Before we go on to construct relativistic physics there's another important
consequence of Lorentz invariance and other important way it teaches us to look
at the universe that we need to come back to.
And we need to understand this thing called the invariant interval.
So these are our Lorentz transformations. When you study them,
you notice that they have the following exciting property.
That if you compute this combination of t and x,
C pri-, c2 squared times t squared,2-x^2 minus X squared in one frame,
or compute c2 squared times t2-x^2 squared in the other frame, you get the
same answer. This is why it's called the invariant
interval. All observers agree on it.
Now this turns out that, to have great physical significance.
In fact, you can derive much of relativity from just this property.
Notice that there are, this breaks up the universe into, three pieces, when you
have a particular event. So I'm going to pick a particular event
to be xt=0. equals t equals zero.
That's right here and now. And remember that we are talking about
now over here and here, all up and down this axis, and what I
notice is first of all, if I set s equal to zero,
that means that C times T is either plus or minus X.
Oh, X is either plus or minus C times T. That means these two trajectories of
beams of light going either to the right or to the left emerging form the event
that we call the x equals to zero or any particular event.
Or, by the way, if you go to negativity, beams of light coming from the left or
from the right, and impinging exactly at t equal zero at x equals zero.
All of these are the regions in space at which
S squared is zero, and of course, it's zero, and we said these light
well, lines are preserved in both, in any,
frame. So it's not surprising that S equals zero in one, frame means S equals
zero in the other frame. these, these two lines are called the
light cone, and they kind of look funny for a cone, but, you can imagine that
they would be an actual cone if space were not one-dimensional but
two-dimensional. Then, instead of minus x squared here, I
would have to add minus y squared minus z squared, and here minus y prime squared
minus z prime squared, and if I only had y and not z, then x squared plus y
squared equals some constant times t2, squared is a cone,
so that, the cone would be rotated this way in the,
if you had a y axis coming out of the board.
if you have z, then this circle should be replaced with a sphere.
That's drawing in more dimensions than I even care to attempt.
so this is what is called the light cone. And light cones are very important.
they describe thrust as I said, the projecteries of beams of light that cross
at some particular event, moving in all possible directions in one dimension
neither move to the left or to the right. Here, s squared to zero along the light
curve. the rest of the universe is divided into
four regions. in this two dimensional case though.
Remember that if the light cone, if we were in a higher dimension and we
returned those circles, then there actually only three regions because you
can move around over here to connect. The entire outside of the light cone is
connected, but the inside of the light cone always
breaks up into two pieces. The inside of the light cone is the
region where C squared T squared is bigger than X squared than either T is
positive or negative so there's a piece of the light cone here and a piece of the
light cone there. Whereas the rest of the universe is where
X squared is bigger than c squared t squared.
So s squared is negative. This is negative, and s squared negative,
in more than two dimensions is one big region.
Okay, what do these mean? Well, let's think about it.
So what it means is this. Let's look at a region where s2 squared
is positive. remembering that s squared is c squared t
squared minus x squared. this means that if I put some event over here at positive
t. since ct is bigger than x, now I'm making
everything positive so I can take the square roots, that means that x over t is
less than the speed of light c. So I can construct an observer, I can
thrown a rock or construct an observer that starts at this event and ends up
here. This is an allowed motion for an
observer, because all world lines of observers are, remember, constrained by v
less than c to be within this region. Oh, so that means two things.
One is, it means that if I want to say, cause something to happen here, I can do
it by throwing a rock at it. I, if I throw the rock, okay, I may need
to throw the rock very fast, but it doesn't need to go faster than light.
The other thing is if instead of throwing a rock I throw a clock, then I have
essentially established a T-prime axis, which goes through both this event, and
that event there is some observer that is present at both.
in that frame I can now draw the corresponding X-prime axis if I want.
But the important thing is, that in that frame, the two events happened at exactly
the same place. Oh, they happened at exactly the same
place that means they both happened at along the t prime axis that's at x prime
equals zero and so s squared is actually c squared time t prime squared where t
prime is the length of time as measured by the clock I threw there.
this is what we call proper time. It's the time between these two events as
measured by a clock that is partis-, that is inertial, and is present at both
events. And this is given in terms of the
invariant interval, which must be positive for there to exist such a clock,
as the root of S squared divided by C. We call that proper time and often denote
it by the Greek letter tau. correspondingly, if I have an event, down
here at negative T, with s squared positive, then that
negative t is still bigger in magnitude than x squared, and that again, if you
can just follow this down here. That means the event is sitting inside
the light cone, and an observer from there, moving at less than the speed of
light can make it to x equals to equal zero.
And therefore you can throw a rock from here and hit somebody there and it again
the time as measured by that rock. If it were a clock would be the copper
time. So we have fancy names for this, these
two parts of the light cone, and they indicate their significance.
The upper part of the light cone with T bigger than zero and S squared positive,
is called the future. It's called the future because these are
all the places where I can throw something at it less than the speed of
light, and if you include the boundary, which is all the plate, the events that
can be reached at the speed of light from my current location.
And if you remember that nothing moves faster than light,
then all of the events here are all of the events on which a decision taken here
could possibly have an in-, influence. I can decide whether or not to throw a
rock that would hit here. I have no way to make a rock get to here.
So, whether this is my future or my past
does, doesn't matter. It's not my causal future, because no
physical influence will be allowed. I cannot throw a rock faster than light,
so I cannot make a rock that was here also be present at this position at that
time. That requires moving faster than light.
Similarly, this part over here is legitimately my causal past.
These are all the events where someone taking the decision there could still get
the information, or the causal effect, to meet at this position X equals T equals
zero. So every event in space time, every point
in this four dimensional thing has a light cone associated to it.
which breaks the space time into three regions,
the future, the past, and the rest.
And remember these are actually connected, so they are really three
regions rather than four. What's the rest?
Well, let's see what the rest is. If s squared is negative, that means you
are outside the light code well for one thing we said no observer can get from
here to there so we're talking now about an event over here.
I can't have an observer for which this is on the t axis, because that observer's
t axis would be tilting more than 45 degrees from the vertical.
But for that reason, of course I can find and observe, find an observer for which
this event lies on his X prime axis. That means I can't make it fall at the
same position as I did before for some observer, as x equals zero, but I can
make it simultaneous. So I can always find some observer for
any event, here in the outside of the light cone.
There is some observer who thinks it's on the x prime axis, who think it's
simultaneous with the event we started with.
And that, for that observer, of course, the time of this event is t prime equals
zero. Here's the speed with which that observer
needs to move. And because T squared is bigger than X
squared, that C squared, D squared is bigger than
X squared, this is bigger. This is less than the speed of light, so
that's legitimate. and, that means that, for that observer,
since t prime is zero, S squared is just negative X prime
squared. So, the root of negative S, S squared,
when this makes sense, when S squared is negative, is the proper distance.
It's the distance between these two events,
as seen but measured by the observer for which they are simultaneous, of which
that determines the speed wi- with which that observer moves.
And now what happens if, instead of this green observer, I consider an observer
moving even faster? So, his world line tilts from the
vertical even more than that of the green observer, and his
x double prime axis tilts from the horizontal even more. So for that
observer notice, this point here, which had a positive value of T has a negative
value of T double prime. the black observer thinks this event
happened after, the exciting meeting of all the observers over here.
The green observer thinks it happened at the same time, and the blue observer
thinks it happened before they met. So it's a darned good thing that no
decision taken here could impact something that happened there,
because, if it could, then the blue observer effect preceded cause.
You can only maintain, in the context of this crazy simul-, relative simultaneity
story. The
construct of a causal theory, if you allow that no material particle, we
already talked about that. No observer, but also no information, no
physical effect can move faster than light because moving faster than light is
precisely tantamount to moving in an acausal way to effect proceeding cause.
As long as you allow for its, it's effect preceding cause, as seen by someone
moving sufficiently fast in the appropriate direction.
The result is that we cannot allow in a relativistic theory any information to
propagate faster than light. this is the origin of the idea of a
cosmic speed limit, the speed of light, not only on material particles but also
on information.