[MUSIC PLAYING] 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.