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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.
Two of the most famous implications of Lorentz transformations are phenomena called “length
contraction” and “time dilation” – and while their names make them sound like they're
two sides of the same coin, they're definitely not.
We'll start with time dilation, which is easier to see – suppose I have a clock with me
that ticks every two seconds.
But if you're moving at a third the speed of light to my left, then from your perspective,
the time coordinates at which my clock now ticks are slightly farther apart – according
to you it takes about 2.12 seconds between each tick, instead of 2 seconds.
Time literally is running slower for me relative to you, because Lorentz transformations, which
represent how relative motion works in our universe, kind of stretch things out a bit.
Likewise, if you have a clock with you that ticks every...
2.83 seconds, then from my perspective it will tick every 3 seconds.
So both of us perceive each other's perception of time as running slow by the same factor
– that is, relative motion causes our perception of the duration of time between events to
become longer, or dilated - “time dilation”.
If you're wondering how it can make sense that we both perceive each other's time as
running slow, well, I have another whole video on that, but in short, it's because our respective
worldlines (which correspond to our own time axes) are rotated relative to each other,
and so we each only attribute a projection of the other person's worldlines' length as
representing movement through time, and the rest as movement through space.
The factor by which intervals are dilated depends on how fast we're moving relative
to each other, v, and the expression looks like this – but it's really just saying
“how much higher up in time is this point after a Lorentz transformation?”
And if you plot the expression, you'll see that for slow speeds, relative time intervals
are roughly equivalent, but the closer you are to light speed, the more relative perception
of times becomes distorted.
Length contraction, on the other hand, is a tad more complicated.
First, we need something with length.
Let's suppose we have a cat whose tail is at position 0 for all time, and whose head
is 600 million meters to the right for all time (remember, each horizontal tick mark
here represents 299,792,458m).
So from my perspective, the cat is 600 million meters long.
However, from your perspective moving at a third the speed of light to the left, the
ends of the cat get stretched out from each other by the Lorentz transformation – which
at first might seem like dilation of distance, not contraction.
And this is indeed true – from your perspective, the distance between the cat's tail at my
time t=6 and the cat's head at my time t=6 is indeed longer: it's now 636 million meters
(dilated by exact same factor as in time dilation).
However, this dilated distance doesn't represent the length of the cat from your blue perspective,
because these measurements of the positions of its head and tail no longer happen at the
same time, and the cat moves in between when the measurements are taken - that's what having
a slanted world line means - changing position as time passes, aka movement.
And if something moves while you're measuring it, that measurement doesn't represent its
length.
So to correctly measure the length of the cat from your perspective, we need to measure
the positions of its front and back at the same time according to your perspective.
Which is this distance here, which is clearly shorter - 566 million meters.
In fact, it turns out it's exactly the inverse factor from the other distance - instead of
multiplying 600 million by 1.06, it's divided by 1.06.
The same thing happens the other way, too: if you have a cat that's stationary in your
perspective, then when I view it from my perspective, I'll measure its length (by measuring the
head and tail at the same time, according to me), as being shorter.
This is the phenomenon we call “length contraction” – the measured lengths of moving things
are shorter than when those things are viewed as not moving.
The precise factor by which lengths are contracted again depends on how fast we're moving relative
to each other, v, and looks like this.
And similar to the case of time dilation, the closer you are to light speed, the more
relative perception of lengths becomes distorted.
So let's recap: time dilation of moving objects is simply the direct effect of Lorentz transformations
stretching consecutive time coordinates apart in time, while length contraction of moving
objects is a combination of the stretching effect of lorentz transformations on spatial
distances (which is kind of like a “distance dilation”) PLUS then changing the times
at which we're comparing things because they were no longer simultaneous.
This is what I meant when I said earlier that time dilation and length contraction aren't
two sides of the same coin: time dilation compares the times of the same events in the
new perspective, and it pairs with distance dilation, which compares the positions of
the same events in the new perspective.
Length contraction, in contrast, compares positions at the same time according to the
new perspective.
So you might be wondering, is there a time version of length contraction, then?
Is “Time contraction” a thing?
Yes, yes it is (though people almost never talk about it and it doesn't have an official
name, but I think it's nice to complete the full picture.
the one missing piece is to compare times at the same position): Let's imagine I've
put a lightbulb at every point in space (even in between where I can attach them to the
time globe) and I turn them all on simultaneously at one time, and then turn them off simultaneously
a little bit later.
From your moving perspective, any particular one of my lights will have its on-off time
interval dilated, of course, but at any particular location in space (like, where you are), the
duration of time between when the lights go on and when the lights go off will actually
be shorter.
Maybe it should be called “duration contraction”!
So, to summarize, when changing to a moving perspective in our universe, there's both
time dilation and length contraction, but there's also distance dilation and duraction
contraction.
These four ideas said aloud as words certainly sound super contradictory and impossible (I
mean, how can the time be both shorter and longer?!), but if you have a spacetime globe
it's easy to understand there are no paradoxes or contradictions – we simply need to be
more careful with our ideas of distance and time intervals when applied to extended objects
in spacetime: do we mean the time between two exact events (time dilation), or the time
between the versions of those events that happen at the same place (duration contraction)?
Do we mean the distance between two events regardless of when they happen (distance dilation),
or the distance between the versions of those events that happen at the same time (length
contraction)?
This is subtle stuff, and words and equations by themselves make these concepts really really
hard to keep straight; but a spacetime diagram doesn't lie.
To get more experience with time dilation and length contraction yourself, I highly
recommend Brilliant.org's course on special relativity.
There, you can do problems that build off what you learned in this video and explore
real world scenarios where it's important to take time dilation and length contraction
into account, like the famous Michelson Morley experiment.
The special relativity questions on Brilliant.org are specifically designed to help you go deeper
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.