Notes on Relativity and Cosmology - Physics Department, UCSB
Notes on Relativity and Cosmology - Physics Department, UCSB Notes on Relativity and Cosmology - Physics Department, UCSB
32 CHAPTER 1. SPACE, TIME, AND NEWTONIAN PHYSICS A β C B γ α Yes, they will. The point is that clock α might actually pass through α as well as shown below. 01 A β 01 C B 01 γ α Now, by assumption T we know that α and β will agree at event A. Similarly, α and γ will agree at event A. Thus, β and γ must also agree at event A. Finally, a proof!!! We are beginning to make progress! Since the time of any event is well defined, the difference between the times of any two events is well defined. Thus, the statement that two events are ‘at the same time in a given reference frame’ is well-defined!! Er,..... But, might two events be at the same time in one reference frame but not in other frames?? Well, here we go again... Second Corollary to T: Any two reference frames measure the same time interval between a given pair of events. Proof: Recall that a reference frame is defined by a set of synchronized clocks. From the first corollary, the time of an event defined with respect to a synchronized set of clocks is well-defined no matter how many clocks are in that synchronized set. Thus, we are free to add more clocks to a synchronized set as we like. This will not change the times measured by that synchronized set in any way, but will help us to construct our proof. So, consider any two events E 1 and E 2 . Let us pick two clocks β X and γ X from set X that pass through these two events. Let us now pick two clocks β Y and γ Y from set Y that follow the same worldlines as β X and γ X . If such clocks are not already in set Y then we can add them in. Now, β X and β Y were
1.4. NEWTONIAN ADDITION OF VELOCITIES? 33 synchronized with some original clock α X from set X at some events B and C. Let us also consider some clock α Y from set Y having the same worldline as α X . We have the following spacetime diagram: E 1 00 11 E 2 00 11 α X α βX C Y 00 11 β Y γ x 00 11 B γ Y Note that, by assumption T, clocks α X and α Y measure the same time interval between B and C. Thus, sets X and Y measure the same time interval between B and C. Similarly, sets X and Y measure the same time intervals between B and E 1 and between C and E 2 . Let T X (A, B) be the time difference between any two events A and B as determined by set X, and similarly for T Y (A, B). Now since we have both T X (E 1 , E 2 ) = T X (E 1 , B) + T X (B, C) + T X (C, E 2 ) and T Y (E 1 , E 2 ) = T Y (E 1 , B) + T Y (B, C) + T Y (C, E 2 ), and since we have just said that all of the entries on the right hand side are the same for both X and Y , it follows that T X (E 1 , E 2 ) = T Y (E 1 , E 2 ). QED In contrast, note that (S) basically states directly that position is well-defined. 1.4 Newtonian addition of velocities? Let’s go back and look at this speed of light business. Remember the 99%c example? Why was it confusing? Let V BA be the velocity of B as measured by A (i.e., “in A’s frame of reference”). Similary V CB is the velocity of C as measured by B and V CA is the velocity of C as measured by A. What relationship would you guess between V BA , V CB and V CA ? Most likely, your guess was: V CA = V CB + V BA , (1.1) and this was the reason that the 99%c example didn’t make sense to you. But do you know that this is the correct relationship? Why should you believe in equation (1.1)? The answer (still leaving the speed of light example clear as coal tar) is because (1.1) follows from assumptions S and T!!! Proof: Let A, B, C be clocks. For simplicity, suppose that all velocities are constant and that all three clocks pass through some one event and that they are synchronized there. The more general case where this does not occur will be one of your homework problems, so watch carefully!! Without Loss of Generality (WLOG) we can take this event to occur at t = 0. The diagram below is drawn in the reference frame of A:
- 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 34 and 35: 34 CHAPTER 1. SPACE, TIME, AND NEWT
- Page 36 and 37: 36 CHAPTER 1. SPACE, TIME, AND NEWT
- Page 38 and 39: 38 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
1.4. NEWTONIAN ADDITION OF VELOCITIES? 33<br />
synchr<strong>on</strong>ized with some original clock α X from set X at some events B <strong>and</strong> C.<br />
Let us also c<strong>on</strong>sider some clock α Y from set Y having the same worldline as<br />
α X . We have the following spacetime diagram:<br />
E 1<br />
00 11<br />
E 2<br />
00 11 α<br />
X<br />
α<br />
βX<br />
C Y<br />
00 11<br />
β<br />
Y<br />
γ x<br />
00 11<br />
B<br />
γ Y<br />
Note that, by assumpti<strong>on</strong> T, clocks α X <strong>and</strong> α Y measure the same time<br />
interval between B <strong>and</strong> C. Thus, sets X <strong>and</strong> Y measure the same time interval<br />
between B <strong>and</strong> C. Similarly, sets X <strong>and</strong> Y measure the same time intervals<br />
between B <strong>and</strong> E 1 <strong>and</strong> between C <strong>and</strong> E 2 . Let T X (A, B) be the time difference<br />
between any two events A <strong>and</strong> B as determined by set X, <strong>and</strong> similarly for<br />
T Y (A, B). Now since we have both T X (E 1 , E 2 ) = T X (E 1 , B) + T X (B, C) +<br />
T X (C, E 2 ) <strong>and</strong> T Y (E 1 , E 2 ) = T Y (E 1 , B) + T Y (B, C) + T Y (C, E 2 ), <strong>and</strong> since we<br />
have just said that all of the entries <strong>on</strong> the right h<strong>and</strong> side are the same for both<br />
X <strong>and</strong> Y , it follows that T X (E 1 , E 2 ) = T Y (E 1 , E 2 ). QED<br />
In c<strong>on</strong>trast, note that (S) basically states directly that positi<strong>on</strong> is well-defined.<br />
1.4 Newt<strong>on</strong>ian additi<strong>on</strong> of velocities?<br />
Let’s go back <strong>and</strong> look at this speed of light business. Remember the 99%c<br />
example? Why was it c<strong>on</strong>fusing?<br />
Let V BA be the velocity of B as measured by A (i.e., “in A’s frame of reference”).<br />
Similary V CB is the velocity of C as measured by B <strong>and</strong> V CA is the velocity<br />
of C as measured by A. What relati<strong>on</strong>ship would you guess between V BA , V CB<br />
<strong>and</strong> V CA ?<br />
Most likely, your guess was:<br />
V CA = V CB + V BA , (1.1)<br />
<strong>and</strong> this was the reas<strong>on</strong> that the 99%c example didn’t make sense to you. But<br />
do you know that this is the correct relati<strong>on</strong>ship? Why should you believe in<br />
equati<strong>on</strong> (1.1)?<br />
The answer (still leaving the speed of light example clear as coal tar) is because<br />
(1.1) follows from assumpti<strong>on</strong>s S <strong>and</strong> T!!! Proof: Let A, B, C be clocks. For<br />
simplicity, suppose that all velocities are c<strong>on</strong>stant <strong>and</strong> that all three clocks pass<br />
through some <strong>on</strong>e event <strong>and</strong> that they are synchr<strong>on</strong>ized there. The more general<br />
case where this does not occur will be <strong>on</strong>e of your homework problems, so watch<br />
carefully!! Without Loss of Generality (WLOG) we can take this event to occur<br />
at t = 0.<br />
The diagram below is drawn in the reference frame of A:
Change language