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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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