You probably know that moving clocks
take slower compared to the ones that
are at rest. Or that clocks that are
closer to a black hole take slower
compared to the ones that are far away.
But there's something even more
important. These are both wrong. I kid
you not. This turns out to be one of the
most misunderstood ideas in all of
relativity. Clocks never tick slower.
Time never slows down. When I first read
about this in a book called Exploring
Black Holes, a highly suggested book on
general relativity, I had the my whole
life was a lie kind of moment. I mean,
what do you mean? This is wrong. Isn't
this at the heart of relativity? And if
time never really slows down, what
exactly is time dilation? It felt like I
was back to square one. Nothing made
sense until now. After going through
that book thoroughly, oh my god, I think
I finally get it. I understand that this
is a big misconception. I've had it for
a long time as well. So by the end of
this video, you and I will move away
from this metaphors and we'll step by
step rediscover from scratch purely
intuitively how physicists actually
think about time dilation. So if you're
ready for this, let's begin. So where do
we start? Well, let's talk to Archable
Wheeler, the co-author of that book,
Exploring Black Holes. And if you don't
know who that is, he's a person who
coined the term black hole. So legendary
guy. All right, Wheeler. Why is it wrong
to say a moving clock runs slower?
Wheeler says because Einstein's
relativity never said that. Wait, what?
Yeah, instead it says you will measure a
moving clock to run slower. And at this
point, I'm disappointed. What? Isn't
that the same thing, Wheeler? What's the
difference? And Wheeler says, "Oh, it's
a big difference." In fact, let me give
you an example. You and your friend are
moving side by side in a car. You glance
over and see yourself overtaking her.
So, you measure her car to be slower.
Does that mean that her car is slower?
And I say, "Of course." That's how
normal people talk. We don't say, "I
measure your car to be moving slower."
We say, "Your car is slower." And
Wheeler says, "Okay, but what if both of
your speedometers read exactly 100 m an
hour?" In fact, what if she sees herself
overtaking you and she measures your car
to be going slower? Then what? Wheeler,
what nonsense is this? You're saying
both of our cars are going at 100 miles
an hour and yet we both see ourselves
overtaking the other person. How does
that make any sense? Whether says,
"Let's look from the top. If both of
your cars were actually going exactly
parallel to each other, yeah, then it
doesn't make any sense. But what if your
cars were having an angle? Now look,
even though both of your cars are moving
at exactly the same speed, from your
perspective, you're overtaking her. You
measure her car to be slower. But from
her perspective, she's overtaking you.
She measures your car to be slower. So
look, Moes, right in front of your eyes.
Just because you measure the other car
to be slower doesn't mean it is slower.
So what's going on over here? Well, both
of you are judging the other person's
motion from an angle. That's why you see
the other person's car to be going
slower even though it's not. Your
perspectives are tilted. Isn't that
incredible? Wheeler says, "Your friend
and you measuring the other person's
clock to be slower are exactly like
these two drivers." I mean, the
situation is so incredibly analogous.
So, just like the car speedometer, both
of your clocks are ticking at exactly
the same rate. Time is passing at the
same rate. Now in the cars example, you
measured the other person to be slower
because you're measuring each other's
motion from an angle in 2D space.
Similarly, here you're measuring each
other's clock to be ticking slower
because you are measuring the motion
from an angle in fourdimensional space
time.
So if you want a full intuition of
what's going on, just like before, we
need a top view, a space-time top view.
we need to draw a space-time diagram. So
let's do that. So let's bring them
together and draw a space time graph. So
over here the horizontal represents the
space and the vertical represents the
time. Let's choose a reference frame in
which they're both moving away from each
other at exactly the same speed. It'll
make things super analogous to our cars.
Okay, so here's what it will look like.
Look, they're going away from each other
at the same speed. This is called their
world lines. And now look at how
analogous the situation is with respect
to the cars. Oh my god. Over here the
cars are moving at exactly the same
speed through space. Here both of us are
going at the same speed through
spaceime.
Here both the cars have traveled exactly
the same distance in space. So let's say
100 miles. Meaning if their autometers
reading was zero here, both of their
autometers over here will be 100. Right
here both of us have traveled exactly
the same distance in spaceime.
Now in spacetime this distance is called
proper time. It's a measure of how much
we have aged or how much our clocks our
personal clocks have ticked. So for
example we can say 3 years. Both of us
have aged exactly by 3 years. So if both
of our clocks were in sync over here and
say it was zero then here it would read
three years. Our clocks are like the
odometers over here. So look clocks are
ticking at the same rate. Time passes at
the same rate for everyone. So just like
how here we are measuring each other's
motion from an angle and that's why we
measure the other person's car to be
going slower. Look, we are measuring
each other's person's motion through an
angle in for spacetime. That's why we
measure the other person's clock to be
ticking slower. Even though clocks are
ticking at a normal rate, time is
passing normally.
Now, this is already starting to blow my
mind. I'm I'm starting to get some
beautiful intuition. got wheeler over
here. I was able to draw my line of
sights and then rotate to look at things
from my perspective and I could clearly
see why I measure her to be behind me
and similarly why I me she measures me
to be behind her. I could I could do
that. Can we do the same thing over
here? Is it possible? And Wheeler says,
"Yeah, we can do that. that we can draw
line of sights and we can rotate and we
can measure from each other's
perspective and this will be super clear
and I'm like all right let's do it let's
let's let's get that intuition okay so
are you ready for it all right let's do
this let's get rid of this let's draw a
line of sights in spacetime and let's
rotate oh my god this
wait what the heck is this
I don't understand what's going on stop.
What is this? And Wheeler says, "Oh, oh,
I forgot. Mahes, you don't have
intuition for space-time rotation,
right?" And I'm like, "What the heck is
a space-time rotation?"
Wheeler says, "All right, all right,
Mahes, let's take a step back. Let's
come back to the cars over here. When we
rotate over here, we follow a circle,
right? Why? Why does rotation over here
follow a circular path?" And I say well
that's what rotation is you know
rotation is circular right and ver says
no it's got something to do with the
geometry what do you mean in normal
space the distance formula is given by
the pythogoras theorem right and we know
that x and y displacements are arbitrary
depend upon your graphs and how your
axes are tilted but the value of this
distance is absolute it's an invariant
it doesn't matter who looks at it
everybody agrees that This is for
example 100 miles. And so look this is a
constant and so we have x^2 + y square
is a constant which is a circle. That's
why the rotations are circular.
In fact think of it this way. If my axis
tilts such that x value increases then
the y-value has to decrease so that the
sum stays a constant. And that's why if
x value increases the y value decreases.
Similarly, if the x value increases
here, the y-value must decrease in such
a way it traces a circle. That's why
rotations are circular.
Okay, that kind of makes sense. Now,
what about in space time? The distance
formula in spaceime is given very
analogous to Pythagoras theorem, but
with a minus sign. And if you're
wondering where that minus sign comes
from, we've talked about that in great
detail in our previous video. I'm not
trying to fish for views over here, but
I want to tell you that this doesn't
come just like that. Um, it can be
rediscovered from scratch step by step.
And I've done that in a previous video.
So if you need that intuition for that,
feel free to check that out.
But for the purpose of this video, we're
just going to borrow that that the
distance formula in spaceime very
similar to Pythagoras, but with a minus
sign. So Wheeler says ma look just like
over here the t value and the x value
are arbitrary they depend upon reference
frame your how your axis are tilted but
the value of this distance proper time
that's an invariant so this time our
equation becomes t ^2 - x² is a
constant. What is that equation? That is
a graph of a hyperola a rectangular
hyperola. Again to get some more
intuition, think of it this way. This
time if x value increases, the t value
should also increase so that the
difference stays a constant. So if the x
value increases, the t- value increases.
And if the x value increases here, the
t- value increases. And so we now get a
hyperola.
So in spacetime, the rotation must
follow the tip of a hyperola.
That is the difference between rotations
in normal space and rotations in
spacetime.
And this is super non-intuitive to begin
with, but once we get some practice with
this, oh boy, everything is going to
fall in place. Okay, but wait a second.
I have a question. You can draw lots of
hyperbolies, right? Hyperboly. Which
hyperbola should we draw over here?
Wheeler says, excellent question.
Remember that the world lines can have a
maximum angle of 45°.
This represents the speed of light.
We're assuming the speed of light is one
giving you an angle of 45°. Since you
cannot go beyond the speed of light,
these hyperboly cannot go beyond the
45°ree lines. In other words, these
45°ree lines form the asymptotes or
tangents to the hyperbole at infinity.
And so only the hyperboly which have
these 45°ree asmtotes are the ones that
we draw over here. And so look our
distance formula in spacetime represents
a family of hyperboly with vertex at the
origin. And similarly the distance
formula in space represents a family of
circles with the center at the origin.
Look at how analogical situation is. Oh
my god.
If you were to draw lines from here all
the way to this particular circle, they
all have the same length because they're
all the radi. So in this particular
case, we can say they all are 100 miles.
And therefore, this circle represents a
100 miles circle, right? The same thing
applies over here.
All the lines drawn from this point onto
the hyperbole are of the same length in
space time. And I say we but they don't
look the same length. They look
different. And Verer says that's the
whole point. This geometry is non-
ukidian. You can't draw non- uklidian
geometry on paper or on screen. It gets
distorted. That is what makes it super
non-intuitive, but at the same time
super duper interesting. But the analogy
is so so amazing that I can use this to
derive insights. I can now say all these
lines even though they look different,
they must all have the same length. And
when I say length, I'm talking about
proper time. So they all for example in
this case have three ears. These lines
are all three years long. And so we can
say this is the hyperola of 3 years. And
so therefore just like how while
rotating over here we follow the tip of
a circle because they all represent 100
miles. While rotating we should rotate
along the hyperola because they all
represent three years. Does that make
sense?
Okay. So, are you ready for the
rotations now? Let's do it slowly. Here
goes. Just like how I have now gone into
the perspective of the red car. I am now
looking at things from my reference
frame. Similarly, we can go back and go
into her reference frame. Similarly, we
go on into her reference frame over
here. Isn't that amazing? So, we've
gotten the first thing down. We now
understand how to do rotations.
But now comes the second thing, line of
sight. How do you draw line of sight?
Well, look, the line of sight is drawn.
Let's get clear all of those things. The
line of sight is drawn tangent to the
circle.
Similarly, over here, we're going to
draw a line of sight tangent to the
hyperola.
But we have a question. Wait a second.
Over here, I know what line of sight
means in space. But what is the meaning
of line of sight in spacetime?
Vither says, great question. Let's go
back to the analogy. So if I were to
look at my line of sight, for example,
what this represents is that all the
stuff that is in front of it, that is
something that I'm yet to cross, right?
I'm continuously moving. So I'm yet to
cross them. So in some sense, they lie
ahead. They lie in my future. And all
the stuff behind, well, I've already
crossed them. So they lie behind me.
They are in my past. And all the stuff
that is right along this line of sight,
they are currently I'm passing them
right now. So that represents my present
or simultaneity.
Wait, why is it called simultaneity?
Well, because all the stuff that is
there right now, for example, if there
are two barrels right now, they're in my
present because I'm currently crossing
them and I'm crossing them at the same
time simultaneously.
So the line of sight really represents
the line of simultaneity. It tells from
my perspective what is future, what is
past and what is simultaneous.
Now guess what? The same thing applies
over here. All the events remember this
is spacetime. So every single point is
an event. So all the events above this
line represents my future. All the
events below represents my past. And all
the events along this line line
represents my simultaneity. So if there
are two supernova explosions over here
in my perspective they are simultaneous.
Now I may not see them simultaneously
because this one is closer. So its light
reaches me before the light from here.
But if I were to do all the calculations
I would eventually conclude, hey these
two happen simultaneously in my frame.
But remember her axis is tilted with
respect to mine. So her definition of
future, past and simultaneity is
different. And therefore, for example,
she will disagree that the two barrels
are simultaneous. She will say, "She
passed through this barrel first, then
she passed through this barrel."
And now imagine how weird their
conversation would be if they didn't if
they couldn't perceive this angle. They
might be thinking they're moving
parallelly. And so one person says,
"Hey, I pass both the barrels at the
same time." The other person says,
"Uh-uh, I pass one barrel first and then
the other one." And they're wondering,
"How is that possible?"
Relativity of simultaneity. But we
understand from the top view. Hey, come
on folks. This whole simultaneity
business is relative. What's important
is that these two barrels are there on
space. Now, whether they are
simultaneous or not depends on your
tilt, depends on the direction in which
you're moving obviously, right? The same
thing applies over here. My friend's
simultaneity axis is tilted. So her
definition of future past and what's
simultaneous is different. In fact,
look, she says the two supernovi
expressions are not simultaneous. This
happened first. It's already in our past
and now this is happening. And so we
disagree on what's simultaneous and
what's not. And we make a big deal out
of it thinking that oh my god relativity
of simultaneity all non-intuitive.
But it's only non intuitive if you think
in terms of space and time separately.
If you were to look at the top view, it
makes perfect sense. The two supernovi
exists on spaceime. That's a fact. Now
whether you call them simultaneous or
not depends upon the angle in which
you're moving in spaceime and based on
that how your simultane lines are
tilted. I don't know about you, but
doesn't this make so much sense? Oh my
god.
So long story short, the lines of line
of sight really represents the line of
simultaneity. And Wheeler says, "Mahes,
so now you have the full picture. You
have the full context. Can you look at
this diagram and see what each one
measures the other clock is doing?" And
I'm like, "Yeah, we can." I mean, look
at this. From my perspective
right now at my present you know it's 3
years but my line of simultane cuts
through her past
and that's why when I measure her clock
I will say it's less than 3 years maybe
about two years so you can see right in
front of your eyes the reason for that
is not because her clock is ticking
slower the reason is because my axis of
simultaneity is tilted My present cuts
through her past. It is so analogous,
right?
And the same thing applies to her as
well from her perspective. Look, her
simultaneity cuts through my past and so
she measures my clock to be behind her.
It's all about our tilts in
simultaneity. That explains it all. So
now let's do a rotation and see from my
perspective her clock is behind and from
her perspective my clock is behind. I am
getting goosebumps over here.
So now Wheeler asks Mahes let's go back
to your original statement. Saying a
moving clock runs slower is equivalent
to saying a car's odometer measured from
an angle runs slower. Do you agree with
this?
No, I don't agree with this. Sure, when
you measure a car's motion from an
angle, you will measure it to be slower.
But saying that it's autoometer is
running, that's absolutely wrong. No,
you measure it to be slower because you
are it's it's its geometry. You're
measuring it from an angle. But saying
its odometer is running slower with
respect to me or its wheels are spinning
slower with respect. No, none of that is
happening. This is wrong. The same idea
holds here, says Wheeler. When an object
is moving in space, what's really
happening is that you're measuring its
motion from an angle in 4D spacetime.
And because our simultaneous are tilted,
we measure the other person's clock
behind us. But it would be wrong to say
that their clocks are running slower.
Absolutely wrong.
And I hope you agree now, the same thing
applies to gravity as well. Black holes,
for example, don't slow down time or
make clocks run slower. No, all it does
is it curves spacetime and that affects
what we think is simultaneous close to
it. At the end of the day, it's a
comment on how simultaneity is changing.
It's a comment on geometry.
That's why these are useful metaphors
for beginners who don't want to think
about hyperbolic geometry. But man, this
is so wrong. It causes so much
misconceptions. We are so much better
off with this picture. It is the most
accurate way to think about time
dilations. It's how physicists actually
think about it. And once we get
comfortable with hyperbolic geometry, it
starts making intuitive sense. This is
the way. This is the way. By the way,
now that we're thinking about relativity
geometrically and not as time slowing
down, all kinds of time related
misconceptions go away. For example, a
common misconception is that, hey, if
you travel at very high speeds, you age
slower, you live longer. Wrong. Time
never slows down. Clocks never tick
slower. If you go close to a black hole,
you age slower. You will live longer.
Wrong. Time never slows down. Clocks
never tick slower. As you enter the
black hole, the rest of the universe
speeds up. And so you see the entire
future history of the universe behind
you. Wrong. Clocks never tick slower or
faster. Time never slows down or speeds
up. All of those things go away. And the
most notorious paradox, the twins
paradox goes away as well. I mean
earlier we would think that hey if my
twin were to go very far away and then
come back I am at rest and since moving
clocks run slower her clock must be
running slower. She must be aging slower
and so she must be younger. But the same
thing applies from her perspective. From
her perspective she is at rest and the
rest of the universe moved away and then
came back. So it must be the rest of the
universe. their clock must have been,
you know, slower and therefore I am the
one who's younger. Paradox. But now it
goes away because we know this statement
itself is wrong. Clocks never tick
slower. Time never slows down. Boom.
Twins paradox goes away.
But wait,
I have a problem.
If time never ticks slower, clocks never
slow down, then we would predict both of
them to be of the same age, right? But
we know that's wrong. Relativity
predicts that she would be younger. How
does that make any sense, Wheeler? So,
description. All right, happy learning. Back to the video. So, Wheeler, how can
people age differently if time passes at
the same rate for everyone? And Wheeler
says, great question. Let's go back to
the car analogy. And I love it. This car
an energy has all the answers. Okay, so
we have two cars moving at the same
speed. That's the point. Moving at the
same speed, taking this path. What if
they decide to meet up at some point,
say over here? So the question is if
their autometer reading is zero over
here then what will be their autometer
reading when they pass through this
point will they be the same or
different? And I say that's a
straightforward question. The blue car
would have a higher odometer rating. It
has traveled a longer distance. But
Wheeler says Mahesh they're both
traveling at the same speed. Their
autometers are running at the same rate
throughout the journey. Then why does
one accumulate more autoometer reading
compared to the other?
Because they take different paths.
Exactly. If you take different paths
from one location to another, you will
accumulate different autoometer
readings. Right? The same thing happens
in space time. If you go back to your
twins uh your the twins case then look
between two events of separating and
then eventually meeting they both take
different paths through spaceime just
like over here when you take different
paths you'll end up with different
autoometer readings. If you take
different paths over here you'll end up
with different autoometer. In other
words your proper time readings.
Therefore, by taking different paths
through space time between two events,
you can end up aging differently.
Okay, that makes sense. All right, so
now let's dig deeper. Wheeler asks,
which of them would have a higher roter
reading? You said I said this one,
right? Why? Because this path is longer.
How do you know that? Come on. This I
can see it. No, the point is that over
here things might be intuitive, but over
here it's not. So, you need to find a
really elegant way to convince yourself
that.
Okay, here's here's how we can do that.
I can draw a line over here because it's
symmetric about this point and compare
these two paths. If I draw a circle
because we live in a circular geometry
over here, then I can immediately see
that hey, look, this path is longer than
this path by this much amount because it
sticks outside the circle. And similarly
I can say this path is longer than this
path. So this total path is longer than
this path and therefore this path is
longer. Boom. And now Wheeler says
Mahesh perfect moment of truth. Can you
apply the same thing over here and
convince yourself of which path is
longer and which is shorter. Instead of
circle you have to draw hyperola. Great
idea to pause the video folks. Moment of
truth. Please try.
Okay. We divided this into two paths,
we'll divide this into two paths as well
and only focus between these two path
lengths. And to do that, instead of
drawing a circle centered here, we're
going to draw a hyperola centered over
here. How do you draw a hyperola? Well,
first you draw your light cones because
that you know light paths because 45°
that will be your asytotes and then you
draw a hyperola and then we'll get rid
of them because that's not necessary.
Okay. So now we can look at this and we
can say hey this path is shorter than
this one by this much amount. Can you
see? Because remember all distances from
this point to the hyperola are equal.
But this one is shorter
and therefore this path is shorter than
this one. And of course the same logic
applies over here. And as a result,
look, the traveling twin took a shorter
path and therefore she accumulated less
distance which is less proper time.
Therefore, she is younger.
Even though clocks are ticking at the
same rate, time is passing at the same
rate, she took a shorter path through
spacetime between these two events.
That's why she is younger. Oh my god.
This is it. This is how you think about
time dilation.
And Vera says, "Now I know you have a
question about her perspective. We'll
get to that. But before we do this, we
can now come up with a principle. See
over here between any two points, if you
take a straight line, that is the
shortest path." Right? If you take any
bent path, that is longer. more bent
longer the path becomes. The same thing
applies over here but in reverse. The
straight line between any two events
that path is the longest path. If you
take a bent world line that path becomes
shorter. If you bend it even more that
path becomes even shorter.
So over here we can say straight paths
is the shortest distance between two
locations. What can we say here? Well
straight paths is the longest distance
between two not location events.
Why longest? Because of the non-
ukidian. That's the weirdest part over
here. The non-ucukidian geometry the
negative sign that we saw earlier in the
Pythagoras theorem.
But what exactly is a straight path over
here? What is the definition of
straight? Well, over here we can say
something is straight. If you're driving
in such a way you don't have to turn
your steering wheel, then you know that
you're taking a straight path. That's
the best way to define straight. Over
here, over here, what is the definition
of straight? When do we say a world line
is straight? If there are no forces
acting on you, if you're not
accelerating, if you have an inertial
path. So we can say inertial path will
always accumulate what is distance
proper time. So inertial path will give
you maximal proper time or maximal aging
between two events.
And so the reason why the earth twin was
is is aging more is because he took that
inertial path from this event to this
one. But the astronaut took a
non-inertial path, a shorter path. So
she aged less.
But we learn. Now let's come to the main
argument. Her path is bent from my
perspective. That's why she's taking a
shorter path appearing younger to me.
But wait, can't I say from her
perspective, she is traveling a straight
world line and it's my path that is
bent. Can't I do that? And so I would
end up being younger. The paradox
remains. Wheeler says, "Okay, let's come
back over here." From the red car's
perspective, it took a straight path,
but the blue one took a longer bent
path. So, can I now say from the blue
car's perspective, it took a straight
path and the red one took a longer path?
No. No, you can't say that. No. I mean,
it's an objective fact that the blue car
took a longer path, right? Like, it
doesn't depend upon your reference
frames. The blue car has traveled more
distance. That's it. That's an objective
fact. This doesn't work. This doesn't
work. And so says in the same way, this
doesn't work.
The reason why it doesn't work is
because those transformations are not
allowed. You can't rotate like that. The
only rotation that's allowed over here
is circular. So if you do a circular
rotation, look, whatever you do, you
can't make that bent path straight. The
fact that this path is bent is an
objective fact. So it doesn't matter how
rotate how you rotate, you'll always end
up with the same fact that this path is
longer. The same thing applies over
here. You can't do any rotations. You
are only allowed to do hyperbolic
rotations over here. And when you do
those hyperbolic rotations, you will
always find that it's this path that is
bent. And therefore, and you'll always
find that the red one is traveling
straight inertial path. So it will
always have maximal aging and the
traveling twin would be younger.
Okay, Wheeler. But wait, if I were to
put a camera on the blue car and follow
it, I could in principle say that it is
taking a straight path and the rest of
the world is the one that's turning,
right? I could say that, right? Yeah,
you can do it, but the outcome will not
change. Why not? I mean, I know it
doesn't, but like why not? Let me see
it. Says, let's do it. So, let's look at
an animation. Okay. So, from the red
car's perspective, this is how the
animation looks like.
Okay. Now, let's focus on the blue car
and let's see what the animation looks
like from here.
Okay. Since the camera is is there on
the blue, we'll assume that the blue car
did not turn is the rest of the world
that turned. So, boom. This is what it
looks like. And look, when the rest of
the world turns, this car teleports from
here to here. Of course, it's not really
teleporting. It's just that the rest of
the world turned. And so now the
animation continues. And sure, by doing
some hacks and mid rotations, you can
make this world line straight. But then
this world line gets cut and the outcome
is the same. This path is longer and
this path is shorter. That fact does not
change whatever you do.
The same thing happens in spacetime. You
can try to make this world line straight
by doing mid-level rotations and
stitching them together like we did in
this case. But then this will get
broken. And whatever you do, the fact
remains that this world line will be
shorter than this one. It's an objective
fact that the red car did not turn their
steering wheel. It's an objective fact
that the blue car turned their steering
wheel. That does not depend on reference
frames. Same is over same is the case
over here. It's an objective fact that I
took a straight world line. I took an
inertial path and she's the one who
accelerated over here. That fact cannot
go away by doing any kind of
transformations. And that's that's how
you know your paradox gets resolved.
Okay. The final question we could be
having is all of this is in flat
spacetime, but what if we go to curve
spacetime? Wheeler, it looks like things
are going to get super complicated,
right? Ander says actually no come back
to the cars over here. Okay. If this was
not a flat space but it was say curved
there was a hill over here or something
like that. The principle still applies.
The person who does not turn their
steering wheel ends up taking the
shortest path and the one who does turn
the steering wheel takes a longer path.
That principle works even when there is
curvature. Right? The only thing that we
need to modify is we need to be careful
about the word straight. Straight
doesn't make any sense anymore because
your world itself is curved. But that's
okay. Our definition of straight is that
your steering wheel does not turn. As
long as you have a path in which the
steering wheel does not turn between two
locations, that is the shortest path. We
call this path geodisics. So we say the
geodisics is the shortest distance
between two locations even in curved
space. Guess what? The same thing
applies over here. Even if this was
there was curvature over here the same
thing would apply. Inertial paths it
will not not look straight. It might get
distorted but the path that is inertial
the path in which there is no forces
acting on it will end up having maximal
aging between the two events. So you can
see why physicists love this way of
thinking because it doesn't matter
whether you're in flat spacetime or
curve spacetime. This principle applies.
is you don't have to think about
gravitational time dilation, inertial
chyntic time dilation. This is a single
principle that applies in general. But
there's a second effect that again we
can see from here.
If you take the distance between two
points on a flat space, notice what
happens to the distance when we add
curvature. It becomes longer. Even
though the car has not turned its
steering wheel, it's going straight. the
path length becomes longer because of
the curvature. Right? So your odometer
will end up accumulating more distance
when it's passing on a curved space.
The same thing applies over here except
it'll be the opposite. If your world
line passes through a strongly curved
space-time area,
your autoometer will accumulate less
proper time. Again, why less? because of
the negative sign.
Which means let's say that there was a
black hole somewhere over here. Let's
say the black hole is far away from the
earth. So it doesn't affect me much. But
the traveling twin comes very close to
the black hole and then goes away. What
happens now? Well, first of all, we know
that that this traveling twin has taken
a non-inertial path to begin with. So
that's a shorter path to begin with. But
now that she goes close to a curved
spacetime, curvature reduces her proper
time. Doesn't increase. It's the exact
opposite. Just like curvature increases
the distance, the curvature over here
reduces the proper time even further.
And therefore now when she comes back,
she comes back even younger
than in the flat space-time um case. So
that's why when you travel close to a
black hole and come back, you end up
becoming super young, not because your
clock ticks slower or anything like
that, but the effect of curvature is
that you end up accumulating even
smaller proper length, proper time along
that path. And again, if you ever get
confused, come back over here. Just like
how when a space gets curved, it adds
more distance. Similarly, when spacetime
gets curved, it reduces your proper time
for the world lines that are close by.
That's that's basically how it happens.
So, putting it all together, how do you
think about time dilation in the most
accurate way without resorting to
metaphors like time slowing down or
clock ticking slower? Well, first start
with the familiar space. If you want to
reach some location ahead of you, you
can take a straight path that is the
shortest distance or you can take any
other bent path curved paths which will
give you longer distance. Right?
Similarly, if you want to reach a future
event in spaceime, um an event could be
something like being on earth one
revolution later. Okay? You can take
different paths. If you don't do
anything, just sit quietly take an
inertial path to that event, you will
age the maximum and that happens to be
about a year. Since most of us take an
inertial path, most of us will end up
aging by a year when one revolution
happens. But relativity teaches us you
don't have to take that path. You can
take noninertial paths like that of the
traveling twin over here. You can go
very far away and accelerate and bend
your world lines. All by going close to
a region where the spacetime is really
curved like a black hole or a neutron
star. Both of this will shorten your
space-time path and it'll allow you to
reach that future event quicker. That's
what relity teaches us. You can reach
those future events quicker. And so, for
example, you can reach that future event
say maybe in not one year but maybe in
just 3 hours of your personal time. And
when that happens, you would have aged
only by 3 hours, but the rest of them
who took inertial paths would have aged
by a year. You end up younger than them.
Not because your time was ticking slower
or not because you are aging different
at a different rate, none of that. But
because you took a shorter path through
spacetime to reach the future event and
those all those other people took a
longer inertial path.
That's how you think about time
dilation.
I'll see you