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