Okay so we're going to take the principle of relativity, and try to understand what
it tells us about physics, and we need a mathematical framework in order to
describe this. And the mathematical framework in which
one discusses relativity is called space time.
This is often introduced as some kind of mystical fourth dimension.
I want to demystify space-time. I want us to get very, very comfortable
with it. So we're going to spend quite a bit of
time getting comfortable with what space-time means.
So, alright, what do we mean by space?
You can get into Kantian philosophy, but what we mean by space is all the
possible places something can be. And we live in a three-dimensional
universe, so all the possible places something can
be are listed as, are labeled by three numbers.
Pick an origin, pick some axes, measure the distance east,
north, and up from that position, and you find all the possible, positions an
object can be, and, the position of an object at a given time is given by these
three numbers. We're studying mechanics.
We're studying the motion of objects. To describe the motion of an object, you
basically let your clock run, and at any given time, you specify where the object
is. So to discuss the motion of an object, you need to compute, or find, or
measure three functions. It's position east and west, it's position north and
south, its position up and down, and, you will remember to the days you were,
doing, high school algebra, that to understand
such functions, you often plot them. You take the T axis.
You plot the function x as a function of t and if you were describing the motion
of an object which started at some point, say x equals zero and moved up the x axis
and then stopped, and then moved a little bit down the x axis and then you stop,
steady okay, then you would have drawn this graph of the function x of t.
This is something we've all done. This taper, this plane in which I've
drawn it, is space time. Space time is the place in which you
clock the motions of objects. Now in this case I was plotting the
motion of an object in one dimension. What would I do if the object were moving
in a two dimensional space? Well then I'd have to specify both the
function x of t and a function y of t. So what I would do, perhaps if I were
graphically talented, which I am not, is I would add an Y axis, and then at any
time, I would plot X by moving my point up and down, Y by moving my point in and
out. And for example, if I study the motion of
that object that was moving in a circle before,
then the motion of the object would be in some circle in this X Y plane.
But as I plot it, as a function of T, that would be like the motion of the
object that we studied before was up and down the X axis.
When I plot it as a function of T, the object would say, start here at T equals
zero, and then move along the circle as a function of T,
and I would be plotting, if I could draw, the sort of helical object.
This is, would be called the world line of an object that is moving in a circle
in the X Y plane. Now, my graphical skills are not that
great, so let's do a slightly better drawing and see what it teaches us.
Here we have our friend and this time a green sphere.
And it's going to be moving at a constant speed around this blue circle.
And what I've done is I've embedded this circle,
which is a two-dimensional space. Motion is in two dimensions, and I've
embedded it in a three dimensional cube where the vertical direction is going to
denote time. It is indeed a curiosity of drawing space
time diagrams that rather than having the T access horizontal, it's our habit to
have the T access be vertical. And so if I plot the motion of this
particle is going to be given by the green dot,
there's going to be a red dot that will move,
that will, at any time, at any vertical position,
be dropped at the place along the circle where the particle is at the time
corresponding to that slice. And so, as I plot this, the green
particle moves around the circle. The red dot tracks it, and the red dot
also is placed at any time, at the vertical heights corresponding to that
time, and the net result is that I construct the helix that I was trying so
unsuccessfully to draw. All this is, is the motion, if you want
the storyboard of the motion of this particle.
Space-time is the place in which you draw graphs of X and Y as a function of T.
So. Let's study some simpler motions in space
time. What we're going to have two dots moving
around in our 2-dimensional space.
One is going to be moving up the x axis and the other up the y axis.
The yellow dot moves slowly and the red dot, as you can see, moves faster.
What do the space time plots of these motions look like?
So again, I plot x and y as a function of t.
I'll have a blue trace for the yellow dot and a green trace for the red dot.
And what we see when we look at the world lines that we have drawn is, of course,
they are plots of where the dots where as a function of time.
And when we tilt it we see that the slow moving yellow dot generated a steeper
trace then the more rapidly moving. Red dot, this is reasonable because over
any interval of time, as between each time we plotted these spheres, the
speedier red dot moved farther along the Y axis than the slower yellow dot managed
to move along the X axis. And we see that the traces, these world
lines, these graphs, for both of these red and yellow dots are straight lines.
They were moving in a straight line and their world lines are straight lines.
That's not exactly the same thing. In this diagram, what we're going to is
we're going to have the same red dot, it's going to move along the y axis in
the same straight line, but it's going to accelerate.
What is that going to do to its world line?
Well, when it's moving slow, its world line will be close to the vertical, and
as it speeds up, its world line will curl away from the vertical, so that the graph
that we draw is no longer a straight line, despite the fact that the motion of
the object was in a straight line. Because it was not moving at a constant
speed, its work line curved. So this is space time.
Space time is the place where you plot R of T.
Now, of course, I plotted a one dimensional motion,
and whenever I'm plotting anything, it'll only be motion in one dimension, because
I can barely plot that. The computer managed to do a decent job
manning, plotting two dimensional motion. If you were really going to plot motion
in a three-dimensional space, you would need a four-dimensional graph.
We don't really have the technology to draw a four dimensional graph
effectively. We can't even imagine it,
but we can mathematically describe it. And really this fourth dimension is
basically the statement that we are drawing three functions at once.
There is nothing fancier about that. And so, the points in space time.
The points in this four dimensional graph.
what are they? The points in space rather places you can
be. The points in space time are all the
places you can be at a given time. These are all the places all the ways
something can happen, what we call given position in space in given time is an
event. It's a more specific statement than a, a
position. So for example Durham at this particular
time is an event. Durham is a location.
Right now my world line passes through Durham, because at this point I am in
Durham at this event. but Durham in 1985 is also the same
location. It's a different event and I was not there.
And so a point in space time is called an event.
It specifies something happening or not happening,
and through all of the possible places and times, you draw a one-dimensional
line. That is the, trajectory of a particle.
So, we do not move in space time.
We do not move in time. these are all ideas that are confusing,
that come from treating time in the same way that we treat space.
What the extra dimension is that we add to space-time is basically the slices of
the movie of our life where you prod-, use the variable T to plot your position
as a function of time. Hopefully that helps demystify this
story. we've learned something about the
geometry of space-time. We've learned that if your trajectory in space is
curved, then your worldline will curve. The worldline is this graph of all the
positions that you've occupied over the course of some time interval.
So, if the object is changing direction of motion in space, then it's worldline
tracking this change will change it's direction as well.
If an object is moving in a straight line, then its worldline will not have to
change direction, but if the object accelerates, the worldline curves towards
or away from the vertical, as the object is slowing or accelerating.
What does that tell us about an object whose world line is straight?
Well an object whose world line is a straight line in space time is an object
that is moving in the fixed direction at a fixed speed.
Aha. What we have is a geometric description
of Newton's first law in space-time. Space-time gives us a nice geometric
description of objects. The, the world line of an object upon
which no force acts is a straight line in space-time.
And conversely, an object whose world line is a straight line in space-time is
an object upon which no force has acted. So space-time is useful.
File that away, we'll come back to it. So, here is that, version of space-time
that I am able to draw. We imagine a one dimensional universe,
objects are free to move to the left or to the right along this X axis.
We plot their motion along X as a function of T.
As usual we have the T axis vertical and so there are some names that you might
give to all things. So all of the things, all of the points
in this graph that are, I've picked some time T=0, so the coordinates of a point,
a event. In this graph is given by its X
coordinate and its T coordinate. X gives you the distance from some
particular point, which I've called X equals zero, at which this, event
occurred, and T gives you the time after some particular time that I decided to
call T equals zero, when this event occurred, so give me an X and a T, and
that tells you if something happened at that position at that time.
You've specified where and when it happened, so all of the X axis here are
all of the events that happened at the time T equals zero.
By convention, I'm going to call that now.
what are all the events that occurred at the position x equals zero?
Those are all the things that naturally would be called here.
So, the t axis might be called here. The x axis might be called now.
And what we saw is that the world lines of objects that are moving at a constant
velocity are going to be straight lines. Your instantaneous velocity is
essentially the slope of your world line. But the slope is measured from the t
axis. So it's a sort of backward slope because
we've put t vertically. So if you take this red world line, that
describes an object moving to the right. It starts at this negative time way over
here to the left and then it moves to the right.
And if you look at this green world line, it describes an object which moves
faster. You could measure your velocity by taking
some time interval. Notice that I've arranged and will often
do that, for both objects to be at the same place at the same time.
At t0, equals zero, they were both at X equals zero.
If these were material objects, they would have crashed into each other with
the green object overtaking the red object and slamming into it.
But they, in this case, pass right through each other and continue moving at
a straight line. And if I take some time, delta t later.
And I measure where these objects have gone.
You can see that in the time, delta t, the red object has gone.
This distance, delta x red, and the green object has gone the larger distance over
here, delta x green. this is telling me that the green object
is moving faster than the red object. This is what we're saying.
What does it mean for an object to have a vertical world line.
A vertical world line means x is a function of t as constant.
The blue object is a stationary object that sits at the same location throughout
time. What does it mean to have horizontal
world? Nobody has a horizontal world line.
A horizontal world line is a collection of all the event that happened at the
same time so just as the X axis is what I call now, the orange axis might be what
you call a little bit later, these are a collection of all of the events along
this line are all the things that happen at a particular time so at this
particular time indicated by the orange line, the blue particle was where it
always was, the red particle was at this position, the green particle was at that
position and so on. I said that space time is the right place
to discuss relativity. Let's see what relativity looks like in
space time. So what I have here is space time as seen
by some observer. Their axis are drawn in black, I will use
a green pen to denote their observations because I don't have a black pen.
So this is the black observers now this is the black observers here.
And I have here a world line that is tilted to the right, that means it's the
world line of something moving from left to right at some constant velocity along
this diagram. And what this is going to be is the node
or the origin as seen by another observer.
The blue observer thinks that this is here, what he thinks of as.
Something that is stationary is going to be something that is moving relative the
first observer. This is precisely the situation that
relativity describes. The world is observed by an observer in
the black purse axes, and the world is observed by someone else who is moving at
a constant velocity relative to the first one.
And the statement is that the laws of physics as observed in the blue system
should be the same as the laws of physics as observed in the black system with the
green notations. And so we have arranged things so that at
the time that the black axis s-, select as T equals zero.
We've picked that time to be the exact instant where along their motion from
left to right the blue observer passed the location of the gr-, black observer.
And moreover, we have set it up, so that at that same time both observers started
their clocks. What this means is that the blue
observer. Considers T equals zero to be the same as
what the black observer considers. So they agree on what now is, and then
they measure time with synchronized precise clocks from that moment on.
And what happens then is that if there's some event going on.
Somewhere over here an event occurs, a birthday party takes place.
Well, then the black observer, measures when and where the event occurred, and he
declares that it occurred at a distance x.
From the place from the origin and at a time t.
What does the, blue observer measure? Well, it's clear that the blue observer
measures a different distance, since, the blue observer is at a different position
at the time the party occurred. He measures the party to occur at this
distance, to the right of his origin, and he also measures the time of.
At which the party occurred. And since they synchronize their clocks,
they agree on the measure of, measurement of time.
So, they measure, for the same birthday party, the same event, the same time.
But they disagree on where the birthday party occurs.
It's not that they disagree, but they describe it with different coordinates.
The position to the right of the axis. Or the origin, at which, the moving
observer, measures the party differs from the pos-, distance from the origin that,
black stationary observer measures the party to be at, by a quantity that is V
T, where V T is simply the distance over here,
by which the two origins differ. And it changes the function of t, because
over time, the blue observer is moving to the right, and can invert these
relations, and figure out how to find the description given by the stationary
observer in black, given the data in the measurements of the moving observer in
blue. And it's, you can solve these linear
equations rather easily for x and t as functions of x prime and t prime.
And it's not surprising that these equations look the same,
except the sign of v has changed. What this means is,
the black observer sees the blue observer moving to the right with speed v.
If the blue observer were to describe what the black observer is doing,
they would find that the black observer is moving to the left with exactly the
same speed. There is not a difference in principle
between these two systems. What they measure is the relative
velocity. V is not some absolute object.
It's the relative velocity. The blue observer is moving say to the
right in this picture relative to the black observer, or equivalently the black
observer is moving to the left relative to the blue observer,
and physics does not select who is right about this.
This is all very logical and am, I am indeed belaboring the obvious, but let's
belabor it one more time. Let's understand what this tells us about
some an important topic called velocity addition.
So here's the same collection of two observers with their here's and now's,
and now I've introduced the world line of some other object here,
and this object is also moving to the right in this picture.
It's moving faster, it turns out, than the blue observer so that it's moving to
the right both respect to the black observer and respect to, with respect to
the blue observer. But the blue observer can measure its
velocity. Notice this is a straight world line,
the motion is inertial motion with a constant velocity.
And as the blue observer measures it this velocity is given by some number in
meters per second, u prime. Now both observes will agree that this
thing is moving. They will not agree on its speed relative
to them. Let's compute the speed with which the
black observer measures, the speed that the black observer measures for this
green object for which the blue observers, observer measures a speed u
prime. Well how do we do that?
This equation describes the motion, in the, blue observers system and, the
coefficient gives us, the velocity. Note that I have selected, particularly
nice world line for the green object, I have set it up so that the green object
meets both observers at the same time that they meet each other.
You will often discuss such things, discussing an object which makes the same
motion, but, meets the observers a little bit later or a little bit earlier, is not
that much more complicated, adds in a necessary algebra step, so we
won't deal with it. And so, how do we transcribe the motion
in as seen by the moving observer to the motion as scene by the black observer
very easy We know the relation, we wrote it in
general, between, the coordinates as observed by the observer in black, and
those as observed by the observer in, blue,
and given X prime and T prime, you can figure out X and T.
Now we need to plug into this, the information we have, which is that X
prime is U prime T prime. So, I plug that in.
U prime T prime plus V T prime. I collect terms and write that as U prime
plus V, T prime. And then I remember that because they
synchronize their clocks, t and t prime are the same thing, and I
reach the conclusion that x is u prime plus vt.
Oh, good. This is the equation of exactly the same
form but describes an object moving at a constant velocity.
Excellent. Inertial motion.
Remember physics should not match, should not change between frames.
If we met one observer who says no force is acting on the green object, the other
observer agrees. No force is acting on the green object.
Its motion, its world line is a straight line in both frames.
Its motion has a constant velocity, and that constant velocity is just obtained
by adding the velocity as measured from the moving observers viewpoint with the
relative velocity between the two observers.
This is not that, anything. I mean, this is not saying anything that you didn't
already know. If you are standing in a train that is
moving 100 miles an hour, and you are walking at one mile an hour in the
direction the train is moving, relative to the ground you're moving at 101 miles
per hour. This is so obvious.
Why did I take so long to discuss it? We'll see why it's worth discussing in a
minute.