This object is at the heart of
understanding relativity. It fully
describes space-time geometry, including
its curvature. It helps us compute with
absurd levels of precision how this
curvature affects time without which GPS
satellites and Google maps wouldn't
work. It helped us discover and
correctly model black holes and event
horizons, gravitational waves, and even
the expansion of the universe itself. I
kid you not, it's that powerful. It's
called the metric tensor. But what
exactly is it? When I tried to read
about it, all I got were these vague
definitions and that it can be written
as a giant matrix with a lot of
components and that it's coariant,
whatever that means. And it's used in
this particular formula which has a lot
of mus and news. Yeah, none of that made
any intuitive sense to me until now.
After going through a lot of resources,
I think I finally have some intuition
behind the math of relativity. And boy
oh boy, it's so cool. And I'm super
excited to share that with you. So in
this video, we're going to pretend we
don't know anything about metric
whatever. And like always, discover
rediscover everything from scratch, step
by step, intuitively. So if you're ready
for this, let's begin. So Einstein,
where do we start? And Ein says, let's
get right to it. Let's start with a
ridiculously simple question. Can you
find the distance between these two
houses? And I say, sure. I mean
horizontally they are three units apart
and vertically they are four units
apart. So from Pythagoras theorem they
must be five units apart. But Ein says
mish what's the physical distance in
kilome? And I say well for that you need
to tell me how big a unit is for example
if one unit is 1 kilm then the distance
is going to be 5 km and Ein says exactly
Moes this is the metric tensor. I mean
there is a small nuance to it which
we'll get back to later but at the heart
of it metric tensor is a mathematical
object that converts from arbitrary
coordinates to real physical distances
and I say that's it that's what a metric
is is basically a conversion factor then
why do we call it a tensor and make and
make a big deal out of it why does it
have all of these components
mahesh why don't we shrink the grid to
half its size now look the horizontal
distance is instead of 3 units, 6 units
and the vertical distance has gone from
4 to 8 units which means according to
Pythagoras this distance is now 10
units. Does that mean that the distance
between the houses has changed? And I
say of course not because look now 1
unit is half a kilometer. So if we
multiply by the metric then we get back
the same answer. And Eisen says look
this means when we shrunk the grid the
metric transformed in such a way that
the physical distance remained invariant
and that lights up a bulb in my head
because that's exactly what a tensor
does. That's the core feature of a
tensor. Its core feature is that when
the coordinates transform the tensor
transforms in such a way that the
underlying physical quantity remains
invariant. So that's why it makes sense
to call t the metric a tensor.
Beautiful, right? Okay. Now that we know
that the metric is a tensor, the next
obvious question we have should be
Einstein, what's the rank of this
metric? And Ein says, Mahesh, for that,
tell me this. How many numbers do you
think you need to fully represent the
metric? And I say, well, just one
number, right? Like we have shown over
here. And smiles and says, well, Mah,
that's the case if you're using a square
grid. But what if you're using a
rectangular grid? And at first I'm like,
why would anybody use a rectangular grid
like this? And I says, we'll get back to
that. But now, how many numbers do you
need? And I say, yeah, well, now one
number is not enough. Now we need two
numbers. One number to represent how big
the unit is in the horizontal. Another
number to represent how big the unit is
in the vertical. So, right, we need two
numbers. And now Mahesh, can you compute
the distances one more time to just
confirm we get the same answer?
All right, let's do this. So in this new
grid, we have 1 2 3 6 units
horizontally, but still four units
vertically. Ooh, that means I cannot
directly use Pythagoras because they
don't have the same units. I better
first multiply with the metric. And now
I get 3 km, 4 km. So obviously this will
be 5 km. Now if I use the Pythagoras
theorem the hypotenuse square equals the
base squar plus the opposite side
squared giving us 5 km. So Ein says look
Mahesh this means when you're using a
rectangular grid you can no longer
directly use Pythagoras. You need to
have some coefficients in front that
represents the scaling factor. It tells
you how the different scales are for
horizontal and vertical. Here the
scaling factor is half square which
is.25 and here the scaling factor is
just one. So remember that we said there
was a nuance over here. Well, it's these
numbers that are the components of the
metric tensor, not these the square of
those values because it's the squares
that become the direct coefficient for
the x² and the y square terms.
Which means we can now generalize
pythagoras theorem for rectangular grid.
So all we have to do is add coefficients
in front of x² and y square. These are
the scaling factors. Again, these are
the components of the metric tensor. And
I ask well wait a second why do you call
it gxx and not just gx? Einson says well
that's because this is the coefficient
of x² term x * x and this is the
coefficient of y square term. Now at
this point I need to apologize. See I
should have actually written it as delta
x and delta y because remember these
represent displacements not coordinates
but I forgot about it and for the rest
of the presentation I've just written it
as x and y. So for the rest of the video
wherever you see X Y or Z for that
matter remember it's actually
displacements not coordinates. Okay end
of apology. All right. So this means we
need two numbers to fully represent the
metric tensor right and Eisen says oh
Mahes that's true if we are dealing with
axes that are always perpendicular but
what if the axis themselves are skewed
like oh no.
So Mahesh, do you think the formula
still works in this coordinates? And I
say, well, no, because we no longer have
a right angle triangle. But it's nothing
we cannot handle actually because we can
always con make constructions and
complete the right angle triangle. So
let's call this as X and Y and forget
about scaling for a second and then
we'll realize we've tackled this while
doing vector addition in high school.
What we should do is we should complete
this triangle and then if this angle is
theta then the base becomes y cos theta
and the opposite side becomes y sin
theta. And look now I can compute the
hypotenuse. So the hypotenuse squared
becomes the base squar plus the opposite
side squared. And so now if I simplify I
get this. And since sin square theta +
cos square theta is just 1 I can write
this as y square. And there you have it.
And Ein says perfect. Now all we have to
do is make this a little bit more
elegant.
How shall we do that? Well, first of
all, you have this 2xy cos theta over
here. Well, we can write this as xy cos
theta plus yx cos theta. And ask why?
Well, because there's a nice ring to it,
right? I mean, you have x² term, then
you have an xy, then you have the yx,
and there's the y square. So, it's just
beautiful. Okay. But now if we add back
the scaling factor, can you imagine what
the general formula looks like? Well,
it's going to be some coefficient * x²
plus some coefficient * xy plus some
coefficient * yx plus some coefficient *
y 2. Which means we can write the most
general formula like this. And by the
way, these two coefficients, as you can
see, are exactly the same thing. It's
just written that way. So it be looks
beautiful and symmetric. And while we're
at it, the last elegance we can do is
instead of writing as x squ, we can just
write as x * x and y * y. And boy oh
boy, look at this. The most general form
of distance formula that will work in
any coordinate system written elegantly.
Isn't that amazing?
Anyways, look mahesh. These four
coefficients now become the components
of the metric tensor. These two value
tells you how much the x and the y axis
are scaling. And these two which are
equal they tell you the angle between
the x and y. So you need a total of four
numbers in general to fully represent
the metric in two dimensions. And I say
well actually it's only three numbers
right? The fourth one is the same. And
yeah three unique numbers the fourth one
is redundant. Remember these two are
exactly the same. It's written just for
the sake of symmetry. So if you write it
as a matrix the matrix should be
diagonally symmetric. For example these
are the values that you can have. So
look these two numbers should always be
the same. And if these numbers become
zero, that means we the the grid is
perpendicular. We get back our
rectangular grid. And if these two
numbers are one, well, we get back our
square grid. We get back our Pythagoras
theorem. So this is the simplest metric
that you can think about. But anyways,
this is what the metric looks like in
two dimensions. Mahesh, what do you
think the metric looks like in three
dimensions? How many total numbers do we
need and how many of them are unique?
And I say well we can just extend this
right we can just write a zed over here
and we get a total of nine terms. Now xx
y and zz they represent the scaling in x
y and z axis and these terms represent
the angle between xy and y z and z x and
of course these three terms must be
redundant with these three terms that
means in total we have only six unique
terms and says perfect that's exactly
right and look because we have written
it so elegantly it's easy to write the
whole distance formula in three
dimensions it's going to be x into x *
gxx x x into y * gxy x into z * gxz and
so on and so forth. So that's why we
write it so elegantly. So beautiful
isn't it? So Mahesh can you now tell
what the rank of the metric tensor is?
And I say well okay in two dimensions uh
the metric has four components two
squared. In three dimensions it has nine
components. So 3 squared. In n
dimensions it'll have n squared
components. So so square that represents
the rank. So it must be a rank two
tensor. And of course another way to see
it is that there are two index to
represent the tensor. So it's a rank two
tensor. And Einstein says, "Perfect. Its
metric tensor is ranked two." But wait,
wait, wait, wait. Now I have a question.
Actually, a really, really, really
silly, silly question. Einstein, how do
I write this in n dimensions? I mean,
notation wise, because we have x, y, z,
and then what? Right? And Ein says,
that's not a silly question at all.
Notations are a big deal. So, let's look
at how we do this. First thing because
we want to write this in n dimensions
instead of writing it as x y and z we
will write it as x1 x2 and x3 that's how
we'll represent our dimensions. Okay. So
that's the first change we will do and
the moment we do that we can now instead
of writing as gxx and gxy we can write
as g11 g12 g13 and so on and so forth.
And the beauty is mahesh look look we
can now write this as a summation isn't
it? We can write this as summation of g
and we can use some letters to represent
the index over here and the indexes will
run from one to n where n represents the
dimension. So let's write it this way.
So s square is the summation and we
usually like to use Greek letters for
the index. So mu and new that's where
the mu and new comes from 1 to n and we
can write as g mu new and x mu * x new.
Beautiful. That's the notation we can
write and extend this to n dimensions.
But now just when everything is going
really well, everything starting to make
perfect sense, Einstein does something
very weird to me. He raises the indices
on the displacement terms.
Einstein, buddy, what did you just do?
Where do we have why do we have x square
and x cq? Where do they come from? And
Einstein says m no, no, no, these are
not powers. There are no there are no
exponents over here. These are still
just indices but instead of writing as a
subscript we write them as a
superscript. And I say why? Well Einson
says remember when we shrunk the grid uh
to half its size the metric also became
half and when we blow up the grid to
twice the size the metric also
transforms to become double. So look the
metric changes in the same way as the
grid. It co-varies with the grid. And so
we call such tensors co-variant tensors.
Covariant tensors are the ones that vary
in the same way as the grid. In
contrast, if you look at the
displacement vectors, when the grid
shrunk to half its size, they became
double. And if the grid becomes double
in size, they became half, which means
they vary in the opposite way as the
grid varies. And so we call such tensors
or such vectors contravariant tensors or
contravariant vectors. Contravariant
means these are the tensors or vectors
that change in the opposite way as the
grid changes. And the convention that we
have adopted is for co-variant tensors
we use the index as a sup sub subscript
like usual. But for contravariant ones,
the ones that vary in the opposite way,
we'll choose the opposite convention.
We'll use them as superscripts. It's
it's just a convention.
And so if you go back to our distance
formula, we can immediately look at the
index and say that hey this is a
co-variant term and these two are
contravariant because they have
superscripts. And because you're
multiplying a coariant with a
contravariant, we end up with invariant
distance. That's what makes the distance
invariant. It it's beautiful if you ask
me. And finally finally, you know,
because we know it's a summation, let's
just get rid of the summation term and
keep it clean. And boom, there you have
it. Beautiful, isn't it? I mean, I am
not much of a notation guy, but I'm a
sucker for elegance, and this is so
elegant. So, that's it. This is the most
general form of the distance formula,
right? And Einstein smiles says, "No,
there's one last generalization left."
Mahesh, what if you use a grid that
looks like this? Oh my god, this is such
a mess. Why would anybody use a grid
like this? I know, I know. So we'll get
to that later on. But anyways, over
here, oh my god, the units are changing
everywhere. The angles are changing
everywhere. Oo, this means the metric is
changing everywhere. At every point, we
have a different metric, which means we
now have a metric field. So how do you
calculate the distance from here to here
if the metric itself varies? Well, we
can break it into small small parts. We
can zoom into small sections and if the
parts are very small we can assume
approximate the metric to be constant in
that particular small section and then
use that metric to calculate the
distances of those tiny parts. In other
words, this formula in general can only
be applied to small patches of space.
This is a differential formula. So I
should have called it as the small
differential element squared and these
are small differential um x and y or you
know coordinate terms displacement
terms. This means in general if we are
calculating it we have to add up all
these individual elements which means we
have to integrate it. This is it in its
full glory the most general way to
calculate distances and we discovered it
all ourselves just by using pythogoras.
I mean, imagine how tearyeyed a
Pythagoras would be if he knew that we
could generalize his formula for n
dimensions and make it work for abstract
coordinates. Oh my god, that is amazing.
And what's more amazing is that we can
discover a lot of concepts just like
this from scratch by using
brilliant.org, the sponsor of this
you use my link to buy a subscription, you'll also get a 20% off. Okay, back to
the video.
Einstein, this is all amazing and I
truly appreciate it, but you still
haven't answered my original question.
Why use this? I mean, I can always
choose to use a square grid and stick to
Pythagoras, right? And Einstein says,
"Yes, Mahes, you're right. When it comes
to flat surfaces, you don't need to use
these curved coordinates with varying
metric. You can just choose to use a
flat grid with a constant metric and
just stick to Pythagoras. But what if
you're dealing with a curved surface
like that of the Earth? Now, even if you
try to put the simplest grid like
latitude and longitude lines, look, now
the distance, the unit keeps changing.
So, you have no choice but to deal with
varying metric. It's impossible to find
a grid where the metric stays the same
everywhere. That's just not possible.
Which means you are forced to use this
general formula to calculating distances
over here. But wait a second. I can I
can draw a map and put a square grid on
it, right? Yes, says Einstein. But you
can never ever make the metric a
constant. This is impossible. Whenever
you do look at maps, the metric keeps
varying one way or the other. That's why
all world maps are distorted because
you're forced to have a varying metric.
It is impossible to draw a world map
which has a constant metric. It's not
possible because the intrinsic geometry
itself is curved.
Okay. So what does it all mean now? And
Eisen says and this is the climax now.
Okay. Ready for it? So Eisen says this
means we now have a way to check if a
spa given space is curved or not. If
someone gives you a coordinate system
with a varying metric and you want to
check whether it's flat or curved just
try to do a coordinate transformation
and see if you can find a metric that
looks like this a constant one. If you
can then that means the space is flat.
Uh this varying metric was just an
artifact of using weird coordinates. But
if you find that it's impossible to
convert it into a constant metric,
regardless of what coordinates you
choose, you just cannot transform it.
Then that means that the varying metric
is not just an artifact of coordinates.
It is built into the geometry. That
means that that space was curved all
along. So look, we have a mathematical
way to check whether a space is curved
or not. Isn't that amazing? And I say,
"That sounds awesome, and I still need
some time to wrap my head around it. But
why do I ever need it?"
I mean, why go through all of this
trouble? I can always look at a space
and tell whether it's flat or curved,
right? His says, "Not really, Mahes. I
mean, look at a cylinder. It looks
curved to you, but you can put a grid
with a constant metric because you can
literally open up a cylinder. So, in
that sense, it is flat. So, you cannot,
you know, it's not that intuitive. But
more importantly, your intuition only
works for two-dimensional surfaces. But
this mathematical tool that we have that
works for any number of dimensions. That
means we have a method at least in
principle to check whether a I don't
know a 10-dimensional abstract space is
flat or curved
from Pythagoras
to the ability to check flatness and
curvature of ndimensional surfaces. I
don't know about you, but that is
insane. I mean tell me tell me that that
is not insane.
But what Einstein does next is even more
insane.
Einstein says if we can apply this to n
dimensions can we just apply it to
fourdimensional space time. This was the
big leap that Einstein had. I mean this
idea of metric and curvature and all of
this existed way before Einstein but
people only applied it to space because
time was an independent universal thing
until special relativity came along.
Einstein showed that it's much better to
model our universe not as three
dimensions of space plus time but
four-dimensional spaceime. And so now
we're about to discover how to take this
idea to its full potential. Are you
ready? Let's jump to spaceime. So
instead of two houses which are very
close to each other, let's consider two
planets far away. So over here both axes
are space. But if you want to jump to
space time, we want to consider one axis
to be space and the other to be time. So
now we have space and time.
Now when we only consider space, what
matters to us is physical distance
between things. For example, the
distance between planets. Let's say
these two planets are four light years
apart. But when it comes to space time,
what's important for us is distance
between events. That's what matters. So
let's consider an event. Let's imagine
there's a spaceship that moves from here
to here at a constant velocity. So our
first event would be the spaceship
leaving this planet. So let's put that
over here at time t=0. And let's say the
spaceship goes from here to here at a
constant velocity and arrives over here
5 years later. So that is the second
event. So the spaceship arrives over
here and this is event two and notice
the event two is 5 years apart in time.
The central mind-boggling thing about
relativity is that just like how when
you think about distances between two
objects like the two houses the
horizontal and the vertical values are
arbitrary. They depend on you know which
coordinate system you use. They change
isn't it? What doesn't change is this
actual physical distance. In a similar
way in space-time the space and time
between events are also arbitrary which
means there could be another observer
moving with respect to us and he will
disagree with these values. He might say
something like hey the space between
events is just one and a half light
years and the spaceship only traveled
for three and a half years and we are
both right. Space and time between
events are arbitrary. They are not real
things. They depend on reference frames.
But even though people disagree on
horizontal and vertical, they all agree
on this physical distance given by this
formula, the Pythagoras. In a similar
way, even though people disagree with
space and time between events, they will
all agree. Everybody will agree on the
space time distance or space time
interval between events. And how do you
calculate that? Well, you calculate very
similar to this except you have a minus
sign. And at first I'm like, wait a
second, where does that minus sign even
come from? Like you're just throwing
that formula at my face. And Ein says,
actually there's a very elegant way to
even rediscover this idea. But that's a
story for another video because it comes
from special relativity and the fact
that, you know, we all move at the speed
of light through spaceime, all of that
stuff. Nothing to do with metric and
stuff. This is special relativity. So
let me know if you want to dig deeper
into what that minus sign is. Where does
it come from? That's also pretty cool.
But for the sake of this video, what's
important to understand is that if you
now plug in the values 5 squar minus 3
square, we get this to be 3. And the
same value we'll get if you plug in 3
1/2 and 1.8. In other words, every
single observer in this universe will
agree that the space-time interval
between these two events is three units.
That is invariant. Everyone agrees on
it.
But wait a second. What exactly is this
three units? How do I think about it?
And essence is again we can rediscover
these ideas ourselves. And we'll do that
in a separate video. But this term
happens to be proper time. Which means
it's the amount it tells us how much an
object has aged when it has gone from
one part, you know, one event to
another. So in other words, the
inhabitants of this spaceship only aged
by 3 years. So even though we saw the
spaceship take 5 years to go from here
to here, the inhabitants only aged by 3
years. This is time dilation. But what's
important is that everybody agrees, even
the other observer will agree that the
folks in the spaceship only aged by 3
years. That is something that's
invariant and that's why we call it the
proper time because everybody agrees on
that. So long story short, the distance
formula here tells you well how much is
the distance between two objects, right?
But the distance formula over here,
Mahesh, it tells you how much an object
has aged when it goes from one event to
another.
very different things but still look at
the formula they are so similar which
means we can draw a parallel between
these two just like over here we can say
that the metric is a constant I can now
put a grid over here and say the metric
over here is also a constant
but of course a small difference is that
there's a minus sign because this
there's a minus sign in my formula but
everything else remains exactly the same
by the way we give this metric a name we
call it the Minowski metric um It's
named after Einstein's professor who he
didn't like very much. Now, of course,
this is just the two-dimensional version
of the metric where we have one
dimension of time and only one dimension
for space. Of course, we live in four
dimensions. So, this is the full
Minowski metric. And again, the minus
sign captures how space behaves
differently than time. And yes, you can
make the time negative and space
positive. It doesn't matter. It's just a
convention.
But the point is just like before when
you had a constant metric we said the
space is flat. Now because we have a
constant metric we can say the space
time is flat and just like before you
are free to choose whatever you know
coordinates you want and if you choose
weird coordinates you'll end up with
weird metric but you can always
coordinate transform it back to Minowski
metric. That's what it means to be flat.
This varying metric was just an artifact
of choosing weird coordinates. However,
Mahesh, what if you find yourself in a
space-time with varying metric which you
can never transform back to Minowski.
Then that means this varying metric is
no longer an artifact of you know weird
coordinates. It is intrinsic to
space-time geometry. It is intrinsically
curved.
This is how you can think about
spacetime being flat and curved. And I
find this amazing and really amazing
because till now I used to think of
spacetime as flat and curved only using
an analogy. You know the 2D analogy of
you know flat and curved. But now my
thinking has been upgraded. We can now
think rigorously. Spacetime is flat if
you can get the Minowski metric. If you
can never transform it back to Minowski,
it's curved. That is insane.
But remember curved spacetime is gravity
which is now encoded by metric. In other
words, it's the metric tensor that
encodes gravity. It encodes the
space-time geometry in it. Wow. But of
course, the obvious ultimate question
for us now is how do you actually do
this? What operation can you do on a
given metric to check whether it can be
turned into Minowski or not? In other
words, what operation should you do on a
metric to check whether you're in flat
spacetime or not? And Einstein says
that's the ultimate question and that's
something we should dig deeper in a
separate video. And by the way, he did
struggle with it a lot. Um, but
fortunately he had an amazing friend
Marcel Gman who was a brilliant
mathematician. So they collaborated
together and they figured it all out.
Einstein realized exactly what are the
different operations you need to do. It
turns out you need to do some calculus
on the metric tensor and they figured it
all out out and eventually Einstein was
able to encode all of this in his famous
field equations. It inputs mass and
energy distribution in spacetime and
then it outputs after a lot of work, if
you're lucky, it'll output the metric.
And this metric encodes the geometry of
spacetime. It encodes curvature. And
when you compute the metric for a simple
point mass, you'll notice that the
metric blows up at a couple of spots.
You could dismiss that as a weird
mathematical glitch or take it
seriously, dig deeper and discover event
horizons and black holes. That's
literally how we did it. And if you set
the proper time to zero, you get the
path that light takes. It's called the
null path. Einstein used exactly this to
predict how starlight would bend near
the sun. And during the 1919 solar
eclipse, Edington measured it and gave
the world the first direct confirmation
of general relativity. It's the same
math we use to simulate these black
holes as well. And if you do a bit of
calculus on the metric tensor, you can
figure out the freef fall paths, the
orbits, the geodics. Einstein did this
and realized that planetary orbits
should actually process slightly and it
perfectly accounted for the Mercury's
orbital procession. And because this
equation gives you proper time, you can
use it to compute how fast clock ticks
at different altitudes. That's exactly
what's needed to correct GPS systems.
Without it, they would be off by
kilometers every single day. But that's
not all. You may even wonder, can the
metric vary with time? Could it
oscillate? Well, then you would
hypothesize gravitational waves. And if
the metric changes continuously with
time, then you can hypothesize the
expansion of the universe itself and
even predict the big bang. The metric is
the heart of general relativity. We'll
dig deep into all of these soon. See
you.