CHAPTER 2 Special Theory of Relativity-part 1

Transcription

1 CHAPTER 2 Special Theory of Relativity-part The Apparent Need for Ether 2.2 The Michelson-Morley Experiment 2.3 Einstein s Postulates 2.4 The Lorentz Transformation 2.5 Time Dilation and Length Contraction 2.6 Addition of Velocities 2.7 Experimental Verification 2.8 Twin Paradox 2.9 Space-time 2.10 Doppler Effect 2.11 Relativistic Momentum 2.12 Relativistic Energy 2.13 Computations in Modern Physics 2.14 Electromagnetism and Relativity It was found that there was no displacement of the interference fringes, so that the result of the experiment was negative and would, therefore, show that there is still a difficulty in the theory itself - Albert Michelson, 1907

2 Newtonian (Classical) Relativity A reference frame is called an inertial frame if Newton laws are valid in that frame.

3 Inertial Frames K and K

4 The Galilean Transformations For a point P In system K: P = (x, y, z, t) In system K : P = (x, y, z, t ) v t x P K K x -axis x-axis

5 The Galilean Transformations

6 The Inverse Relations Step 1. Replace with Step 2. Replace primed quantities with unprimed and unprimed with primed

7 The Transition to Modern Relativity Although Newton s laws of motion had the same form under the Galilean transformation, Maxwell s equations did not keep the same form. (ex) the speed of light always c 1/ 0 0 In 1892, Hendrik Lorentz and George FitzGerald proposed a radical idea that that solved the electromagnetic problem: Space was contracted along the direction of motion of the body. A precursor to Einstein s theory. In 1905, Albert Einstein proposed a fundamental connection between space and time by introducing two postulates to reconcile the problem and define Special Relativity. -> He proposed that space and time are not separated and that Newton s laws are only an approximation.

8 2.1: The Apparent Need for Ether The wave nature of light suggested that there existed a propagation medium called the luminiferous ether or just ether. Ether had to have such a low density that the planets could move through it without loss of energy It also had to have an elasticity to support the high velocity of light waves

9 An Absolute Reference System The Ether frame was proposed as an absolute reference system in which the ether was stationary and the speed of light was constant c. In the other systems, the speed of light would indeed affected by the relative speed v of the reference ether frame, such as c + v. However, Maxwell s equations don t differentiate between these two systems, since the speed of light is given by c 1/ 0 0 The Michelson-Morley experiment was an attempt to show the existence of ether.

10 2.2: The Michelson-Morley Experiment Albert Michelson ( ) was the first U.S. citizen to receive the Nobel Prize for Physics (1907). He built an extremely precise device called an interferometer to measure the minute phase difference between two light waves traveling in mutually orthogonal directions. Differences in speed of light moving in different directions through the ether would be detected as a phase difference at the detector.

11 The Michelson Interferometer

12 Accuracy the time difference can be shown to be (page 23): exp( i ) t ~ radian

13 Michelson s Conclusion Michelson-Morley experiment

14 The Lorentz-FitzGerald Contraction

15 2.3: Einstein s Postulates Albert Einstein ( ) was only two years old when Michelson reported his first null measurement for the existence of the ether. At the age of 16 (1895) Einstein began thinking about the form of Maxwell s equations in moving inertial systems. In 1905, at the age of 26, he published his startling proposal about the principle of relativity, which he believed to be fundamental.

16 2.3: Einstein s Postulates With the belief that Maxwell s equations must be valid in all inertial frames, Einstein proposes the following postulates: 1) The principle of relativity: The laws of physics are the same in all inertial systems. There is no way to detect absolute motion, and no preferred inertial system exists. 2) The constancy of the speed of light: Observers in all inertial systems measure the same value for the speed of light in a vacuum.

17 Revisiting Inertial Frames and the Re-evaluation of Time In Newtonian physics we previously assumed that t = t Thus with synchronized clocks, events in K and K can be considered simultaneous Einstein realized that each system must have its own observers with their own clocks and meter sticks Thus, events considered simultaneous in K may not be in K.

18 The Problem of Simultaneity Frank at rest is equidistant from events A and B: Frank sees both flashbulbs go off simultaneously. Mary is moving (K ) Frank is at rest (K) Mary, moving to the right with speed observes events A and B in different order: Mary sees event B, then A.

19 The Lorentz Transformations The special set of linear transformations that: 1) preserve the constancy of the speed of light (c) between inertial observers; 2) account for the problem of simultaneity between these observers Lorentz transformation equations: Frank Mary

20 Lorentz Transformation Equations Relativistic factor 1 1 when v 0

21 2.4 Derivation of the Lorentz Transformations Use the fixed system K and the moving system K At t = 0 the origins and axes of both systems are coincident with system K moving to the right along the x axis. A flashbulb goes off at the origins when t = 0. According to postulate 2, the speed of light will be c in both systems and the wavefronts observed in both systems must be spherical. K K ct 21

22 Derivation (con t) Spherical wavefronts in K: Spherical wavefronts in K : ct Note: these are not preserved in the classical transformations with Let x ' ( x vt) so that x ( x ' vt '), where the parameter must be close to 1 for v c. By Einstein's first postulate that the laws of physics by the same in both systems, '=. By Einstein's second postulate that the speed of light is c in both systems, xct, x' ct'. ct ' x ' ( ct vt) and ct x ( ct ' vt ') t t v c t t v c t t v c v c 2 ' (1 / ) and '(1 / ) ' '(1 / )(1 / ) 1 1 v / c 2 2

23 Finding a Transformation for t Recalling x = (x vt) substitute into x = (x + vt) and solving for t we obtain: which may be written in terms of β (= v/c):

24 Complete Lorentz Transformations Including the inverse (i.e v replaced with v; and primes interchanged) xvt xvt x ' xvt v / c 1 y' y z' z 2 2 tvx/ c tvx/ c vx t' ( t ) v / c 1 c v c x x' vt' y y' z z' vx ' t (' t ) 2 c 1) If v << c, i.e., β 0 and 1, Galilean transformation. 2) Space and time are now not separated. 3) For non-imaginary transformations, the frame velocity cannot exceed c. ( is real)

25 2.5: Time Dilation and Length Contraction Consequences of the Lorentz Transformation: Time Dilation: Clocks (T 0 ) in K run slow with respect to stationary clocks (T) in K. T T 0 T 0 1 v / c 2 2 Moving clocks appear to run slow Length Contraction: Lengths (L 0 ) in K are contracted with respect to the same lengths (L) stationary in K. L L L v c 0 1 / Moving objects appear contracted in the direction of the motion

26 Time Dilation To understand time dilation the idea of proper time must be understood: The term proper time,t 0, is the time difference between two events occurring at the same position in a system as measured by a clock at that position. Same location (spark on then off )

27 Time Dilation Not Proper Time spark on then spark off Beginning and ending of the event occur at different positions

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29 He also measures the round trip time in his frame (K) T 0 = t 2 t 1 = 2L/c Note, T 0 = T 0 Now, he observes the round trip of the light in moving frame (K ) Looking at one of the triangles: T T v / c T Note T T 0 moving clocks (K ) run slow as measured by stationary observers (K).

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32 Length Contraction To understand length contraction the idea of proper length must be understood: Frank (K) and Merry (K ) have a meter stick at rest in their own system such that each measures the same length at rest. The length as measured at rest is called the proper length. Frank (K) measures his proper length: L x x 0 r l Merry (K ) measures her proper length: Mary L ' x' x' 0 r l Note that L L ' 0 0 What is the moving stick length (L) measured by Frank at rest? Frank

33 Length Contraction Using the Lorentz transformations of x x' x v / c Mary x' x' x x r l r l SInce L ' 0 L L L ' 0 0 L L L v c 0 1 / x ' xvt vx t' ( t ) 2 c Frank Therefore L L 0 Moving objects appear contracted in the direction of the motion to stationary observers.

34 v = 0.8c Moving objects appear contracted in the direction of the motion to stationary observers. v = 0.9c v = 0.99c v = c 34

35 A Gedanken Experiment to Clarify Length Contraction At t = 0, the flashbulb goes off. At t = t 1, the light reaches mirror after travelling the distance of: At t = t 2, the light returns back to the left end of the stick after travelling the distance of: L L After rearranging, t1, t2 t1 c v c v ct L vt 1 1 ct ( t) Lvt ( t) Total time measured by Frank for light to round trip the stick: L L 2L 2 T= t1( t2 t1) cv cv c T 2 L / c From time dilation of T T, and L L L0 L 0 vt ( t) 2 1

36 Length Contraction Result Moving objects appear shorter to stationary observers. Equivalently Distances appear shorter to moving objects.

37 Length Contraction Result Moving objects appear shorter to stationary observers. Equivalently Distances appear shorter to moving objects.

38 2.6: Addition of Velocities

39 Note: although relative motion of the frames is along x, the y and z components of velocity are also effected.

40 The Lorentz Velocity Transformations In addition to the previous relations, the Lorentz velocity transformations for u x, u y, and u z can be obtained by switching primed and unprimed and changing v to v:

41 speed does not exceed c. Apparent from Lorentz transforms that c is speed limit.

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44 2.8: Twin Paradox The Situation i. One of twins decides to travel out to a distant planet at high speed and return to Earth. ii. The second remains on the Earth. iii. Which twin is younger when they meet up? The Paradox i. When traveling twin returns, stationary twin reasons that the travelers clock has run slow and as such must return younger. ii. However, traveling twin claims that it is the earthbound twin that is in motion and consequently his clock must run slow i.e. earthbound younger. iii. Who s observations are correct?

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46 2.9: Spacetime Spacetime diagrams were first used by H. Minkowski in Often called Minkowski diagrams Represent events in terms of the usual spatial coordinates x, y, and z and a fourth coordinate ict. To represent events in spacetime use Minkowski diagrams. i 1 Paths in Minkowski spacetime are called worldlines.

47 Particular Worldlines (Light line) (Light line)

48 Worldlines and Time World line of Mary (K ) in Frank frame (K) x events which happen simultaneously for one observer (Mary), happen at different times for the other (Frank).

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50 Spacetime diagrams Basics 45 o V c When V c; y( ct) x c xvt y ct x V y ' 1 when V c V c When V c; c xvt y ct x V c y ' 1 V

51 Spacetime diagrams In Galilean t B t ct A The two events A and B occur at the same time in both frames. t' t x x x' x vt tan v c v c

52 In special relativity Mary ct ct Frank x x

53 In special relativity Length contraction: Time dilation: L0 L L 0 T T T 0 0

54 Spacetime diagram for Twin Paradox Distance to star system: 8 light year (ly) Speed of spaceship: 0.8 c Flank thinks the round trip time = 2 x (8 ly/0.8 c) = 20 years But, Frank measures Mary s clock ticking more slowly He think Mary s travel time is only years

55 Invariant Quantities: Spacetime Interval Mary Frank s Or, since all observers see the same speed of light, then all observers, regardless of their velocities, must see spherical wave fronts. s 2 = x 2 c 2 t 2 = (x ) 2 c 2 (t ) 2 = (s ) 2

56 Spacetime Invariants If we consider two events, we can determine the quantity s 2 between the two events, and we find that it is invariant in any inertial frame. The quantity s is known as the spacetime interval between two events. 2 s

57 Spacetime Invariants There are three possibilities for the invariant quantity s 2 : 1) s 2 = 0: x 2 = c 2 t 2, and the two events can be connected only by a light signal. The events are said to have a lightlike separation. 2) s 2 > 0: x 2 > c 2 t 2, and no signal can travel fast enough to connect the two events. The events are not causally connected and are said to have a spacelike separation. 3) s 2 < 0: x 2 < c 2 t 2, and the two events can be causally connected. The interval is said to be timelike.