Hey, Spacetimers, two weeks ago we issued a challenge question.
Which of two particles, one orbiting
around the outside of a planet and one
going straight through the middle,
reaches the other side first?
There was a Newtonian and an Einsteinian version
of the challenge.
Let's start with the answer to the Newtonian version.
The key to the Newtonian challenge
was getting an expression for the gravitational force
on the particle that's falling through the planet
when it's a distance r from the center of the planet.
As it turns out, if you use the hints that I
gave in the original video, it works out
to be proportional to that distance.
Now there's another force that comes up
in elementary physics that's also
proportional to the distance of a particle from an equilibrium
point.
Namely, the force that a spring exerts on a block.
Now, in the case of a block on a spring,
you can work out an expression based
on those proportionality constants
for how much time it takes to do a complete oscillation.
Now, by drawing a formal algebraic analogy
between the gravitational equation on the one hand,
and the spring equation on the other,
you can find analogous quantities
and figure out an expression for the period of oscillation
of a particle falling through a planet.
Namely, how much time it takes to go and come back.
Cut that in half, and you know how much time it
takes to reach the other side.
Now, independent from what I just said,
you can also work out an expression
for the orbital period of a particle moving
under the planet's gravity in a circular orbit right
at the surface.
If you compare that expression to the expression
that you get for the oscillatory period of a particle going
through the center, you find that algebraically, they're
identical.
This is kind of a coincidence of the specific situation
of a uniformly dense planet and Newtonian mechanics.
But nonetheless, it means that the half periods are also
the same, and it turns out that the particles tie.
Now the only reason I know that's the answer
is that I actually did out the algebra.
A lot of you tried to submit answers
that had some verbal intuitive explanation, but none of them
really held water.
You have to do the algebra in this case,
because this is very specific to the situation
of a uniformly dense planet.
Change the density profile even a little bit,
and the answer changes.
So you can see a complete algebraic solution,
and I encourage you to look at it
by going down to this link, which is also
down in the description.
But a lot of you did submit algebraically correct
complete solutions, from which we selected five randomly
to win a T-shirt.
So if you see your name passing by on the screen below,
you're one of our five winners.
Congratulations.
Please send an email to pbsspacetime@gmail.com
that has your full name, your full mailing
address, and your preferred T-shirt size in American
sizing-- small, medium, large, extra large, like that.
We'll send one out to you as soon as we can.
Good work and congratulations to everyone who
submitted a correct response.
For the Einsteinian challenge, we had fewer than five people
submit correct responses.
Your names are scrolling down below.
Congratulations, send us an email with your specifications
and we'll get you your prize.
Now explaining the answer to the Einsteinian challenge
is a bit more involved because we don't know
the geometry from the outset.
You have to--
[LASER HUMS]
What was that?
[SUSPENSEFUL MUSIC]
[LASER HUMS]
(DEEP VOICE) Gabe, I am your replacement.
Matt O'Dowd, is that you?
(NORMAL VOICE) Yeah.
It's me.
Oh, so you're the new host of Space Time?
Yeah.
But why'd they give me all this?
It's to see--
[LASER HUMS]
--if you're ready.
[LASERS HUM]
Question one.
In 15 billion years, what will you
find at the center of the solar system?
Well, assuming the collision with Andromeda
doesn't disrupt our planetary system,
it'll be a white dwarf remnant, and after the sun
ejects its outer layers, there's a planetary nebula.
[LASERS HUM]
Question two.
Which quasar is the most distant object ever observed?
Trick question.
It's not a quasar at all.
It's a gravitationally-lensed protogalaxy.
Wait, really?
I didn't know that.
[LASERS HUM]
Question three.
What's the answer to the Einsteinian version
of the two-particle challenge question?
Have you even worked that out yet?
Actually, no.
[LASERS SHUT DOWN]
I haven't done it yet, but I could use some help.
Let's go to the white board.
You need a marker?
No, I got one.
Awesome.
So here's where I am so far.
You start with the general spherical symmetric metric.
And the effective potential formulation
for radio [INAUDIBLE].
And from there--
[MUSIC PLAYING]
OK.
So we evaluate these integrals.
See which one's bigger.
And that tells us who wins the race.
I think that's it.
So ends up winning the race?
Well, the answer to that question,
along with all the math that we used to figure it out,
can be accessed at this document with this link
that we also have down in the description.
Why aren't we just telling you the answer on the air?
Well, I think it's really important that you actually
go and look at the math.
You know, physical intuition is incredibly important
and you can go a really long way verbally talking
through a physics problem.
But in the end, sometimes you just got to do the math.
And that's doubly true in general relativity,
where the ground is always shifting under your feet.
In every problem, the rules of geometry change.
You don't know how clocks work.
And you pretty much have to start from scratch.
There's just no way around it.
Some of you might be daunted by the general relativity
solution.
But remember, the Newtonian solution is also there.
And it's a really good idea just to look at the [INAUDIBLE].
At least to see how this process goes.
I couldn't agree more.
Guys, it's time for me to go.
I've had a great run here, but you're in good hands with Matt.
These are the Spacetimers.
Spacetimers, this is Matt O'Dowd.
And it's unfortunately time for me to go.
I'm really excited to be taking over this show.
We're going to be doing some extremely cool topics
with the same rigor intensity that you've been used to.
And as a bonus, I talk slower than Gabe.
So I'll see you guys in a couple weeks for a fresh episode
of Space Time.
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