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Relativity

Chapter 37

Relativity is an important subject that looks at the measurement of where and when events take place, and how these events are measured in reference frames that are moving relative to one another.

In this Chapter we will explore with the special theory of relativity (which we will refer to simply as "relativity"), which only deals with inertial reference frames (where Newton's laws are valid). The general theory of relativity looks at the more challenging situation where reference frames undergo gravitational acceleration.

In 1905, Albert Einstein stunned the scientific world by introducing two "simple" postulates with which he showed that the old, common-sense ideas about relativity are wrong. Although Einstein's ideas seem strange and counter-intuitive, e.g., rate at which time passes depends on the speed of reference frame, these ideas have not only been validated by experiment, they are being used in modern technology, e.g., global positioning

satellites.

37-

Some references on relativity

- The original papers of Einstein
- N. David Mermin
- American Journal of Physics65, 476-486 (1997) and 66, 1077-1080 (1998)
- http://people.ccmr.cornell.edu/~mermin/homepage/minkowski.pdf
- It's About Time:Understanding Einstein's RelativityN. David Mermin
http://press.princeton.edu/titles/8112.html

- Others

About that speed

- conventional foot (ft) = 0.3048 m.
1 foot (f) = 0.299792458 m. (1.64% smaller)

1 f/ns = 299,792,458 m/s = c, speed of light.

(ns = nanosecond = 10-9 sec)

The Postulates

1. The Relativity Postulate: The laws of physics are the same for observers in all inertial reference frames. No frame is preferred over any other.

2. The Speed of Light Postulate: The speed of light in vacuum has the same value c in all directions and in all inertial reference frames.

Both postulates tested exhaustively, no exceptions found!

37-

The Ultimate Speed

Fig. 37-2

Experiment by Bertozzi in 1964 accelerated electrons and measured their speed and kinetic energy independently. Kinetic energy →∞ as speed → c

Ultimate Speed→Speed of Light:

37-

Testing the Speed of Light Postulate

If speed of light is same for all inertial reference frames, then speed of light emitted by a source (pion, p0) moving relative to a given frame (for example, a laboratory) should be the same as the speed light that is emitted by a source that is at rest in the laboratory).

1964 experiment at CERN (European particle physics lab): Pions moving at 0.9975c with respect to the laboratory decay, emitting two photons (g).

The speed of the light waves (g-rays) emitted by the pions was measured always to be c in the lab frame (not up to 2c!)→same as if pions were at rest in the lab frame!

37-

Measuring an Event

Fig. 37-3

Event: something that happens, can be assigned three space coordinates and one time coordinate

Where something happens is straightforward, when something happens is trickier (for example the sound of an explosion will reach a closer observer sooner than a farther observer.

Space-Time Coordinates

1. Space Coordinates: three dimensional array of measuring rods

2. Time coordinate: Synchronized clocks at each measuring rod intersection

How do we synchronize the clocks?

All clocks read exactly the same time if you were able to look at them all at once!

Event A: x=3.6 rod lenghts, y=1.3 rod lengths, z=0, time=reading on nearest clock

37-

The Relativity of Simultaneity

If two observers are in relative motion, they will not, in general, agree as whether the two events are simultaneous. If one observer finds them to be simultaneous, the other generally will not.

Simultaneity is not an absolute concept but a rather relative one, depending on the motion of the observer.

Sam observes two independent events (event Red and event Blue) occurring at the same time, Sally, who is running at a constant speed with respect to Sam also observes these two events. Does Sally also find that the events occurred at the same time?

WARNING: When we speak of observers like Sam and Sally, we are referring to the entire space-time coordinate system (frame of reference) in which each is at rest. The observer's location within their frame of reference does not affect the relativistic physics that we discuss here.

37-

A Closer Look at Simultaneity

Fig. 37-4

- Events Blue and Red same distance from Sam and Sally,
- Sam at rest→the light from two events reaches him at same time →he concludes that the two events occurred at the same time (in his frame).
- Sally is moving to right→sees the light from Red event before the light from Blue event. Distance from Sally to B' and R'same and light travels at c from both events towards Sally →Event Red must have occurred at an earlier time (in her frame)!
- What would a third stationary observer, Bill, standing to the right of Sam observe?

37-

The Relativity of Time

The time interval between two events depends on how far apart they occur in both space and time; that is, their spatial and temporal separations are entangled.

Fig. 37-5

37-

The Relativity of Time, cont'd

When two events occur at the same location in an inertial reference frame, the time interval between them, measured in that frame, is called the proper time interval or the proper time. Measurements of the same time interval from any other inertial reference frame are always greater.

In previous example, who measures the proper time?

Speed Parameter:

Lorentz factor:

37-

The Relativity of Time, cont'd

Fig. 37-6

Lorentz factor g as a function of the speed parameter b

37-

Two Tests of Time Dilation

1. Microscopic Clocks. Subatomic particles called muons are unstable and decay (transform into other particles). The average time from when a muon is produced to when it decays (Dt) depends on how fast the muon is moving.

Muon stationary in lab (production and decay in same place, at muon itself) Dt0=2.200 ms

If muon is moving at speed 0.9994c with respect to the lab (production and decay in different places in the lab frame) the lifetime measured by laboratory clocks will be dilated

37-

Two Tests of Time Dilation, cont'd

1. Macroscopic Clocks. Super precision atomic clocks (large systems) flown in airplanes b~7x10-7 (Hafele and Keating in 1977 within 10%, and U. Maryland a few years later within 1% of predictions) repeated the muon lifetime experiment on a macroscopic scale

If the clock on the U. Maryland flight registered 15.00000000000000 hours as the flight duration, how much would a clock that stayed on earth (lab frame) have measured for the duration? More or less? Does it matter whether airplane returns to same place?

37-

The Relativity of Length

The length L0 of an object in the rest frame of the object is its proper length or rest length. Measurement of the length from any other reference frame that is in motion parallel to the length are always less than the proper length.

37-

Does a moving object really shrink?

Fig. 37-7

Must measure front and back of moving penguin simultaneously to get its length in your frame. Let's do this by having two lights flash simultaneously in the rest frame when the front and back of the penguin align with them.

In penguin's frame, your measurements did not occur simultaneously. You first measured the front end (light from front flash reaches moving observer first as in slide 37-7) and then the back (after the back has moved forward), so the length that you measure will appear to be shorter than in the penguin's rest frame.

37-

Proof of Eq. 37-13

Sam is sitting on bench at train station. Using a tape measure, Sam determines the length of the station in his frame, which is the proper length L0. Sally is sitting on a train that passes through the station. What is the length L of the train station that Sally measures?

v

According to Sam, Sally moves through the station (time interval between passing point A and then point B, different places in Sam's frame) in time Dt=L0/v :

Sally

Train

B

A

Sam

length of train station

For Sally, the platform moves past her. She passes points A and B at the same place in her reference frame (proper time) in time Dt0:

37-

The Lorentz Transformation

Fig. 37-9

How are coordinates x, y, z, and t reporting an event in frame S related to the coordinates x', y', z', and t' reporting the same event in moving frame S'?

Gallilean Transformation Equations

Origins coincide at t = t' =0

Lorentz Transformation Equations

37-

The Lorentz Transformation, cont'd

What about S coordinates in terms of S' coordinates?

Switch from one frame to the other by letting v→ -v

What about position and time intervals for pairs of events?

37-

Doppler Effect for Light

Let f0 represent the proper frequency (frequency in the source's rest frame)

If source and detector moving towards one another b → -b

Note: Unlike Doppler shift with sound, only relative motion matters since there is no ether/air to be moving with respect to.

Low Speed Doppler Effect

For b<<1

Same as for sound waves

37-

Doppler Effect for Light, cont'd

Astronomical Doppler Effect

Proper wavelengthl0 associated with rest frame frequency f0.

Replacing b=v/c and using l-l0 = |Dl| = Doppler shift

37-

Doppler Effect for Light , cont'd

Fig. 37-12

Transverse Doppler Effect

Classical theory predicts no Doppler shift observed at point D when source S is at point P.

For low speeds (b<<1)

Transverse Doppler effect another test of time dilation (T=1/f)

Proper periodT0=1/f0)

37-

Doppler Effect for Light , cont'd

The NAVSTAR Navigation System

v1

f03

f01

v3

vairplane

v2

f02

Given v1, v2, v3, f01, f02, f03, and measured f1, f2, f3, can determine vairplane,

37-

A New Look at Momentum

relativistic expression using Dt=Dt0g, where the time Dt0 to move a distance Dx is measured in the moving observer's frame

37-

A New Look at Energy

Table 37-3 The Energy Equivalents of a Few Objects

Object Mass (kg) Energy Equivalent

Electron ≈ 9.11x10-31 ≈ 8.19x10-14J (≈ 511 keV)

Proton ≈ 1.67x10-27 ≈ 1.50x10-10J (≈ 938 MeV)

Uranium atom ≈ 3.95x10-25 ≈ 3.55x10-8J (≈ 225 GeV)

Dust particle ≈ 1x10-13 ≈ 1x104J (≈ 2 kcal.)

U.S. penny ≈ 3.1x10-3 ≈ 2.8x1014J (≈ 78 GWh)

Mass energy or rest energy

37-

A New Look at Energy , cont'd

The total energy E of an isolated system cannot change

Total energy

37-

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