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
