March 7, 2011

1. March 7, 2011 Special Relativity Reading: Chapter 4, Rybicki & Lightman

2. PRELIMINARIES History > SR developed out of a need to reconcile classical, Newtonian mechanics, Maxwell’s Equations and the laws of EM > SR does not deal with gravitation, only electromagnetic forces Inertial Coordinate Systems A key concept in SR is the definition of inertial reference frames: A reference frame is inertial if every particle within it which is initially at rest remains at rest, and every test particle in motion continues that motion without change in speed or direction. Then in SR  All the laws of physics are the same in every inertial inertia frame. “Principle of Relativity”

3. Example: An inertial reference frame is a space ship in free fall near the Earth • A free particle at rest in the vehicle remains at rest • A particle given a gentle push moves across the vehicle in a straight • line with constant speed. • Why? Because the particle and the ship are both falling with the • same acceleration towards the Earth • The space ship is only an inertial reference system when it is • small enough that differences in the accelerations over its size can be • ignored. • If particles didn’t have the same acceleration regardless of size, • shape, composition, etc. then an observer inside the space ship • would notice a relative acceleration among different particles. •  some of the particles at rest would not remain at rest, others would.

4. A sufficiently large space ship would not be an inertial reference frame, because the acceleration by gravity is not uniform across it. from Spacetime Physics by Taylor & Wheeler

5. “The laws of physics are the same in every inertial reference frame” • This means that if you derive a law of physics in one frame, you can apply • it in another. • Both the form of the laws of physics and the numerical values of the • physical constants that the laws contain are the same in every frame. • All inertial frames are equivalent in terms of every law of physics •  the laws of physics cannot provide a way to distinguish one • inertial reference frame from another. • Although the laws of physics are the same in every frame, • measured quantities (like the time between events A and B, the distance • measured between two points, even the force) will not be the same. • Only the mathematical form of the laws of physics are the same.

6. Note: Special relativity can deal with accelerated motion, but only in inertial frames

7. Events The concept of an “event” is crucial in SR. An “event” is something that “happens” at a specific location in space (x,y,z) and at a specific time, t The key question in SR is: How do the coordinates x,y,z,t of a given event relate to the coordinates of the same event as measured by an observer in another inertial coordinate system. One imagines a lattice-work grid to measure x,y,z and a bunch of “clocks” which are somehow synchronized by pulses of light, say, and then can record when a particular “event” happens at point x,y,z These things seem intuitively obvious, but what is meant by a measurement at a point x,y,z,t is crucial.

8. “The Observer” The word “observer” is a shorthand way of speaking of the whole collection of recording clocks associated with one inertial frame.

9. For simplicity, we will define our inertial coordinate systems for two observers in such a way that the origins are chosen and the clocks synchronized in such a way that when the origins coincide, the clocks read zero. The speed between the two systems is parallel to the x-axis. y' y v x' x z' z So we want to connect the coordinates of some event in one frame (x,y,z,t) to the coordinates of the same event in the other frame, (x‘,y‘,z‘,t‘)

10. Before SR, the transformation was obvious: The Galilean Transformation However, the fact that physical constants are the same in all frames implies that the SPEED OF LIGHT (c) is the SAME IN ALL FRAMES: A pulse of light along the x-axis in the “unprimed” system travels in time In the “primed” system, it also travels since light travels at c in all frames But: So the Galilean transformation predicts

11. Instead, one can show that the proper transformation is the Lorentz Transformation. where or letting The reasons why the transformation takes this form are discussed in texts about SR, but note that since we do satisfy the “pulse of light” argument, since transform correctly

12. The inverse of is which looks like the original, except that “primed” and “unprimed” variables are interchanged, and v is replaced by -v

13. Note: deviations from Galilean transformations become large when Our every-day experience involves Some implications of what this means:

14. K frame: unprimed K’ frame: primed Length Contraction Consider a rigid rod of length at rest in frame K’ What is the length in frame K? where x2and x1 are the positions of the ends of the rod measured at the same time t in the K frame. The rod appears shorter by a factor

15. This is symmetric: K sees a rod which appears shorter by a factor of 1/γ K’ would see the same rod held up by an observer in the K frame contracted by 1/γ This is because the observers in the 2 frames would not agree that the ends of the stick were measured at the same time in both frames.

16. Time Dilation Suppose a clock at rest at the origin of the K’ frame measures an interval of time What time interval does an observer in frame K measure? In K’ we have x’=0, so the time interval measured in K is The moving clock appears to have slowed down. Similarly, K’ will think K’s clocks have slowed down.

17. Relativity of Simultaneity Events which are simultaneous in one frame are NOT simultaneous in any other frame. All the clocks in your inertial frame seem to be going at the same rate, but clocks in the moving frame differ from one another depending on their LOCATION. So at a given t in the K frame, must be some fixed value for all clocks in the K’ frame Thus the greater x’ is, the smaller t’ is. This effect is the explanation for most “paradoxes” in SR.

18. To illustrate, Imagine we have two clocks in the K’ frame at positions A and B, separated by length L’. A flashbulb, located exactly in the middle, goes off. A and B are instructed to set their clocks to t’=0 when the flash reaches them, which in the K’ frame, happens at exactly the same time L’ K’ What does K see?

19. What does K see? K v v t=0 At t=0, the flash goes off, and the wavefront starts to expand B A At t=tA, it reaches Point A, and A resets his clock at t’ = 0 t=tA A B Sometime later, at t=tB, the wavefront reaches B, and he resets his clock, also at t’ = 0 t=tB B A Thus, to an observer in the K frame, A has reset his clock BEFORE B reset his clock.... whereas in K’, the clocks were reset SIMULTANEOUSLY.

20. Cosmic-Ray Mesons: Observed time dilation Bruno Rossi & D.B. Hall 1941, Phys Rev, 59, 223 Cosmic Rays impinging on the top of the Earth’s atmosphere produce μ-mesons which decay via with some half-life. neutrinos • The mesons travel down through the Earth’s atmosphere at v ~ c • From the half-life for decay, one can estimate how many should be • seen at sea-level vs. high in the atmosphere, given the travel time. • Many more are seen at sea level than expected •  their radioactive “clock” appears to be running slow.

21. GPS: Observed time dilation • GPS system consists of a network of 24 satellites in high orbits around • the Earth. • Each satellite has an orbital period of about 12 hours, and an orbital • speed of about 14,000 km/hour • The satellite orbits are arranged so that at least 4 and sometimes as • many as 12 satellites are visible from any point on Earth. • Each satellite carries an atomic clock which “ticks” with an accuracy of • 1 nanosecond. (Actually they each carry 2 cesium clocks and 2 • rubidium clocks). • A GPS receiver determines your position by comparing time of arrival • signals from a number of different satellites, and thus figuring • out how far you are from each satellite -- “triangulation.”

22. The inexpensive hand-held GPS receivers can determine your position • to 5 or 10 meters. To achieve this accuracy you need to know the • clock ticks to an accuracy of 20-30 nanoseconds. Military applications • and airliners have more accurate GPS receivers. • Special relativity: the clocks on board the satellites will run slower than • atomic clocks on the ground by 7 microseconds per day because of time • dilation. • General relativity: The clocks on board the satellites run FASTER than • atomic clocks on the ground by 45 microseconds a day because the • curvature of space-time from the Earth’s gravitational field is LESS • farther from the surface. • If you don’t take these two effects into account then you would • be as much as 10 km off from the build up of errors each DAY. • To solve this basically the number of cesium atom “ticks” per second is • redefined for the clocks on the satellites so that “one second” on the satellite • is the same as “one second” on the ground. • (on the ground a CS-133 clock frequency is 9,192,631,770 Hz)

23. TRANSFORMATION OF VELOCITIES Suppose a particle has velocity in frame K’ What is its velocity in frame K? y y’ K’ K v x x’ Lorentz Transform

24. Differentials  so...

25. Note: These formulae can be used to add velocities e.g. Suppose In Galilean transforms, you expect But, using the Lorentz-transform formulae sums of velocities are ALWAYS < c

26. TRANSFORMATION OF ACCELERATIONS Similarly, one can ask how accelerations transform from K to K’ velocity so

27. Recall So

28. Similarly, can show So ax and ay are related to ax’ and ay’ by some fairly complicated expressions involving not only v, but also ux’ and uy’ If a body is instantaneously at rest in the K’ frame, then ux’ = uy’ = 0 and ax is diminished by γ3 ay is diminished by γ2

29. Transformation of Angles (Aberration of Star Light) We saw how ux, uy, uz transformed to ux’, uy’, u’z for v = relatively velocity of frames along x-axis. We could also write u in terms of the components parallel and perpendicular to v, where v is in some arbitrary direction with respect to x,y Then y’ y u u’ x x’

30. The angle where The interesting case is u=u’=c then

31. Or, in terms of cosθ: also so

32. Stellar Aberration Discovered by James Bradley in 1728 Bradley was trying to confirm a claim of the detection of stellar parallax, by Hooke, about 50 years earlier Parallax was reliably measured for the first time by Friedrich Wilhelm Bessel in 1838 Refn: A. Stewart: The Discovery of Stellar Aberration, Scientific American, March 1964 Term paper by Vernon Dunlap, 2005

33. Because of the Earth’s motion in its orbit around the Sun, the angle at which you must point a telescope at a star changes A stationary telescope Telescope moving at velocity v

34. Analogy of running in the rain

35. As the Earth moves around the Sun, it carries us through a succession of reference frames, each of which is an inertial reference frame for a short period of time.

36. Bradley’s Telescope With Samuel Molyneux, Bradley had master clockmaker George Graham (1675 – 1751) build a transit telescope with a micrometer which allowed Bradley to line up a star with cross-hairs and measure its position WRT zenith to an accuracy of 0.25 arcsec. Note parallax for the nearest stars is ~ 1 arcsec or less, so he would not have been able to measure parallax. Bradley chose a star near the zenith to minimize the effects of atmospheric refraction. .

37. The first telescope was over 2 stories high, attached to his chimney, for stability. He later made a more accurate telescope at his Aunt’s house. This telescope is now in the Greenwich Observatory museum. Bradley reported his results by writing a letter to the Astronomer Royal, Edmund Halley. Later, Brandley became the 3rd Astronomer Royal.

38. Vern Dunlap sent this picture from the Greenwich Observatory: Bradley’s micrometer

40. In 1727-1728 Bradley measured the star gamma-Draconis. Note scale

41. Is ~40 arcsec reasonable? The orbital velocity of the Earth is about v = 30 km/s Aberration formula: (small β) (1)

42. Let Then α is very small, so cosα~1, sinα~α, so (2) Compare to (1): we get Since β~10^4 radians  40 arcsec at most

43. BEAMING Another very important implication of the aberration formula is relativistic beaming That is, consider a photon emitted at right angles to v in the K’ frame. Suppose Then

44. So if you have photons being emitted isotropically in the source frame, they appear concentrated in the forward direction.