introducing special relativity philip freeman james ball
TRANSCRIPT
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: [email protected]
•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