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  • 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 movingthat 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 youhow 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

  • workin 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 neverflip overto 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 iswhich of course

  • jives with the wholeconstant speed of lightthing.

  • 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 lightwhich 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 wholenothing can accelerate to light speedthing.

  • 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.

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

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B1 speed pellet relative spacetime moving death

Relativistic Addition of Velocity | Special Relativity Ch. 6

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    林宜悉 posted on 2020/03/28
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