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.
