Michelson Interferometer: An Optical Instrument for Accurate Length Measurements

1. Chapter 26 Relativity

2. Relative Motion (Galilean Relativity) Chapter 3 Section 5 http://www.physics.mun.ca/~jjerrett/relative/relative.html

3. Michelson Interferometer Chapter 25 Section 7

4. Michelson Interferometer • The Michelson Interferometer is an optical instrument that has great scientific importance • It splits a beam of light into two parts and then recombines them to form an interference pattern • It is used to make accurate length measurements

5. Michelson Interferometer, schematic • A beam of light provided by a monochromatic source is split into two rays by a partially silvered mirror M • One ray is reflected to M1 and the other transmitted to M2 • After reflecting, the rays combine to form an interference pattern • The glass plate ensures both rays travel the same distance through glass Active Figure: The Michelson Interferometer

6. Measurements with a Michelson Interferometer • The interference pattern for the two rays is determined by the difference in their path lengths • When M1 is moved a distance of λ/4, successive light and dark fringes are formed • This change in a fringe from light to dark is called fringe shift • The wavelength can be measured by counting the number of fringe shifts for a measured displacement of M • If the wavelength is accurately known, the mirror displacement can be determined to within a fraction of the wavelength

7. Luminiferous Ether Classical physicists (Maxwell, Hertz, etc.) compared electromagnetic waves to mechanical waves Mechanical waves need a medium to support the disturbance (air, water, string, etc.) The luminiferous ether was proposed as the medium required (and present) for light waves to propagate Present everywhere, even in empty space Massless, but rigid medium Could have no effect on the motion of planets or other objects

8. Verifying the Luminiferous Ether Associated with the ether was an absolute frame of reference in whichlight travels with speed c The Earth moves through the ether, so there should be an “ether wind” blowing If v is the speed of the “ether wind” relative to the Earth, the observed speed of light should have a maximum (a), minimum (b), or in-between (c) value depending on its orientation to the “wind”

9. Michelson-Morley Experiment First performed in 1881 by Michelson Repeated under various conditions by Michelson and Morley Designed to detect small changes in the speed of light By determining the velocity of the Earth relative to the ether

10. Michelson-Morley Equipment An interference pattern was observed The interferometer was rotated through 90° • Used the Michelson Interferometer • Arm 2 is initially aligned along the direction of the earth’s motion through space • Should observe small, but measurable, shifts in the fringe pattern as orientation with the “ether wind” changes Active Figure: The Michelson-Morley Experiment

11. Michelson-Morley Results Measurements failed to show any change in the fringe pattern No fringe shift of the magnitude required was ever observed The addition laws for velocities were incorrect The speed of light is a constant in all inertial frames of reference Light is now understood to be an electromagnetic wave, which requires no medium for its propagation The idea of an ether was discarded

12. Relativity I Sections 1–4

13. Basic Problems The speed of every particle of matter in the universe always remains less than the speed of light Newtonian Mechanics is a limited theory It places no upper limit on speed It breaks down at speeds greater than about 10% of the speed of light (v > .1c) Newtonian Mechanics becomes a specialized case of Einstein’s Theory of Special Relativity When speeds are much less than the speed of light v<<c

14. Galilean Relativity Choose a frame of reference Necessary to describe a physical event According to Galilean Relativity, the laws of mechanics are the same in all inertial frames of reference An inertial frame of reference is one in which Newton’s Laws are valid Objects subjected to no forces will move in straight lines

15. Galilean Relativity, cont. A passenger in an airplane throws a ball straight up It appears to move in a vertical path This is the same motion as when the ball is thrown while standing at rest on the Earth The law of gravity and equations of motion under uniform acceleration are obeyed

16. Galilean Relativity, cont There is a stationary observer on the ground Views the path of the ball thrown to be a parabola The ball has a velocity to the right equal to the velocity of the plane The law of gravity and equations of motion under uniform acceleration are still obeyed

17. Galilean Relativity, final The two observers disagree on the shape of the ball’s path Both agree that the motion obeys the law of gravity and Newton’s laws of motion Both agree on how long the ball was in the air Conclusion: There is no preferred frame of reference for describing the laws of mechanics

18. Galilean Relativity – Limitations Galilean Relativity does not apply to experiments in electricity, magnetism, optics, and other areas Results do not agree with experiments According to Galilean relativity, the observer S should measure the speed of the light pulse as v+c Actually observer S measures the speed as c What is the problem?

19. Albert Einstein 1879 – 1955 1905 published four papers: Brownian motion Photoelectric effect 2 on Special Relativity 1916 published theory of General Relativity Searched for a unified theory Never found one

20. Einstein’s Principle of Relativity Resolves the contradiction between Galilean relativity and the fact that the speed of light is the same for all observers Postulates The Principle of Relativity: All the laws of physics are the same in all inertial frames The constancy of the speed of light: The speed of light in a vacuum has the same value in all inertial reference frames, regardless of the velocity of the observer or the velocity of the source emitting the light

21. The Principle of Relativity The results of any kind of experiment performed in one laboratory at rest must be the same as when performed in another laboratory moving at a constant velocity relative to the first one No preferred inertial reference frame exists It is impossible to detect absolute motion with respect to an absolute frame of reference

22. The Constancy of the Speed of Light Been confirmed experimentally in many ways A direct demonstration involves measuring the speed of photons emitted by particles traveling near the speed of light Confirms the speed of light to five significant figures Explains the null result of the Michelson-Morley experiment – relative motion is unimportant when measuring the speed of light We must alter our common-sense notions of space and time

23. Consequences of Special Relativity In relativistic mechanics There is no such thing as absolute length There is no such thing as absolute time Events at different locations that are observed to occur simultaneously in one frame are not observed to be simultaneous in another frame moving uniformly past the first In Special Relativity, Einstein abandoned the assumption of simultaneity

24. Thought experiment A boxcar moves with uniform velocity v Two lightning bolts strike the ends Flashes leave points A’ and B’ on the car and points A and B on the ground at speed c Simultaneity – Thought Experiment • Observer O is midway between the points of lightning strikes on the ground, A and B • Observer O’ is midway between the points of lightning strikes on the boxcar, A’ and B’

25. The light signals reach observer O at the same time He concludes the light has traveled at the same speed over equal distances Observer O concludes the lightning bolts occurred simultaneously Simultaneity – Results

26. Simultaneity – Results, cont By the time the light has reached observer O, observer O’ on the car has moved The light from B’ has already moved by observer O’, but the light from A’ has not yet reached him The two observers must find that light travels at the same speed Observer O’ concludes the lightning struck the front of the boxcar before it struck the back (they were not simultaneous events)

27. Simultaneity – Summary Two events that are simultaneous in one reference frame are in general not simultaneous in a second reference frame moving relative to the first That is, simultaneity is not an absolute concept, but rather one that depends on the state of motion of the observer In the thought experiment, both observers are correct, because there is no preferred inertial reference frame

28. Time Dilation, Moving Observer The vehicle is moving to the right with speed v A mirror is fixed to the ceiling of the vehicle An observer, O’, at rest in this system holds a laser a distance d below the mirror The laser emits a pulse of light directed at the mirror (event 1) and the pulse arrives back after being reflected (event 2)

29. Time Dilation, Moving Observer Observer O’ carries a clock She uses it to measure the time between the events (Δtp) The p stands for proper She observes events 1 and 2 to occur at the same place Light travels distance 2d = cΔtp The time interval Δtp is called the proper time The proper time is the time interval between events as measured by an observer who sees the events occur at the same position You must be able to correctly identify the observer who measures the proper time interval

30. Time Dilation, Stationary Observer Observer O is a stationary observer on the Earth He observes the mirror and O’ to move with velocity v By the time the light from the laser reaches the mirror, the mirror has moved to the right • The light must travel farther with respect to O than with respect to O’

31. Observer O carries a clock He uses it to measure the time between the events (Δt) He observes events 1 and 2 to occur at different places Events separated by distance vΔt Light travels distance cΔt Time Dilation, Stationary Observer

32. Time Dilation, Observations O and O’ must measure the same speed of light The light travels farther for O The time interval, Δt, for O is longer than the time interval for O’, Δtp Observer O measures a longer time interval than observer O’ by the factor gamma Active Figure: Time Dilation

33. Time Dilation, Example The time interval Δt between two events measured by an observer moving with respect to a clock is longer than the time interval Δtp between the same two events measured by an observer at rest with respect to the clock For example, when observer O’, moving at v = 0.5c, claims that 1.00 s has passed on the clock, observer O claims that Δt =  Δtp= (1.15)(1.00s) = 1.15 s has passed – Observer O considers the clock of O’ to be reading too low a value – “running to slow” A clock in motion runs more slowly than an identical stationary clock v O’ O

34. Time Dilation – Equivalent Views Initial View: Observer O views O’ moving with speed v to the right and the clock of O’ is running more slowly Equivalent View: Observer O’ views O as the one who is really moving with speed v to the left and the clock of O is running more slowly The principle of relativity requires that the views of the two observers in uniform relative motion must be equally valid and capable of being checked experimentally

35. Time Dilation – Generalization All physical processes slow down relative to a clock when those processes occur in a frame moving with respect to the clock These processes can be chemical and biological as well as physical Time dilation is a very real phenomena that has been verified by various experiments

36. Time Dilation – Verification Muons are unstable particles that have the same charge as an electron, but a mass 207 times more than an electron Muons have a half-life of Δtp = 2.2 µs when measured in a reference frame at rest with respect to them (a) – unlikely to reach the Earth’s surface. Relative to an observer on earth, muons should have a longer lifetime of Δtp =  Δtp (b) – likely to reach surface A CERN experiment measured lifetimes in agreement with the predictions of relativity

37. Length Contraction The measured distance between two points depends on the frame of reference of the observer The proper length, Lp, of an object is the length of the object measured by someone at rest relative to the object The length of an object measured in a reference frame that is moving with respect to the object is always less than the proper length This effect is known as length contraction

38. Length Contraction – Equation Length contraction takes place only along the direction of motion Active Figure: Length Contraction

39. Length Contraction, Example The length between two points L measured by an observer moving with respect to a ruler is shorter than the length Lp between the same two points measured by an observer at rest with respect to the ruler For example, when observer O’, moving at v = 0.5c, claims that a length of 1.00 m is measured by a ruler, observer O claims that L = Lp /= (1.00 m)/(1.15) = 0.87 m is the measured length between the two points – Observer O considers the length of O’ to be “contracted” A ruler in motion is contracted compared to an identical stationary ruler v O’ O