We've talked
before about flat spacetime here,
but before we can graduate to the curved version,
and general relativity, we need a stronger foundation
in spacetime geometry.
So today, on "Space Time," it's spacetime.
To begin, here's a heads up.
Today's episode will reference information
from three of our earlier episodes.
To avoid getting lost, you should
pause me and watch them now if you have not already done so.
All right?
Here we go.
In the episode on curvature, we saw
that tiny patches on the surface of a sphere
look like the Euclidean plane.
Since we know how to draw straight line segments
in a plane, then we know how to do that in tiny patches
on the sphere.
So draw tiny straight segments over a series of tiny patches,
join them, and voila, you have a geodesic on the sphere.
Eventually, we want to do the same thing in curved spacetime.
However, tiny patches of curved spacetime don't look Euclidean.
They look like flat spacetime, which although not curved,
still has a geometry that doesn't always agree
with our visual intuitions.
So we won't know what to do in each tiny patch
unless we first understand what straight line, tangent vector,
and parallel mean in flat spacetime.
Clarifying that is the goal of today's episode.
Now, we're not going to do a complete treatment
of special relativity or all aspects of flat spacetime
geometry.
We're not even going to get close.
That's fascinating stuff that you
can learn about from other links in the description.
But I only have a few minutes.
And I need to stay on task.
You guys ready?
Awesome.
Let's get started.
Our principle tool for exploring flat spacetime geometry
will be something called a spacetime diagram
for representing physical events.
Here's how it works.
Pretend the world has no gravity, Newtonian
or otherwise.
I'm not glued to Earth's surface.
The moon doesn't orbit Earth.
In fact, there is no Earth since Earth
is held together by gravity.
This gravity-free world is what flat spacetime describes.
To record when and where events in this world happen I've
got a clock to tell time, nice.
And I've got an infinitely long stick with tick marks
on it attached to me at the x equals zero mark,
my own personal x-axis.
My clock and my axis together make
a frame of reference, which should also have y and z-axes,
but I want to keep things visually simple.
To represent this set up in a diagram, let's
copy my x-axis onto a blackboard and add a vertical axis
to show the time on my clock.
Actually, that vertical axis is showing the distance ct
that light travels per tick of my clock, which
is interchangeable with clock ticks
since I know the speed of light.
I know that seems like an awkward way to record time,
but you'll see in a minute why it's convenient.
Now recall from our earlier flat spacetime episode
that points on this blackboard are not
locations in a two-dimensional physical space.
Rather, they are events, each of which
occurs somewhere along my axis in a one-dimensional physical
space and at some moment according to my clock.
Is my frame of reference inertial?
Well, let's see.
I release a ball.
And it just hangs there.
So yes, inertial frame.
All right, now, some weirdo in a red shirt carrying
his own clock and x-axis approaches me
from the left at constant speed.
He passes me just as my clock reads zero.
And at that same moment, I shoot a photon from a laser pointer
to the right.
And, oh yeah, this guy's towing a monkey.
Anyway, say I plot the values of ct on my clock
as the photon passes different marks on the x-axis.
I get a nice 45 degree line thanks
to the funky vertical axis units I was using.
If I do the same for the guy in red, I also get a line.
But that one is more vertical since he
moves slower than the photon.
In the same amount of time on my clock,
he passes fewer marks on my x-axis.
Those lines that we just drew link all the events
at which the photon and the red guy respectively are present.
They're called world lines.
And they represent the entire histories
of things, things like that monkey.
He's perched at a fixed spot on the red guy's x-axis.
And so, he moves at the same constant speed
that the red guy does.
So the monkey's world line is also a line parallel
to the red guy's world line.
What about me?
I am at x equals 0 for every event at which I am present.
So my world line is vertical.
It coincides with my time axis.
But it's only vertical for my frame of reference.
Red guy also has a frame of reference.
And from his perspective the diagram looks like this.
See, according to him, the photon also
moves rightward at speed c.
We agree about the speed of light,
so its world line looks the same as in my diagram.
But red guy says that I'm moving not stationary.
I'm moving to the left.
So my world line isn't vertical.
It points slightly backward.
Instead, red guy's world line and the monkey's world line
are vertical in his diagram since they're
both stationary from this perspective.
Now, on either diagram, I could also have drawn lines that are
more horizontal than 45 degrees, but they wouldn't be world
lines because to be present at two events represented
by points on such a line an observer or a photon would have
to be moving faster than light, which normal objects
and photons cannot do.
Now, spacetime diagrams are great for visualizing
cool phenomena like time dilation,
or length contraction, or disagreements between observers
about event sequence.
And that's fun, but it would take us off course.
Instead, I just want to use these diagrams to establish
how parallel transport works in flat spacetime.
Because here's the thing, the answer
is not clear a priori since you can't trust your eyes
in spacetime diagrams.
Watch this.
These highlighted points in my diagram
and in the red guy's diagram correspond to the same events.
So those are the same points.
And thus, these are the same line segments in spacetime.
Yet, between diagrams, their visual length, and the angle
between them, have changed.
If you throw in the perspective of a pony flying to the left
superfast looking at the same events, different again.
So what's going on?
Well, spacetime diagrams preserve the spacetime interval
between points with its weird minus sign,
not the Pythagorean Euclidean notion of distance
that seems to be hard wired into our eyes and brain.
So while these diagrams help quasi-visualize things,
spacetime doesn't really look like this.
Now, that's not surprising, but it
is disorienting since we rely so much on our eyes.
But notice that visually parallel lines do
remain visually parallel in all three diagrams.
The visual separation between lines
may change, as does their tilt, but they're always parallel.
So the visual criterion for parallelism
works since it's either met or not
met in every frame of reference.
It also happens to work for vectors.
So now, we are finally in a position
to state some things that may have seemed visually obvious,
but that I don't think really were.
For starters, this spacetime really is flat.
Parallel lines stay parallel.
Second, the world lines of the red guy, the monkey,
the photon, and me are all straight.
All of their tangent vectors remain tangent
when parallel transported.
So all of them are geodesics.
However, not all world lines are geodesics.
Suppose I see a car approaching me, slowing down, turning
around, and speeding away.
A tangent vector to its world line
doesn't stay tangent under parallel transport.
So it's not a geodesic.
And this is interesting.
In Newtonian mechanics, we distinguish
inertial and noninertial observers dynamically
by using the floating ball test.
But in spacetime, we can also distinguish those classes
of observers geometrically.
Inertial observers have geodesic world lines and noninertial
ones don't.
And that's kind of the whole point
of talking about spacetime in the first place.
By representing dynamical phenomena
from the physical world as geometric objects and relations
in a tenseless mathematical space,
we can discover facts about physics
just by exploring geometry in that space.
It's kind of cool.
I want to wrap up by going back tangent vectors for a second
because I haven't told you yet what they represent physically.
In a more standard drawing of motion over time,
like what you might see in a physics 101 class,
tangent vectors to trajectories represent
velocities, how fast you're going and in what direction.
But that interpretation doesn't work on a spacetime diagram.
Think about it.
Motion is relative.
In my frame, my own velocity is zero,
but in the red guy's frame it isn't.
So ordinary velocity would not be a frame invariant geometric
vector in spacetime.
Also, things don't move through spacetime.
It's not this kind of space.
Instead, we want to represent some aspect
of dynamical motion over time through space
as a static geometric object.
The question is how do we do that?
Well, without justification, which
would take us too far off topic, here's what turns out to work.
Instead of tracking changes in the monkey's position
on my axis with respect to ticks of my clock--
that's ordinary velocity-- I'm going
to track two hybrid quantities instead,
the monkey's position on my axis relative to the monkey's clock
and the time on my clock relative to the monkey's clock.
That, it turns out, if I stack it,
is the tangent vector to the monkey's world line.
It's called the monkey's 4-velocity,
even though that's a bit of a misnomer
since there's no motion through spacetime.
Just a term.
And more interestingly, the length of that vector, at least
in the sense of spacetime interval length,
is minus the speed of light squared.
In fact, every observer's 4-velocity
always has a length of minus the speed of light squared, even
the accelerating car's 4-velocity.
So if we call the spacetime length of a 4-velocity vector
a spacetime speed, then the world line
of every inertial observer is a constant-speed straight line.
That's what I've been meaning all along.
And accelerated observer's world lines
are constant-speed non-straight lines.
Chew on all that because it's our departure
point for talking about curved spacetime in the next episode.
And none of this is obvious.
So I hope you see now why I took this detour.
I know it's a lot to take in, but you've
got a week to mull it over before we plunge head-first
into curved spacetime.
send noodz