when relativistic effects would become significant for a rocket with constant acceleration.

1. "E1: Start the simulation and check when relativistic effect become noticable. Try this for different accelerations."
2. "E3: Reflect about the scaling of 'earthly' happenings. Take as an example a rocket with constant acceleration of 10 times that of earthly gravitation g = 9,8 m/s2. How long will it take till relativistic effects become noticeable? How far away will the object be at that time?"

Target Audience and Educational Value

This simulation applet is a valuable educational tool for individuals learning about special relativity, particularly the concepts of spacetime diagrams, worldlines, the speed of light as a limit, and the divergence between classical and relativistic predictions at high speeds. The interactive nature of the simulation allows for hands-on exploration of these abstract concepts, making them more intuitive. The suggested experiments further encourage deeper engagement and critical thinking.

Technical Details

The simulation is built using JavaScript and HTML5, making it accessible through web browsers without the need for additional plugins. It is part of the Open Educational Resources / Open Source Physics @ Singapore project, indicating a commitment to free and open access to educational materials. The EasyJavaScriptSimulations Library is mentioned in the licensing information, suggesting it might have been used in the development.

Potential Uses

• Classroom demonstrations in physics courses covering special relativity.
• Interactive learning tool for students to visualize and understand worldlines and relativistic effects.
• Self-study resource for individuals interested in learning about the basics of special relativity.
• Foundation for further discussions on more advanced topics in relativity.

This briefing document provides a comprehensive overview of the "Worldline and Special Relativity Theory JavaScript Simulation Applet HTML5" resource. It highlights the core concepts presented, the functionality of the simulation, and its potential educational applications.

 

Worldline and Special Relativity Theory Study Guide

Quiz

1. In the context of special relativity, why is it useful to represent motion as a worldline on a Minkowski diagram?
2. Explain the significance of scaling the time axis by the speed of light (ct) in a Minkowski diagram. What advantage does this offer?
3. Describe the appearance of a light signal originating from the origin on a space-time diagram (Minkowski diagram) that uses ct as the ordinate. What is this structure called when including two spatial dimensions?
4. According to classical mechanics, what would be the trajectory of an object under constant acceleration in space-time? How does special relativity differ in its prediction for such an object?
5. What is a fundamental postulate of special relativity regarding the speed of light, and how does this postulate affect the possible speeds of other objects?
6. In the simulation, what are the key differences observed between the "classical" (black wheel) and relativistic (magenta point) worldlines of an accelerating object, especially at higher speeds?
7. Explain the concept of time dilation from the perspective of a resting observer viewing a moving object approaching the speed of light.
8. What does the light cone represent in a space-time diagram, and what is the implication for events lying outside this cone with respect to the origin?
9. Describe how the simulation allows users to explore the relationship between acceleration and the divergence of classical and relativistic worldlines.
10. According to the source, when do relativistic effects become significant for moving objects, and how does the scaling of the axes relate to our everyday experiences?

Quiz Answer Key

1. Representing motion as a worldline on a Minkowski diagram, with time as the abscissa and position as the ordinate, is useful in special relativity because it visualizes events in four-dimensional spacetime and highlights the interconnectedness of space and time, especially for objects moving at relativistic speeds. This framework allows for a clear depiction of concepts like time dilation and length contraction.
2. Scaling the time axis by the speed of light (ct) gives both the time and space axes the same dimension of length. This simplifies the geometric interpretation of spacetime intervals and makes it easier to visualize the invariant nature of the speed of light, as light signals appear as lines at a consistent 45-degree angle.
3. A light signal passing the origin on a Minkowski diagram with ct as the ordinate appears as a straight line at a 45-degree angle to the axes. When including two spatial dimensions, this forms a cone known as the light cone, with the origin at its apex, representing the boundary of causal influence.
4. Under classical mechanics, an object under constant acceleration would follow a parabola in space-time, indicating no upper limit on its achievable speed. Special relativity predicts that an accelerated object can only approach, but never reach, the speed of light, resulting in a hyperbolic worldline in space-time.
5. A fundamental postulate of special relativity is that the speed of light in a vacuum is constant for all inertial observers, regardless of the motion of the light source. This implies that no observed object with mass can travel at or faster than the speed of light, as this would violate this fundamental principle.
6. At lower speeds, the classical and relativistic worldlines in the simulation coincide, both appearing parabolic. However, at higher speeds, they diverge significantly. The classical worldline (gray parabola) continues to increase in gradient, exceeding the speed of light, while the relativistic worldline (blue hyperbola) approaches the gradient of the light cone, indicating that the object's speed approaches but never reaches c.
7. From the perspective of a resting observer, a time scale in a moving object is dilated, meaning time appears to pass more slowly for the moving object as its speed increases relative to the observer. This is one of the key effects predicted by special relativity and becomes more pronounced as the object approaches the speed of light.
8. The light cone in a space-time diagram represents the paths that light can take from a single event at the apex. Events that have a causal connection to the origin must lie within the future light cone (for future events) or the past light cone (for past events). Events outside the light cone are causally disconnected from the event at the origin, meaning no information or influence can travel between them faster than light.
9. By adjusting the acceleration slider, users can observe how different constant accelerations affect the curvature of both the classical and relativistic worldlines. They can see that higher accelerations lead to a quicker divergence between the two paths, demonstrating when relativistic effects become more noticeable for a given acceleration.
10. Relativistic effects become noticeable when the speed of an object is no longer small compared to the speed of light. The scaling of the axes, where one unit of the space axis corresponds to a light-second (approximately 300,000 km), illustrates that the high speeds required for significant relativistic effects are far beyond the velocities encountered in most everyday "earthly" happenings, which are confined to the immediate vicinity of the origin on such a scale.

Essay Format Questions

1. Discuss the fundamental differences in how classical mechanics and special relativity describe the motion of objects under constant acceleration. How does the concept of a "worldline" help to illustrate these differences?
2. Explain the significance of the speed of light as a universal speed limit in the context of special relativity. How is this limit visually represented in a Minkowski diagram using the concept of the light cone, and what are its implications for causality?
3. Analyze the role of the Minkowski diagram in understanding the interconnectedness of space and time in special relativity. How does the choice of coordinates (time vs. position on the axes) differ from classical physics, and why is the ct scaling important?
4. Based on the provided simulation and the principles of special relativity, discuss why the classical notion of unlimited acceleration is not physically realistic. Explain the behavior of a relativistically accelerating object as it approaches the speed of light from the perspective of both a resting observer and an observer moving with the object.
5. Consider the scaling of space and time in the simulation (where 1 unit of x is a light-second). Reflect on the implications of this scaling for our understanding of when relativistic effects become significant in real-world scenarios. Use examples to illustrate your points.

Glossary of Key Terms

• Worldline: A trajectory or curve representing the path of an object through four-dimensional spacetime. Each point on the worldline corresponds to a specific event in space and time.
• Special Relativity: A theory of the structure of spacetime. It deals with the relationship between space and time in the absence of gravity and is based on two postulates: the laws of physics are invariant in all inertial frames of reference, and the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.
• Minkowski Diagram: A two-dimensional graph used in special relativity to visualize spacetime, typically with time (or ct) on one axis and spatial position on the other. It allows for the geometric interpretation of relativistic effects.
• Event: A specific occurrence that happens at a particular point in space and at a particular moment in time. A point on a worldline represents an event.
• Speed of Light (c): The speed at which light propagates through a vacuum, approximately 299,792,458 meters per second. It is a fundamental constant in physics and the ultimate speed limit for any object with mass.
• Light Cone: A geometric representation in spacetime that shows the possible future and past locations of light rays emanating from a single event. It separates spacetime into regions that are causally connected and causally disconnected from that event.
• Classical Mechanics: The physics of motion based on Isaac Newton's laws, which describes the motion of macroscopic objects at speeds much smaller than the speed of light. It treats space and time as separate and absolute.
• Acceleration (b): The rate at which the velocity of an object changes with respect to time. In the simulation, it is a constant value that can be adjusted.
• Time Dilation: A phenomenon in special relativity where time passes differently for observers in relative motion. A moving clock is observed to tick more slowly than a stationary clock.
• Space-time: The unified four-dimensional continuum consisting of three spatial dimensions (length, width, height) and one temporal dimension (time), in which all physical events occur.

Sample Learning Goals

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For Teachers

Worldline and Special Relativity Theory JavaScript Simulation Applet HTML5

Instructions

Acceleration Slider

Toggling Full Screen

Play/Pause and Reset Buttons

Research

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Video

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 Version:

Other Resources

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FAQ: Worldline and Special Relativity

What is a worldline in the context of physics?

A worldline is the trajectory of a single point object through four-dimensional spacetime. It depicts the path an object takes through three spatial dimensions (x, y, z) and one time dimension (t). Each point on the worldline represents an event, which occurs at a specific location in space and at a specific moment in time.

Why is spacetime visualized differently in special relativity compared to classical physics?

In classical physics, we typically plot position against time, with position on the vertical axis (ordinate) and time on the horizontal axis (abscissa). However, in special relativity, it's habitual to use a Minkowski diagram, where time (often multiplied by the speed of light, ct) is on the abscissa and spatial position is on the ordinate. This representation becomes particularly useful when dealing with objects moving at speeds approaching the speed of light, as it treats space and time on a more equal footing. Using 'ct' makes both axes have the dimension of length, simplifying the visualization of spacetime intervals.

What is the significance of the speed of light (c) in special relativity and worldlines?

The speed of light is a fundamental constant in special relativity. A key principle is that light travels at a constant velocity c in a vacuum for all observers, regardless of their relative motion or the motion of the light source. On a spacetime diagram where 'ct' is plotted against position, the worldline of a light signal passing through the origin appears as a straight line at a 45-degree angle to the axes. This forms the boundary of the light cone. No object with mass can travel faster than light, meaning the worldline of any such object must always remain within its future light cone.

What is the light cone and why is it important?

The light cone, originating from an event, represents the boundary in spacetime that defines the region causally connected to that event. The future light cone encompasses all events that can be causally influenced by the event at its apex, while the past light cone includes all events that could have causally influenced the event at its apex. Since nothing can travel faster than light, the worldlines of all objects and causal influences must lie within these cones.

How does the worldline of an object moving under constant acceleration appear in classical mechanics versus special relativity?

In classical mechanics, an object under constant acceleration, starting from rest, would have a parabolic worldline in spacetime. There is no theoretical limit to the speed it can achieve. However, in special relativity, an object under constant acceleration will have a hyperbolic worldline. As the object accelerates, its speed approaches the speed of light, but never reaches it. From the perspective of a stationary observer, the object's mass increases and its acceleration decreases as it gets closer to the speed of light.

How does the simulation illustrate the difference between classical and relativistic worldlines?

The simulation shows a classical object (black wheel) and a relativistic object (magenta point) both starting at the origin with the same constant acceleration. The classical object follows a parabolic path (light gray), eventually crossing the light cone (red lines) and exceeding the speed of light. In contrast, the relativistic object follows a hyperbolic path (blue), initially coinciding with the classical path at low speeds but then diverging. The relativistic worldline approaches and runs parallel to the light cone, demonstrating that the object approaches, but never reaches, the speed of light.

When do relativistic effects become significant and noticeable in the context of everyday life?

Relativistic effects become significant only when objects move at speeds that are a considerable fraction of the speed of light. In everyday life, the speeds we encounter are vastly smaller than the speed of light. Therefore, on the spacetime scale where one unit of distance corresponds to a light-second (approximately 300,000 km), most "normal life" happenings are confined to a very small region near the origin of the spacetime diagram, and the classical and relativistic predictions are virtually indistinguishable.

What are some experiments or thought exercises suggested by the simulation?

The simulation suggests several experiments: (E1) Observing when relativistic effects become noticeable for different accelerations. (E2) Observing the light cone originating from a moving object and considering how a stationary observer would perceive light signals emitted forward or backward. (E3) Reflecting on the scaling of earthly events, such as a constantly accelerating rocket, and estimating when relativistic effects would become noticeable. This involves calculating the time and distance classically to reach a significant fraction of the speed of light and then considering the limitations imposed by relativity.