Chapter 12 Electrodynamics and Relativity Ether 12.1 The Special Theory of Relativity Properties of the ether: Since the light speed c is enormous, the ether had to be extremely rigid. So it did not impede the Ether: Since mechanical waves require a medium to motion of light. For a substance so crucial to electro- propagate, it was generally accepted that light also require a magnetism , it was embarrassingly elusive. Despite the medium. This medium, called the ether, was assumed to peculiar property just mentioned, no one could detect its pervade all mater and space in the universe. ghostly presence. “Absolute” frame: The Maxwell’s equation was inferred Efforts to detect the ether: Michelson inspired by the that the speed of light should equal c only with respect to Maxwell took the problem of detecting the ether as a ether. This meant that the ether was a “preferred” or challenge. He developed his interferometer and used it to try “absolute” reference frame. to detect the earth’s motion relative to the ether. The result were not conclusive. ∂ 2 E 1 ∂ 2 E − = 0 ∂x2 c2 ∂t 2 1 2 The Michelson-Morley Experiment The Michelson-Morley Experiment (II) Michelson and Morley wanted to detect the speed of the Parallel: earth relative to the ether. If the earth were moving relative to L L (2L / c) the ether, there should be an “ether wind” blowing at the T = 0 + 0 = 0 same speed relative to the earth but in the opposite direction. 1 (c − v) (c + v) (1− v2 / c2 ) Michelson-Morley interferometer: Use light speed Perpendicular: variation to verify the existence of ether. 2L (2L / c) T = 0 = 0 2 (c2 − v2 )1/ 2 (1− v2 / c2 )1/ 2 L v2 ∆T = T −T ≅ 0 ( ) 1 2 c c2 3 4 The Michelson-Morley Experiment (III) The Two Postulates The two postulates in the theory of special relativity are: Using v=30 km/s, the expected shift was about 0.4 fringe. 1. The principle of relativity: All physical laws have the Even though they were able to detect shifts smaller than same form in all inertia frames. 1/20 of a fringe, they found nothing. 2. The universal speed of light: The speed of light in free space is the same in all inertial frames. It does not depend Possibilities: on the motion of the source or the observer. z The ether was dragged with the Earth. z No ether. Both postulates are restricted to inertial frames. This is why the theory is special. z Constant light speed. •The principle of relativity extends the concept of covariance from mechanics to all physical laws. •The constancy of the speed of light is difficult to accept at first. All the experimental consequences have confirmed its 5 correctness. 6 Some Preliminaries Some Preliminaries (II) Event: Event is something that occurs at a single point in A reference frame is assumed space at a single instant in time. to consist of many observers Observer: An observer is ether a person, or an automatic uniformly spread through the device, with a clock and a meter stick. Each observers can space. Each observer has a record events only in the immediate vicinity. meter stick and a clock to make measurements only in the Reference frame: A reference frame is a whole set of immediate vicinity. observers uniformly distributed in space. The frame in which an object is at rest is called its rest frame. To synchronize four equally spaced clocks, a signal is sent Synchronization of clocks: It is extremely important to out by clock A to trigger the other clocks---each of which has define precisely what is meant by the time in a given been set ahead by the amount of time it takes to travels from reference frame. This requires a careful procedure for the A to the given clock. synchronization of clocks. 7 8 Relativity of Simultaneity Relativity of Simultaneity How can we determine whether two events at different (another example) locations are simultaneous? Two events at different locations are simultaneous if an Two events that are simultaneous in one inertial observer midway between them receives the flashes at the system are not, in general, simultaneous in another. same instant. Relativity of Simultaneity: Spatially separated events that are simultaneous in one frame are not simultaneous in another, moving relative to the first. 9 10 Geometry of Relativity: Time Dilation Time Dilation (II) How does the relative motion of two frames affects the Now let us find the time interval recorded in the frame S, in measured time interval between two events? which the clock has velocity v. The time interval ∆t in frame S measured by two observers A and B at different positions. ∆∆tt ()cLv⋅=+⋅22 () 2 2L 220 τ = 0 c 2L 1 Tt=∆ =0 ⋅() c 1/− vc22 1 TT==γγ0 where A proper time, τ, is the time interval between two events as 1/− vc22 measured in the rest frame of a clock. In this frame both events occur at the same position. (Note: properÆown), Note that we have used c as the speed of light in both 11 frames---in accord with the second postulate. 12 Time Dilation (III) Time Dilation (IV) Since γ>1, the time interval T measured in frame S (by two Another example: clocks) is greater than the proper time, T0, registed by the clock in its rest frame S’. The effect is called time dilation. h τ = (proper time) Two spatially separated clocks, A and B, record a greater c time interval between two events than the proper time recorded by a single clock that moves from A to B and is hvt22+∆() 1 h present at both events. ∆=tt ⇒∆= cc1/− vc22 1 ∆=t γτ, where γ = 1 1/− vc22 γ = 1− v2 / c2 Moving clocks run slow. 13 14 Example of Time Dilation Geometry of Relativity: Length Contraction Experimental evidence (muon decay): Consider a rod AB at rest in frame S, as shown below. The The reality of time dilation was verified in an experiment distance between its ends is its proper length L0: performed in 1941. The proper length, L0, of an object is the space interval between its ends measured in the rest frame of the object. Rest frame at ground: An elementary particle, the muon (µ), decays into other particle. The particle decay rate is −t /τ N = N0e where τ = 2.2 µs is called the mean lifetime. Moving frame at the upper atmosphere: Another source of generating muon is bombarded with cosmic ray protons. The muon generated with this method has the speed of v=0.995c. The mean lifetime is 10 times longer than their cousins that decay at rest in the laboratory. 15 16 Length Contraction (II) Length Contraction (III) Another example: An observer O’ in Frame S’, which moves at velocity v relative to frame S, can measure the rod’s length. By recording the interval b times at which O’ passes A and B. The measurements in the two frames are cτ Frame S : L0 = ∆x = v∆t 1 LL00= ( : proper length) ⇒ L = L0 2 Frame S′: L = ∆x′ = v∆t′ γ Lvt+∆ Lvt −∆ ∆=tt12, ∆= 12cc LL L1 ∆=tt, ∆= ⇒∆=∆∆= ttt+ 2 12cv−+ cv 12 c1/ − v22 c cc111 Lt=∆==γτ L 22γγγ220 Moving objects are shortened. 17 18 Effects of Length Contraction (I) Length Contraction Effects (II) distortion rest frame v = 0.0 c v = 0.5 c muon frame v = 0.95 c v = 0.99 c 19 20 The Twin Paradox The Barn and Ladder Paradox Nothing in the theory of relativity catches the imagination more than the so-called twin paradox. Twin A stays on earth while twin B travels at high speed to a nearby start. When B returns, they both find that A has aged more than B. The paradox arises because of the apparent symmetry of before farmer’s view the situation: In B’s frame, it is A that leaves and returns, so one should also find that B has aged more than A. Who’s right? A > B ? What’s going on? B > A ladder’s view 21 22 The Lorentz Transformation The Addition of Velocity The laws of electromagnetism are not covariant with respect ′ ′ ′ ′ to the Galiliean transformation. However with Lorentz dx = γ (dx + vt ) = γdt (ux + v) transformation they are covariant. The space and time are vdx′ u′v dt = γ (dt′ + ) = γdt′(1+ x ) related shown as follows: c2 c2 Rest frame x′ = γ (x − vt) Taking the ratio of these equations we find u′ + v vx u = x ′ x 2 t = γ (t − 2 ) 1+ u′v c c x Moving frame x = γ (x′ + vt′) A extreme case c + v vx′ when u′x = c, we have ux = = c ′ 1+ cv c2 t = γ (t + 2 ) c 23 24 Covariant Vector, Contravariant Vector, The Structure of Spacetime: (i) Four-vectors and Invariant Quantity v 0123 aaaaa= xctxxxyxz≡===, , , , and β = the covariant vector (row): a µ ()0123 c µ 0123 ≡−()aaaa xxx001=−γβ(), a0 112 xxx=−γβ() 1 the Lorentz transformations µ µ a 22 the contravariant vector (column): a a = xx= a2 33 3 xx= a xx00γγβ− 00 invariant quantity under Lorentz transformation xx11−γβ γ 00 3 the Einstein 0 µµv summation convention b ==Λ xxv 22∑ 1 xx0010 v=0 3 µµ µµ0123b ab==− ab a a a a abµ = abµ 330001 µµ∑ ()2 xx v=0 b 3 b the Lorentz transformation matrix 00 11 22 33 0 0 11 2 2 3 3 25 =−ab + ab + ab + ab =− ab + ab + ab + ab 26 The Invariant Interval Space-Time Diagrams (Minkowski Diagrams) 01 23 01 23 Two event A and B occurs at ( xxxxA ,AAA , ,) and ( xxxx BBBB, , , ) 22 2 µ µµ lightlike (I == 0, ct d ) the displacement 4-vector: ∆≡xxxA −B µ 22 2 the interval between two events: I ≡∆ xxµ ∆ =− ctd + timelikeIctd<> 0 (22 2 ) spacelike Ictd><0 (22 2 ) lightlikeIctd== 0 (22 2 ) timelike (I <> 0, ct22 d 2) 22 2 27 spacelike (Ictd>< 0, )? Occurs at the same time.