Reference Frames, pt. 2

Reference Frames, Part 2 Relativity and Astrophysics Lecture 03 Terry Herter

Office Hours: Mon. 2:00-4:00, Tues. 11:00-12:00

Outline  

Local Character of Frames Inertial (Free Float) Reference Frames  

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Test Particles Synchronizing clocks The Ideal Observer Some worked problems from end of chapters 1 and 2 of Spacetime Physics

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Reading

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Problem Assignment: Due Wed. 9/09/09

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Definition Note – only applies of limited regions of time and space

Spacetime Physics: Chapter 2 1-4, 1-10, 1-12, 2-2, 2-4, 2-5 (see Lecture/Reading Schedule) Will try to post problems “early” as we cover material in lecture Reference Frames, pt. 2

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Reference Frames, pt. 2

Some physics you should know 

Newton’s equation F  ma

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Newton’s Law of Gravity F

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Force = mass  acceleration

GMm r2

Gravitational force between two objects of masses M and m is proportional to their masses and the inverse square of the distance, r, between them.

Distance, velocity, and acceleration d  vt v  at 1 d  at 2 2

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distance = velocity  time (constant velocity) velocity = acceleration  time (constant acceleration) For constant acceleration

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Local Character of Frames 

A “perfect” free float frame is one that you cannot tell you are moving in space or falling in a gravity field. 

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A free-fall frame near the Earth is not perfect 

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The railway car is now drop vertically. Particles will now move 2 mm apart by impact.

Answers: 

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20 m railway car released from 315 m on surface of Earth takes 8 seconds to impact – particles at either end will move 1 mm closer by impact.

Example 2: 

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Earth

Near the Earth, gravity is not uniform

Example 1: 

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Particles will not move relative to one another (“on their own”) within your measurement accuracy.

Your experiment may not care about this small motion (not enough sensitivity) Or make the frame smaller or time shorter so it doesn’t affect your experiment Or go to space, far away from Earth

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Earth

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Inertial Reference Frame “Inertial” reference frame

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Defines a region of spacetime in which particles remain at rest (or in motion without changing speed or direction) relative to one another within some specified level of accuracy. Also called “Lorentz” (or “free-float”) reference frame.

This test can be carried out entirely within the freefloat frame.

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You don’t need to look outside the room.

Notes:

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NO experiment is done with infinite accuracy! Only general relativity (the theory of gravitation) describes motions in unlimited regions of spacetime – using a series of adjacent free-float frames.

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Synchronize Your Watches 

Events occur in spacetime 

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How can we determine this? 

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To record these we need both a location and a time Layout your 3D coordinate grid (say at 1-meter intervals) and place a clock at each lattice point. We want all clocks to read the “same time” for observers in this reference frame, i.e., the clocks should be synchronized.

Synchronizing the clocks.  

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Pick one clock as the “reference clock” Send out a flash of light (say at midnight) from reference. As the light pulse passes by, set each clock to the light travel time (distance) the clock is from the reference clock. 

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That is if a clock is 5 meters away, it should read 5 meters at the instant the flash goes by.

The time of an event is taken to be the time on the clock nearest it.

Reference clock

The clock records the time and place of the event for later readout and analysis. We can choose the grid size to match the accuracy we need – small for particle physics large for the solar system. Reference Frames, pt. 2

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Ideal Observer 

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We shall often refer to “observer” which is just a shorthand for the measurements associate with the collection of clocks in the inertial reference frame. The “observer” is the person who collect the data from the recording clocks. 

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It may take the observer a long time to actually collect this data due to the finite speed of light. 

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The location and time of an event is recorded by the nearest clock.

For instance, our inertial frame may cover many light-years of interstellar space. So we refer to this person as the ideal observer because in practice it may be difficult for a single real observer to do this.

Space probe example 

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A probe landing on Triton (a moon of Neptune) is 242 light-minutes away. We could not command it real time but could certainly play back the landing data in great deal (at least 242 minutes later).

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Measuring Particle Speeds  

Conventionally speed (or velocity) is measured in meters/sec. However, when we measure time in meters of light-travel time, speed is expressed in meters of distance covered per meter of time. meters of distance covered by particle meters of time required to cover that distance particle speed in meters/sec  speed of light in meters/sec

particle speed 

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That is, we can express speed as a fraction of the speed of light

v  

Here, vconv is the velocity in conventional (m/sec) units. Note: 

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vconv  c

Your textbook uses v here whereas many authors use  for the dimensionless velocity. Reference Frames, pt. 2

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Reference Frames, pt. 2

Moving Frames 

Suppose you have two inertial frames in space  

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Is either one of these frames special? 

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A laboratory frame A rocket frame – moves at constant velocity relative to the laboratory frame – no acceleration They pass through one another. No – either one could be labeled as the laboratory frame! All overlapping inertial frames are equivalent. No frame owns an event, even the one that “causes” it (such as a firecracker being lit and set off). Particles will move in straight lines in all inertial reference frames.

Events are primary. Reference frames are secondary. Reference Frames, pt. 2

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Worked Examples  

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Problem SP 1-9 Problem SP 2-1

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Problem: SP 1-9 

Trips to Andromeda by rocket  

Andromeda galaxy is ~ 2106 light-years distant from Earth as measured in an Earth-linked frame. Could you travel to Andromeda in your lifetime?  Treat Andromeda and Earth as points and neglect motion

between the two. 

a) Trip 1. Your one-way trip takes a time 2.01106 year (in Earth frame) to cover the distance of 2.00106 light-years. How long does the trip last as measured in your rocket frame? 

We use invariance of spacetime interval

interval2  2.01106 2  2.00 106 2 t

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2 rocket

0

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Earth Frame Rocket Frame

So we have trocket = 200,000 years

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Problem: SP 1-9 (cont’d) 

b) What is your rocket speed on Trip 1 as measured in the Earth frame? Express it as a fraction of the speed of light. 

Note if your rocket move 1/2 the speed of light it takes 4106 years to cover 2106 light-years, so

v 

For trip 1 we then have v

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2 106 light  years 1  4 10 6 years 2

2.00 10 6 lyr  0.995 2.0110 6 yr

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Problem: SP 1-9 (cont’d) 

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c) Trip 2. Your one-way trip takes a time 2.001106 year (in Earth frame) to cover the distance of 2.00106 light-years. How long does the trip last as measured in your rocket frame? What is the rocket speed as a fraction of the speed of light? Travel time in rocket frame t rocket  2.0012  2.000 2 106 yrs

Rocket Frame

 63,000 yrs 

Speed of rocket v

2.000 10 6 lyr  0.9995 2.00110 6 yr

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Earth Frame

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Problem: SP 1-9 (cont’d) 

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d) Trip 3. Suppose you want the rocket time to be only 20 years for a one-way trip. What is your speed for this trip? Use the spacetime interval 2 2 2 t rocket  tearth  d gal

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Now tearth 

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Spacetime Interval

d gal

 t

v

2 d gal

v 

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 1

2 1 t rocket  1  5 10 11  0.99999999995 2 2 d gal

v2

d

2 d gal

2

2 rocket

2 gal

2 2 d gal  t rocket

So that v

1 2 2 d gal 1  t rocket

1  z n  1  nz A2290-03

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z  1 14

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Problem: SP 2-1 

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You are riding in an elevator that is show upward out of a cannon. Think of the elevator after it leaves the cannon and is moving freely in the gravitational field of the Earth. Neglect air resistance. 

A) While the elevator is still on the way up, you jump from the “floor” of the elevator. Will you 1) fall back to the “floor”, 2) hit the ceiling, or 3) do something else? If so, what?

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B) You wait to jump until after the elevator has passed the top of its trajectory and is falling back towards Earth. Will your answers to part (A) be different? If so, what happens?

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C) How can you tell when the elevator reaches the top of its trajectory? Reference Frames, pt. 2

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