Relativistic Addition of Velocity | Special Relativity Ch. 6 - YouTube

In our universe, when you change from a non-moving perspective to a moving one, or vice versa,

that change of perspective is represented by a what's called Lorentz transformation,

which is a kind of squeeze-stretch rotation of spacetime that I've mechanically implemented

with this spacetime globe.

Lorentz transformations keep the speed of light the same for all perspectives, since

that's an experimentally verified fact of our universe.

For example, let's say I'm not moving – that is, I'm at the same position at all times,

and you're moving a third the speed of light to my right, and you turn on a flashlight.

Then that light will move at the speed of light, c, or about 300 million m/s, which

is drawn as a 45° line on this spacetime diagram.

And viewed from your perspective, you're not moving (aka you're at the same position at

all times) but the light ray still travels at the speed of light.

In fact, viewed from ANY moving perspective, the light ray always moves along a 45° line

on a spacetime diagram (at least one with the axes scaled like this).

So light speed plus your speed equals light speed - it's almost more like what happens

when you add something to infinity than adding together two finite numbers.

But what about speeds slower than light speed?

What if you're traveling at 60% the speed of light to the right, and you shoot a death-pellet

that is itself going 60% the speed of light to the right relative to you – how fast

is it going from my perspective?

The intuitive answer to this question is that if the death-pellet is going 180 million meters

per second to the right relative to you, and you're going 180 million meters per second

to the right relative to me, then the death-pellet must be going 360 million meters per second

to the right relative to me, which is faster than light.

And which is wrong.

In our universe, velocities don't simply add up when you change perspective.

They almost do for things moving much slower than light (which I'll explain in a bit) but

in general that's not how our universe behaves.

Here's a spacetime diagram from your perspective of you shooting a death-pellet to the right

at 50% the speed of light - that is, taking 4 seconds to go as far as light would in 2

seconds.

And here's what happens when we shift to my perspective, from which you are moving to

the right at 50% the speed of light.

The death-pellet is still moving to the right relative to you, still moving really darn

fast, but it's not moving as fast as light - its worldline is not quite

a 45° line.

And while stuff going 60% the speed of light is kind of reaching the limits of what the

spacetime globe can reasonably display, if you shoot a death-pellet at 60% the speed

of light and then we shift to my perspective from which you're going 60% the speed of light,

the death-pellet still isn't going faster than light.

And it can't be, which you can kind of get a feeling for from how Lorentz transformations

work – in our universe, when you change from one moving perspective to another, your

perception of spacetime squeezes and stretches along the 45° lines that represent the speed

of light, and this can only rotate worldlines to angles that are between those 45° lines.

Stretching out a line on a rubber sheet makes the line's angle approach the direction of

stretching, but never “flip over” to be pointing the other way.

So even if we shot a death-pellet going 60% the speed of light FROM a death-pellet going

60% the speed of light FROM a death-pellet going 60% the speed of light and so on, the

final speed would be close to but not quite the speed of light, because of how relative

velocities combine in our universe.

This is one of the consequences forced upon us by the constancy of the speed of light:

in a universe (like ours) where changes of velocity don't change the speed of light,

then changes of moving perspective can never make other velocities change from a relative

speed less than the speed of light, to a relative speed equal to or greater than light.

If we have an object moving at a speed v relative to your perspective, and you're moving relative

to me with speed u, then the equation that describes precisely what speed the object

is moving relative to my perspective is

v frommyperspective equals v fromthemovingperspective plus u over 1+v fromthemovingperspective times

u all over c squared.

You'll notice that if you put in c, the speed of light, for one of the velocities, the equation

always gives the answer c back, no matter what the other velocity is – which of course

jives with the whole “constant speed of light” thing.

And you'll notice that if both velocities are less than the speed of light, then the

equation always gives back an answer less than the speed of light – which is what

we were describing earlier about relative speeds never adding up to a speed faster than

light.

Which of course jives with the the whole “nothing can accelerate to light speed” thing.

And you'll notice that if both velocities are a lot lot smaller than the speed of light,

then the v times u divided by c squared term in the bottom is essentially zero, and so

the whole thing is essentially v+u – this is the sense in which, for slow speeds, velocities

DO simply add together.

But not for speeds close to light speed; our universe is more subtle than that.

For a deeper look into how to compare relativistic velocities, I highly recommend heading over

to Brilliant.org's course on special relativity.

There, you can explore custom scenarios that build off the topics in this video to get

an intuitive understanding of the mathematics of relativistic velocity addition - like how

to warn earth of an incoming relativistic alien invasion.

The special relativity questions on Brilliant.org are specifically designed to help you take

the next step on the topics I'm including in this series, and you can get 20% off of

a Brilliant subscription by going to Brilliant.org/minutephysics.

Again, that's Brilliant.org/minutephysics which gets you 20% off premium access to all

of Brilliant's courses and puzzles, and lets Brilliant know you came from here.