Outline - Feb. 8, 2010

• Postulates of special relativity

• State of Motion

• Reference Frames

• Consequences of c = constant

• Time dilation and length contraction

• See Chapter S2 (pgs. 425-442)

Einstein’s Relativity Leads to Fascinating and Non-Intuitive Results

What we mean by “motion” is relative!

Time is not absolute, it is relative!

Distance is not absolute, it is relative!

Understanding all starts with Newton’s First Law (N1 = the law of inertia) and TWO observers, one moving in a straight line at constant speed with respect to the other.

Special Relativity (1905)

Q: What’s “special” about Special Relativity?

A: There are no accelerations (no gravity)

All reference frames in Special Relativity are “inertial reference frames”; the observers are not accelerating and Newton’s first law (N1) is observed

N1: an object will remain in a state of uniform motion in a straight line or in a state of rest until acted upon by a force.

But what is a state of rest? What is a state of motion?

(These are not silly questions…)

It’s all about relative motion

We need two observers, one moving with respect to the other in a straight line at a constant speed:

1. A passenger on a train, traveling on a straight track and a constant speed with respect to the ground

2. A bystander standing next to the tracks, watching the train go by

Passenger: bottle of wine stays at rest unless it is acted upon by a force (N1)

Bystander: bottle of wine is moving with constant speed in a straight line (N1)

Both observers see N1 in action, both are in “inertial reference frames”, but the very idea of MOTION is a relative concept.

v

dining car

Reciprocity

Passenger on train has an equal right to say that she is at rest and the world is moving by her; everything remains the same (just moving in the opposite direction) from her perspective.

Neither person has a valid claim to be “at rest” because the ideas of “rest” and “motion” are relative!

“I am in a state of motion”

and

“I am in a state of rest”

are NOT valid statements.

Postulates of Special Relativity

1. The laws of nature are the same in all inertial frames of reference

2. The speed of light (in vacuum) is the same in all inertial frames of reference

Unsettling Consequences of Special Relativity

1. Measurements of time are relative (time dilation)

2. Measurements of distance are relative (length contraction)

Simultaneity is Relative!

Imagine the world’s longest, fastest train car (600,000 km long and moving at 100,000 km/s). At each end of the train car is a sensor that will open the door

when light strikes it.

c = 300,000 km/s, so light can travel half the length of the train car in 1 second

Put a lamp at the center of the train car, and have a passenger turn it on.

What will the passenger on the train car say about when the doors open?

Passenger’s Reference Frame

Simultaneity: Passenger vs. Bystander The passenger says that both doors open simultaneously, and they do so 1 second after the lamp was switched on.

The bystander says that 1 second after the lamp was switched on, the light (photons) have traveled 300,000 km up and down the track and the lamp has moved 100,000 km from its location when it was turned on.

The bystander says the back door opens before the front door, because the light catches up to the back of the train before the front of the train!

Bystander’s reference frame

Clocks and Rulers

Give our two observers, the passenger and the bystander, identical clocks and identical rulers and ask them to measure time and distance in two experiments.

tp = time measured by passenger

tb = time measured by bystander

dp = distance measured by passenger

db = distance measured by bystander

The observers are in a state of uniform, relative motion in a straight line. Will they agree on time and distance?

Experiment 1: How long does it take for light to travel from the floor of the train, bounce off a mirror on the ceiling and return to the floor?

Reference frame: passenger on the train

time = (distance traveled)/speed

total distance = 2h

speed of light = c

tp = 2h/c

Experiment 1: bystander’s reference frame

The bystander sees the entire apparatus move past at constant speed.

The light goes straight up and down in the passenger’s reference frame, but not in the bystander’s reference frame!

Freeze-frames from impossibly fast high-speed camera

start end

Length of the path of the light = 2d, according to the bystander

Height of triangle = h

tb = 2d/c, but d > h so tb > tp (!!!!!!!!!!!!!!!)

The moving clock is running slow relative to the stationary clock.

The bystander says that the clock on the train is losing time compared to his own clock.

Measurements of the passage of time are not absolute, they are RELATIVE!

We know b = vtb is the length of the base of the triangle as seen by the bystander. Using Pythagorus it’s easy to show

d2 = h2 + (b/2)2 = h2 + (vtb/2)2

Remember we had tp = 2h/c and tb = 2d/c. Patience and a few lines of algebra will then give us

The Boost (Lorentz) Factor

The boost factor is always GREATER than 1 because v is always LESS than c (nothing physical can travel at the speed of light or faster)

What about distance/length?

Measure the distance between the trees using speed and time:

distance = speed x time

db = vtb is what the bystander measures

dp = vtp is what the passenger measures

But we know tb > tp, so db > dp!

The passenger measures a shorter distance than does the bystander: LENGTH CONTRACTION

We know:

So it’s easy to show:

The bigger is (v/c), the bigger is the boost factor and the SHORTER objects will appear in the HORIZONTAL direction

Let’s take a ride on the T down Comm Ave…

v << c

v = 0.5c

v = 0.75c

v = 0.85c

v = 0.95c

If you lived life near the speed of light, the world would seem very different…

1. Clocks in uniform motion with respect to you would appear to run slow (they lose time compared to your clock)

2. An object moving past you would appear shorter/narrower along the direction of relative motion, compared to when you observe the object to be at rest

Is there anything our two observers CAN agree on?“spacetime interval”

Our two observers disagree on the separate values of x and t, but

they will agree on the value ofs (if they simply

substitute their own measured values into the

formula).

s is known as the “spacetime interval”.

Back to ‘39…

How fast would a passenger (on a spaceship) have to travel such that only 1 year passes for him, but 100 years have passed on earth?

Solution: v = 0.99995c = 299,985 km/s

Wait a minute!!

By reciprocity, can’t the space traveler say the earth is in motion and, therefore, when the earth “gets back to him”, everybody on earth should be younger than him because their clocks ran slow?!

Nope, but the explanation is subtle…

“I am in a state of motion” is not a valid statement.

“My motion has changed” **is** a valid statement!!

Reciprocity only applies to two inertial reference frames, one moving with respect to the other, that never change.

The space traveler goes through multiple changes of reference frame (Earth, trip out, trip back, Earth), including accelerations/decelerations.

OK, so is any of this for real?

Yep. But to understand the test, we have to go one step farther and include GRAVITY.

Is any of this practical?

Can you really travel far into your future if you want to?

Practicalities(mass and energy)

Suppose you have two identical bricks, one of which is on a railway platform (and stationary with respect to you), and one of which is on a flatbed railway car that is moving at a constant velocity with respect to you.

The instant the railway car passes you by, you give both bricks a shove with an identical FORCE.

N2 says F = ma, so an acceleration must occur.

BUT: time is running more slowly on the railway car, so the force felt by the brick on the car lasts for less time than for the brick on the railway platform (say, 1 billionth of a second versus 2 billionths of a second)!

F = ma

If the force is applied for less time to the brick on the railway car, that means there is less affect on the moving brick’s velocity (i.e., less acceleration).

The only way 2 identical forces can give rise to different effects (different accelerations) is if the masses are different!

The faster an object is traveling, the greater is its mass (compared to when it is at rest):

mmove = mrest

Traveling fast takes whopping amounts of energy…

The faster you go, the more your mass increases, and the more energy you need to make you go faster!

Suppose an astronaut has a mass of 75 kg.

How much energy would it take to get up to speeds where she could travel into her future?

v Energy (Joules)

0.5c 1.15 1.0x1018

0.87c 2.03 6.9x1018

0.99995c 100.0 6.7x1020

For comparison, the annual US energy consumption is about 1020 Joules.

Note that we haven’t even given our astronaut a space suit (about 80 kg) or a space ship (mass of space shuttle is 100,000 kg)

Minute Paper

A few sentences on one of the following:

* something you found particularly interesting today

* something you found particularly confusing today

* questions on things from today that you would like to know more about

Be sure to PRINT your name legibly