# Relativistic Addition Of Velocity | Special Relativity Ch. 6

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.

Video source: https://www.youtube.com/watch?v=R5oCXHWEL9A