special relativity –part ii - university of rochesterkdouglass/classes/ast102/lectures/... ·...

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SPECIAL RELATIVITY – PART IIHOMEWORK 2 ON WEBWORK – DUE TOMORROW (WEDNESDAY) AT 7PM

February 4, 2020Astronomy 102 | Spring 2020 1

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SPECIAL RELATIVITY – PART IIThe Lorentz transformation and the Minkowski absolute interval

The mixing of space and time (spacetime) and the relativity of simultaneity: several examples of the use of the absolute interval

Experimental tests of special relativity


Michaelson-Morley experiment, mounted on a stone slab that is floating in mercury.

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SPECIAL-RELATIVISTIC TIME DILATION


𝑉Frame 2

Frame 1

∆𝑡$

∆𝑡$

∆𝑡%

∆𝑡% =∆𝑡$

1 − 𝑉$

𝑐$

= 𝛾∆𝑡$ (The clock is at rest in Frame 2.)

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Example: A clock with a second hand is flying by at half the speed of light. How much time passes between ticks?

Consider the clock to be at rest in Frame #2 (again, its rest frame) and us to be at rest in Frame #1:

∆𝑡% =∆𝑡$

1 − 𝑉$

𝑐$

=1 s

1 − 0.5𝑐𝑐

$

=1 s

1 − 14

= 1.15 s


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MORE DILATION

A clock, ticking once per second when at rest, flies past us at speed 0.995𝑐. How long between ticks, in our reference frame?


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Observer measures the intervals between ticks on a moving clock, using her own clock for comparison.

∆𝒕𝟏 would always be 1 second if Galileo’s relativity applied.

𝑽 ∆𝒕𝟏0 1 s

10 km/s 1 s

1000 km/s 1.000006 s

100 000 km/s 1.061 s

200 000 km/s 1.34 s

290 000 km/s 3.94 s


Clock∆𝑡$ = 1 s between ticks

𝑉

Her results:

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SPECIAL-RELATIVISTIC VELOCITY ADDITION – 𝑥


𝑉Frame 2

Frame 1

𝑣$

𝑣%

𝑣% =𝑣$ + 𝑉

1 + 𝑣$𝑉𝑐$Note that velocities can be positive or negative.

𝑣$

𝑥

𝑣$ is positive 𝑣$ is negative

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Example: Observer #2 is flying east by Observer #1 at half the speed of light. He rolls a ball at %7𝑐 toward the east. What will Observer #1 measure for the speed of the ball?

𝑣% =𝑣$ + 𝑉

1 + 𝑣$𝑉𝑐$=

13 𝑐 + 1

2 𝑐

1 +13 𝑐

12 𝑐

𝑐$

=56 𝑐76=57 𝑐


(east)

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𝑉Frame 2

Frame 1

𝑣$ =%7𝑐, east

Observer measures the speed of a ball rolled at %

7𝑐 in a

reference frame moving at speed 𝑉, using her surveying equipment and her own clock for comparison.

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SPECIAL-RELATIVISTIC VELOCITY ADDITION – 𝑥Her results for the speed of the ball, for several values of the speed 𝑉 of Frame #2 relative to Frame #1 (all towards the east):


𝑽 𝒗𝟏0 100,000 km/s 100,000 km/s

10 km/s 100,008.9 km/s 100,010 km/s

1000 km/s 100,888 km/s 101,000 km/s

100,000 km/s 179,975 km/s 200,000 km/s

200,000 km/s 245,393 km/s 300,000 km/s

290,000 km/s 294,856 km/s 390,000 km/s

If Galileo’s relativity applied

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Example: Observer #2 is flying east by Observer #1 at half the speed of light. He rolls a ball at %7𝑐 toward the west. What will Observer #1 measure for the speed of the ball?

𝑣% =𝑣$ + 𝑉

1 + 𝑣$𝑉𝑐$=

−13 𝑐 + 12 𝑐

1 +−13 𝑐

12 𝑐

𝑐$

=16 𝑐56

=15𝑐


(east)

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VELOCITY ADDITION PRACTICE

Observer #2, flying east at 𝑉 = 0.995𝑐, rolls a ball at 0.995𝑐 west. How fast does the ball appear to Observer #1 to travel?


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CONSEQUENCES & PREDICTIONS OF EINSTEIN’S SR

• Length contraction (Lorentz-Fitzgerald contraction)

• Time dilation

• Velocities are relative, except for that of light, and cannot exceed that of light

• Spacetime mixing: “distance” in a given reference frame is a mixture of distance and time from other reference frames

• Simultaneity is relative.

• Mass is relative.

• There is no frame of reference in which light can appear to be at rest.

• Mass and energy are equivalent.


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EQUATIONS FOR LENGTH CONTRACTION, TIME DILATION, AND VELOCITY ADDITION IN ONE DIMENSION


𝛾 =1

1 − 𝑉𝑐

$

∆𝑥% =∆𝑥$𝛾

∆𝑡% = 𝛾∆𝑡$

𝑣% =𝑣$ + 𝑉

1 + 𝑣$𝑉𝑐$

You need to understand how to use these formulas.

∆𝑥%, ∆𝑡%, 𝑣%

Frame 1

∆𝑥$, ∆𝑡$, 𝑣$

𝑉Frame 2

∆𝑥$

∆𝑡$

𝑣$

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A QUESTION ON LENGTH CONTRACTION

Your friend Tim can throw a baseball at 99.9% of the speed of light. You watch the baseball fly past you; because of length contraction, it looks like

A. A tiny sphere

B. A thin rod, the baseball’s diameter in length

C. A thin disk, same diameter as the baseball

D. It did before

From our point of view, which way is the baseball oriented relative to its motion?

A.

B.

C.


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LORENTZ TRANSFORMATION: MIXING OF TIME AND SPACE


𝑦%

𝑥%

Frame 1

𝑥$, 𝑡$

𝑥%, 𝑡% “Event” (say, a firecracker exploding)

Observers’ coordinate systems coincide (𝑥% = 𝑥$) at 𝑡% = 𝑡$ = 0. They both see an event, and report where and when it was.

𝑦$

𝑥$

Frame 2

𝑉

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LORENTZ TRANSFORMATION: MIXING OF TIME AND SPACE

The position and time at which the observers see the event are related by the following equations:

𝑥% =𝑥$ + 𝑉𝑡$

1 − 𝑉$

𝑐$

= 𝛾 𝑥$ + 𝑉𝑡$ 𝑡% =𝑡$ +

𝑉𝑥$𝑐$

1 − 𝑉$

𝑐$

= 𝛾 𝑡$ +𝑉𝑥$𝑐$

• Position in one reference frame is a mixture of position and time from the other frame.

• Similarly, time in one reference frame is a mixture of position and time from the other frame.

• The mixture is generally called spacetime.

(We will not be using these equations on homework or exams.)


Lorentz transformation

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MINKOWSKI ABSOLUTE INTERVAL: MIXING OF TIME AND SPACE


𝑦%

𝑥%

Frame 1

∆𝑥$, ∆𝑡$

∆𝑥%, ∆𝑡%

Each observer sees both events and reports the distance ∆𝑥 and time interval ∆𝑡 between them.

Two events (say, two firecrackers exploding)

𝑦$

𝑥$

Frame 2

𝑉

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MINKOWSKI ABSOLUTE INTERVAL: MIXING OF TIME & SPACE

The distance and time interval each observer measures for the two events turn out to be related by

∆𝑥%$ − 𝑐$∆𝑡%$ = ∆𝑥$$ − 𝑐$∆𝑡$$

Thus, the quantity

Absolute interval = ∆𝑥%$ − 𝑐$∆𝑡%$ = ∆𝑥$$ − 𝑐$∆𝑡$$

= ∆𝑥$ − 𝑐$∆𝑡$

is constant; the same value is obtained with distance ∆𝑥 and time interval ∆𝑡 from any single frame of reference.

• This can be derived directly from the Lorentz transformation.

• We will be using this formula for the absolute interval.


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NOMENCLATURE REMINDER

• By 𝑥%, we mean the position of an object or event along the 𝑥-axis, measured by the observer in Frame #1 in his or her coordinate system.

• By ∆𝑥%, we mean the distance between two objects or events along 𝑥, measured by the observer in Frame #1.

• By 𝑡%, we mean the time of an object or event, measured by the observer in Frame #1 with his or her clock.

• By ∆𝑡%, we mean the time interval between two objects or events, measured by the observer in Frame #1.


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MINKOWSKI ABSOLUTE INTERVAL

In other words:

Absolute interval = distance in Frame 1 $ − 𝑐$ time interval in Frame 1 $

= distance in Frame 2 $ − 𝑐$ time interval in Frame 2 $

Usually, we will have Frame #1 at rest (that makes it “our frame”), and Frame #2 in motion.

Events that occur simultaneously (∆𝒕 = 𝟎) in one frame may not be simultaneous in another; simultaneity is relative.


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GEOMETRIC ANALOGY FOR THE ABSOLUTE INTERVAL

Absolute distance (on a map) is covered the same for men and women, even though they take different paths and have different coordinate systems.

The direction the men call North is a mixture of the women’s north and east. The direction the women call North is part north, part west, according to the men.


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GEOMETRIC ANALOGY FOR THE ABSOLUTE INTERVAL

Absolute distance (on the map) is governed by the Pythagorean theorem:

Absolute distance = distance north $ + distance east $

Note the similarity (and the differences) to the Minkowski absolute interval in special relativity:

Absolute interval = distance $ − 𝑐$ time interval $


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EXAMPLE OF THE RELATIVITY OF SIMULTANEITY


Firecrackers detonated simultaneously according to the driver (“you”)

V

Observer (“I”) watches the firecrackers go off and record the time of each explosion.

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EXAMPLE OF THE RELATIVITY OF SIMULTANEITYYour spacetime diagram My spacetime diagram


(𝒙𝟐) (𝒙𝟏)

(𝒕𝟐)

(𝒕𝟏)

My results: I see the rearmost firecracker detonate first, followed by all the others in sequence.

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EXAMPLE OF THE RELATIVITY OF SIMULTANEITYYour spacetime diagram My spacetime diagram


(𝒙𝟐) (𝒙𝟏)

(𝒕𝟐)

(𝒕𝟏)

My results: I see the rearmost firecracker detonate first, followed by all the others in sequence.

∆𝒙𝟐

∆𝒕𝟐 = 𝟎

∆𝒙𝟏

∆𝒕𝟏

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CAR-AND-FIRECRACKER EXPERIMENT #2Car: 1 km long, moving at 1.62×10T km/s; car backfires (event B), then firecracker on front bumper detonates (event D)

Absolute interval is 0.8 km according to each observer, even though the time delay is 2×10UV s according to the observer in the car and 4.51×10UV s for the observer at rest.


∆𝒙𝟐

∆𝒕𝟐

∆𝒙𝟏

∆𝒕𝟏

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OTHER SIMPLE USES OF THE ABSOLUTE INTERVAL

Suppose that you (at rest) were told that events B and D happened simultaneously as viewed from the car, and you measured that they appeared to be separated by 1.19 km along the road (instead of 1 km: longer, because of the motion of the car). Are they simultaneous as seen from rest?

The formula for Absolute Interval can be rearranged:

∆𝑡% =1𝑐

∆𝑥%$ − ∆𝑥$$ + 𝑐$∆𝑡$$ =1.19 km $ − 1 km $ + 299792 km/s $ 0 s $

299792 km/s

= 2.14×10UV s


No

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Suppose that you (at rest) saw the events B and D happen simultaneously, separated by 0.84 km along the road (shorter than 1 km: this is exactly the setup for Lorentz length contraction). What was the time interval between events B and D as seen in the car?

Again, we rearrange the formula for the absolute interval:

∆𝑡$ =1𝑐

∆𝑥$$ − ∆𝑥%$ + 𝑐$∆𝑡%$ =1 km $ − 0.84 km $ + 299792 km/s $ 0 s $

299792 km/s

= 1.80×10UV s


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Suppose the events B and D both happen at the same spot on the car (the tailpipe, for example), separated by 2×10UV s as seen from the car. You see them to be 2.38×10UV s apart (longer: this is exactly the setup for time dilation). How far apart along the road do events B and D appear to you to be?

Again, we rearrange the formula for the absolute interval:

∆𝑥% = ∆𝑥$$ − 𝑐$∆𝑡$$ + 𝑐$∆𝑡%$

= 0 km $ − 299792 km/s $ 2×10UV s $ + 299792 km/s $ 2.38×10UV s $

= 0.387 km


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RECOGNIZE WHAT CONCEPT TO USE

What kind of problem is this? What formula should you use?

One type of radioactive particle decays in 2×10UV s on average, if it is at rest. How long does it take if it is moving at 0.995𝑐?

A. Length contraction

B. Time dilation

C. Velocity addition

D. Absolute interval


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RECOGNIZE WHAT CONCEPT TO USEWhat kind of problem is this? What formula should you use?

In my car, 1 km long and moving at 99% of the speed of light, I flash my headlights and taillights simultaneously; you see the flashes delayed by 4×10UV s. How far apart along the road are the spots when the flashes appeared to you to occur?


B. Time dilation




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RECOGNIZE WHAT CONCEPT TO USE

What kind of problem is this? What formula should you use?

I throw a meter stick so that it moves parallel to its length; it looks to you to be only half a meter long. How fast is it moving, relative to us?


B. Time dilation




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EXPERIMENTAL TESTS OF RELATIVITY

Einstein’s theories of relativity represent a rebuilding of physics from the ground up.

• New postulates and assumptions relate the most basic concepts, such as the relativity of space and time and the “absoluteness” of the speed of light.

• The new theory is logically consistent and mathematically very elegant.

• The new theory contains classical physics, as an approximation valid for speeds much smaller than the speed of light.

• Still: it would be worthless if it did not agree with reality (i.e. experiments) better than classical physics.


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EXPERIMENTAL TESTS OF SCIENTIFIC THEORIESNo scientific theory is valid unless:

• It is mathematically and logically consistent

• And its predictions can be measured in experiments

• And the theoretical predictions are in precise agreement with experimental measurements

Example: Prediction by the “old” and “new” relativity theories for the speed of a body accelerated with constant power. Both theories are valid for low speeds, but only special relativity is valid over the whole range of measurements.


0.01 0.1 1 10 100 1 103 1 1040

1 1010

2 1010

3 1010

4 1010

5 1010

6 1010

Galilean relativitySpecial theory of relativityMeasurements .

Time since start (sec)Sp

eed

(cm

/sec

)

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EXPERIMENTAL TESTS OF SCIENTIFIC THEORIES

What constitutes a valid scientific experiment?

• Measurements are made with accuracy (the size of potential measurement errors) sufficient to test the predictions of prevailing theories

• And the accuracy can be reliably estimated

• And the measurements are reproducible: the same results are obtained whenever, and by whomever, the experiments are repeated


0.01 0.1 1 10 100 1 103 1 1040

1 1010

2 1010

3 1010

4 1010

5 1010

6 1010

Galilean relativitySpecial theory of relativityMeasurements .

Time since start (sec)

Spee

d (c

m/s

ec)

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EXPERIMENTAL TESTS OF SPECIAL RELATIVITY

• Michelson-Morley type experiments: speed of light always the same in all directions in all inertial reference frames, to extremelyhigh accuracy. Repeated many times, and not just in Cleveland.

• Nuclear reactors/bombs: mass-energy equivalence (𝐸 = 𝑚𝑐$) via fission


Sagnac interferometer (spins while emitting light that travels in a loop)

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EXPERIMENTAL TESTS OF SPECIAL RELATIVITYHigh-energy accelerators used in elementary particle physics

• Radioactive particles are seen to live much longer when moving near light speeds than when at rest (direct observation of time dilation).

• Heavy ions get denser (length contraction) as they move faster, and the EM field becomes more intense in the direction of motion.

• Though accelerated particles get extremely close to the speed of light, none ever exceed it.

• Particles become more massive the faster they travel (𝐸 = 𝑚𝑐$)


Relativistic heavy ion collider at BNL

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special relativity –part ii - university of rochesterkdouglass/classes/ast102/lectures/... ·...

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