Tensor Calculus 25 - Geometric Meaning Ricci Tensor/Scalar (Volume Form)
TLDRThis video delves into the geometrical implications of the Ricci curvature tensor and scalar, building upon the concept of sectional curvature introduced in a previous video. It explains how the Ricci tensor measures volume changes along geodesics, utilizing both an orthonormal basis approach and a new method involving the volume element derivative. The video also corrects a mistake regarding dimensionality in a previous example and explores the relationship between the Ricci scalar, volume deviation in curved spaces, and the algebraic properties of these tensors.
Takeaways
- π The script continues a discussion on the geometrical meanings of the Ricci curvature tensor and the Ricci scalar, building on a previous video's introduction to sectional curvature and its implications on geodesic behavior.
- π It corrects a mistake from a previous video regarding the dimensionality of a ball used to illustrate changes in volume due to Ricci curvature, emphasizing the importance of using a two-dimensional disc for such examples.
- π The Ricci tensor is explored through a new approach involving the volume element derivative, applicable in any basis, not just orthonormal ones.
- π The volume form, denoted by the lowercase Omega symbol, is introduced as a tensor that quantifies the volume enclosed by a set of vectors, with its numerical output determined by the determinant of a matrix formed by the vectors' components in an orthonormal basis.
- 𧩠The concept of the Levi-Civita symbol is explained as a tool for calculating the volume of parallelograms formed by vectors, with its properties and use in summations to find determinants highlighted.
- π The script explains how to calculate the volume enclosed by vectors in non-orthonormal bases by involving the determinant of the forward matrix or the square root of the metric tensor matrix determinant.
- π The covariant derivative of the volume form is shown to be zero, indicating that the rules for measuring volumes remain consistent along geodesics, despite actual volumes potentially changing.
- π The Ricci tensor is derived from the second covariant derivative of the volume form, revealing how it quantifies the rate of volume change due to spatial curvature along geodesics.
- π’ The Ricci scalar is discussed as a scalar obtained from the summation of the Ricci tensor, indicating how the volume of a ball in curved space deviates from that in flat space.
- π The video concludes with a detailed example of how the Ricci scalar relates to the surface area of a sphere compared to a flat disk, illustrating the impact of positive curvature on volume measurements.
- π The script promises further exploration of the algebraic properties of the Ricci tensor and scalar in a subsequent video, hinting at a deeper dive into their mathematical characteristics and implications.
Q & A
What is the Ricci curvature tensor and what does it measure?
-The Ricci curvature tensor is a measure of the curvature of space that quantifies how volumes change as they move along geodesics. It sums up the sectional curvatures in every direction of an orthonormal basis, providing an overall sense of how volumes expand or contract due to the space's curvature.
What is the relationship between sectional curvature and Ricci curvature?
-Sectional curvature measures how neighboring geodesics converge or diverge due to the curvature of space. The Ricci curvature is the sum of all sectional curvatures in every direction of an orthonormal basis, giving a comprehensive measure of volume change.
How is the Ricci tensor related to the volume change of a ball moving along geodesics?
-The Ricci tensor indicates how the volume of a ball changes as it moves along geodesics in a curved space. A positive Ricci curvature implies that volumes tend to shrink, while a negative Ricci curvature suggests that volumes tend to expand.
What is the volume form and how is it used to calculate volumes in different bases?
-The volume form, denoted by the lowercase Omega symbol, is a tensor that gives the volume enclosed by a set of vectors. It is calculated by taking the determinant of the matrix with the components of each vector arranged in columns in an orthonormal basis. For non-orthonormal bases, the volume form is given by the square root of the determinant of the metric tensor times the Levi-Civita symbol.
What is the significance of the covariant derivative of the volume form being zero?
-The covariant derivative of the volume form being zero signifies that the rules for measuring volumes do not change along geodesics, even though the volumes themselves may change due to the curvature of space.
How does the Ricci tensor arise from considering the volume element derivative?
-The Ricci tensor arises from taking two covariant derivatives of the volume form along a geodesic path. This approach to the Ricci tensor works in any basis and leads to the understanding that the Ricci tensor measures how volumes change due to the curvature of space.
What is the Ricci scalar and how does it relate to the volume of a ball in curved space?
-The Ricci scalar is a scalar obtained by summing over the Ricci tensor with one upper index and one lower index. It indicates how much the size of a ball in curved space deviates from the standard volume of a ball in flat space, given the same radius.
What is the significance of the Ricci scalar being positive or negative in the context of space curvature?
-A positive Ricci scalar indicates that the space has a tendency for volumes to be smaller than expected for a given radius, as seen on a sphere. Conversely, a negative Ricci scalar suggests that volumes are larger than expected for the same radius, which could be the case in negatively curved spaces.
How does the Ricci tensor differ from the sectional curvature in terms of measuring the effects of space curvature?
-While sectional curvature measures the convergence or divergence of neighboring geodesics in a specific plane, the Ricci tensor provides a more comprehensive measure of how volumes change as they move along geodesics in all directions of space.
What is the connection between the Ricci tensor and the metric tensor in the context of volume change?
-The components of the volume form tensor, which is used to calculate volumes in different bases, include the square root of the determinant of the metric tensor matrix. This connection shows that the metric tensor plays a crucial role in understanding how volumes change in different coordinate systems.
Outlines
π Introduction to Ricci Curvature and Scalar
This paragraph introduces the concepts of Ricci curvature and Ricci scalar, continuing from a previous video. It discusses the sectional curvature and how it measures the convergence or divergence of geodesics due to space curvature. The Ricci curvature is defined as the sum of sectional curvatures in an orthonormal basis, and it's related to the volume change of a ball moving along geodesics. The presenter corrects a previous mistake regarding the dimensionality of a ball in a curvature example and emphasizes that the Ricci curvature tracks changes in the size of a ball as it travels along geodesics. A new approach to understanding the Ricci tensor through the volume element derivative is introduced, applicable in any basis, not just orthonormal.
π Understanding the Volume Element and its Derivative
The paragraph delves into the concept of the volume element, denoted by the lowercase Omega symbol, which is a tensor representing the volume enclosed by a set of vectors. It explains how to calculate the two- and three-dimensional volume using the volume form and the determinant of a matrix formed by the components of vectors in an orthonormal basis. The use of the Levi-Civita symbol is introduced for a more compact representation of the volume calculation. The paragraph also discusses how to handle non-orthonormal bases by involving the determinant of the forward matrix or the Jacobian in the calculation. The importance of the metric tensor and its relation to the volume change when changing bases is highlighted.
π Exploring the Covariant Derivative of the Volume Form
The focus shifts to the covariant derivative of the volume form and its implications for volume changes along geodesics. It is shown that the covariant derivative of the volume form is zero, indicating that the rules for measuring volumes remain constant along geodesics, even though the volumes themselves can change. The paragraph explains the process of parallel transporting vectors along a geodesic path and how this affects the volume enclosed by these vectors. The multi-linearity properties of tensors are used to prove that the covariant derivative of the volume form equals zero, which is a key result leading to the introduction of the Ricci tensor.
π Ricci Tensor and Volume Change in Curvature Space
This paragraph explores the relationship between the Ricci tensor and how it quantifies volume changes due to space curvature. It describes the process of taking two covariant derivatives of an arbitrary volume spanned by vectors, which leads to the Ricci tensor. The paragraph explains how separation vectors, representing the distance between neighboring geodesics, are used in this process. The second derivative of the volume is shown to contain a term proportional to the original volume, with the Ricci curvature as the constant of proportionality. This term distinguishes volume changes due to curvature from those that can occur in flat space due to geodesic alignment.
π Ricci Scalar and Deviation of Volume from Flat Space
The paragraph introduces the Ricci scalar, which is derived from the Ricci tensor and provides information about how the volume of a ball in curved space deviates from that in flat space. It discusses the concept of comparing volumes by radius and explains how the surface area of a bowl shape on a sphere differs from that of a flat circle. The Taylor series expansion is used to illustrate the relationship between the areas of these shapes, with the Ricci scalar appearing in the second-order term. The implications of positive and negative Ricci scalar values on the volume of a ball are explained, with examples of how the volume can either increase or decrease depending on the curvature.
π Conclusions and Further Exploration of Ricci Properties
The final paragraph summarizes the two main interpretations of the Ricci tensor: one based on sectional curvature and the other on the derivative of the volume form. It emphasizes that the Ricci tensor measures volume changes along geodesics due to curvature and does not provide information about shape changes. The Ricci scalar is again highlighted as an indicator of how volume deviates from the flat space expectation for a given radius. The paragraph concludes with a mention of future content on the algebraic properties of the Ricci tensor and scalar, inviting viewers to look forward to the next video in the series.
Mindmap
Keywords
π‘Ricci curvature tensor
π‘Ricci scalar
π‘Sectional curvature
π‘Geodesics
π‘Orthonormal basis
π‘Volume element
π‘Levi-Civita symbol
π‘Covariant derivative
π‘Metric tensor
π‘Riemann curvature tensor
Highlights
The video continues the discussion on the geometrical meanings of the Ricci curvature tensor and the Ricci scalar, building upon the concepts introduced in the previous video.
Sectional curvature, which measures how neighboring geodesics converge or diverge due to space curvature, is explained using an orthonormal basis.
The Ricci curvature is defined as the sum of all scalar curvatures in every basis vector direction, indicating volume change as a ball moves along geodesics.
A correction is made regarding the dimensionality of a ball in the context of Ricci curvature, emphasizing the importance of using a two-dimensional ball for accurate representation.
A new approach to understanding the Ricci tensor is introduced, which involves considering the volume element derivative and works in any basis, not just orthonormal.
The volume element, or volume form, is described as a tensor that gives the volume enclosed by a set of vectors, with a focus on its numerical output.
The concept of the volume form is explored through orthonormal basis vectors, which always form a box with volume one.
The volume form's numerical output is explained using the determinant of a matrix formed by vector components in an orthonormal basis.
The use of the Levi-Civita symbol in calculating the volume of parallelograms formed by vectors is discussed, providing a compact way to compute determinants.
The method of changing basis and its implications on volume calculation, using the forward matrix or Jacobian determinant, is explained.
The relationship between the determinant of the metric tensor and the volume change when changing basis is established.
The volume form tensor's components in a non-orthonormal basis are derived, incorporating the square root of the metric tensor determinant.
The covariant derivative of the volume form is shown to be zero, indicating that the rules for measuring volumes remain constant along geodesics.
The second derivative of the volume along a geodesic path is explored, revealing its relationship with the Ricci tensor and volume change due to curvature.
The Ricci tensor is interpreted as indicating how volumes change in size as they move along geodesics in space, specifically due to curvature.
The Ricci scalar is discussed as a measure of how the volume of a ball in curved space deviates from that in flat space, with implications for understanding space curvature.
The video concludes with a summary of the two interpretations of the Ricci tensor and the significance of the Ricci scalar in understanding space curvature's impact on volume.
Transcripts
in this video I'm going to continue
talking about the geometrical meanings
of the Ricci curvature tensor and the
Ricci scalar I started this discussion
in the last video number 24 and the link
to that video is in the description in
that previous video we introduced the
idea of sectional curvature which
measured how neighboring geodesics
either converged together or diverged
apart due to the curvature of space we
also showed how in an orthonormal basis
we could define the Ricci curvature
which is the sum of all scalar
curvatures in every basis vector
direction I also showed that we can
compute the Ricci curvature using the
Ricci tensor I also showed how the Ricci
curvature could tell us how the volume
of a ball is changing as it moves along
geodesics in this example since the
Ricci curvature is zero the volume of
the ball isn't changing I want to
correct a mistake I made here since
there are only two curvature directions
in this example which are the e1 and e2
basis vectors I should have been using a
two dimensional ball which would be a
flat 2d disc the Ricci curvature would
be measuring the change in area of this
disc as it spreads out or contracts in
the e1 e2 directions I apologize for
getting the number of dimensions wrong
but the basic idea that I explained is
still correct the Ricci curvature still
does track the change in size of a ball
as it travels along geodesics in space
now this approach to the Ricci tensor
using sectional curvature is nice but it
only works in an orthonormal basis in
this video I'm going to talk about a new
approach to the Ricci tensor which
arises from considering the volume
element derivative and this approach
works in any basis so first off what is
the volume element or volume form we
denote the volume form by the lowercase
Omega symbol the volume form is a tensor
that gives us the volume enclosed by a
set of vectors for example we can see
that the a and B vectors here can be
used to form a two-dimensional
parallelogram if we want to know the
two-dimensional volume or the area of
this parallelogram
this would be given by the volume form
Omega acting on the a and B vectors
similarly the volume of this
parallelogram shape formed by the x y&z
vectors is omega of x y&z now how do we
figure out the actual numerical output
of the volume form let's start with a
simple example if we are given a set of
orthonormal basis vectors these
orthonormal basis vectors will always
form a box that has volume one the
output of the volume form is always 1
for example the two-dimensional volume
or area of this square formed by a
2-dimensional orthonormal basis is equal
to 1 because orthonormal vectors have
length 1 by definition and similarly the
volume of this cube formed by a 3d
orthonormal basis is also equal to 1 now
what about the volume created by vectors
that are not orthonormal like the
vectors U and W here well if we expand U
and W into components in an orthonormal
basis we can compute the volume of this
parallelogram given by Omega of U and W
by computing the determinant of this
matrix with the components of each
vector arranged in columns this is
because these vector components arranged
in a matrix basically form a linear map
which takes us from the unit square to
this parallelogram if you've taken a
first linear algebra course I'm hoping
you remember that the change in volume
that results from a linear map is given
by the determinant of the matrix for
that linear map so since this matrix
containing the vector components forms a
linear map from this square of area 1 to
this parallelogram the area of the
parallelogram should be the determinant
of this matrix so taking the determinant
of this matrix we find that the area of
this parallelogram is u 1 times W 2
minus u 2 times W 1 another way of
writing this more compactly is by doing
a summation with this epsilon symbol
which is called the levy chavita symbol
the levy chavita symbol in 2d has
indexes I J when the indexes are one two
the levy chavita symbol equals plus 1
when the indexes are 2 1 the levy
chavita symbol equals negative 1 and
when the indexes are repeated such as 1
1 or 2 2 the symbol equals 0 so if we
expand the summation of the levy chibita
symbol with the U and W components we
get four terms the 1 1 and 2 2 terms go
to 0 because they have repeated indexes
the one to levy chibita symbol is
positive 1 and the 2 1 levy chibita
symbol is negative 1 we can see that
this gives us the expected formula for
the determinant of this matrix we can
get the volume created by a set of 3
vectors in 3 dimensions in a similar way
we get the components of the vectors in
an orthonormal basis and arrange them in
this matrix then take the determinant of
this 3 by 3 matrix we can also use the
three dimensional levy chibita symbol in
a summation to get the answer for the
determinant the levy chavita symbol in
3d is a little more complicated than in
2d we get plus 1 for the levy chibita
symbol if the ijk indexes are an even
permutation of 1 2 3 we can see some
examples of even permutations here an
even permutation just means we start
with 1 2 3 and then do an even number of
swaps between the digits for example 2 3
1 is an even permutation because we get
it by starting with 1 2 3 and then
swapping the left and middle indexes
then swapping the middle and right
indexes to swaps is an even number of
swaps so it's an even permutation the
levy chavita symbol is negative 1 for
odd permutations 1 3 2 is an odd
permutation because we get it by
starting with 1 2 3 and then swapping
the right and middle indexes 1 swap is
an odd number of swaps so we
-1 and similar to 2d if there are any
repeated indexes in the levy to be
dissemble we just get 0 such as with 1 1
3 or 2 2 2 you can check for this
summation with 27 terms only 6 of them
are not 0 and they result in the correct
formula for the determinant of this 3x3
matrix so the lesson here is that the
levy chibita symbol is a useful tool for
getting the volume of the parallelogram
shape formed by a set of vectors all we
do to get the volume is to take the levy
chavita symbol and summit with all the
components of the vectors that form the
parallelogram shape although please note
that this method only works if the
vector components are measured in an
orthonormal basis now what if we're
working in a basis that isn't
orthonormal well let's think about what
it means to change basis we already know
that the volume of the shape forms by
two vectors U and W is the determinant
of this matrix this is very similar to
what happens when we change from an old
basis e1 e2 to a new basis e1 tilde and
e2 tilde in previous videos I've labeled
the coefficients that change from one
basis to another with the letter F for
forward these coefficients can be put
together in a matrix which in previous
videos I've called the forward matrix
the volume created by the new basis
vectors would then just be the
determinant of the matrix with the F
coefficients if we're dealing with
curvilinear coordinates where the basis
vectors are partial derivative operators
the volume change we get by moving from
an old basis to a new basis is the
determinant of this matrix of partial
derivatives also known as the
determinant of the Jacobian so the
important takeaway for this slide is
that the volume enclosed by a set of
basis vectors is given by the
determinant of the forward matrix or
equivalently the determinant of the
Jacobian if our coordinate system is
curvilinear now recall that the
components of the metric tensor in
given bases are just the dot products of
the basis vectors and if we expand the
new basis vectors in terms of the old
basis vectors using summations with the
forward F coefficients we get this and
these dot products are just the
components of the metric tensor in the
old basis now if we take the determinant
of both sides of these equations we can
write the new basis metric tensor
determinant as a product of the
determinant of F the determinant of F
and the determinant of the old basis
metric tensor matrix this is because the
determinant of a product of matrices is
the same thing as a product of the
determinants and since the old basis is
orthonormal the metric tensor matrix is
the identity matrix and so this
determinant is equal to 1 in the
previous slide we said that the volume
change we get when we change basis is
the determinant of the F matrix but as
we've shown the square of the
determinant of F is equal to the
determinant of the metric tensor matrix
in the new basis and so the volume
change is equal to the determinant of F
but it is also equal to the square root
of the determinant of the metric tensor
in the new basis and if we're dealing
with curvilinear coordinates we can come
to the same conclusion that the volume
change which is normally the determinant
of the Jacobian can also be written as
the square root of the determinant of
the metric tensor matrix so the volume
enclosed by a new set of basis vectors
can be written in several different ways
including the determinant of the forward
matrix the determinant of the Jacobian
or the square root of the metric tensor
matrix determinant so let's say we have
two vectors U and W written in terms of
a basis which is not orthonormal how do
we get the volume Omega of U and W the
volume formula has two parts first the
square root of the determinant of the
metric tensor matrix in the new basis
this computes the change in volume that
we get from changing from an orthonormal
basis to this new non orthonormal
second we multiply that by the
determinant of the matrix of vector
components as measured in the new basis
this takes care of the volume change we
get by building up the U and W vectors
from the new basis vectors e1 tilde and
e2 tilde
if we write this determinant using the
levy chavita symbol we get this formula
and this combination of the square root
of the metric tensor and the levy
chavita symbol gives us the components
of the volume form tensor Omega so in an
orthonormal basis the levy chibita
symbol alone gives us the components of
the volume form but in an arbitrary non
orthonormal basis we also need the
square root of the metric tensor matrix
determinant from that basis to get the
correct components of the volume form so
we've learned about a number of tensors
in this series like the metric tensor
Riemann tensor and Ricci tensor and we
know that when they act on vectors to
get the output values we just do
summations with the tensor components
the volume form is yet another tensor
and the components we use to do the
summation are the determinant of the
metric tensor times the leve chavita
symbol so now that we understand what
the volume form is let's take two
covariant derivatives of the volume form
along some geodesic path this will give
us the second rate of change of a volume
formed by a set of basis vectors as we
move along a geodesic path we'll find
that taking this derivative will lead us
to the Ricci tensor the first thing we
need to do is show that the covariant
derivative of the volume form is zero
this doesn't mean that volumes don't
change as they move along geodesics we
saw in the previous video that volumes
can in fact change along geodesics what
this does mean is that the rules for
measuring volumes don't change along
geodesics so to prove this we start with
a geodesic path and by taking a set of
vectors that form a box we then parallel
transport these vectors along the path
we've
in previous videos that the levy chavita
connection has the property of metric
compatibility this means that when we
parallel transport vectors along a path
using the levy chibita connection the
lengths of the vectors stay the same and
the angles between the vectors stay the
same this means that when we parallel
transport these vectors the volume
created by these vectors will not change
in size therefore the covariant
derivative of this volume is zero when
we apply this covariant derivative we
are actually applying it to a product of
four terms the volume form and each of
these three vectors using product rule
we get four terms each with the
covariant derivative applied to a
different part of the formula now since
these vectors a B and C are all parallel
transported this means that their
covariant derivative is zero by
definition and since Omega is a tensor
and a multi linear map when one of the
inputs is zero that means that the
entire output is zero this can be proven
using the multi linearity properties of
a tensor so we're left with the
conclusion that the covariant derivative
of the volume form must be equal to zero
in other words the derivative of the
volume form components along this path
are zero so knowing this let's look at
what happens when we apply two covariant
derivatives to an arbitrary volume
spanned by some vectors we know that the
volume of this shape formed by the u W
and T vectors is just the summation of
the volume form components summed with
the vector components now recall
previously when thinking about curvature
we looked at the separation vector
between two neighboring geodesics now
when you look at these three vectors
that create the volume u W and T I want
you to think of these as separation
vectors each keeping track of the
separation distance with its own
neighboring geodesic path so I'm going
to rename u W and T
as s1 s2 and s3 since they are all
separation vectors for some geodesic I'm
also going to replace the summation
indexes with mu1 mu2 and mu3 I realize
this notation might be a bit confusing
but it basically just means that up here
I is the first summation index mu 1 J is
the second summation index mu 2 and K is
the third summation index finally I'm
going to rewrite the product of these
three separation vector components using
this product notation where we do the
product of D separation vector
components where D is the dimension of
the space and if we take two covariant
derivatives of the volume along this
geodesic since the volume is a scalar
this is the same thing as the ordinary
derivative along the geodesic path using
the path parameter lambda and recall
that since the volume form has a
derivative of zero we can take the
volume form components and put them
outside the derivative so let's apply
this first derivative to the product of
the S vector components you should know
by now that using product rule on a
product gives us a sum of terms where
every term has the derivative applied to
a different factor I'm denoting the
derivative here by an overhead dot we
can do the same thing here when we take
the derivative of this product by
pulling the S J components outside of
the product and applying a dot to take
the derivative there is a summation over
J here so this means that we get a sum
of terms where in every term a different
SJ component gets the derivative this is
very similar to what we've written out
here where in each term a different
component gets the derivative dot now
for the second derivative we can think
of it as doing product rule on this
factor and then this factor the first
factor is easy we just apply the
derivative to s J dot and get sjws
for the second factor we can use these
same reasoning as above and pull out
another index out in front
SK this time and apply a dot for the
derivative and removing the K index from
the product now recall that taking two
derivatives of a separation vector along
a geodesic just gives us this negative
Riemann tensor term this means that this
separation vector component with two
derivatives equals this summation with
the Riemann tensor components and the s
and V vector components this s and V are
given in the Y and Zed summations and
this V is given in the X summation so
now we can sub this expression in for
the S double dot now here notice in this
product that all the MU in Dex's are
used except form you J and recall in the
Levy chavita symbol that if there are
any repeated indexes it goes to zero now
this S term out here has a superscript y
that's involved in a summation so it can
take on many different values like mu 1
mu 2 and so on but if we use mu 1 or mu
2 and it's already in the leve chavita
symbol the term will go to 0 because
there's a repeated index this means that
the only term that will end up being
nonzero in the summation occurs when y
equals mu J because mu J does not
already occur in this product so we can
set y equals mu J because all other
versions of y in this summation go to 0
and now since this product doesn't
include the J index we can simply take
the S component with the J index and put
it back in the product so we get this
where the product goes from I equals 1
to D without skipping any indexes but
you'll notice that all of this right
here is really just the original formula
that defined the volume of the
parallelogram shape we were originally
talking about it has the components of
the volume form and the product of
all these separation vector components
so all of this is just equal to the
volume capital B and notice that since
the Riemann tensor components have a
summation on the upper index and the
lower middle index this is actually just
the definition of the Ricci tensor XZ
and of course this summation just gives
us the Ricci curvature in the direction
of the e vector so the conclusion after
all of this is that the Ricci tensor
tells us how volumes change in size as
we move around in space along geodesics
the second order derivative of the
volume has a term that is proportional
to the original volume with the Ricci
curvature being the constant of
proportionality now you might be
wondering about this other term here
that involves the first derivative of
the separation vector components recall
in the last video that I said geodesics
spread apart for two different reasons
when they accelerate apart this is due
to the curvature of the space but when
they move apart at a constant velocity
this can occur in flat space because of
the way the geodesics are angled so this
first term in the second derivative with
the Ricci tensor detects how volumes
change due to the curvature of the space
while the second term measures the
volume changes that can occur in flat
space because of the aligning of
geodesics so the conclusion is that the
Ricci tensor tells us how volumes change
specifically because of the curvature of
the space were in when we're traveling
along geodesics recall that for the
sphere the Ricci curvature is always
positive and with this negative sign
here this means that volume is moving
along geodesics on the sphere will
always shrink in volume because the
geodesics on the sphere are always
converging the last thing I'll talk
about is the Ricci scalar which is the
scalar we get when we do a summation on
the Ricci tensor with one upper index
and one lower index and recall that
raising an index involves doing a
summation with the inverse metric
in the last video I said that the Ricci
scalar tells us how much the size of a
ball in curved space deviates from the
standard volume of a ball in flat space
so in 2d the standard flat disk with the
given circumference will contain an area
equal to PI R squared and this area is
less than the surface area of an
upside-down bowl shape that sits on the
top of a sphere even though the bowl has
the exact same circumference so given
this circular circumference we don't
actually know how much area will fit
inside the answer could change depending
on the curvature of space there's the
completely flat space area but the area
will increase slightly if this circle
sits on a very large ball which causes
the area to bulge a little and as the
sphere gets smaller and smaller the
amount of curvature increases and the
amount of surface area contained within
the perimeter also increases in the edge
cases when we have flat space the
surface area is PI R squared and in the
case where the circle is the boundary on
the equator of a sphere the surface area
is 2 PI R squared so we can see that the
surface area inside the circle can
double if the curvature of the space is
large enough now when it comes to the
Ricci scalar were generally interested
in comparing ball sizes by their radius
not by their circumference and when I'm
talking about the radius I don't mean
this distance I mean the radius as seen
from the point of view of someone
traveling from the North Pole of the
ball out to the edge of the circle so
this radius is really an arc length
given by the radius of the ball times
this angle Phi so let's start by getting
the surface area of this upside-down
Bowl shape on a sphere of radius capital
script R we can compute this by taking a
collection of thin disks of radius Rho
and integrating over their thickness D s
we know from trigonometry
tree that row is equal to the sphere
radius times the sine of this angle
theta we also know that the thickness of
each disc is equal to this tiny arc
length D s which is equal to the spheres
radius times the small angle D theta so
subbing those formulas into our area
integral and taking the constants out in
front we get to PI capital R squared
times the integral of sine theta D theta
from zero to Phi and remember Phi is the
angle measured from the vertical to the
edge of the upside-down Bowl shape the
antiderivative of sine is negative
cosine and subbing in the limits of
integration we get cosine of 0 which is
1 and since these negative signs cancel
out we end up with this formula for the
surface area of the bowl and recall that
we said that the radius of this circle
as seen by someone walking from the
North Pole to the edge of the bowl is
given by this arc length lowercase R and
by the familiar arc length formula since
lowercase R equals capital R Phi and
this means that the angle Phi is given
by the radius of the bowl as seen by
someone walking from the North Pole
divided by the radius of the entire
sphere so this here is the formula for
the surface area of the upside-down Bowl
as a function of the bowls radius as
measured using the walking distance from
the North Pole now let's compare the
surface area of this bowl shape against
the standard flat space area of a circle
PI R squared we can immediately cancel
these pies and to handle this cosine I'm
going to expand it as a Taylor series
where we get a summation of all the even
powers of little R over big R with
alternating positive and negative signs
this positive one and this negative one
cancel and if we distribute this
negative sign into the brackets
containing the Taylor series all these
signs end up
next I'll distribute this term out in
front to all the terms inside the Taylor
series in the first term we can see that
all the numerators cancel with all the
denominators and were left with just one
in the second term some of the powers
cancel and we're left with 1 over 24
times 2 over big R squared times little
R squared and I'm not going to bother
with the higher-order terms because they
aren't as important you might be
wondering why I'm keeping this 2 here
instead of writing 1 over 12 that will
become clear soon so now given that we
know the metric tensor and the Ricci
tensor for the sphere of radius capital
R let's compute these spheres Ricci
scalar so we have a double summation
over i and j for the metric and Ricci
tensor x' that gives us four terms but
two of them go to zero in the remaining
terms these sign terms in the numerator
and denominator cancel and we're left
with two over big R squared so we can
see that the Ricci scalar does appear in
the second order term in the Taylor
series expansion between the area of the
bowl on the sphere and the area of a
circle in flat space because this ratio
is a little less than 1 it means that
the surface area of the bowl is slightly
less than the area of a flat disc given
the same radius little R so this might
be a bit confusing but just to review
the sphere has positive curvature and
that means that for a given boundary
circle the surface area on the sphere
given by this bowl shape is greater than
the area of the flat circle with the
same circumference but for a given
radius the surface area of the bowl
shape on the sphere is less than the
area of the circle in flat space with
the same radius so when we think of
positive curvature it can either mean
more area for the same circumference or
it can mean less area for the same
radius in a negatively curved space it
would be the opposite where we get less
area for the same circumference or more
area for the same radius so this is what
the Taylor series looks like for the
ratio of circle areas in the specific
case where our curved space is a sphere
for general curved spaces in d
dimensions the ratio of the volumes has
this Taylor expansion where the
second-order coefficient is the Ricci
scalar divided by 6 times the number of
dimensions plus 2 you might ask where
this formula comes from but
unfortunately I don't know
I tried searching for a proof online but
I wasn't able to find a proof that I
understood so unfortunately I can't
prove this for you if I do find a proof
that's understandable I may upload
another video about it but for now all I
can do is give you some links in the
description to the complex proofs that I
don't understand
so to summarize these past two videos
we've learned two interpretations of the
Ricci tensor and how it tracks volume
changes along geodesics one
interpretation involves sectional
curvature and the other interpretation
involves the derivative of the volume
form the sectional curvature tracks how
neighboring geodesics either converge
with positive curvature or diverge with
negative curvature by measuring the
second rate of change of a separation
vector when we look at all these
sectional curvatures along geodesics in
every direction in an orthonormal basis
we can add all these sectional
curvatures together to get the Ricci
curvature the Ricci curvature tells us
the overall volume change as a ball
travels along geodesics the Ricci
curvature does not tell us about how the
ball changes shape but it will tell us
how the ball changes volume the Ricci
curvature can be calculated using the
components of the Ricci tensor for the
second interpretation we learned that
the volume of a box formed by a set of
vectors is given by
the volume form acting on those vectors
and the components of the volume form
tensor is given by the square root of
the determinant of the metric tensor
matrix multiplied by the levy chibita
symbol we found that the second
derivative of the volume due to the
curvature of space is given by the Ricci
tensor term which computes the Ricci
curvature finally we discussed the Ricci
scalar which tells us how the volume of
a given radius and curved space deviates
from the volume of a ball with the same
radius and flat space if the Ricci
scalar is positive the volume will be
smaller than expected for a given radius
as seen on the sphere if the Ricci
scalar is negative the volume will be
bigger than expected for that radius in
the next video I'll talk about the
algebraic properties of the Ricci tensor
and Ricci scalar as always if you enjoy
my videos and would like to leave a tip
you can check the description for a link
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