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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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# 相対論的速度の付加｜特殊相対論 第6章 (Relativistic Addition of Velocity | Special Relativity Ch. 6)

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林宜悉 に公開 2021 年 01 月 14 日