Special RelativityTopics

Frames of referenceEinstein’s development of special relativityProper time and lengthRelativistic timeDeath of simultaneityRelativistic lengthRelativistic momentumVelocity additionMatter-energy equivalenceCan you break the speed limit?Muons, twins, and antimatterEinstein the man

MotivationDiscover the origins of the Special Theory of Relativity.

1

A short mathematical explorationBefore we start talking about Special Relativity, I want to get the hardest bit of math out of the way so it won’t be a distraction later on.

Say hello to gamma:

It makes sense to track this function, using “v/c” as the variable.

v/c = β

2

A short mathematical exploration

β (v/c) γ 0.1 1.005

0.3 1.0480.5 1.1550.7 1.4000.8 1.6670.9 2.2940.95 3.2030.97 4.1130.99 7.0890.999 22.366

3

That was the hard stuff!

If you’re still worried about the math, don’t be. You’ve already passed the hard stuff—and really, it wasn’t that bad, was it?

The challenge with Einstein is the mind-bending change in perspective it requires.

4

Understanding reference framesRecall our definitions of reference frames, from the first week of class.

An inertial reference frame is one in which Newton’s First Law applies:

LAW #1: An object at rest stays at rest, unless acted upon by an external, unbalanced force.

Similarly, an object in motion continues in motion, unless acted upon by an external, unbalanced force.

5

Understanding reference framesLet us examine more carefully the idea of inertial, and noninertial reference frames.

We will time travel to 1960, Ontario Canada, to the studios of Drs. Hume and Ivey for guidance.

6

Einstein beginsEinstein began his ruminations with the observation that, whether they were correct or not, Newton’s Laws did not depend upon your location in the Universe, orientation, or even velocity. Said differently, one cannot detectmotion.

Newton’s Laws were equally true in every inertial reference frame.The same could also be said for Maxwell’s equations for light, or every law of physics, for that matter!

Example: Newton’s second law is not:

F=ma(1+V/Vz + X/Xsc)

Vz=vertical speed out of disk of galaxy;Xsc=distance from the center of the Virgo Supercluster.

7

Einstein begins

Einstein also noted that everyone seemed to measure the same value for the speed of light. He explained this by simply adopting it as an assumption.

This is how Einstein interpreted Michelson’s inability to detect the lumeniferous aether: He said that light always travelled at the same speed, and that ultimately, there was no need to invoke a mysterious aether!

8

Einstein’s postulatesThe assumptions of relativity, therefore, are:

1. The laws of physics work for all observers. (or, absolute motion cannot be detected)

2. The speed of light (c) is the same for all observers.

A modern perspective is to say that the constancy of “c” is a law of physics, thus reducing the number of assumptions from two to one:

The laws of physics work the same for everyone.

9

Einstein’s postulates

1. The laws of physics work for all observers. (or, absolute motion cannot be detected)

2. The speed of light (c) is the same for all observers.

On first glance there is nothing here that seems particularly difficult to accept, but weird things await us.

Much of the weirdness from Einstein’s theory comes from the finite speed of light.

Indeed, Einstein claims to have been haunted for years by the question, “What do you see if you travel with a photon?”

10

Einstein’s postulates1. The laws of physics work for all observers.

(or, absolute motion cannot be detected)2. The speed of light (c) is the same for all observers.

Consider simple velocity addition…

Consider two observers. One is stationary, the other is moving at 0.5c.

A pulse of light passes them…what does each see?

Both observers measure the pulse of light traveling at “c”. 11

Annus mirabilis: 1905On a Heuristic Viewpoint Concerning the Production and Transformation of Light

Explaining the photoelectric effect via the particle nature of light.

On the Motion of Small Particles Suspended in a Stationary Liquid, as Required by the Molecular Kinetic Theory of Heat

A study of Brownian motion, essentially solidifying the concept of atoms. Previously, atoms had been thought of as useful conceptual tools, that might not have corresponding physical reality.

On the Electrodynamics of Moving BodiesSpecial relativity.

Does the Inertia of a Body Depend Upon Its Energy Content? Simply stated, the formulation of E=mc2.

The 1922 telegram by the Swedish Secretary of Sciences clearly stated Einstein’s Nobel Prize was for his accomplishments, but that the theory of relativity was not one of them!

12

Being properBefore we proceed, it is important to become familiar with the relativistic use of the adjective “proper.”

Proper Distance, Proper LengthA distance or length is said to be “proper” if (and only if)it is measured in the frame of reference in which the object being measured is not moving.

Proper Time IntervalA time interval is said to be “proper” if (and only if) the time interval is measured at the same place (in some reference frame).

13

Examples of propernessSuppose road-side Rhonda is standing at a roadside bus stop, accompanied by her handi-dog.

Road-side Rhonda can measure the proper length of a parked car, or how long it takes for her dog to fall asleep (the proper time).

Meanwhile, Bertha is on a bus, and is about to have lunch.

Bus-rider Bertha can measure the proper length of the moving bus, and how long it takes for her to eat her lunch (the proper time.

Examples of propernessBut…Road-side Rhonda cannot measure the moving bus’s proper length, nor the proper time for bus-rider Bertha to eat her lunch.

Similarly, bus-rider Bertha cannot measure the parked car’s proper length, nor the proper time for road-side Rhonda’s dog to nap.

This raises the question: what happens when you measure things in a reference frame moving with respect to yours?

The light clockLet us follow Einstein in another thought experiment.

We start with a highly simplified, but highly impractical clock. Imagine two mirrors, separated by a distance (D). How long (∆t′) does it take for a photon to make a round trip?

Use the Rate Equationv = ∆x/∆t→ c = 2D/∆t′ → ∆t′ = 2D/c

Notes:1. The prime symbol is on ∆t′ because it is traditional.2. ∆t′ is measured from the same bottom point, so it is a proper time. 16

The light clockSuppose, now, that the light clock is moving to the right at velocity (v) with respect to a stationary observer. The clock moves from A to B to C.

What does the stationary observer, sitting by the clock dials, measure as the clock-box zips to the right?In the time (∆t) it takes for the beam to take the round trip, it travels to the right as it moves up and down. The beam does all this, of course, at the speed of light. 17

The light clock

In half of the time for the entire round trip—(∆t/2)—the photon travels from the bottom mirror to the top mirror. In this time, it travels to the right, from position “A” to position “B”.

The horizontal distance is ∆x = v(∆t/2).18

Relativistic effect #1: time dilation

During this time (∆t/2), the photon travels (at c) along the long side of the triangle, covering a distance c(∆t/2).

Since:L2 = Δx2 + Δy2

Therefore:(cΔt/2)2 = (vΔt/2)2 + D2

Solving for Δt…19

Solving for Δt :

Since: and: ∆t′ = 2D/c (from before)

THEN….

20

Relativistic effect #1: time dilation

∆t = γ∆t′ - or - ∆t′ = ∆t/γ

Relativistic effect #1: time dilation

What does this mean?

The proper time interval measured by a person moving with the clock is ∆t′.

A second person, watching the clock zip by, experiences the time interval γ∆t′.

If the person with the clock experiences 10 seconds, the second person experiences more than 10 seconds!

Time, in a frame of reference that is moving with respect to yours, progresses at a slower rate!

Examples:The ISS’s speed ≈ 7706 m/s (0.0000257c).

Astronauts on the ISS experience time at 0.99999999967× the rate you do.

21

∆t = γ∆t′ - or - ∆t′ = ∆t/γ

Relativistic effect #1: time dilation

What about speeds near the speed of light? Consider a round trip to α Centauri (8.6 LY). How long does it take?

v/c Δt′/Δt Time (Earth) Time (Astronaut)0.1 0.995 86.0 y 85.7 y0.5 0.866 17.2 y 14.9 y0.8 0.600 10.8 y 6.5 y0.95 0.312 9.1 y 2.8 y0.99 0.141 8.7 y 1.2 y0.999 0.045 8.6 y 0.38 y = 4.6 months

22

Relativistic effect #2: relativistic simultaneityWhat does it mean for two events to be “simultaneous”?

It means the two events happened at the same time.

Suppose a double-lightning bolt hits the ground in two places.

To an observer at the midpoint between the strike points, the flashes from both events will arrive at the same time.

The observer declares that the two lightning strikes occurred simultaneously.

Now, let’s make the situation a little more complicated.23

Relativistic effect #2: relativistic simultaneitySuppose a double-lightning bolt hits the front and back ends of a moving train car, and as a result the wheels instantly spark and scorch the tracks.

An observer on the ground, at the midpoint between the scorch marks, sees the light from both flashes at the same time.

This observer declares that the two lightning strikes occurred simultaneously.

24

Relativistic effect #2: relativistic simultaneityBut…to the observer on the train, the situation is different. The train is moving to the right, so the passenger moves towards the photons in front of him, at the same time those photons are moving towards him.

Flash!

Photons from right received.

Photons from left received.

The passenger says the lightning struck the front of the train before it strikes the back.25

Relativistic effect #2: relativistic simultaneity

26

L

What the trainside observer sees

What the train passenger sees

Relativistic effect #2: relativistic simultaneityHow much of an effect is this?

If the two lightning strikes are separated by a distance L, and the train has a velocity of “v”, the passenger will measure a time (∆t) between arrival times of the pulses, given by:

27

L

Relativistic effect #2: relativistic simultaneityWho is correct? Were the events really simultaneous, or did one really happen before the other?

They both are! Remember, in relativity, both reference frames are equally valid.

Consider two spaceships passing each other in the opaque and mysterious Mutaran Nebula. One ship is struck by two asteroid chunks. Whether the strikes were simultaneous or not, depends upon your frame of reference.

28

Relativistic effect #3: relativistic lengthThe endpoints of a train station are marked by points at x1 and x2.

The proper length of the train station Lo, measured by a dog by the station, is given by Lo = x2 - x1

A person on a train moves by at velocity (v).

The dog measures the time (Δt) for this person to travel from x1 to x2, and so: Lo = v(Δt)

NOTES: The Δt is for events at different locations, so it is not a proper time This is similar to the light clock moving sideways in time Δt. 29

Relativistic effects on lengthFrom the perspective of the train passenger, the station is sliding by with velocity (v).

The moving passenger measures the station to move past in the proper time interval Δt′.

This means that the length of the station from the perspective of the passenger is: L′ = v(Δt′).

Since (from before): Lo = v(Δt) then…Lo/L′ = v(Δt)/v(Δt′) = Δt/Δt′

30

Relativistic Effects on LengthFrom the previous page:

Lo/L′ = v(Δt)/v(Δt′) = Δt/Δt′

But from our observations on time dilation, where moving reference frames have been shown to have a slower rate of time-flow:

∆t = γ∆t′ - so - Lo/L′ = γ -or -

L′ =Lo/γ

This is called Lorentz contraction.

Examples: Lo = 10 mv/c L′

0.1 9.95 m0.5 8.66 m0.8 6.00 m0.95 3.12 m0.99 1.41 m0.999 0.45 m 31

Relativistic Effects on LengthWhat does this mean? Does the object REALLY become shortened?

No—there are no forces acting upon the object to shorten it.

Furthermore, consider a massive neutron star—if you shifted into a frame of reference moving rapidly, you would see the neutron star become compressed in one dimension. If you insisted the neutron star really were compressed, you could force it to become a black hole! This is obviously not possible.

This business of Lorentz contraction, ultimately comes from a disagreement in where the front and back ends of a travelling object lie, in a simultaneous moment of time, that you must use when you measure its length.

L′ =Lo/γ

32

THE EquationThe most famous result of relativistic physics is:

E=mc2

This means that matter and energy are both aspects of the same thing: mass-energy. The energy from this equation is called its “rest energy.”

Mass-energy can be switched back and forth in form, like ice to water to steam. This powers stars and nuclear power plants.

We saw hints that matter and energy were interchangeable when we saw that both entities have wavelengths, and were subject to particle-wave duality.

33

Momentum and EnergyMomentum (p) Energy (E)

For Newton mv ½mv2

For Einstein: γmv γmc2

This means that as objects move faster and faster, the amount of momentum (or energy) needed to accelerate it to even greater speeds is harder and harder.

To move a particle to the speed of light would require infinite energy and infinite momentum. You can’t do this.

The speed of light is a cosmic limit.

34

Relativistic Velocity AdditionJust to round things out… What happens when you add two velocities?

Newton Einstein v1 + v2

Suppose a rocket was moving at a velocity V, and it shot a missile forwards at the same velocity. How fast would the missile move past you?

V Newton Einstein 0.1 c 0.2 c 0.198 c 0.5 c 1.0 c 0.800 c 0.8 c 1.6 c 0.976 c 0.95 c 1.9 c 0.999 c 0.99 c 1.98 c 0.9999 c 0.999 c 1.998 c 0.999999 c

35

Einstein’s Interpretation of the Speed LimitEinstein: Matter and energy is interchangeable. Furthermore, matter and photons all travel through space-time at the speed of light.

The rate you travel through space-time is divvied into your spatial velocity, and your temporal velocity.

A stationary object (zero spatial velocity) has all its motion through space-time expressed in the time component. It travels through time at 1 sec/sec.

A moving object has some of its space-time motion in its spatial velocity, and has less to dedicate to its time component. It travels through time at less than 1 sec/sec.

A photon has all its space-time motion concentrated in its spatial velocity, and has no time to dedicate to a time component. Photons do not experience time.

Nothing can exceed a spatial velocity of “c”, because that would be more speed than anything/everything has.

36

Antimatter

37

Recall the relativistic equations for momentum and energy? p = γmv E = γmc2

Fiddling with these, you will find the following:(pc)2 + (mc2)2 =

In other words, the total energy (E) is a combination of its momentum (p) and its rest energy (mc2).

In 1928, Paul Dirac noted that when there is no motion (p=0), this equation can be solved two ways:

E = +mc2

E2 = (mc2)2

E = -mc2

Since energy is a positive value, and c2 must be positive, the second solution predicted “negative mass,” or antimatter. The positron was discovered by Carl Anderson in 1932. Antihydrogen and antihelium-3 have been created artificially.

E2

pc

mc2E

Superluminal MotionAmazingly, in active galactic nuclei jets, blobs of matter have been observed coming out at velocities greater than the speed of light! Velocities up to about 10c have been seen!

What is really going on…The blobs are coming almost directly towards us. So in our frame, the blob is almost keeping up with the light it emits.

The light from every episode of the blob’s entire life reaches us at almost the same time.

As the blob moves sideways, we receive all of its sideways-moving images at nearly the same time!

38

Cherenkov Radiaton

In transparent matter (water, air, gas), photons travel at the speed Vp:

Vp = c/n

where n = the index of refraction.

Air H2O Glass1.0003 1.33 ~1.6

High-energy electrons from reactors pass through the water used to shield the reactor core. Many travel at speeds higher than Vp. As they disturb the electric fields of the H2O molecules, they emit a forward-facing shock cone of photons (mostly UV) that slows the particle to sub-light speeds.

This is called Cherenkov radiation.

Pretty cool.39

Calculations: Relativistic EffectsAn object is travelling at 1×108 m/s with respect to you.

Calculate β and γ.

If this object is 10 m long, how long does it look to you?

An object is travelling at 2.5×108 m/s with respect to you.

Calculate β and γ.

In the time this object experiences 60 seconds, how much time do you experience?

40

Calculations: Spaceship ProbesA spaceship travelling at 1×108 m/s (with respect to you) fires a probe that leaves the ship at a launch speed of 2.5×108 m/s. How fast does the probe travel through space, as viewed by you?

41

Calculation: Spaceship HeadlightsA spaceship travelling at an extremely high speed (v) shoots out a beam of light. How fast does the beam of light travel, as viewed by a person on the ground?

42

Calculations: Photons FlyNote that if you watch two photons travelling in opposite directions, as they pass each other, they have a relative velocity of 2c.

But what do the photons see? (i.e., v1 = v2 = c)

43

Cosmic particles strike the troposphere (h=7000 m), forming π-mesons, which then decay into high-energy muons (v=0.998c).

Muons decay in about 2×10-6 sec. How far can they travel?

Non-relativistic calculationD = v×t = (0.998) × (3×108m/s) × (2×10-6 sec) ≈ 600 m.

But they are easily detected at the Earth’s surface. How do they reach the surface before they decay???

Relativistic calculationD = v×(γt) = (0.998) × (3×108m/s) × (15.8) × (2×10-6 sec) ≈ 9500 m.

What do muons “see”? They see a Lorentz-contracted atmosphere:

= 15.8

Datm′ = Datm/γ = 7000 m/15.8 = 440 m

Calculation: Muons

44

Twin ParadoxSpacestation-Sally and Rocket-Rhonda are both 20 years old.

Rhonda flies at 0.8c to a point 8 LY away. She reaches the point, then immediately returns, flying again at 0.8c.

Spacestation-Sally’s PerspectiveRocket-Rhonda flew away for 10 years (t = 8 LY/0.8c), and then back for 10 years—a total of 20 years. Spacestation-Sally is therefore 40 years old when Rocket-Rhonda returns.

But because of time dilation, Rocket-Rhonda aged only 20/γ years (12 y) during her high velocity trip. When the two twins reunite, one’s age is 40, and the other’s is 32.

45

Twin ParadoxRocket-Rhonda’s PerspectiveTraveling at 0.8c, the distance to the turn-around point is only 4.8 LY—the 8 LY is Lorentz contracted (i.e., 8/γ LY). It takes only 6 years (4.8 LY/0.8c) for each leg of her trip—hence she is 20+6+6=32 years old upon her return. This agrees with Spacestation-Sally’s perspective.

But….in Rocket-Rhonda’s frame, Spacestation-Sally is the one who is moving.

Indeed, during her 12-year trip, Rocket-Rhonda observes Spacestation-Sally to age only 7.2 years (12/γ years). Upon their reunion, 32-year-old Rocket-Rhonda expects to find her sister to be only 20+7.2 = 27.2 years old. Imagine her surprise, when she meets a woman who is 40! 46

Twin ParadoxThe situation seems to be symmetric—each twin sees the other age more slowly.

The solution to the twin paradox is to notice that Rocket-Rhonda accelerates at the turn-around point of her travel. Accelerating reference frames are not treated by special relativity. The situation is not symmetric.

47