In middle school geometry, I learned
that the distance between any two points
in space comes from the Pythagoras
theorem. But turns out in space time,
the formula for the distance between two
points is almost the same except with a
minus sign. Why is there a minus sign?
What's going on? The answer that I most
commonly got was that, hey, that's
because space and time are fundamentally
different. Yeah, that didn't explain
anything. So, I looked into the math,
which turns out to be equally
unsatisfactory. So, I thought I would
never understand. I would never get a
true intuition behind what's going on
over here until now. After combining the
insights from these two incredible books
on relativity, my god, it clicked. It
made perfect sense.
So, in this video, we're going to
pretend we don't know anything about
spacetime or this minus sign or whatever
and start from scratch. And not only
will we rediscover all the ideas
intuitively and visually like we always
do, but by the end of the video, you
will find it obvious that the distance
formula in spacetime must have a minus
sign. Otherwise, the concept of time
itself wouldn't make sense and our
universe would cease to exist.
So, if you're ready for this, let's
begin. So, Einstein, where do we start?
Einstein says, "Mahes, a space shuttle
is traveling from Earth to Proxima
Centauri at say about 80% the speed of
light and a light beam passes by. What
is the speed of light with respect to
the ship?" We may already know the
answer to this, but if you want to
understand things deeply, we have to
pretend we don't know anything and try
and rediscover things from scratch. So,
let's use the good old Newtonian
mechanics.
If I wait for 1 second on my clock, the
shuttle travels.8
C. Let's make C equals 100 units. So,
the ship travels 80 units. Meanwhile,
light itself travels 100 units. So,
after 1 second, light is only 20 units
ahead. So, the speed of light relative
to the ship is just 20 units/s.
But Ein says, Mahesh, this creates a big
problem. I'm like, what problem?
Remember in physics being at rest and
moving with a constant velocity are
exactly the same things look from
different perspective. That's the
cornerstone of Newtonian relativity.
No experiment can ever tell you whether
you are really at rest or really moving
at constant velocity motion because that
distinction is meaningless. And I'm like
yeah I get that. Good. Now think about
this. If the people on the ship measure
the speed of light to be anything
different than C, then the laws of
electromagnetism is different from them.
We have now an experiment that gives you
a different result because you're
moving. So they would know that they're
truly moving. And that breaks the
Newtonian relativity.
Now, if you aren't convinced, and you
shouldn't be, and if you have more
questions, and you should, then our
previous video digs deep into it in
excruciating detail, okay? So feel free
to check that out. But long story short,
the speed of light must be the same for
all inertial observers.
Okay, but then how does it work out over
here, Einstein? And Ein says, well,
Mahes, what if moving things shrink in
the direction of motion? Then the ship
and everything inside, including their
rulers, would be shorter compared to
ours. And so they would now measure a
distance that is larger than what we'd
expect.
And similarly, what if moving clocks
ticked slower? Then their clocks would
be ticking slower compared to ours,
which means less than 1 second would
have passed for them.
So look, Mahesh, light covers a much
larger distance in a much smaller time
in their perspective. So with the right
math, the speed just comes out exactly C
for all inertial observers.
That's how Einstein got lane contraction
and time dilation. Incredible, right?
But wait, Einstein, are you saying that
the ship is physically shrinking and
that their clocks are somehow physically
slowing down? And Einstein says, "No,
that's an old, outdated idea."
The physicist Henrik Lawrence had
already written down these effects
mathematically. That's why these are
called Laurens transformations. But his
motivation was quite different. He like
many others at the time believed that
light travels through a medium called
ether and it filled the entire universe.
But there was a problem. See, if ether
fills the entire universe, then moving
through space should produce a backward
ether wind. And Earth is moving through
space. So, we should be able to detect
ether wind, too. But even the most
sensitive instruments couldn't detect
it. So, Lawrence argued there is ether
wind. It's just that the wind alters the
properties of materials and physically
makes them shorter and slows down our
clocks just in the right way to hide
ether. So that means Lawrence's
motivation for these transformations
were purely to save ether theory, not to
revolutionize physics or anything. But
Einstein says, hey, claiming that
something exists but you cannot detect
it is not very scientific. If you can't
measure it, you can't detect it, then
maybe it doesn't exist.
In his 1905 paper on relativity, look,
he writes, "The introduction of a
luminopheris ether will prove to be
superfluous." I love the words they use.
So Einstein did what nobody dared to. He
said, "Let's get rid of ether entirely
and deal with the consequences." And
what's the consequence? These effects
aren't physical distortions at all. Just
like how motion and velocity are
relative, observer dependent, space and
time themselves must be relative. That
was the paradigm shift. But what does it
really mean for space and time to be
relative?
Einstein says, look, from our reference
frame, this shuttle covers about four
light years in 5 years. But someone
moving relative to us will find that the
ship only covered 1.8 lighty years for
example in just 3 and 1/2 years and the
people inside the ship they will record
something else all together. And the
wild part is we are all correct. So look
space and time are relative to
observers.
But Einstein why why does our universe
behave this way? And Ein says, well,
let's look at what he actually wrote
here. He writes, these two postulates
suffice for the attainment of a simple
and a consistent theory of the
electronamics of moving bodies based on
Maxwell's theory for stationary bodies.
So basically Einstein says speed of
light is the same for all and space and
time are relative and that's that. There
is no deeper explanation
or is there?
This is where Einstein's old math
professor from Zurich, Herman Minowski,
comes in. He looks at his students paper
and says, "Hm, maybe there is a deeper
explanation behind this." And I'm like,
Minowski, what's the deeper explanation?
Minowski says, "Let's suspend a toy
space shuttle in a square frame with
lights casting shadows on the horizontal
and the vertical walls." Okay, Mahes.
Now, what would happen to the shadows if
the frame tilts?
Well, the shadow links will change
obviously, but what if there was a
shadow person who thinks their universe
is just 2D and these shadows are real
objects? What would they conclude?
Oh, see where is going with this? They
wouldn't notice the frame tilting.
They would just conclude horizontal and
vertical lengths are relative.
And Vinci says exactly. They would talk
about dilation and contraction and
whatnot, but you and I see the full
picture. The world is 3D and the
shuttle's length has not really changed.
It's only the frame that tilted changing
the shadow lengths. It's just geometry.
So, Mowski asks, "What if space and time
are just like the horizontal and the
vertical walls? What if the lengths and
the times we measure are just shadows of
a deeper reality of fourdimensional
spaceime? Then we can explain link
contraction and time dilation and all
the effects of relativity geometrically.
Oh man, this is so incredible. And this
is why Minskowski famously said, and I
quote, "Henceforth, space by itself and
time by itself are doomed to fade away
into mere shadows, and only a kind of
union of the two will preserve an
independent reality." Gosh, these quotes
are epic.
Okay, now you and I might still be
wrapping our heads around what's going
on over here, so let's take a step back.
I mean usually when we take a relativity
course we just jump straight into
spacetime take it for granted and start
solving problems and stuff right but
look at the journey how it all unfolded
I mean it first started with Einstein
making a big leap big conceptual leap
about rejecting absolute space and time
then his professor Mowski took a second
leap probably even bigger rejecting
space and time altogether as just
shadows of a 4D space-time geometry this
is why I love to learn and rediscover
things all by myself. It's so enriching.
And one way to do that for free is by
the QR code. All right, happy learning. Back to the video. So, how do you think
Einstein reacted to his professor's uh
idea of space-time geometry? Well,
apparently he made remarks like, "Since
the mathematicians have invaded the
theory of relativity, I do not
understand it myself anymore. These
mathematicians with their
fourdimensional space are getting on my
nerves."
Yeah, Einstein and probably a lot of
others were not really a fan. They
thought this was absolutely unnecessary.
So, Mowski, your idea of spacetime is
radical, but at this point it's just an
analogy. What convinces you that it is
the ultimate reality? And Mowski says,
"Mah, answer this. Do you think that
these shadow people could somehow figure
out that they are just seeing shadows,
2D shadows of the actual 3D object?" And
I'm like, "No, because they're confined
to their 2D universe, right?"
No. What if Menowski says, "What if they
looked carefully? They'd notice
something strange. Whenever the
horizontal shadow shrinks, the vertical
one stretches."
And maybe a clever shadow mathematician
comes along and discovers that if you
square their lengths and add them up,
that total always stays the same
regardless of the of how the frame
tilts. It would be an invariant
that would make them go, "Oh my god,
this can't be just a coincidence. Maybe
we are just looking at 2D projection of
an underlying 3D world." So, Mowski
asks, "Mah, what if space and time could
be combined in a similar way to get an
invariant, something that everyone
agrees on? Then we too would be like,
"Oh my god, that can't be a mere
coincidence. Maybe we are just looking
at 3D projections, shadows of an
underlying 4D world, spacetime."
Okay, this is interesting. We're getting
somewhere. But how do we discover this
invariant of the supposed 4D spaceime?
Well, let's get back to the analogy.
This time, what if we just hung two Lego
bricks? If they're far apart, just like
before, tilting the frame changes the
shadow distances.
But if they're at the same spot, then
the shadow distance is always zero.
Tilting doesn't change it anymore.
So look, if both the horizontal and
vertical shadow distance is zero in one
frame, then they are always zero.
So Moes, what do you think is equivalent
of this for spaceime?
And I'm like, well, this means if we
have events that are at the same place
and the same time, then everyone would
agree on it.
Exactly, says Minowski. If two bombs
exploded at different places, one after
the other, those events are separated
both in space and time. But when you
change frames, our theory is that the
space-time axis tilt. That's why those
separations stretch or shrink. So our
friend in another frame may see the
explosion happen much closer in space
and time. In fact, she might even
conclude the time separation is zero.
For her, the explosion is simultaneous.
But if both bombs go off at the same
place and same time, in that case the
separation in space and time is zero.
Tilt the frame. however you want. Zero
stays zero. So, everyone agrees on that.
Events that happen at the same place and
time are absolute.
Awesome. So, we now have something that
everyone agrees on. Whenever we have
events that happen at the same place and
time, that's an invariant. Everybody
agrees on it. But how does that help us
in our original scenario?
Suppose the space shuttle has a clock on
board. When it leaves earth, you see the
clock reads zero. Now those two events
departure and the clock reading zero is
happening at the same place in time.
That means everyone agrees on that. And
when the shuttle reaches Proxima Centuri
if the onboard clock reads say 3 years
again everyone agrees on that. So look
Mahesh even though we all disagree on
the amount of distance and time that we
have measured with our own rulers and
clocks we all agree the people in the
spaceship aged by 3 years.
In other words the time elapsed on the
ship that is the invariant. Now because
everyone agrees on it we call it the
proper time. The symbol we use is the
Greek towel. Think of it as the time
measured by a clock that's present at
both events. That's important. Like for
example, the ship's clock. In contrast,
our clocks are not present at both
events. We instead rely on our
coordinates. So these are coordinate
times and coordinate distances. They
depend on reference frames. They are the
shadows. But the proper time in some
sense is real. It's like the actual
distance between the two hanging Legos.
It represents the true distance in
spaceime. But wait a second here we just
looked at the clock that was on board
and that's how we could figure out what
the proper time was between the two
events. But in general events don't come
with a clock you know next to it. So in
general how do we connect our coordinate
measurements that we can do coordinate
space coordinate time and proper time.
How do we do this? How do we do this
intuitively within Kowski? And then
Kowski says well Mahesh now to find this
connection we have to use Lawrence
transformations and do a bit of math.
There isn't a way to do this intuitively
or is there Louis Epstein who wrote the
book relativity visualized one of my
favorite books thinks otherwise? He
makes the last conceptual leap. Are you
ready for this? Epstein says Mahes
everything in relativity can be
visualized intuitively using just one
fact. What is that? Everything in this
universe, including you and me, are
traveling through spaceime at the speed
of light. The reason we can't go faster
than light is because we can't go slower
than light. The only speed everything
can ever move at is the speed of light.
And I'm like, okay, that sounds
absolutely ridiculous. What do you mean,
Epstein?
All right, Mahesh. Right now, this
shuttle isn't moving through space, but
it is moving through time. So, it's
moving in spaceime. And in spacetime,
the speed will always be C, the speed of
light. We'll choose C to be one. Okay.
Which means right now that velocity
vector is pointed completely in time
direction.
Wait, so you're saying that right now
that the shuttle is traveling through
time at the speed of light? What does
that even mean?
That means Epston says for every second
in my clock, the clock on the ship goes
ahead by 1 second.
Epston
defines the speed through time as the
time elapsed on its clock, meaning its
proper time per unit time on our clock,
the coordinate time. And here that's 1 /
1. So it's just one. So the speed
through time is one. Speed of light, it
basically means its clock is ticking at
full speed, the same rate as our clock.
Also notice it's projecting a fulllength
shadow. So we see the ship to be its
full length in space.
But what does this whole thing look like
from the perspective of our moving
friend?
Well, since our friend is moving, his
frame is tilted so that he sees the ship
moving to the right. So look Mahesh in
his frame that vector is tilted but its
total magnitude still stays the same one
and that tilt explains
everything. When 1 second passes on his
clock look less than 1 second passes on
the ship's clock.
The ship's speed through time is reduced
because part of its velocity is now
spent moving through space. That is time
dilation.
But notice nothing physically has
changed. All that changed is the
coordinates. The frame tilted.
But wait, there's more.
Look at the projection on the space now.
It's shorter. That's why our friend
measures the ship itself to be shorter.
Length contraction. Incredible, right?
And the faster it moves, the more
extreme these effects get. Finally, when
the vector tilts almost horizontal,
almost all its velocity is through
space. The ship is practically frozen in
time. Its length and time intervals are
infinitely distorted. This is why light
travels at sea through space because its
velocity is completely pointed in space.
Thinking of our universe as a
four-dimensional spaceime is starting to
make so much sense. Oh my god. Take
about Epstein and of course Minkowski.
Quick note though, Epstein's diagrams
are not standard space-time diagrams.
Okay, look. We are drawing coordinate
space versus proper time. Usually when
we say spacetime diagram and 45°ree
light cones and all of that stuff, we
draw coordinate space versus coordinate
time.
But who cares whether it's standard or
not? The ultimate question is can we put
it all together mathematically? Can we
find a connection between coordinate
space coordinate time and the proper
time? Epstein, how do we do this?
Mahesh, let's decompose this velocity
vector into horizontal and vertical. The
hypotenuse over here represents its
total velocity through space time. This
is the velocity through time and this is
the velocity through space. So how do we
connect this? And I'm like well
Pythagoras velocity through space time
squar should equal velocity through
space 2 plus velocity through time
squared. So let's simplify. We know
velocity through space time is always 1.
What is velocity through space? And I'm
like, that's just dx over dt. How much
it moves through space per unit
coordinate time. Now, what is velocity
through time? This is the important one,
but it's very similar. It's how much it
moves through its own time, proper time,
per unit coordinate time. So, d tow by
dt. Plugging in and simplifying, what do
we get? What do we get? Oh, here it is.
Here it is. The proper time squared
equals the coordinate time squared minus
the coordinate space squared. We have
rediscovered
the ultimate equation connecting all
three. This is the distance formula in
spacetime.
Okay, I'm sure we have a ton of
questions now, but my first step
whenever I rediscocover or look at a new
equation is to do a sense check. And I
do that by plugging in some numbers. So
going back to our space shuttle,
remember from our perspective, we saw
that it covers a distance of about four
light years in 5 years. Now those are
our coordinate measurements, Minowski
shadows. But if you plug in, we now find
the proper time to be 3 years that we
saw, right? Okay. Meaning the people on
the ship are only 3 years old, which
means for them the journey only took 3
years. Exactly like what we saw. But
let's check our friend's perspective.
She measured the ship covering just one
and a 1.8 light years in only 3 and 1/2
years. Different shadows, right? Because
her frame is tilted. But now she plugs
into the same rule. And look, she too
agrees that the folks on the ship have
aged by 3 years. So we will disagree on
our on our own measurements of distance
and time because those are just
coordinate shadow lengths which depend
on our frames which are tilted. But the
proper time 3 years is something that we
all agree on. It is the true interval
between these two events. It's the
invariant quantity. It's the actual
distance in some sense between the two
events in spacetime.
So can we now answer the question where
does the minus sign come from? The
absence diagram actually helps us see it
from the ship's frame. It's not moving.
So its velocity in spaceime is directed
upwards only through time. So at time t
equals0 earth leaves it and 3 years
later proxima century arrives. So it
takes them 3 years. But if we look from
our perspective, the frame is tilted.
But the speed in spaceime is always one.
So look, three years are not enough. We
need to wait more about a total of 5
years. And from our friend's
perspective, that tilt is slightly less.
So for her, the ship only takes about 3
and 12 years.
Which means, look, Mahesh, when we
change frames, it's the vertical
distance that stays the same. That's the
invariant. The hypotenuse and the base
are the shadows.
But in normal space, when we tilt the
frame, the hypotenuse is the invariant.
That's why in normal space, the formula
for the invariant ends up with a plus
sign, but it ends with a minus sign in
space time. And since these invariant
quantities are real distances in normal
space the distance formula has a plus
sign and in spacetime the distance
formula ends up with a minus sign.
That's where it comes from. Here the
distance is just the proper time. But
why does it work out this way? What is
the core reason?
In normal space you can move at any
speeds you want. And there is no speed
uh speed restriction. But in spaceime,
you can only ever move with one speed,
the speed of light. See, as a result,
the faster you move through space, the
slower you move through proper time,
that's what the minus sign is saying.
So, putting it all together, when people
usually say that, hey, the minus sign is
because space and time are fundamentally
different. No, that's not accurate. What
they really mean is that spacetime is
fundamentally different from space. How
different? Two reasons. In space, it's
the hypotenuse that ends up becoming the
invariant. But in spacetime, it's the
vertical leg that ends up becoming the
invariant. And in space, you can move at
any speeds you want. You have no
restrictions over there. But in
spaceime, you can only ever move with
just one specific speed, the speed of
light. These two together as we saw
makes sure that we end up with a minus
sign in our distance formula.
But I promised you full intuition. So
here's the last question. What if
hypothetically
our universe had a plus sign in the
space-time interal formula? What would
the universe look like? Well, remember
in our universe the minus sign comes
from the fact that the total speed
through spaceime is always one, the
speed of light. So the speed through
space or time can never exceed one.
Meaning nothing can ever move faster
than speed of light in space and no
clocks can ever tick faster than well
what my clock is ticking right now. But
if there was a plus sign then after
rearranging look we see that that
restriction goes away. For example the
speed through space can be two and that
would make the speed through time three.
Meaning we could see things traveling at
twice the speed of light and they would
be aging twice as quickly as we do. The
speed through space and time could be as
big as possible. There would be no more
restrictions.
Isn't that awesome? No. Because if
signals could travel at infinite speeds,
then changes any changes physical,
chemical, biological, could all happen
instantly. In other words, the entire
history of the universe from the
beginning to the end would happen
instantly. The universe wouldn't make
sense anymore.
So look, it's the negative sign that
preserves the speed limit. It ensures
signals cannot travel at infinite
speeds. It ensures that changes don't
happen instantly but evolve over time.
And this allows our universe to
meaningfully evolve. The minus sign
gives the universe a meaning.
Wow.
So although Einstein initially dismissed
Minowsky's ideas, eventually he came
around. In fact, when he struggled for
years to reconcile gravity with
relativity, he realized the missing
piece had been in front of him all
along.
He later writes, I realize that only
through Minowsk's four-dimensional
formalism does the theory become
sufficiently general and clear. So
Einstein not only embraced his idea, he
took it to an entirely new level,
creating general relativity, the most
successful model of the universe we have
even today.
So although Minowski unfortunately never
lived to see it, the very idea that
Einstein had dismissed as unnecessary
became the foundation for his greatest
achievement.
I'll see you soon.