Spacetime and Geometry

I am reading Spacetime and Geometry : An Introduction to General Relativity – by Sean M Carroll. The blog contains answers to his exercises, commentaries, questions and more.

List of Answers to Exercises in Spacetime and Geometry : An Introduction to General Relativity – by Sean M Carroll

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Self imposed exercises

9 comments:.

homework 1 vectors and relativity answer key

Firstly, Thank u for your answer, I think there's something wrong with equation 42 in your chapter 3 exercise 4(b) answer, the basis of a vector should be the transformation of the down index, instead of the up index, and then the basis of the dual vector is also wrong.I am a beginner of GR. If there is any mistake, I hope you can correct me. Thank you.

Thanks Chun! I think you are right. I have added your comment as not in the document.

hello dear. I need the answer of exercise 4 for chapter6. So thanks

After a 10-year break, I decided to pick up GR again, using Carrol's book this time. Do you intend to make your solutions complete? If so, we can work together.

I see that you have not provided an answer to Ex. 13, Chapter 1. The gist of the exercise is that the extra term is a total derivative and therefore won't affect equations of motion. In fact, it is a so-called Chern-Simons term. For some other more interesting non-Abelian Gauge theories, such a term can have global topological consequences (i.e. instantons). For EM in 3+1 dimensions, it has no consequence.

If you are still interested in 2.9. https://www.physicsforums.com/threads/differential-forms-integration-exercise.378255/

For posterity, Exercise 2.11 is quite easy for those who have studied QFT. Here is a brief sketch. (a) The action has to have 0 mass-dimension. $A^{(3)}$ as a 3-form needs to be integrated over a 3-volume, of which one dimension is time. So the object itself must have two spatial dimensions. (b) The dual of a now 4-form gauge field is a 7-form field. So We need to integrate over a seven-sphere. The conservation part is a direct consqeuence of Stokes's theorem for forms. (c) Following the same reasoning as (a), the answer is 5 since tilde-A is a 6-form. (d) Only F and *F are allowed to appear in the Lagrangian. The reasoning is exactly the same as in vanilla EM.

I picked up Carroll to refresh some of my skills in GR (I'm working on my own solution set just for my own sake). These have been very helpful for checking and giving a nudge here and there. Just wanted to say thanks for posting them!

Hey Matt, I'm a rookie and working on GR. If by any chance you could share your solutions, I would be extremely grateful as I am really struggling to solve some of these

General Relativity -- PHYS-GA 2060

Homework 1 Vectors And Relativity

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Homework 1 vectors and relativity – Embark on an enthralling journey with Homework 1: Vectors and Relativity, where we unravel the intricate relationship between space, time, and motion. Delve into the fundamental concepts of vectors, the language of physics, and explore how they intertwine with the enigmatic realm of relativity.

Get ready to witness the bending of space-time, the dilation of time, and the contraction of lengths, as we delve into the heart of Einstein’s groundbreaking theories.

Prepare to navigate a world where the familiar laws of physics take on new dimensions, and where the boundaries of our understanding are pushed to their limits. Join us as we uncover the captivating secrets of vectors and relativity, unlocking a deeper appreciation for the fabric of our universe.

Introduction to Homework 1 Vectors and Relativity

Homework 1 delves into the fundamental concepts of vectors and relativity, laying the groundwork for further exploration in physics. Vectors, mathematical entities with both magnitude and direction, play a crucial role in describing physical quantities such as displacement, velocity, and force.

Relativity, on the other hand, challenges our classical understanding of space and time, introducing concepts like time dilation and length contraction.

Through this homework, you will gain a deeper understanding of these foundational concepts and develop essential problem-solving skills in vector analysis and relativity.

Mathematical Foundations

In the realm of physics, vectors and relativity play a pivotal role in understanding the nature of space, time, and motion. Vectors, mathematical entities with both magnitude and direction, provide a powerful tool for representing physical quantities like displacement, velocity, and force.

Relativity, on the other hand, challenges our classical notions of space and time, introducing the concept of spacetime and the effects of relative motion.

To delve into the world of vectors and relativity, we begin with a review of vector algebra, exploring the fundamental operations of dot and cross products. These operations allow us to manipulate vectors and extract valuable information about their relationships and orientations.

Vector Algebra

  • Dot Product: The dot product, denoted by A·B, calculates the scalar quantity representing the projection of vector A onto vector B. It provides a measure of the parallelism or anti-parallelism between the two vectors.
  • Cross Product: The cross product, denoted by A×B, results in a vector perpendicular to both A and B. It represents the area of the parallelogram formed by the two vectors and is used to calculate quantities like torque and angular momentum.

Moving on to the realm of relativity, we encounter the Lorentz transformations, a set of equations that describe how the coordinates of an event change when viewed from different inertial frames of reference. These transformations have profound implications for vector quantities, altering their magnitudes and directions.

Lorentz Transformations

The Lorentz transformations introduce the concept of spacetime, where space and time are intertwined into a single entity. They reveal that the speed of light is constant for all observers, regardless of their motion, and that the passage of time and the measurement of distances are relative to the observer’s frame of reference.

The impact of the Lorentz transformations on vector quantities is significant. For example, the velocity of an object as measured by an observer in one frame of reference will differ from the velocity measured by an observer in a different frame of reference.

Similarly, the length of an object can appear different depending on the observer’s motion.

The mathematical foundations of vectors and relativity provide a powerful framework for understanding the physical world. By manipulating vectors and applying the Lorentz transformations, we gain insights into the nature of space, time, and motion, paving the way for further exploration in the realm of physics.

Applications in Physics

Vectors play a crucial role in describing physical quantities, particularly in the fields of mechanics and electromagnetism. They provide a concise and effective way to represent both the magnitude and direction of these quantities.

Velocity and Acceleration

In mechanics, velocity and acceleration are vector quantities. Velocity describes the rate of change of an object’s position, while acceleration describes the rate of change of velocity. Both velocity and acceleration have both magnitude and direction. The magnitude of velocity is speed, and the magnitude of acceleration is often called simply “acceleration.”

Force is another vector quantity. It is a push or pull that acts on an object, causing it to accelerate. Force has both magnitude and direction. The magnitude of force is measured in newtons (N), and the direction of force is indicated by a vector.

Relativity is a theory of space and time developed by Albert Einstein. It has profound implications for our understanding of the universe and has revolutionized our understanding of many physical phenomena.

Time Dilation and Length Contraction

Two of the most famous predictions of relativity are time dilation and length contraction. Time dilation refers to the phenomenon where time appears to pass more slowly for objects moving at high speeds. Length contraction refers to the phenomenon where objects appear to be shorter when moving at high speeds.

Problem-Solving Techniques

Solving vector problems in relativity requires a systematic approach. Here’s a step-by-step guide to help you:

Vector Representation

Represent vectors using their components (e.g., Cartesian or polar coordinates) or as linear combinations of basis vectors. Understand the relationships between vector components and their magnitudes and directions.

Coordinate Transformations, Homework 1 vectors and relativity

Relativity involves changing between different frames of reference. Learn how to apply coordinate transformations (e.g., Lorentz transformations) to vectors to account for changes in space and time.

Relativistic Invariants

Certain quantities, such as the spacetime interval, remain constant under Lorentz transformations. Identify and utilize these invariants to simplify problem-solving.

Problem-Solving Exercises

Practice solving vector problems in relativity. Start with simple scenarios and gradually increase the complexity. Examples include:

  • Finding the relative velocity between two moving objects.
  • Calculating the time dilation experienced by a moving observer.
  • Determining the length contraction of a moving object.

Visualization and Representation

To enhance comprehension of vectors and relativity, interactive tables and graphics can be employed. These visual aids provide a tangible representation of abstract concepts, making them more accessible and intuitive.

Color-coding and annotations can further clarify the relationships between vectors and the effects of relativity. For instance, using different colors to denote different vector components or highlighting key points in diagrams can aid in understanding.

Interactive Table

An interactive table can display vectors in a dynamic manner, allowing users to visualize their properties and transformations. This table can include columns for vector components, magnitude, and direction, with the ability to modify these values and observe the corresponding changes in the vector’s representation.

Graphic Representations

Graphic representations, such as vector diagrams or spacetime diagrams, can illustrate the concepts of relativity. Vector diagrams can depict the addition and subtraction of vectors, while spacetime diagrams can visualize the effects of time dilation and length contraction.

Extensions and Applications: Homework 1 Vectors And Relativity

The concepts explored in this homework assignment provide a solid foundation for further exploration in the realm of vectors and relativity.

Extending the scope of this assignment, one could delve into the fascinating realm of special and general relativity, where vectors play a crucial role in describing the behavior of space, time, and gravity.

Special Relativity

Special relativity focuses on the behavior of objects moving at speeds close to the speed of light. In this framework, vectors are used to represent four-dimensional spacetime, where time and space are intertwined.

  • Lorentz transformations: Vectors can be used to describe the transformation of spacetime coordinates between different inertial frames moving at constant relative velocities.
  • Time dilation: Vectors can illustrate how time slows down for objects moving at high speeds.
  • Length contraction: Vectors can demonstrate how objects appear shorter when moving at high speeds.

General Relativity

General relativity extends special relativity by incorporating gravity into the picture. In this theory, spacetime is no longer flat but is curved by the presence of mass and energy.

  • Curvature of spacetime: Vectors can be used to represent the curvature of spacetime, which affects the motion of objects and the propagation of light.
  • Gravitational lensing: Vectors can explain how light is bent as it passes through curved spacetime, leading to gravitational lensing effects.
  • Black holes: Vectors can be used to describe the geometry of spacetime around black holes, where gravity is so strong that nothing, not even light, can escape.

Resources for Further Exploration

  • Einstein’s Theory of Relativity: https://www.einstein-online.info/en/spotlights/relativity
  • Special Relativity: https://physics.aps.org/tags/special%20relativity
  • General Relativity: https://physics.aps.org/tags/general%20relativity

Answers to Common Questions

What are vectors?

Vectors are mathematical objects that have both magnitude and direction. They are used to represent physical quantities such as velocity, acceleration, and force.

What is relativity?

Relativity is a theory that describes how space and time are related to each other. It was developed by Albert Einstein in the early 20th century.

How are vectors used in relativity?

Vectors are used to represent physical quantities in relativity. For example, the velocity of an object can be represented as a vector.

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