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  • Physics is all about the motion of thingshow planets and stars move, how electrons

  • and protons move, how the movement of molecules results in emergent properties like temperature,

  • and so on.

  • The role of relativity, in physics, is to study how that motion looks from different

  • perspectives.

  • Here I'm usingrelativityin a general sense, to mean from any different possible

  • perspective, moving, accelerating, or otherwise; special relativity in particular is concerned

  • with how motion looks just from a limited, orspecial,” set of perspectives.

  • But either way, relativity (special or not) is about how the motions of things look from

  • different perspectives.

  • Like, if you were looking at the earth and the moon, depending on where you were and

  • how you were moving, it might look like the moon is moving around the earth in a giant

  • circle, or back and forth on a straight line, or that the earth and moon together are tracing

  • out a spiralling path through space.

  • But if the motion of the earth and moon can be described in such different ways, what

  • does any one of these descriptions actually tell us about the earth and moon?

  • Is one of themrightand the otherswrong”?

  • Is there some preferred perspective for observing the earth and moon that gets closer to the

  • true description of what's happening?

  • It's the goal of relativity to answer these kinds of questions.

  • In fact, relativity can essentially be summed up as two basic ideas:

  • 1.

  • To figure out how objects and their motion look from different perspectives, and

  • 2.

  • To notice which properties of objects and motion don't look different from different

  • perspectives.

  • We've already given an example of number 1, with different ways the motion of the earth

  • and moon can look from different perspectives.

  • Number 2, the idea of finding things that don't look different from these perspectives

  • that's a little trickier.

  • In the earth and moon case, for example, all three perspectives appear quite different.

  • But after a while, you might notice that regardless of the perspective, the maximum physical distance

  • between the earth and the moon appears to be the same.

  • So you might sayaha!

  • There's something that's independent of perspective!

  • Maybe it's a fundamental property of the earth-moon system, and not just an artifact of my particular

  • point of view!”

  • And this is why relativity is so important in physics: by studying what changes and what

  • doesn't about a physical system when you change your perspective, you are zeroing in on universal

  • truthsliterally.

  • Facts that remain true from many perspectives throughout the universe (like, perhaps that

  • the distance between the earth and the moon is 384399 kilometers), are literally more

  • universal than a fact that only holds true at a single place and time (like, that the

  • angle between the moon and earth is 150 degrees).

  • Relativity is a way of thinking that helps you to evaluate how universal a given truth

  • is.

  • Ok, enough philosophizing.

  • To make all this tangible, we need a rigorous way of describing moving things and of describing

  • changes to how that motion looks when you change your perspective.

  • We're ultimately going to build up to special relativity, which has to do with motion over

  • time, but we'll start with non-moving things just to get a sense of how intuitive relativity

  • can be.

  • You're probably familiar with specifying the position of a cat on a plane using xy coordinates:

  • this cat is three tick marks to the right of our point of reference, and two tick marks

  • up, so we say it's at position x=3 and y=2, which typically gets written as just a pair

  • of numbers like (3,2).

  • However, (3,2) is not a universal truth – I mean, it's just based on where I'm standing,

  • and how I'm oriented.

  • But over here, where you are, maybe you're rotated by 30 degrees, and you made the tick

  • marks closer together, and suddenly the cat is at a different position: x=9, y=9.

  • Even though the cat hasn't moved.

  • In fact, it's possible to specify the cat's position using any x and y values we want,

  • depending on our point of reference: which corresponds mathematically to where we put

  • our axes and how we orient and scale them.

  • So clearly, specifying the position of something is not a universal truth.

  • Or, in relativity parlance, “position is relative.”

  • A more universal, or absolute, truth can be found if you have two cats: let's say they're

  • at x=0, y=0, and x=5, y=0.

  • And I'm going to stop drawing a person at the origin point of the axes from now on,

  • but you should remember that the axes we use represent a particular perspective and orientation

  • from which we measure things.

  • Ok, so The distance between these two cats is clearly 5 – they're at the same y value

  • and their x values differ by 5.

  • If we move and rotate our point of reference now, the cats are at positions... uh, x=1,

  • y=1 and x=5, y=4.

  • So they differ by 4 in the new x direction and 3 in the new y direction.

  • But the overall distance between the cats, which we can find using the pythagorean theorem,

  • is the square root of 4 squared plus 3 squared, which is the square root of 25 which is 5.

  • Which is the same distance we calculated with the original axes!

  • This turns out to be a general truth: on a plane, the distance between two things doesn't

  • change if you change your perspective just by shifting your point of reference or your

  • orientation.

  • I like to think about this as similar to how if I take a piece of paper, and slide it around

  • and rotate it, I haven't actually changed anything on the piece of paper.

  • Or, in relativity parlance, “distances are absolute.”

  • The geometric intuition for this is that you can move your axes around, slide them up and

  • down, and rigidly rotate them, without affecting your description of the distance between two

  • things.

  • If you like, we can make this mathematically precise by calling the original coordinates

  • x and y, and the new coordinates x new and y new; then when we've slid the x axis an

  • amount Delta x (technically called a “translation by Delta x”), we say that x new=x-Delta

  • x, and when we slide the y axis by an amount Delta y (technically called a “translation

  • by Delta y”) we say that y new=y-Delta y.

  • The minus sign is there because if you slide your origin point closer to something, its

  • new x and y coordinates will be smaller.

  • Changes of orientation are a little fancier, but it's really just some geometry: if you

  • re-orient the x and y axes counterclockwise by an angle theta, the new coordinates look

  • like x new=x times cos theta minus y times sin theta and y new = y times cos theta plus

  • x times sin theta.

  • If you want a fun algebra exercise, you can use these equations (or even their 3D counterparts!)

  • to check that indeed that the distance between two points doesn't change when you slide or

  • rotate your axes.

  • But the messiness of all the details here really clouds the simplicity of what's going

  • on.

  • The important geometric idea I want you to remember is that rotating and sliding axes

  • doesn't change the distance between two points.

  • However, the distance between two points does change if we're allowed to change the spacing

  • of the tick marks!

  • If when we change our axes we also double the tick marks, then the distance between

  • the cats becomes 10, not 5.

  • Turns out distance, measured in numbers, is not so universal...

  • But there is a more universal truth!

  • Suppose we have a stick that's 1 tick mark long (according to the original axes) – conventionally

  • this thing might be called a “meterstick; and now we can saythe two cats are 5 sticks

  • apart.”

  • When we again move and rotate our axes and change the spacing of the tick marks, the

  • cats are again 10 tick marks apart, but the stick is also now 2 tick marks long, so the

  • distance between the cats is still 10/2, or 5 sticks.

  • This is an example of an even more general physical truth: the distance between two things,

  • measured in terms of another physical thing, doesn't change when you change your perspective

  • by shifting your point of reference or orientation or the spacing of your tick marks.

  • In relativity parlance, we'd say thatthe ratio of two distances is absolute.”

  • Basically, if you want to actually describe a distance, you can't just specify a number,

  • like, I'm five away from youyou have to say what you're measuring distance in terms

  • of, and what number of those things your distance is equal to.

  • This is kind of a subtle point and is very important if you're interested in metrology,

  • the study of measurement and units.

  • But because it doesn't really play a major role in special relativity, from now on I'm

  • going to be a bit sloppy and just assume that whenever we're talking about distances, we're

  • talking about distances not as numbers but in terms of some reference distance, like

  • meters, or cats, or whatever.

  • And the same will apply to times: when we talk about a time interval, we'll assume it's

  • a time interval in terms of some reference time, like the second (which is based off

  • of how long it takes a certain electromagnetic wave to vibrate once).

  • Which brings us to the motion of objects over time!

  • To describe a moving object, it's customary to use a horizontal coordinate axis for the

  • left-right x position, but instead of using the vertical axis to represent height y, we

  • use it to represent time t.

  • So for something not moving, something that stays at the same position x at time t=0,

  • t=1, t=2, and so on, we draw a straight vertical line through x.

  • For something moving one meter per second to the right, we draw a line that goes one

  • meter to the right for every second that transpires vertically.

  • It's important to note that we're not saying that the object is moving through 2D space

  • along a 45 degree linethe object is moving purely 1-dimensionally along the x axis, and

  • we're just showing those different one-dimensional positions as time passes.

  • This wholetime on the vertical axisthing can also be a bit weird at first since

  • in most other situations you've probably encountered time plotted on a horizontal axis; but vertical

  • time has its merits and it's convention at this point, so it's worth getting used to.

  • I like to think of each horizontal line as representing a different snapshot of a scene.

  • We could show the snapshots one after another as time actually passes, of course, but it's

  • useful to be able to see all of the snapshots at once, so if we display each snapshot at

  • a consecutive vertical position, we get a nice representation in a single static image

  • of motion that normally takes place over time.

  • This geometric way of representing motion over time is called a “space-timediagram,

  • and it's so central to intuitively understanding relativity that it's worth doing a few more

  • examples.

  • Though if you feel comfortable with space-time diagrams, world-lines, and so on, by all means,

  • skip ahead.

  • Say we have a cat attached to a spring, bouncing back and forth, left and right.

  • If we plot this motion on a spacetime diagram, as time passes we see the cat move left and

  • right, leaving behind a trace in the shape of a sine wave.

  • On the other hand, if we're given a spacetime diagram and want to recover the motion of

  • the cat, we simply slide the diagram downwards at a constant rate and move the cat left and

  • right so that it follows along the traced-out path.

  • This is important: a traced-out path in a spacetime diagram is a faithful recording

  • of an object's motion.

  • And these paths are calledworld-lines,” presumably because they show where in the

  • world the object has gone (though byworldwe often meansolar systemoruniverse”).

  • Any particular point on a worldline has coordinates t and x, which we write as a pair telling

  • us for time t what position x the object was located.

  • So far we've just been representing one-dimensional motion on our spacetime diagramsjust

  • one spatial direction the object is moving in, and then time as the vertical axis.

  • If we want to use a spacetime diagram to represent motion in two dimensions, like the moon orbiting

  • the earth, we actually need three dimensions to do so: the two horizontal directions for

  • the moon and earth to move in, and the vertical direction to trace out the snapshots as time

  • passes.

  • Pretty cool, huh!

  • But if you have multiple particles moving complicatedly, this can get really messy on

  • a 2D screen.

  • And it's physically impossible to make a full spacetime diagram for three dimensional motion,

  • because you would need four spatial dimensions to do so: three horizontal directions for

  • the spatial motion, and a vertical direction for time.

  • Which of course is impossible in our universe with its measly three spatial dimensions.

  • So physicists normally restrict their spacetime diagrams to just one or two spatial dimensions,

  • and time going vertically.

  • So, how does relativity work with spacetime diagrams?

  • That is, now that we know how to describe motion geometrically, how do changes in perspective

  • affect that description?

  • Let's take as an example me staying put right at x=0, and a cat moving one meter per second

  • to the right away from me, starting at time t=0.

  • It may not surprise you to notice that when you slide the x axis to the left or right,

  • the particular x positions of me and the cat at any particular time change, but the distance

  • between us doesn't.

  • At time t=0 we're still 0 meters apart, at time t=2 we're 2 meters apart, and so on.

  • So you can slide the x axis back and forth however you like.

  • Similarly, if you slide the time axis up and down, the absolute time when the cat starts

  • moving away from me changes, but time intervals don't change: the cat still takes 2 seconds

  • to get 2 meters away from me.

  • So you can slide the t axis up and down, and distances in time are left unchanged.

  • And if we have 2-dimensional motion, then changes in the orientation of the two spatial

  • axes don't change the distances between objects at any particular time: essentially, you can

  • re-orient the xy axes however you like.

  • So the relativity we applied to purely spatial diagrams applies pretty well to space-time

  • diagrams, too.

  • To summarize the major takeaways: relativity is about understanding how changes in perspective

  • do or don't affect motion.

  • For objects in space, we can describe motion over time geometrically using spacetime diagrams,

  • and spacetime diagrams can help us see how changes of perspective affect how the motion

  • of objects looks.

  • Like how changing your position and orientation correspond to sliding the axes around and

  • rigidly changing the orientation of the spatial axes, and yet these transformations don't

  • change the spatial distance between two points at the same time or the temporal distance

  • between two points at the same location in space.

  • However, all of this relativity is static: and by that I mean that we haven't yet talked

  • about how motion looks from a perspective that is itself moving.

  • That's ultimately the key to special relativity, and will be the subject of the next video.

  • If you'd like to play around with some spacetime diagrams yourself, I highly recommend checking

  • out thepropagation of lightinteractive quiz on Brilliant.org, this video's sponsor.

  • This quiz is seriously coolit uses spacetime diagrams to guide you through how astronomer

  • Olemer deduced that the speed of light must be finite just by observing the orbit

  • of Jupiter's moon Io.

  • It's a super clever idea, and the quiz does a great job of using spacetime diagrams to

  • help visualize the situation and guide you through calculating the speed of light yourself.

  • In fact, this quiz is part of a whole course on Special Relativity that Brilliant has available

  • at brilliant.org/MinutePhysicsSpecialRelativity, and doing problems like theirs after watching

  • videos like mine is a great way to practice and really internalize the ideas of special

  • relativity.

  • If you decide to sign up for premium access to all of their courses and quizzes, you can

  • get 20% off by going to Brilliant.org/minutephysics, or even better, go to brilliant.org/MinutePhysicsSpecialRelativity,

  • which lets Brilliant know you came from here and takes you straight to their relativity

  • course.

Physics is all about the motion of thingshow planets and stars move, how electrons

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時空圖|狹義相對論第二章 (Spacetime Diagrams | Special Relativity Ch. 2)

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    林宜悉 發佈於 2021 年 01 月 14 日
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