Notes on Relativity and Cosmology - Physics Department, UCSB
Notes on Relativity and Cosmology - Physics Department, UCSB Notes on Relativity and Cosmology - Physics Department, UCSB
38 CHAPTER 1. SPACE, TIME, AND NEWTONIAN PHYSICS Principle of Relativity: The Laws of Physics are the same in all inertial frames. This understanding was an important development. It ended questions like ‘why don’t we fall off the earth as it moves around the sun at 67,000 mph?’ Since the acceleration of the earth around the sun is only .006m/s 2 , the motion is close to inertial. This fact was realized by Galileo, quite awhile before Newton did his work (actually, Newton consciously built on Galileo’s observations. As a result, applications of this idea to Newtonian physics are called ‘Galilean Relativity’). Now, the Newtonian Physics that we have briefly reviewed worked like a charm! It lead to the industrial revolution, airplanes, cars, trains, etc. It also let to the prediction and discovery of Uranus and Pluto, and other astronomical bodies. This last bit is a particularly interesting story to which we will return, and I would recommend that anyone who is interested look up a more detailed treatment. However, the success of Newtonian physics is a story for other courses, and we have different fish to fry. 1.8 Homework Problems 1-1. Suppose that your car is parked in front of a house. You get in, start the engine, and drive away. You step on the gas until the speedometer increases to 30mph, then you hold that reading constant. Draw two spacetime diagrams, each showing both the house and the car. Draw one in your own frame of reference, and draw the other in the house’s frame of reference. Be sure to label both diagrams with an appropriate scale. 1-2. Derive the Newtonian addition of velocities formula v CA = v CB + v BA for the case shown below where the 3 objects (A, B, and C) do not pass through the same event. Note that, without loss of generality, we may take the worldlines of A and B to intersect at t = 0.
1.8. HOMEWORK PROBLEMS 39 A B C t=0 (If you like, you may assume that the velocities are all constant.) Carefully state when and how you use the Newtonian postulates about time and space. 1-3. To what extent do the following objects have inertial frames of reference? Explain how you determined the answer. Note: Do not tell me that an object is inertial “with respect to” or “relative to” some other object. Such phrases have no meaning as ‘being inertial’ is not a relative property. If this is not clear to you, please ask me about it! a) a rock somewhere in deep space. b) a rocket (with its engine on) somewhere in deep space. c) the moon. 1-4. Which of the following reference frames are ‘as inertial’ as that of the SU campus? How can you tell? a) A person standing on the ground. b) A person riding up and down in an elevator. c) A person in a car going around a curve. d) A person driving a car at constant speed on a long, straight road. 1-5. Show that Newton’s second law is consistent with what we know about inertial frames. That is, suppose that there are two inertial frames (A and B) and that you are interested in the motion of some object (C). Let v CA be the velocity of C in frame A, and v CB be the velocity of C in frame B, and similarly for the accelerations a CA and a CB . Now, suppose that Newton’s second law holds in frame A (so that F = ma CA , where F is the force on C and m is the mass of C). Use your knowledge of inertial frames to show that Newton’s second law also holds in frame B; meaning that F = ma CB . Recall that we assume force and mass to be independent of the reference frame.
- Page 1: Notes on Relativit
- Page 4 and 5: 4 CONTENTS 2.3 Homework Problems .
- Page 6 and 7: 6 CONTENTS 8 General Relativity and
- Page 8 and 9: 8 CONTENTS
- Page 10 and 11: 10 CONTENTS While I have worked to
- Page 12 and 13: 12 CONTENTS way you deal with almos
- Page 14 and 15: 14 CONTENTS but this does not mean
- Page 16 and 17: 16 CONTENTS 0.3 Coursework and Grad
- Page 18 and 19: 18 CONTENTS d) include references t
- Page 20 and 21: 20 CONTENTS However, your challenge
- Page 22 and 23: 22 CONTENTS 0.4 Some Suggestions fo
- Page 24 and 25: 24 CONTENTS Course Calendar The fol
- Page 26 and 27: 26 CHAPTER 1. SPACE, TIME, AND NEWT
- Page 28 and 29: 28 CHAPTER 1. SPACE, TIME, AND NEWT
- Page 30 and 31: 30 CHAPTER 1. SPACE, TIME, AND NEWT
- Page 32 and 33: 32 CHAPTER 1. SPACE, TIME, AND NEWT
- Page 34 and 35: 34 CHAPTER 1. SPACE, TIME, AND NEWT
- Page 36 and 37: 36 CHAPTER 1. SPACE, TIME, AND NEWT
- Page 40 and 41: 40 CHAPTER 1. SPACE, TIME, AND NEWT
- Page 42 and 43: 42 CHAPTER 1. SPACE, TIME, AND NEWT
- Page 44 and 45: 44 CHAPTER 2. MAXWELL, E&M, AND THE
- Page 46 and 47: 46 CHAPTER 2. MAXWELL, E&M, AND THE
- Page 48 and 49: 48 CHAPTER 2. MAXWELL, E&M, AND THE
- Page 50 and 51: 50 CHAPTER 2. MAXWELL, E&M, AND THE
- Page 52 and 53: 52 CHAPTER 2. MAXWELL, E&M, AND THE
- Page 54 and 55: 54 CHAPTER 2. MAXWELL, E&M, AND THE
- Page 56 and 57: 56 CHAPTER 3. EINSTEIN AND INERTIAL
- Page 58 and 59: 58 CHAPTER 3. EINSTEIN AND INERTIAL
- Page 60 and 61: 60 CHAPTER 3. EINSTEIN AND INERTIAL
- Page 62 and 63: 62 CHAPTER 3. EINSTEIN AND INERTIAL
- Page 64 and 65: 64 CHAPTER 3. EINSTEIN AND INERTIAL
- Page 66 and 67: 66 CHAPTER 3. EINSTEIN AND INERTIAL
- Page 68 and 69: 68 CHAPTER 3. EINSTEIN AND INERTIAL
- Page 70 and 71: 70 CHAPTER 3. EINSTEIN AND INERTIAL
- Page 72 and 73: 72 CHAPTER 3. EINSTEIN AND INERTIAL
- Page 74 and 75: 74 CHAPTER 3. EINSTEIN AND INERTIAL
- Page 76 and 77: 76 CHAPTER 3. EINSTEIN AND INERTIAL
- Page 78 and 79: 78 CHAPTER 3. EINSTEIN AND INERTIAL
- Page 80 and 81: 80 CHAPTER 3. EINSTEIN AND INERTIAL
- Page 82 and 83: 82 CHAPTER 3. EINSTEIN AND INERTIAL
- Page 84 and 85: 84 CHAPTER 3. EINSTEIN AND INERTIAL
- Page 86 and 87: 86 CHAPTER 4. MINKOWSKIAN GEOMETRY
1.8. HOMEWORK PROBLEMS 39<br />
A<br />
B<br />
C<br />
t=0<br />
(If you like, you may assume that the velocities are all c<strong>on</strong>stant.) Carefully<br />
state when <strong>and</strong> how you use the Newt<strong>on</strong>ian postulates about time <strong>and</strong><br />
space.<br />
1-3. To what extent do the following objects have inertial frames of reference?<br />
Explain how you determined the answer. Note: Do not tell me that an<br />
object is inertial “with respect to” or “relative to” some other object. Such<br />
phrases have no meaning as ‘being inertial’ is not a relative property. If<br />
this is not clear to you, please ask me about it!<br />
a) a rock somewhere in deep space.<br />
b) a rocket (with its engine <strong>on</strong>) somewhere in deep space.<br />
c) the mo<strong>on</strong>.<br />
1-4. Which of the following reference frames are ‘as inertial’ as that of the SU<br />
campus? How can you tell?<br />
a) A pers<strong>on</strong> st<strong>and</strong>ing <strong>on</strong> the ground.<br />
b) A pers<strong>on</strong> riding up <strong>and</strong> down in an elevator.<br />
c) A pers<strong>on</strong> in a car going around a curve.<br />
d) A pers<strong>on</strong> driving a car at c<strong>on</strong>stant speed <strong>on</strong> a l<strong>on</strong>g, straight road.<br />
1-5. Show that Newt<strong>on</strong>’s sec<strong>on</strong>d law is c<strong>on</strong>sistent with what we know about<br />
inertial frames.<br />
That is, suppose that there are two inertial frames (A <strong>and</strong> B) <strong>and</strong> that you<br />
are interested in the moti<strong>on</strong> of some object (C). Let v CA be the velocity<br />
of C in frame A, <strong>and</strong> v CB be the velocity of C in frame B, <strong>and</strong> similarly<br />
for the accelerati<strong>on</strong>s a CA <strong>and</strong> a CB .<br />
Now, suppose that Newt<strong>on</strong>’s sec<strong>on</strong>d law holds in frame A (so that F =<br />
ma CA , where F is the force <strong>on</strong> C <strong>and</strong> m is the mass of C). Use your<br />
knowledge of inertial frames to show that Newt<strong>on</strong>’s sec<strong>on</strong>d law also holds<br />
in frame B; meaning that F = ma CB . Recall that we assume force <strong>and</strong><br />
mass to be independent of the reference frame.
Change language