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Well, we're finally here.
A synopsis of general relativity that
builds on these previous four episodes.
If you haven't seen them then pause me now,
go watch them in order, and meet me
back here after the music to hear about curved spacetime.
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Newton's and Einstein's dispute over gravity
comes down to competing notions of what constitutes
an inertial frame of reference.
Newton says that a frame on earth's surface is inertial,
and relative to that frame, a freely falling apple
accelerates down because it's pulled
by a gravitational force.
But Einstein says, nuh-uh, it's the apple's frame
that behaves like a frame in deep space.
So the apple's frame is inertial and the Earth frame is actually
accelerating upward.
You just get a false impression of a gravitational force
downward for the same reason that a train car accelerating
forward gives you a false impression
that there's a backward force.
So who's right?
Well, between our gravity illusion episode
and your comments, we've seem that Einstein's position
seems internally inconsistent.
Remember that inertial frame in deep space?
Well, the apple accelerates relative to it.
So if inertial frames define the standard of non-acceleration,
how can both of those frames be inertial?
Today we're finally going to show how curved spacetime makes
Einstein's model of the world just as self consistent
as Newton's.
Step one is to express both Newton's and Einstein's
viewpoints in geometric spacetime terms,
since that's the only way to compare them
in a reliably objective way.
Remember, humans experience the world and talk
about the world dynamically, as things
moving through space over time.
But even in a world without gravity,
we already know that clocks, rulers, and our eyes
can all mislead us.
So to be sure we're talking about real things as opposed
to just artifacts of our perspective,
we have to translate dynamical statements
into tense-less statements about static geometric objects in 4D
spacetime.
Let's start with Newton.
He says that spacetime is flat.
Jut think about it, on the flat spacetime diagrams of inertia
observers, the world lines of other inertial observers
are straight, indicating constant spatial velocity.
This captures Newton's idea that inertial observers
shouldn't accelerate relative to other inertial observers.
Newtonian gravity would just be an additional force
we introduced, like any other force, that
would cause some world lines to become curved, i.e, spatially
accelerated.
This is a bit oversimplified, but for today it'll do.
Now for Einstein's position.
This is actually more subtle, and it'll
be easier to explain if I first set up an analogy using
our old friend the two dimensional ant on the surface
of the sphere.
A tiny patch at the equator looks like a plane.
And within that patch, two great circles both look straight.
But suppose the ant believes that he
lives on an actual plane, and decides
to draw an xy grid on a large patch of the sphere,
with its x-axis along the equator
and the y-axis along longitude line.
Relative to this grid, the second grade circle
looks bent, so the ant concludes that it's not a geodesic.
But you see the ant's mistake, right.
His grid is distorted.
You can't put a big, rectangular grid on a sphere
without bunching it up.
Try it withs some graph paper and a basketball.
It doesn't work.
Stated another way, a sphere can accommodate
local Euclidean grids in tiny patches, but not global ones.
So the ant can use his axis as rulers and protractors
within a patch, but not between patches.
Flat space definitions of straightness
apply over small areas, but not big ones.
OK, Einstein's position is that Newton is making
the same mistake as the ant.
Inertial frames, that means axes plus clocks,
are the spacetime equivalent of the ant's xy grid.
If spacetime is curved, then those frames
are only valid over tiny spacetime patches.
So when an observer in deep space
says that the falling apple is accelerating,
he's pushing his frames past the point of reliability,
just like the ant did.
In other words, global inertial frames
don't exist in spacetime.
However, global inertial observers do.
They're observers that have no forces on them.
Their world lines will be geodesics,
and their axis and clocks can serve as local inertia frames,
provided that we think of them as being
reset in each successive spacetime patch.
And by the way, pictures like this
are not intended to make literal visual sense.
On the contrary, they're designed to break
your excessive reliance on your eyes so that your brain becomes
more free to accept what reality isn't.
Remember, no one can really see or draw spacetime.
There is no spoon.
Now the world line of a following apple
turns out to be a geodesic.
It has no forces on it, so there's
no need to invent gravity.
OK, but what about two apples in a falling
box, like at the end of our gravity illusion episode?
Remember, they get closer as the box falls.
Now according to Newton, that happens because the apples
fall radially instead of down.
But according to Einstein, it happens
because the apples are on initially parallel geodesics
that, since spacetime is curved, can and do cross
just like on the sphere.
In contrast, the world line of a point on Earth's surface
is not a geodesic.
It has a net force on it, and it's really accelerating.
So does that mean that Earth's surface
has to be expanding radially?
Well, be careful.
In order to compare distant parts of the Earth,
you'd need a single frame that extends
across spacetime patches.
But that frame can't be inertial.
So any conclusions you base on it
have to be interpreted with a heavy grain of salt.
OK, so Einstein's gravity-free curved space
time sounds like it's self consistent.
But then again, so does Newton's flat spacetime picture
that has gravity injected as a kicker.
So once again, which of them is right?
The answer is, whoever agrees better with experiments.
And there's over a century of experiments to refer to.
Now, we haven't fully fleshed out all of general relativity
yet, but there's one experimental fact
that I can use to show you that space time must be curved,
just based on what we've seen in this series of episodes so far.
It's a cool argument, originally presented over
50 years ago by physicist Alfred Schild, and it goes like this.
Fire a laser pulse from the ground floor
of a building up to a photon detector on the roof.
Now wait five seconds and then do it again.
On a flat spacetime diagram the world lines of those photons
should be parallel and congruent.
Without making any assumptions about how gravity effects
light, that would be true even if it turned out
that gravity slowed photons down and bent their world lines,
since both photons would be affected identically.
Now if spacetime is flat, then clocks on the ground
and on the roof should run at the same rate.
They're both stationary.
Thus, the vertical lines at the ends of the photon world lines
should also be parallel and congruent.
But if you actually do this experiment
you find the photons arrive on the roof
slightly more than five seconds apart.
The excess time is less than a second,
but any discrepancy means that clocks
are running at different rates.
In which case, the opposite sides of this parallelogram
aren't congruent.
And that's geometrically impossible
if spacetime is flat.
Thus, the very existence of gravitational time dilation,
regardless of its degree, requires
that spacetime be curved.
And that means game over for Newton.
In fact, to the extent that we can speak about space and time
separately at all, most of the everyday effects on earth
that Newton would attribute to gravity
are due to curvature in time.
The 3D space around Earth is almost exactly Euclidean.
Those pictures that you see of Earth deforming a grid the way
a bowling ball deforms a rubber sheet,
or even the pictures we sometimes use on this show,
they all suggests spatial curvature only,
so they're somewhat misleading.
Remember, a frame consists of axes and clocks.
And around Earth, spacetime curvature
manifests itself in clocks much more than in rulers.
So even though it's hard to visualize,
it's curved time that makes the free fall orbits of satellites
looks spatially circular in frames of reference that cover
too big a space time patch.
So why is spacetime curved in the first place?
Unfortunately, the math gets heavier here
and good analogies are harder to come by.
But here's the flow chart level answer.
Consider a region of spacetime.
And remember, that means a collection of events, not just
locations.
Its curvature in geodesics are determined
by how much energy is present at those events via set
of rules called, no surprise, the Einstein equations.
So for example, say you stick the energy distribution
of the sun into the Einstein equations and turn a crank.
What comes out is a map of the geodesics in the sun's
spacetime neighborhood.
Now when you translate those geodesics
into 3D spatial and temporal terms, what you find
is planetary orbits or spatially straight,
radially inward trajectories along which
you would see spatial speed increase,
or pretty much anything else that you would otherwise
attribute to a gravitational force.
It's pretty amazing.
I want to conclude with a question once asked by one
of our viewers, Evan Hughes.
If there's no gravity and gravity is not a force,
then why do we keep using that word?
Well, physicists are still human.
As far as I know, most of us have no special ability
to visualize or directly experience 4D spacetime.
So we often think in Newtonian gravitational terms,
because it's easier, and because the resulting
errors are usually small.
We just remind ourselves that it's just a crutch that we
have to use with caution.
But even when people are referring
to relativity or string theory or whatever,
it's just a lot easier to say the word gravity
than say curvature of four dimensional spacetime.
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