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Einstein said that gravity is not a force.

Instead, he said, it's a manifestation

of spacetime curvature.

Sounds great.

Now what's curvature?

In general relativity, objects that fall or orbit aren't being

pulled by a gravitational force, they're

simply following straight line constant speed

paths in a curved spacetime.

Now anyone can say those words at a party to sound cool,

but what do they actually mean?

Well, for a complete answer you can read this 1,200 page

behemeth.

Sorry, there's just no way around that.

But over the next few episodes I'm

going to try to give you a sense of the answer,

a flow chart level view of the relevant concepts

and how they add up to the idea that there simply

is no force of gravity.

We actually started this campaign

in our "Is Gravity an Illustion?" episode.

If you haven't seen it yet, pause and click

here to watch it right now.

Otherwise, what I'm about to say will make no sense.

You all done?

Awesome.

In that episode we noted objections

to Einstein's viewpoint, many of which

you echoed in the comments.

Now ultimately, the way around those objections

is to realize that if the world is a curved spacetime,

then the familiar meanings of terms like a constant velocity

straight line and acceleration will become ambiguous.

We'll be forced to redefine them,

and once we do there's no longer going

to be an inconsistency with saying that falling frames are

inertial, even though they accelerate

relative to one another.

Our goal in this series of videos

is to explain that last statement,

and to explain how it lets you account

for the motion we observe even if there's

no Newtonian force of gravity.

But we need to lay some groundwork first,

so we're going to spread this out over three parts.

In part one we're going to put physics aside and focus

on geometry, specifically on what we really

mean by straight line and by flat verses curved

mathematical spaces.

In part two we'll acquaint ourselves

with the specific geometry of 4D flat spacetime, which

is already weird, even without curvature present.

And finally, in part three we'll put curvature and spacetime

together to tie up all the loose ends

that we raised at the end of our gravity illusion episode.

We'll end up seeing that all the supposedly gravitational

effects on motion can be accounted for just

by the geometry of spacetime.

Now I have to break things up like this,

otherwise there will be too many logical gaps

which defeats the purpose of talking about this at all.

And since you guys, as a collective audience,

asked for this topic I want to try to do it justice.

You guys ready?

OK, buckle up.

Today is part one, that's straight lines

and curved spaces with no physics, just geometry.

Let's start with this picture of the flat Euclidean 2D plane

from high school math class.

Intuitively, we know that curve number one,

joining points A and B in the diagram is straight,

and curve number two is not.

But how do we know that?

See, if we want to do geometry on arbitrary

spaces like on the surface of a sphere or a saddle

or on some funky hillside, that's not a vacuous question.

And as you'll see in a minute, saying

that it's the shortest path from A to B

doesn't work as a general answer.

However, here's what does work.

Draw a tiny vector with its tail at point A.

You can slide that vector from point A

to point B along curve one or along

curve two while keeping it parallel

to its original direction.

This operation is called parallel transporting a vector

along a curve.

OK, now draw a vector at point A,

specifically that's tangent to curve one

and parallel transport that vector to B along curve one.

At every point along the way it remains tangent to curve one.

In contrast, if we take a vector tangent to curve two

and parallel transport it to B along curve two,

it does not remain tangent to curve two at all points.

So it looks like we have our definition.

A curve is straight if tangent vectors stay tangent

when they're parallel transported along that curve.

Mathematicians realized a long time ago

that this definition generalizes very nicely

and it's also very useful.

For example, picture an ant confined

to the surface of an ordinary sphere

with no concept of or access to the direction off the surface.

From the ant's two dimensional confined perspective,

curve one between A and B is straight.

Just look at it.

The vector tangent to curve one at point A

remains tangent all along curve one

as we parallel transport it to point B.

But that's not true along curve two, which is

why curve two is not straight.

Now, from the ambient three dimensional perspective,

you could say that those tangent vectors aren't really

staying parallel and that neither of our curves

is really straight, but the ant, who's very flat,

can't look in three dimensions anymore than we

can look in four dimensions.

Its entire universe is that spherical surface,

and it requires criteria for parallel,

tangent, and straight that it can apply solely

within that two dimensional space.

Here's how the ant can do that.

Over tiny regions of the sphere the ant

can pretend that it's on a plane,

and it can use planar definitions

of parallel and tangent.

So parallel transporting a tangent vector

means breaking up a curve into a gazillion microscopic little

steps and applying planar rules for parallel and tangent

over each step.

Once the ant does that over lots of curves joining A and B,

it finds that the tangent vector will remain tangent only

along a particular curve, a segment of a great circle.

That segment is called a geodesic,

and piecewise it's straight.

By the same process you can find geodesics

on a saddle or a hillside or in three dimensional spaces.

Now note that a geodesic is not always the shortest

curve between two points.

That piece of our great circle that

points the opposite direction is also straight, even though it's

not the shortest curve joining A and B. In fact,

in some spaces that have weird distance formulas,

like flat spacetime, geodesics are sometimes the longest

curves between two points.

So the shortest path rule for straightness

doesn't generalize, but the tangent vector

parallel transport rules does.

And in other curved spaces, multiple straight lines

can join the same two points.

As a result, the notion of distance between two points

is ambiguous in a curved space.

All we can talk about is the length of curves

and their straightness or lack thereof.

All right, now that we know what it means for a line in a given

space to be straight, let's figure out

what it means for an entire space to be curved.

Intuitively, we know a plane is flat

and that a sphere is curved.

But as before, let's ask why.

Again, we can end up defining curvature

using parallel transport.

Here's how.

Parallel transport a vector from A to B

along two different curves.

If the result you get is the same, same vector at point B,

then your space is flat, otherwise it's curved.

Here's an alternate way of thinking about it.

Parallel transport a vector around a closed curve starting

at A and going all the way back to A.

If you end up with the same vector you started with,

your space is flat.

If not, curved.

Now you may have heard an alternate definition

of curvature that involves parallelism.

Namely, take two nearby parallel geodesics

and extend them indefinitely.

If they always remain parallel, your space is flat.

But if those geodesics start converging or diverging

at any point, then the space is curved.

It's not obvious, but that definition

turns out to be logically equivalent to the one I already

gave.

Each one implies the other.

Note that this notion of curvature

does not always agree with your 3D visual intuitions.

For instance, the surface of the cylinder is flat.

If you draw some lines and vectors

on a flat sheet of paper and roll it into a cylinder

you can verify for yourself that parallel lines, indeed,

remain parallel.

Now those lines might close on themselves,

but locally, snippet by snippet, geometry and straightness

and tangency and parallelism all work

just like they do in the plane.

The difference between the cylinder and the plane

in topology, i.e. in the connectedness

of different regions of the space.

Topology is global, but geometry and curvature are local.

Different concepts.

Now in a three dimensional space you

can test curvature the same way we've been describing.

Just move a vector parallel to itself around a circle.

If you end up with the same vector you started with space

is flat, if not, it's curved.

If you think that the vector may have shifted

by less than you can measure, just use a bigger circle

or do lots of loops around the original circle

until the shifts accumulate to a level that you can measure.

So is the three dimensional space around Earth curved?

Well, it turns out the answer is yes,

but it's really hard to measure.

And 3D curved space isn't what explains away

gravity, it's four dimensional curved spacetime.

Why is the spacetime part so critical?

To understand that, we need to get a better grip

on how geometry works in flat spacetime.

And remember, even without curvature,

that geometry is super weird.

Let me give you an example.

In flat spacetime that line has a length of zero,

and these two lines are perpendicular.

You see what I'm talking about?

It's weird.

But I'm getting ahead of myself.

Flat spacetime geometry is part two, which is next week.

To prepare for that, you should watch our episode "Are Space

and Time an Illusion?"

Watch it like 10 times.

I'm not fishing for views here.

You should watch as many videos about special relativity

as you can no matter who's made them.

This is for your benefit to prime your brain.

This stuff is really unintuitive,

so every little bit of osmosis helps.

In the meantime, you can put your questions about geodesics

and curved mathematical spaces down in the comments below.

I'll do my best to address them during the week

and on the next episode of "Spacetime."

Last week we asked whether Australia would ever

get a White Christmas in order to discuss

the calendar, the seasons, and their connection

to Earth's orbit.

Here's what you guys had to say.

But first, quick comment about the leap second video.

I got something wrong in there and didn't want

that misinformation out there.

So we'll re-shoot it soon and then

the link will be working again.

Now to you comments.

Jordan Filipovski, MaybeFactor, and Sharfy

pointed out that most of Australia

doesn't get snow, even in winter.

And several others pointed out that some parts of Australia

do get snow on Christmas, even though it's summer down there.

Look, I'm not a complete climate ignoramus,

I understand all this.

Northern Sweden got snow in June of 2012 too

and ourparentsareourlips said that in central Oregon

it once snowed in July.

I wasn't trying to be that literal.

Australian white Christmas was just

a motif for talking about reversal

of the seasons relative to the calendar.

I did learn something new about Aussie Christmas

though from JakeFace0 and QuannanHade, namely

that Santa's sleigh is already pulled

by six white boomers, or older white furred kangaroos.

Who knew?

Ali Muzaffar and Marko Nara asked

whether geomagnetic reversal, which

happens every half million to million years,

might also reverse the seasons.

I don't think the answer is well understood,

but since magnetic pole reversal wouldn't affect

Earth's orbit or the tilt, any effects on the seasons

would be indirect.

Indigo said that tracking time in the future

might become a challenge if you have to consider

relativistic effects.

Actually, that's already an issue, even today.

GPS breaks if time dilation isn't taken into account.

Time also runs at different rates

at different locations on Earth that have different elevation.

So since those discrepancies are measurable with atomic clocks,

this has to be taken into account when you calibrate time

systems, or for instance, when you

measure the rate at which the Earth slows its rotation.

Finally, Jose Catlett and Dennis Ryan left us with a fun fact

about leap years.

In the Gregorian calendar we add February 29,

but in the Julian calendar that proceeded it

we simply doubled February 24 on leap years.

Now I'm not sure when exactly the transition occurred

to sequential numbering of days even on leap years,

but I'd love to find out.

So if you happen to know, please go back

to the Australia episode and leave a comment

in Jose Catlett's thread.

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