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Hey Crazies.
We all know light takes time to get places, right?
It does travel at 300 million meters per second.
That's 30 centimeters per nanosecond or about 670 million miles per hour.
Light could travel around the earth 7.5 times in a single second,...
well, assuming you could get it to go in a circle.
It takes us days to get to the moon, but light can get there and back in about 2.5 seconds.
No one is saying it isn't fast.
But those numbers are still finite.
We could easily imagine numbers that are larger...
1 billion meters per second, for example.
Why can't light go that speed?
Well, because we think about speeds wrong.
We tend to think of speed as a rate of change,
specifically a time rate of change.
If a car is traveling down the road, it's changing location as time passes.
We can take the distance it traveled
and divide by the time it took to travel that distance.
That measurement could be in meters per second,
or feet per second, or kilometers per hour, or miles per hour,
or something like that:
a unit of distance over a unit of time.
So what's the problem with that?
It fails at significant fractions of the speed of light.
We might think cars are pretty fast.
They're a lot faster than walking or running.
Planes and spacecraft or even faster.
But none of those are moving anywhere near the speed of light.
Even tiny subatomic particles, while fast, are traveling slower than light.
The speed of light appears to be the fastest speed.
Fast fast!
But what if we're in a rocket traveling at 90% the speed of light and...
and we launch a projectile at 11% the speed of light?
Yeah, wouldn't that be 101% the speed of light?
Nope! It'd be 92% the speed of light. What?!
I told you our intuitions fail at speeds that fast.
If you're in a car driving at 30 miles per hour and you throw a ball at 15 miles per hour,
our expectations say someone standing on the ground should see that ball going at 45 miles per hour.
It's a relatively simple calculation.
You just add the numbers together, right?
Wrong!
Eh, well, I mean, it works in this specific example.
But, in general, it's wrong!
The equation actually has a big denominator.
It's just that for the normal human experience, the denominator is effectively one.
For the rocket example, it's larger, which lowers the result from 101% to 92%.
Why is that denominator even there?
Because we insist on speed being distance over time, for some reason.
Like we saw earlier, we treat speed as a rate of change.
If we take a bunch of snapshots at various moments, we can graph distance against time.
The slope or incline of this line would be the speed.
For another car moving slower, the line would be shallower:
smaller slope, smaller speed.
For a third car moving faster, the line would be steeper:
larger slope, larger speed.
But what if instead of using the slope, we use the angle?
A car that isn't moving at all is going to make a horizontal line on this graph.
Let's measure all of our angles from there.
Infinite speed, also known as magical teleportation, would be a vertical line.
That's 90 degrees from the horizontal, which should be the biggest angle possible.
All other speeds between zero and infinity would be measured as an angle between zero and 90.
Faster objects would have a bigger angle,
slower objects a shallower angle.
Can you just make up a measurement like that?
Sure. Why not? All measurements are made up.
Science, do whatever works.
As long as your definitions are logically consistent, you're fine.
This new way of measuring speed is logically consistent.
It just isn't what you're used to.
And believe it or not, it happens to be the better choice.
Remember our previous example?
We were in a car traveling at 30 miles per hour and we threw a ball at 15.
Here's the speed of the car and here's the speed of the ball, each represented as an angle.
The speed of the ball relative to someone standing on the ground is just the two angles added together.
The angles just add, which is the rule we originally wanted speed to follow.
Unfortunately, traditional speed isn't measured as the angle of these lines.
It's measured as the slope of these lines.
Those slopes only add together when the angles are small.
In other words, when the speeds are slow.
If we're talking about rockets and projectiles going at significant fractions of the speed of light,
then slopes do not simply add together.
That means traditional speeds don't simply add together.
But the angles still do.
To avoid confusion, for the rest of this video
I'm going to refer to that new angle speed as rapidity.
That way I can stop saying traditional speed and just say speed.
So, long story short, speeds only add when they're small.
Rapidity always adds.
The problem with rapidity is that it's not measured in typical speed units.
It's measured in angle units like degrees or, better yet, radians;
since that's a unitless unit.
Our intuitions are not built on that, but we can convert.
Trigonometry is the math that relates the sides of triangles to their angles.
It's literally the entire point of trig.
The slope of this path forms a right triangle,
and trig tells us the slope is given by sine over cosine,
which is just tangent.
And we have a nifty identity to help us add those together.
Since tangent is just the slope and slope is just the speed,
we can easily rewrite this with speeds instead.
The angles may add together, but the slopes do not.
(Typing Noises)
That doesn't match the equation from earlier, you big doofus.
Okay. Okay. You may have noticed a minor discrepancy in the formulas I gave you today.
Here's the one I just gave you and here's the one from earlier in the video.
They're close, but there's a minus sign in one denominator and a plus sign in the other.
The one with the plus is the correct one.
Why did you get the minus then?
Because I made an assumption I shouldn't have.
Can we graph distance against time? Absolutely.
Can we use angles to measure speed? Yes, and we should.
We just use the wrong kind of trigonometry:
Euclidean trigonometry.
That's the kind that obeys the normal rules of geometry.
You know, like the angles in a triangle always add up to be 180 degrees.
But when we graph distance against time, what we have is spacetime.
Space may be Euclidean, but spacetime is not.
It's hyperbolic,
which means we need hyperbolic trig functions.
Ahh! Hey! I'm just doing my job.
All right! Who made Whacker Clone?
Rule 7: Only Nick touches the cloning machine.
Where was I?
Right. Hyperbolic trig.
You know how regular trig has the unit circle?
It's kind of popular in the math crowd.
It's just a circle with a radius of one.
You can find all the trig functions on it: Sine, cosine, tangent...
even the cotangent, secant, and cosecant are there.
The sine and cosine are special though, because they form the triangle for any point on the circle.
But a unit circle? That's, like, so 4 millennia ago.
How about the unit hyperbola instead?
If we pick a point on this thing, we can still make a triangle.
We can even label the sides sine and cosine if we want.
But we'll add an H on the end so we remember they're hyperbolic.
Because let's not be totally ridiculous.
Anyway, this fixes our sign problem.
If we swap out the tangents in here for hyperbolic tangents, we get a plus sign in the denominator.
Isn't that a bit hand-wavy? Yeah, maybe a little.
Like, how do we know spacetime is hyperbolic?
Oh, because of the speed of light!
The speed of light is everything!
Speeds added together for the car example, but not the rocket example.
Why not? The speed of light.
More accurately called the speed of causality,
it's exactly 299,792,458 meters per second.
Not only is this the maximum speed at which two points in space can communicate,
it's also a universal constant.
It's exactly this number for all observers everywhere, no matter what.
This is not a guess. It's an observation.
Hyperbolic geometry has the same behavior.
These paths are called asymptotes.
They're the lines that all these hyperbolas approach, but never quite reach.
That sounds a lot like the speed of light.
If we set the slope of those lines to be the speed of light,
we get a model that matches reality very well.
Now, traditionally, the axes are reversed in spacetime diagrams,
but I'm going to break with tradition to stay consistent throughout this video.
If you want to draw them reversed, you can. I'm not your dad.
What matters is the result and that's the same either way.
Not only does this steadily accelerating rocket never reach that diagonal speed, that slope;
any rocket traveling with any acceleration has the same limit.
Speeds along those asymptotes are a universal constant, just like the speed of light.
But why is that constant an upper limit?
Because, hyperbolically, it's infinite.
What? What?! What?!?!
I know. I know. It sounds crazy, but hear me out.
Let's look at the rapidity instead. That's the angle with the horizontal.
The faster the rocket goes, the bigger the rapidity gets,
but only up to a limit: that asymptote, the path of light.
That rapidity limit might look like 45 degrees, but it's not.
Remember, this isn't Euclidean.
It's the hyperbolic tangent that relates rapidity to speed.
The C is the speed of light, which is just there because traditional units suck.
For something traveling at exactly the speed of light, like light,
this side would be equal to one.
That makes the rapidity infinite, not 45 degrees.
The rapidity of light is infinite.
There are no rapidities faster than light.
Why can't light go at a billion meters per second?
Because a billion meters per second doesn't exist.
Speeds that fast simply don't make sense in the context of reality,
which is an insight we never would have gotten without going through all this work.
If you want to dive a little deeper into it, today's sponsor Brilliant can help.
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It will also let Brilliant know you heard about them from me, which helps out the channel.
So, do you want to switch to measuring speed in angle units?
You know you do.
Thanks for liking and sharing this video.
A special thanks goes out to my Patreon patrons and YouTube members like Aleksey Bykov
for all their generous support.
Don't forget to subscribe if you'd like to keep up with us.
And until next time, remember, it's okay to be a little crazy.
For everyone complaining about the Drake Equation, it was never meant to be scientifically rigorous.
It was just meant to start a conversation at a conference.
And, if my comments section is any indication, that's something the Drake Equation is very good at.
Anyway, thanks for watching.