西南交通大学：《大学物理》课程教学资源（讲稿，双语）CHAPTER 25 the speclal theory of relativity

UNIVERSITY PHYSICS II CHAPTER 25 825.1 reference frames and the classical Galilean relativity 1. what is reference frame A reference frame consists of (1)a coordinate system and (2)a set of synchronized clocks distributed throughout the coordinate grid and rest with respect to it. 3

1 1.what is reference frame A reference frame consists of (1) a coordinate system and (2) a set of synchronized clocks distributed throughout the coordinate grid and rest with respect to it. x z y §25.1 reference frames and the classical Galilean relativity

8 25 1 reference frames and the classical Galilean relativity 「T证 825.1 reference frames and the classical Galilean relativity A reference frame has three spatial coordinates and one time coordinate (x,y, z, t). Four dimensional space-time Inertial reference frames Reference frame noninertial reference frames synchronized clocks

2 §25.1 reference frames and the classical Galilean relativity A reference frame has three spatial coordinates and one time coordinate (x, y, z, t). Four dimensional space-time synchronized clocks: l l A B O Inertial reference frames Reference frame noninertial reference frames §25.1 reference frames and the classical Galilean relativity

8 25 1 reference frames and the classical Galilean relativity 2. Some fundamental concepts P Event is something that happens at a particular place and instant. >Observers belong to particular inertial frames of reference, they could be people, electronic instrument, or other suitable recorders y Relativity is concerned with how an event described in one reference frame is related to its description in another reference frame. That is how the coordinates and times of events measured in one reference frame are related to the coordinate, time, and corresponding physica quantities in another reference frame. 825.1 reference frames and the classical Galilean relativity The special theory of relativity is concerned ith the relationship between events and physical quantities specified in different inertial reference frames >The general theory of relativity is concerned with the relationship between nts and physical quantities specified in any reference frames >Transformation equations are that indicate how the four space and time coordinates specified in one reference frame are related to z the corresponding quantities pecified in another reference frame

3 ¾Relativity is concerned with how an event described in one reference frame is related to its description in another reference frame. That is how the coordinates and times of events measured in one reference frame are related to the coordinate, time, and corresponding physical quantities in another reference frame. 2. Some fundamental concepts ¾Event is something that happens at a particular place and instant. ¾Observers belong to particular inertial frames of reference, they could be people, electronic instrument, or other suitable recorders. §25.1 reference frames and the classical Galilean relativity ¾The special theory of relativity is concerned with the relationship between events and physical quantities specified in different inertial reference frames. ¾The general theory of relativity is concerned with the relationship between events and physical quantities specified in any reference frames. ¾Transformation equations are that indicate how the four space and time coordinates specified in one reference frame are related to the corresponding quantities specified in another reference frame. ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ′ ′ ′ ′ ⇔ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ t z y x t z y x §25.1 reference frames and the classical Galilean relativity

8 25 1 reference frames and the classical Galilean relativity Standard geometry we use for the special theory of relativity has two inertial reference frames called s and s’, with their x-andx’- coordinate axes collinear. Imagine collections of clocks distributed at rest throughout each respective frame; the clocks all are set to O s when the two origins coincide. Observer is at rest in frame s 825.1 reference frames and the classical Galilean relativity Observer is at rest in frame sy

4 ¾Standard geometry we use for the special theory of relativity has two inertial reference frames called S and S’, with their x- and x’- coordinate axes collinear. Imagine collections of clocks distributed at rest throughout each respective frame; the clocks all are set to 0 s when the two origins coincide. Observer is at rest in frame S. §25.1 reference frames and the classical Galilean relativity Observer is at rest in frame S’. §25.1 reference frames and the classical Galilean relativity

8 25 1 reference frames and the classical Galilean relativity 3. Classical Galilean relativity Othe Galilean time transformation equation In classical physics time is a universal measure of the chronological ordering of events and the time interval between them Watches in fast sports cars. airplanes those at rest on the ground; the time interval o spacecraft, and oxcarts tick at the same rate between two events and the rate at which time passes are independent of the speed of the moving clock; they are same everywhere. 825.1 reference frames and the classical Galilean relativity @the Galilean spatial coordinate transformation equations T=I x'ex t=t u=u y=y Z =Z

5 3. Classical Galilean relativity 1the Galilean time transformation equation In classical physics time is a universal measure of the chronological ordering of events and the time interval between them. Watches in fast sports cars, airplanes, spacecraft, and oxcarts tick at the same rate as those at rest on the ground; the time interval between two events and the rate at which time passes are independent of the speed of the moving clock; they are same everywhere. t = t′ §25.1 reference frames and the classical Galilean relativity 2the Galilean spatial coordinate transformation equations z z y y x x vt ′ = ′ = ′ = − t = t′ u u v r r rO r r r r r r ′ = − ′ = − §25.1 reference frames and the classical Galilean relativity

8 25 1 reference frames and the classical Galilean relativity @the Galilean velocity component transformation equations dr dr d(x'+vt) u= dt u'+y The velocity components along a direction perpendicular to the motion are the same in two u2=u standard inertial reference fr antes 825.1 reference frames and the classical Galilean relativity @the Galilean acceleration component transformation equations d'x d(x'+vt) dt a;=a,The acceleration component I =a are the same in the two inertial eference frames 6

6 3the Galilean velocity component transformation equations u v t x vt t x u t x u x x x = ′ + ′ + = = ′ ′ ′ = d d( ) d d , d d z z y y x x u u u u u u v = ′ = ′ = ′ + The velocity components along a direction perpendicular to the motion are the same in two standard inertial reference frames. §25.1 reference frames and the classical Galilean relativity 4the Galilean acceleration component transformation equations x x x a t x vt t x a t x a = ′ ′ + = = ′ ′ ′ = 2 2 2 2 2 2 d d ( ) d d , d d z z y y x x a a a a a a ′ = ′ = ′ = The acceleration components are the same in the two inertial reference frames. §25.1 reference frames and the classical Galilean relativity

8 25 1 reference frames and the classical Galilean relativity = n=n =a a=a, F na=n( I'=F Descriptions of what happens as a result of the laws of mechanics may different from one inertial reference frame to another, R but the laws o mechanics are the l same 825.2 the need for change and the postulates of the special theory 1. Why we need relativity theory troubles with our ideas about time The pion T or n created in a high-energy particle accelerator is a very unstable particle. Its lifetime at rest is 26.0 ns; when it moves at a speed of v=0.913c, an average distance of D=17.4 m are observed before decaying in the laboratory. We can calculate the lifetime in this case by D/v=63.7 ns, it is much larger than the lifetime at rest Such an effect cannot be explained by Newtonian physics

7 F ma m a F a a m m = = ′ ′ = ′ = ′ = ′ r r r r r r z z y y x x a a a a a a ′ = ′ = ′ = Descriptions of what happens as a result of the laws of mechanics may different from one inertial reference frame to another, but the laws of mechanics are the same. §25.1 reference frames and the classical Galilean relativity §25.2 the need for change and the postulates of the special theory 1. Why we need relativity theory? 1troubles with our ideas about time The pion π+ or π- created in a high-energy particle accelerator is a very unstable particle. Its lifetime at rest is 26.0 ns; when it moves at a speed of v= 0.913c, an average distance of D=17.4 m are observed before decaying in the laboratory. We can calculate the lifetime in this case by D/v=63.7 ns, it is much larger than the lifetime at rest. Such an effect cannot be explained by Newtoneian physics!

825.2 the need for change and the postulates of the special theory @troubles with our ideas about length Suppose an observer in the above laboratory placed one marker at location of pions formation and another at the location of its decay. The distance between the two markers is measured to be 17.4 m Another observer who is traveling along with ion at a speed of u=0.913c. To this observer. the distance between the two markers showing the formation and decay of the pion is(0.913c)(26,0×1039)=7.1m. Two observers who are in relative motion measure different values for the same length interval 825.2 the need for change and the postulates of the special theory @troubles with our ideas about speed Light signal showi A throwing ball If the observer a throws a ball at superluminal speed. the result observed by Light signal showing A th observer O will Light signal showi B catching ball be that the g the ball before the observer a throwing it 8

8 2troubles with our ideas about length Suppose an observer in the above laboratory placed one marker at location of pion’s formation and another at the location of its decay. The distance between the two markers is measured to be 17.4 m. Another observer who is traveling along with pion at a speed of u=0.913c . To this observer, the distance between the two markers showing the formation and decay of the pion is (0.913c)(26.0×10-9s)=7.1 m. Two observers who are in relative motion measure different values for the same length interval. §25.2 the need for change and the postulates of the special theory 3troubles with our ideas about speed If the observer A throws a ball at superluminal, speed, the result observed by observer O will be that the observer B get the ball before the observer A throwing it. §25.2 the need for change and the postulates of the special theory

s25.2 the need for change and th postulates of the special theory Troubles with our ideas about energy Electron-positron Electron and positron are initially toward one another at a very low spee Electron and positron have annihilated one another radiation and give a radiation The walls of the container Ae absorbed the radiation and increasing the internal energy of this system 825.2 the need for change and the postulates of the special theory troubles with our ideas about light Electromagnetism provides some subtle clues that all is not right with CHilean relativity. a-E aE. aB a-B =0E or at =0E0 at =3.0×108m/s √AnEo This result is not dependent of the speed v of the source of the waves in the equation!

9 4troubles with our ideas about energy ∆Eint radiation + − e e Electron-positron Electron and positron are initially toward one another at a very low speed. Electron and positron have annihilated one another and give a radiation. The walls of the container absorbed the radiation and increasing the internal energy of this system. §25.2 the need for change and the postulates of the special theory 5troubles with our ideas about light Electromagnetism provides some subtle clues that all is not right with Glilean relativity. 2 2 2 0 0 2 2 2 2 0 0 2 or t B x B t E x Ey y z z ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ µ ε µ ε 3.0 10 m/s 1 8 0 0 = = × µ ε c This result is not dependent of the speed v of the source of the waves in the equation! §25.2 the need for change and the postulates of the special theory

825.2 the need for change and the postulates of the special theory Experiment is performed in reference frame S: C Experiment is performed in reference frame s which is moving at speed v in the same direction as light is moving 825.2 the need for change and the postulates of the special theory Therefore the galilean transformation equations between two inertial reference frames need to be modified to accommodate the observed invariance of the speed of light. The new transformation equations, called the lorentz transformation equations together with their implications and consequences, constitute the special theory of relativity 2. The postulates of special theory of relativity @the speed of light in a vacuum has the same numerical value c in any inertial reference frame, independent of the motion of the source and observer 10

10 Experiment is performed in reference frame S: c c Experiment is performed in reference frame S’ which is moving at speed v in the same direction as light is moving: c v r c §25.2 the need for change and the postulates of the special theory Therefore the Galilean transformation equations between two inertial reference frames need to be modified to accommodate the observed invariance of the speed of light. The new transformation equations , called the Lorentz transformation equations, together with their implications and consequences, constitute the special theory of relativity. 2. The postulates of special theory of relativity 1the speed of light in a vacuum has the same numerical value c in any inertial reference frame, independent of the motion of the source and observer. §25.2 the need for change and the postulates of the special theory