introducing special relativity philip freeman james ball

Introducing Special RelativityPhilip Freeman

James Ball

Imagine a world where scientists have not realised that their planet is a sphere.

They have a pretty good idea of how physics works… they think.

Doing Physics on Planet RelativityThere are just a few little problems...

• Some little navigation glitches.• Objects don’t follow perfect parabolas (over long

distances)• Etc….

An Experiment on Planet RelativityTwo intrepid scientists, James and Philip, decide to investigate.

They will observe two rockets and measure heights above the horizon.

A B

How do you measure?To understand height etc. we need to look at how we measure.

James carefully uses a weight (plumbob) to determine up and down.

Frame of ReferenceNow he uses the vertical line to create a horizontal line, making a coordinate system:

We call this a frame of reference. You can remember that this is just what he’d do if he was hanging a picture!

y

x

A B

Two Frames of Reference:

Philip does just the same thing to create his own frame.

Notice that you (with your knowledge of spheritivity) can see something they don’t know!

Measuring the Rockets:A B

Philip and James measure different up/down (and east/west) distances. What’s wrong?

AB B

A

Who’s right? Why does it matter?

A B

AB

AB

Which rocket is higher?

Concept Question:

Who is right?A)PhilipB)JamesC)Both Philip and JamesD)Neither of them

A B

AB

AB

Different directions for “UP”• The thing James and Philip didn’t recognise is

that they are both right, but they are talking about different things!

• They have different directions for “up” and therefore disagree.

Comparing Metre Sticks:Philip and James decide to compare their frames

They compare their metre sticks. What will they find?

Concept Question:

What will they find when they compare metre sticks?A)Philip’s will find that James’s stick is short and she’ll find his is long.B)James will find that Philip’s stick is short and he’ll find hers is long.C)Both Philip and James will find the other’s stick is short.D)Both Philip and James will find the other’s stick is long.

Vertical & Horizontal According to Philip

Philip sees James’s metre stick is tilted, so it doesn’t measure just “vertical”... It ‘mixes in’ horizontal components. Only PART of his measurement is up/down.

L = L0 cos

Vertical & HorizontalAccording to James

James sees the same thing from his ‘frame of reference’. Philip’s metre stick is tilted. His measurements are mixed up!

James’s Frame

L = L0 cos

Each thinks the other’s metre stick is “shrunk”!

• How can Philip think James’s metre stick is shrunk, while James thinks Philip’s is?

• We’ll see this same theme as we look at relativity… which of course is what this story is pointing towards!

Vertical & Horizontal

L = L0 cos L = L0 cos

Each thinks the other has mixed together up/down with east/west so only a component of their metre stick is in the right direction.Because of the way they point, each sees the other’s vertical measurements as “Shrunk” (by a factor of cos )

The relativity of “up”• The idea that “up” depends on the observer solves

this sort of paradox, and many others (eg: why don’t people on the other side of the world fall off?)

• The fact that up varies is an important clue to the nature of space itself (not to mention the shape of their planet, and what gravity is, and so on…!)

Here endeth the parable

We’re Planet Relativity

• We were “planet relativity” about 100y ago.• Physicists were starting to feel that they’d

pretty much gotten everything sewn up and were near the end of physics (hmmm sound familiar?)

• There were just a few little things, at the interfaces between fields, that were a bit messy.

Like what?

• Mechanics

• Electromagnetism

• Thermodynamics

Little contradiction to do with velocity of light in a vacuum

Little contradiction to do with black body radiation (but that’s another story)

Contradiction? I don’t see no contradiction!

No big deal! We can fix this…

• Speed of light must just refer to the speed of light compared to its medium.

• No medium? There has to be a medium = ether.

• But problems still pile up + can’t find the Earth’s motion through the ether.(Michaelson Morley – but actually not very significant in development of relativity)

Enter a wild card

• Enter a young physicist, very irreverent, keeps getting in trouble for attitude problems… but with a gift. Guy name of ALBERT EINSTEIN.

• Einstein’s gift can be seen as being able to say “Let’s just go with it and see what happens!”

• What if BOTH statements are right?

What if?…

Postulates & Time Dilation

Concept Test:What are the two postulates of relativity in everyday language?

Record on your whiteboards (to share)

the twopostulates of special relativity in your own words.

1. All motion is relative= The laws of physics are the same in all inertial frames. = no experiment in a closed laboratory can detect that laboratories constant motion

2. The speed of light is absolute= The speed of light has the same value for all observers regardless of the relative motion of source or observer

So… what does that imply?• Light is constant so let’s use it to make

measuring instruments, eg a Light Clock

Tick!

Tock!counter

What if I have a moving light clock?Your clock My clock

The picture shows the motion of light in your (stationary) and my (moving) clock.Is there any problem with this?

Concept TestHow do I know the light travels at an angle (as shown)?

A)It won’t… light travels the same way regardless of the source, so it will go straight up and miss the top mirror!B)I have to aim the light source diagonally to make the clock work, so in setting up the clock I adjusted for its motion.C)The light is coming from a moving source, so it has both its original motion and a sideways component.D)If it didn’t stay in the mirror then I’d be able to tell I was moving.

Light in my clock goes farther!The faster I go the further light travels in my clock between ticks!

One tick of your clock One tick of my clock

3.0m

Clock moves

> 3.0m

light

Numbers Symbols No Calculation

light

Sample calculation: (let’s try this out!)

• Suppose I’m going at 86.6% the speed of light… How much longer does it take for my clock to tick?

1. How much time does the light take to travel 3.0m in your clock (how long for your clock to tick?)

ans: 10 ns

Your clock

3.0mlight

My clock

vt

lightx

• I’m going at 86.6% the speed of light (relative to you)

2. How far do I move forward during one tick of your clock?

ans: 2.6m

Your clock My clock

3.0mlight

My clock

vt

lightx

• I’m going at 86.6% the speed of light (relative to you)

3. How far has the light travelled in the time for one of your ticks?

ans: 3.0m

Your clock

3.0mlight

My clock

2.6m

ct x

• I’m going at 86.6% the speed of light (relative to you)

4. How much of the distance to the mirror has the light crossed? (x)

ans: 1.5m

Your clock

3.0mlight

My clock

2.6m

3.0mx

• The light in my clock has only crossed part way.

The light is only half way across after one of your ticks. So, when my clock ticks how many times will your clock have ticked?

ans: one tick of my clock = two ticks of yours.

Your clock

3.0mlight

My clock

2.6m

3.0m1.5m

Moving Clocks run slow!• If my clock is moving relative to yours then I

measure time differently!

• In this case MY clock goes “tick” once when your clock goes tick twice. You measure 20ns when I measure 10ns. You measure 2.0 hours when I measure 1.0 hour and so on. My light clock is running slow (by a factor of 2!)

• What will OTHER types of clocks I own show, if you compare them with my light clock?

Concept testWhat does the light clock running slow imply about all other clocks?

A) They will match the light clock, otherwise we could use the difference to tell we were moving.

B) They will match the light clock, because all clocks use electromagnetic forces, so they are affected the same way as light.

C) They will match the light clock, because we calibrate all our clocks using the equivalent of a light clock (definition of the second).

D) Each type of clock will be affected differently, so there is no way to define “the right time”.

How much are things slowed? Let’s calculate symbolically.

Your clock

ctyou

light

My clock

vtyou

lightctme

ctyou

Moving clocks are slow by a factor of • The moving clock is slowed by a factor of

This is the most important factor for elementary calculations in Special Relativity

The faster the motion the greater the slowing:

% of light speed 0 1.00

10 1.0120 1.0230 1.0540 1.0950 1.1560 1.2570 1.4080 1.6790 2.2995 3.2099 7.09

If I am moving at 0.995c (=10) then you see my time passing at 1/10th speed. You see my clock run slow, including my watch running slow, my heart beating slow, my thoughts going slow… my life is just slowed down by this factor.

You see my clock as slow by a factor of What will I see if I look at you?

A. Your clock is running fast by a factor of B. Your clock is running slow by a factor of C. Your clock is running at normal speedD. It depends on how fast your frame is moving (compared to a

frame at rest)

Can this be right?• You saw me moving, and therefore MY clock

was slow. Your clock My clock

Who’s moving though?

• But what do I see?My clockYour clock

I’m the one who’s at rest, YOU are moving!YOUR clock is the one that is slow!

Whose clock is slow?• I think YOUR clock is running slow.• YOU think MY clock is running slow.• Does this sound familiar?

Different directions• Philip thought James’s metre stick was too short,

and James thought Philip’s was.

• This was because their metre sticks were pointed in different directions. They had different directions for “up”.

• The same is true for the observers comparing clocks. We each think the other’s clock is slow because we have different directions for TIME!

Space and Time Spacetime

Which way does your clock

point?

• Spacetime diagrams:

space

time

Like a traditional position-time diagram BUT time goes vertically by convention.

So as time passes things are ‘copied up’:

space

time

spacetim

eSame point in space at different times

Standing Still Running

Different points in space at different times

1)

1)

2)

3)

1)

2)

3)

An Example: • Let’s show how these diagrams work using a diagram

of a story you may have heard at some point.

This is the spacetime story of Little Red Riding Hood.

Once upon a Spacetime…

space

time

Grandma’s house

Red Riding Hood

Big Bad Wolf

“World Lines”

Red’s house

The time axis• Time axis = same point at different times

space

time

1) I’m here

2) Still here

3) Yep, right here

For example, Grandma’s house

Moving at constant velocity• An observer moving (relative to you) seems to

you to have a tilted world-line:

space

time

Position at time (1)

Position at time (2)

Position at time (3)

But the moving observer sees YOU as moving and themselves as at rest!

space

time

1) I’m here

2) Still here

3) Haven’t budged

Nobody is moving relative to themselves:

• But your “moving” observer is standing still relative to themselves:

space

time

1) I’m here

2) Still here

3) Haven’t budged

Their world-line is their time axis!

The moving observer’s time axis points in a different direction!

• Motion is just having your time axis at an angle in spacetime:

Tim

e ax

is 1

What is all time for one observer is partly time and partly space to another!

Tim

e ax

is 2

It’s about time, it’s about space

• If time is changed then space must be affected too, otherwise we wouldn’t agree about the speed of light!

• If I measure a shorter time than you then I must measure the distances as shorter too.

• You see my moving clocks slow (by )• I see your moving metre sticks shrunk (by )

Space and Simultaneity

Tilted space axis and

simultaneity

• Since we see that spaces are also altered by the “Rotation Factor” we might expect that the space axis of a moving frame is rotated just as the time axis is.

• We can see that this is exactly true by considering what that space-axis is.

“At the same time”• The time axis is made up of all the points

which are at the same place (at different times).

• Similarly the space axis is made up of all the points which are the same time (at different places)

Simultaneous Events• We need to ask “when do we know two things at

different places happen “at the same time”?• Clearly we need some signal.

A flash of inspiration:

• One approach is to send a light signal to two observers equally distant from a central point. Since the light travels equal distances in equal times, if they both start their watch when they see the flash then we know they started them at the SAME INSTANT.

Here a signal goes from a central point to two ends of a traditional train car:

For a moving frame, however, it’s a bit trickier…

• In a moving train car for example, you will still see the light as moving at speed c in all directions (one of the postulates, remember)

• But the back of the train is moving toward the source, so it will reach the light first, while the front end is going away so the light must ‘catch up’ to it, taking longer.

If you observe the train car as moving then the signal does NOT

reach the ends “at the same time”!

Train Car is moving this way

The ‘stationary’ observer sees two the “now”s at different times:

time

space

Now!

Now!

Mov

ing

obse

rver

(stationary observer’s frame in blue)

• Notice that the two “now” events are at both different places and different times for the blue observer

Tilted Space Axis:tim

e

Now!Now!

space

• To the “moving” red frame the two events (light reaching the back and light reaching the front) happen at the same time (define the space axis).

• So the two points are on a space axis (same time).

Tilted Space Axis:tim

e

space

Now!

Now!

time

space

• We can now add this space axis to our diagram from the point of view of our blue observer:

The moving observer’s time and space axes are both tilted!

Rotated Frames

Spacetime Rotations and

x' x

t t' c

x'

Changing axes:tim

e

rest

time

space

• The time and space axes are tilted as seen by an observer in a different frame.

• The rotation of the axes moves them TOWARD the diagonal line.

• The diagonal line is equally time and space… it is the path of light,(AKA the light cone)!

time

space

fastfaster

Additional: what’s special about the

speed of light

SpacetimeNotice that the rotated axes move in toward the diagonal. That diagonal line is the ‘light speed line’.

x

t t'

c

x'

stationary

moving

Space-like and time-like

x

tc

RotationsVertizontal: The different directions of ‘up’ were because of a regular ‘circular’ rotation:

Spacetime:The different directions of time are a rotation too, but not a circular one. They are a hyperbolic rotation:

y

x x

t

Special Relativity is the discovery that the geometry of space is hyperbolic

All of the results of special relativity can be derived and calculated using hyperbolic geometry / spacetime diagrams (but you don’t HAVE to do it that way!)

x

t

Eg: to add two velocities find their hyperbolic angles and add the angles, then turn back into a velocity. And so on.

Extra: see an example of

velocity addition

Conclusion

Key points and example of application

• The main idea here is that the results of relativity are not strange or magical, but are the result of a single simple geometric fact:

If you see something as moving that means its time axis is pointed in a different direction in spacetime than your axis points!

x'

x

t t'c

x'

• Light speed is not special because it is ‘light’ but because it is the dividing line between space and time.

• The value of 2.9979458108 m/s is a conversion factor.

• In this sense all motion is at light speed (everyone is going into the future at one second per second)

x

t c

• Space and time are not absolute as separate things, they are components of a single unified thing: spacetime.

• Observers whose time and space axes are in different directions see each others measurements as ‘mixed up’ because what is all time or all space for one observer is a mix of the two for another observer.

A B

AB

A B

• We can convert from one frame to another using a ‘rotation factor’ which allows us to adjust for the different directions.

is always greater than or equal to 1

To use know what is longer/shorter:

• Moving clocks run slow (by a factor of )• Moving metre sticks shrink (by a factor of )• Moving ‘masses’ are ‘increased’ (mass-energy)

(by a factor of )

An example:• Suppose I travel to Planet Relativity from Earth

you see:

I see:

Moving clock runs slow reads a short

time

Moving metre stick is shortened planet is close & trip takes a short

time.  

Example calculations

Where to slip this in (being subversive again)

• Kinematics? • Waves?

Einstein Simplified

E/M? Other?

Extra:A Relativity

LAB!

If you have questions I would be delighted to discuss this with you!

Email: PhilipF@sphericalcows.net

•I hoped to include the twin paradox discussion in this power point, but did not have time before going to press… again email me and I’ll send that to you too!

•Philip Freeman, July 2012