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
A well known cheater and liar |
Self imposed exercises
9 comments:.
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
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Answer to Question #201071 in Mechanics | Relativity for karen
Given the vectors A = 8i + 4j -2k N B = 2j + 6k m C = 3i -2j + 4k m Calculate the following: a. 𝐀 ∙ 𝐁 b. The orthogonal component of B in the direction of C c. The angle between A and C d. A x B e. a unit vector λ that is perpendicular to both A and B f. 𝐀 x 𝐁 ∙ 𝐂
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A resultant vector is
equal to the sum of one vector.
equal to the sum of two or more vectors.
equal to the difference of one vector.
What is the resultant of these two vectors?
What is the resultant vector of the two labeled vectors?
What is the magnitude of the resultant vector, r?
What is the resultant of the two vectors shown above?
A vector includes ______ and ________.
direction, magnitude
magnitude, units
direction, units
scalars, scalars
Which of the following describes the vector above?
45 cm at 30 degrees North of East
45 cm at 30 degrees East of North
45 cm North East
A man drives 50 m north and then 50 m east. What quadrant will the man end up in?
A man drives 30 km south and 5 km east. What quadrant does the man wind up in?
A girl gallops 9 m west and 80 m north. What quadrant will the girl end up in?
- 13. Multiple Choice Edit 2 minutes 1 pt A hiker walks 5 kilometers due north and then 7 kilometers due east. What is the magnitude of the resultant vector? 2 km 8.6 km 12 km 12.6 km
If vector A has an x component of 12 units and a y component of 7 units, what is the magnitude of vector A?
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Homework 1 Vectors And Relativity
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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Mastering Physics Solutions Chapter 1 Introduction To Physics. Owner hidden. ... Jun 5, 2018 — Download. More info (Alt + →) Mastering Physics Solutions Chapter 3 Vectors In Physics. Owner hidden. ... Mastering Physics Solutions Chapter 29 Relativity. Owner hidden. Jun 26, 2018 ...
1.TRODUCTION IN 1 Concept Summary . 2. Homework Problems . 9. General Relativity in a Nutshell . 11. 2.EVIEW OF SPECIAL RELATIVITY R 13 Concept Summary . 14. Box 2.1verlapping IRFs Move with Constant Relative Velocities O . 19. Box 2.2nit Conversions Between SI and GR Units U . 20. Box 2.3ne Derivation of the Lorentz Transformation O . 21
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.
General Relativity Fall 2019 Homework 1 solutions Exercise 1: de ning property of Lorentz transformations ... (otherwise we could construct 4 linearly-independent vectors in R2!). This means, explicitly, eT 0X = 0(12) eT 0 T X 0 = 0: (13) From the rst condition, we nd X 0 = (0;a;b;c)T, where the coe cients cannot all be zero. This implies that
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Find step-by-step solutions and answers to Physics - 9780131371156, as well as thousands of textbooks so you can move forward with confidence. ... Section 4.1: Vectors in Physics. Section 4.2: Adding and Subtracting Vectors. Section 4.3: Relative Motion. ... Section 27.1: The Postulates of Relativity. Section 27.2: The Relativity of Time and ...
1 2 gt2; r B(t) = 1 2 gt2; (8) and both have non-relativistic velocities v= gt˝c. Alice emits the rst photon (with velocity c) at time t= t 1, and the second photon at time t= t 1 + t A. (i) Show that Bob receives the rst photon at time t= T 1 which is given by the implicit formula h+ 1 2 gt2 1 c(T t) = 1 2 gT2 1: (9) 1
These answers are to be used to check against your solutions. Your homework should show all of your work, not just the answers! Section 8.1 - Geometric Vectors 31. h = 1.60 cm; v = 2.05 cm 32. h = 0.99 cm; v = 0.99 cm 33. h = 2.04 cm; v = 0.51 cm 34. h = 2.82 cm; v = 1.03 cm 40. a) h = 0.92 N; v = 1.18 N b) h = 0.31 N; v = 1.47 N 41. h = 36 ...
Exercise 3. Consider a 1 + 1 representation of the sub-spaces with two co-ordinate systems (t;x) and (u;v). Draw in the two spacetimes the worldline of a particle with velocity _x:= dx=dt= 0. Draw in the two spacetimes the worldline of a particle with velocity _x:= k(x= kt) with k<1. Interpret the results. Solution 3.
5. !=<1,1> and !=<−1,2>. The vector we need is <−4,5>. To get these components as a combination of ! and !, we need to find a and b such that !∙1+!∙−1=−4 and !∙1+!∙2=5. Solving this system gives !=−1 and !=3, so the vector is 3! −!. 7. The component form is <6cos45°,6sin45°> = <32,32>. 9.
Find step-by-step solutions and answers to College Physics - 9780134724744, as well as thousands of textbooks so you can move forward with confidence. ... Section 1.6: Vectors and Motion: A First Look. Page 26: Questions. ... Velocities of Objects in Special Relativity. Section 27.9: Relativistic Momentum. Section 27.10: Relativistic Energy ...
Given the vectors A = 8i + 4j -2k N B = 2j + 6k m C = 3i -2j + 4k m Calculate the following: a. 𝐀 ∙ 𝐁 b. The orthogonal component of B in the direction of C c. The angle between A and C d. A x
Answer to Solved A First Course of General Relativity, second edition. | Chegg.com
A man drives 30 km south and 5 km east. What quadrant does the man wind up in? A girl gallops 9 m west and 80 m north. What quadrant will the girl end up in? A hiker walks 5 kilometers due north and then 7 kilometers due east. What is the magnitude of the resultant vector? If vector A has an x component of 12 units and a y component of 7 units ...
Homework - Vector Dynamics Package - Section 2.1; October 3/4 - Vectors and Kinematics Test September/October - Ultimate Kinematics Practice. Ultimate Kinematics Answer Key; September 27-October 2 - Nerf Gun Projectile Lab September 25/26 - Projectile Motion. Homework - Textbook p.60 #34-42, 44-49; September 20/21 - Vector Practice Day ...
2 Al walked at a constant speed of 1.5 m/s. Ben jogged at 3 m/s. Complete the table to show Al's velocity for each part of their journey. Al's velocity Ben's velocity A to B 1.5 m/s north B to C C to D D to A 3 m/s west 3 The table below shows some things that can be measured. Tick the boxes to show if each one is a vector or a scalar quantity.
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.