Tensor Calculus 25 - Geometric Meaning Ricci Tensor/Scalar (Volume Form)

eigenchris
25 Oct 201932:12
EducationalLearning
32 Likes 10 Comments

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
  • πŸ“š 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.
  • πŸ” 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.
  • πŸ“ The Ricci tensor is explored through a new approach involving the volume element derivative, applicable in any basis, not just orthonormal ones.
  • πŸ“ 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.
  • 🧩 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.
  • πŸ”„ 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.
  • 🌐 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.
  • πŸ“ˆ 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.
  • πŸ”’ 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.
  • 🌍 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.
  • πŸ“š 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
00:00
πŸ“š 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.

05:02
πŸ“ 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.

10:02
πŸ” 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.

15:05
🌐 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.

20:08
πŸ“˜ 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.

25:09
πŸ”‘ 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
Upcoming Content
Ricci Scalar Significance
Interpretations of Ricci Tensor
General Case
Taylor Series Expansion
Sphere Example
Volume Deviation
Definition
Ricci Curvature Interpretation
Riemann Tensor Relation
Separation Vectors
Second Derivative of Volume
Covariant Derivative of Volume Form
Change of Basis
Levi-Civita Symbol
Non-Orthonormal Basis
Orthonormal Basis
Volume Element
Correction of Dimension Error
Volume Change
Ricci Curvature
Sectional Curvature
Introduction
Conclusion and Future Discussion
Ricci Scalar
Ricci Tensor and Volume Change
Volume Element Derivative Approach
Ricci Curvature Tensor and Scalar
Geometric Meanings of Ricci Curvature and Scalar
Alert
Keywords
πŸ’‘Ricci curvature tensor
The Ricci curvature tensor is a measure of the amount by which the volume of a small 'ball' in curved space deviates from that of a ball in flat space as it moves along geodesics. It is integral to understanding the geometric properties of space in the context of general relativity. In the video, the Ricci curvature tensor is discussed in relation to how it can be computed and its implications for the changing volume of a ball as it travels along geodesics.
πŸ’‘Ricci scalar
The Ricci scalar is derived from the Ricci curvature tensor and is a single number that summarizes the amount of curvature in a particular point of a space. It is used to quantify how much the volume of a region in curved space differs from that in flat space. The video explains that the Ricci scalar can indicate whether the volume in a curved space is greater or lesser than what would be expected in flat space for a given radius.
πŸ’‘Sectional curvature
Sectional curvature is a measure of the curvature of a 2D surface within a higher-dimensional space. It quantifies how much two neighboring geodesics converge or diverge. The video script discusses how sectional curvature is used to define the Ricci curvature in an orthonormal basis, which is a way to understand how space curves in different directions.
πŸ’‘Geodesics
Geodesics are the shortest paths between two points in a curved space, analogous to straight lines in flat space. They are crucial in the study of curvature as they represent the paths along which objects would naturally move in the absence of external forces. The video explains how the Ricci curvature can indicate how volumes change as they move along these geodesics.
πŸ’‘Orthonormal basis
An orthonormal basis is a set of vectors that are mutually perpendicular and have a magnitude of one. It is used in the context of the video to define the Ricci curvature in a way that is independent of the coordinate system. The script mentions that the Ricci curvature can be computed using the Ricci tensor in an orthonormal basis.
πŸ’‘Volume element
The volume element, denoted by the lowercase Omega symbol in the script, is a tensor that describes the volume enclosed by a set of vectors. It is used to calculate areas or volumes spanned by vectors in a given space. The video explains how the volume element is used to understand how the volume of a shape changes as it moves along geodesics.
πŸ’‘Levi-Civita symbol
The Levi-Civita symbol, also known as the permutation symbol, is a mathematical tool used to compute determinants in a more compact form, especially in the context of tensor analysis. The video script uses this symbol to calculate the volume of parallelograms formed by vectors in an orthonormal basis and to derive the components of the volume form tensor.
πŸ’‘Covariant derivative
The covariant derivative is a generalization of the derivative that is used in curved spaces. It takes into account the curvature of the space and is essential for defining and working with tensors. In the video, the covariant derivative of the volume form is discussed, showing that it is zero, which implies that the rules for measuring volumes remain consistent along geodesics.
πŸ’‘Metric tensor
The metric tensor is a fundamental concept in differential geometry that encodes the geometry of a space. It is used to measure distances and angles in a coordinate-free manner. The video script explains how the metric tensor's components are related to the dot products of basis vectors and how it is used in the calculation of the volume form and Ricci tensor.
πŸ’‘Riemann curvature tensor
The Riemann curvature tensor is a measure of the curvature of a space in four dimensions, capturing how much the space departs from being flat. It is used in the video to explain how taking two covariant derivatives of a separation vector along a geodesic yields a term involving the Riemann tensor, which is related to the Ricci 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
00:00

in this video I'm going to continue

00:01

talking about the geometrical meanings

00:03

of the Ricci curvature tensor and the

00:05

Ricci scalar I started this discussion

00:07

in the last video number 24 and the link

00:10

to that video is in the description in

00:11

that previous video we introduced the

00:14

idea of sectional curvature which

00:16

measured how neighboring geodesics

00:18

either converged together or diverged

00:20

apart due to the curvature of space we

00:24

also showed how in an orthonormal basis

00:26

we could define the Ricci curvature

00:28

which is the sum of all scalar

00:29

curvatures in every basis vector

00:31

direction I also showed that we can

00:34

compute the Ricci curvature using the

00:37

Ricci tensor I also showed how the Ricci

00:40

curvature could tell us how the volume

00:42

of a ball is changing as it moves along

00:44

geodesics in this example since the

00:46

Ricci curvature is zero the volume of

00:49

the ball isn't changing I want to

00:52

correct a mistake I made here since

00:54

there are only two curvature directions

00:56

in this example which are the e1 and e2

00:58

basis vectors I should have been using a

01:01

two dimensional ball which would be a

01:03

flat 2d disc the Ricci curvature would

01:06

be measuring the change in area of this

01:08

disc as it spreads out or contracts in

01:11

the e1 e2 directions I apologize for

01:15

getting the number of dimensions wrong

01:16

but the basic idea that I explained is

01:18

still correct the Ricci curvature still

01:21

does track the change in size of a ball

01:23

as it travels along geodesics in space

01:26

now this approach to the Ricci tensor

01:29

using sectional curvature is nice but it

01:31

only works in an orthonormal basis in

01:34

this video I'm going to talk about a new

01:36

approach to the Ricci tensor which

01:38

arises from considering the volume

01:40

element derivative and this approach

01:42

works in any basis so first off what is

01:46

the volume element or volume form we

01:49

denote the volume form by the lowercase

01:51

Omega symbol the volume form is a tensor

01:54

that gives us the volume enclosed by a

01:56

set of vectors for example we can see

02:00

that the a and B vectors here can be

02:02

used to form a two-dimensional

02:04

parallelogram if we want to know the

02:07

two-dimensional volume or the area of

02:10

this parallelogram

02:11

this would be given by the volume form

02:13

Omega acting on the a and B vectors

02:17

similarly the volume of this

02:19

parallelogram shape formed by the x y&z

02:23

vectors is omega of x y&z now how do we

02:28

figure out the actual numerical output

02:30

of the volume form let's start with a

02:33

simple example if we are given a set of

02:36

orthonormal basis vectors these

02:38

orthonormal basis vectors will always

02:40

form a box that has volume one the

02:43

output of the volume form is always 1

02:46

for example the two-dimensional volume

02:49

or area of this square formed by a

02:52

2-dimensional orthonormal basis is equal

02:55

to 1 because orthonormal vectors have

02:58

length 1 by definition and similarly the

03:02

volume of this cube formed by a 3d

03:04

orthonormal basis is also equal to 1 now

03:08

what about the volume created by vectors

03:10

that are not orthonormal like the

03:12

vectors U and W here well if we expand U

03:16

and W into components in an orthonormal

03:19

basis we can compute the volume of this

03:22

parallelogram given by Omega of U and W

03:25

by computing the determinant of this

03:27

matrix with the components of each

03:29

vector arranged in columns this is

03:33

because these vector components arranged

03:35

in a matrix basically form a linear map

03:38

which takes us from the unit square to

03:41

this parallelogram if you've taken a

03:44

first linear algebra course I'm hoping

03:46

you remember that the change in volume

03:48

that results from a linear map is given

03:51

by the determinant of the matrix for

03:53

that linear map so since this matrix

03:56

containing the vector components forms a

03:58

linear map from this square of area 1 to

04:02

this parallelogram the area of the

04:04

parallelogram should be the determinant

04:06

of this matrix so taking the determinant

04:09

of this matrix we find that the area of

04:12

this parallelogram is u 1 times W 2

04:15

minus u 2 times W 1 another way of

04:20

writing this more compactly is by doing

04:22

a summation with this epsilon symbol

04:24

which is called the levy chavita symbol

04:27

the levy chavita symbol in 2d has

04:30

indexes I J when the indexes are one two

04:34

the levy chavita symbol equals plus 1

04:36

when the indexes are 2 1 the levy

04:40

chavita symbol equals negative 1 and

04:42

when the indexes are repeated such as 1

04:45

1 or 2 2 the symbol equals 0 so if we

04:49

expand the summation of the levy chibita

04:52

symbol with the U and W components we

04:55

get four terms the 1 1 and 2 2 terms go

04:59

to 0 because they have repeated indexes

05:01

the one to levy chibita symbol is

05:04

positive 1 and the 2 1 levy chibita

05:07

symbol is negative 1 we can see that

05:10

this gives us the expected formula for

05:12

the determinant of this matrix we can

05:15

get the volume created by a set of 3

05:18

vectors in 3 dimensions in a similar way

05:20

we get the components of the vectors in

05:23

an orthonormal basis and arrange them in

05:26

this matrix then take the determinant of

05:29

this 3 by 3 matrix we can also use the

05:33

three dimensional levy chibita symbol in

05:35

a summation to get the answer for the

05:37

determinant the levy chavita symbol in

05:40

3d is a little more complicated than in

05:42

2d we get plus 1 for the levy chibita

05:46

symbol if the ijk indexes are an even

05:49

permutation of 1 2 3 we can see some

05:53

examples of even permutations here an

05:55

even permutation just means we start

05:57

with 1 2 3 and then do an even number of

06:01

swaps between the digits for example 2 3

06:05

1 is an even permutation because we get

06:07

it by starting with 1 2 3 and then

06:10

swapping the left and middle indexes

06:12

then swapping the middle and right

06:14

indexes to swaps is an even number of

06:18

swaps so it's an even permutation the

06:21

levy chavita symbol is negative 1 for

06:23

odd permutations 1 3 2 is an odd

06:27

permutation because we get it by

06:29

starting with 1 2 3 and then swapping

06:32

the right and middle indexes 1 swap is

06:35

an odd number of swaps so we

06:37

-1 and similar to 2d if there are any

06:42

repeated indexes in the levy to be

06:44

dissemble we just get 0 such as with 1 1

06:47

3 or 2 2 2 you can check for this

06:51

summation with 27 terms only 6 of them

06:54

are not 0 and they result in the correct

06:57

formula for the determinant of this 3x3

07:00

matrix so the lesson here is that the

07:04

levy chibita symbol is a useful tool for

07:06

getting the volume of the parallelogram

07:09

shape formed by a set of vectors all we

07:12

do to get the volume is to take the levy

07:14

chavita symbol and summit with all the

07:17

components of the vectors that form the

07:19

parallelogram shape although please note

07:22

that this method only works if the

07:24

vector components are measured in an

07:26

orthonormal basis now what if we're

07:29

working in a basis that isn't

07:31

orthonormal well let's think about what

07:34

it means to change basis we already know

07:37

that the volume of the shape forms by

07:39

two vectors U and W is the determinant

07:42

of this matrix this is very similar to

07:45

what happens when we change from an old

07:47

basis e1 e2 to a new basis e1 tilde and

07:52

e2 tilde in previous videos I've labeled

07:55

the coefficients that change from one

07:57

basis to another with the letter F for

08:00

forward these coefficients can be put

08:02

together in a matrix which in previous

08:04

videos I've called the forward matrix

08:07

the volume created by the new basis

08:10

vectors would then just be the

08:12

determinant of the matrix with the F

08:14

coefficients if we're dealing with

08:16

curvilinear coordinates where the basis

08:19

vectors are partial derivative operators

08:21

the volume change we get by moving from

08:24

an old basis to a new basis is the

08:26

determinant of this matrix of partial

08:28

derivatives also known as the

08:30

determinant of the Jacobian so the

08:33

important takeaway for this slide is

08:35

that the volume enclosed by a set of

08:37

basis vectors is given by the

08:39

determinant of the forward matrix or

08:41

equivalently the determinant of the

08:44

Jacobian if our coordinate system is

08:46

curvilinear now recall that the

08:49

components of the metric tensor in

08:51

given bases are just the dot products of

08:53

the basis vectors and if we expand the

08:56

new basis vectors in terms of the old

08:59

basis vectors using summations with the

09:01

forward F coefficients we get this and

09:05

these dot products are just the

09:07

components of the metric tensor in the

09:09

old basis now if we take the determinant

09:13

of both sides of these equations we can

09:16

write the new basis metric tensor

09:18

determinant as a product of the

09:20

determinant of F the determinant of F

09:22

and the determinant of the old basis

09:25

metric tensor matrix this is because the

09:28

determinant of a product of matrices is

09:30

the same thing as a product of the

09:32

determinants and since the old basis is

09:35

orthonormal the metric tensor matrix is

09:38

the identity matrix and so this

09:40

determinant is equal to 1 in the

09:43

previous slide we said that the volume

09:45

change we get when we change basis is

09:47

the determinant of the F matrix but as

09:51

we've shown the square of the

09:53

determinant of F is equal to the

09:54

determinant of the metric tensor matrix

09:56

in the new basis and so the volume

09:59

change is equal to the determinant of F

10:02

but it is also equal to the square root

10:04

of the determinant of the metric tensor

10:06

in the new basis and if we're dealing

10:09

with curvilinear coordinates we can come

10:11

to the same conclusion that the volume

10:14

change which is normally the determinant

10:16

of the Jacobian can also be written as

10:19

the square root of the determinant of

10:21

the metric tensor matrix so the volume

10:24

enclosed by a new set of basis vectors

10:26

can be written in several different ways

10:28

including the determinant of the forward

10:31

matrix the determinant of the Jacobian

10:33

or the square root of the metric tensor

10:36

matrix determinant so let's say we have

10:40

two vectors U and W written in terms of

10:43

a basis which is not orthonormal how do

10:46

we get the volume Omega of U and W the

10:50

volume formula has two parts first the

10:53

square root of the determinant of the

10:55

metric tensor matrix in the new basis

10:57

this computes the change in volume that

11:00

we get from changing from an orthonormal

11:02

basis to this new non orthonormal

11:06

second we multiply that by the

11:09

determinant of the matrix of vector

11:11

components as measured in the new basis

11:13

this takes care of the volume change we

11:16

get by building up the U and W vectors

11:18

from the new basis vectors e1 tilde and

11:21

e2 tilde

11:23

if we write this determinant using the

11:25

levy chavita symbol we get this formula

11:28

and this combination of the square root

11:31

of the metric tensor and the levy

11:33

chavita symbol gives us the components

11:35

of the volume form tensor Omega so in an

11:40

orthonormal basis the levy chibita

11:42

symbol alone gives us the components of

11:44

the volume form but in an arbitrary non

11:48

orthonormal basis we also need the

11:51

square root of the metric tensor matrix

11:53

determinant from that basis to get the

11:55

correct components of the volume form so

11:59

we've learned about a number of tensors

12:01

in this series like the metric tensor

12:03

Riemann tensor and Ricci tensor and we

12:06

know that when they act on vectors to

12:08

get the output values we just do

12:10

summations with the tensor components

12:12

the volume form is yet another tensor

12:15

and the components we use to do the

12:17

summation are the determinant of the

12:19

metric tensor times the leve chavita

12:21

symbol so now that we understand what

12:25

the volume form is let's take two

12:27

covariant derivatives of the volume form

12:30

along some geodesic path this will give

12:32

us the second rate of change of a volume

12:34

formed by a set of basis vectors as we

12:37

move along a geodesic path we'll find

12:40

that taking this derivative will lead us

12:42

to the Ricci tensor the first thing we

12:46

need to do is show that the covariant

12:48

derivative of the volume form is zero

12:50

this doesn't mean that volumes don't

12:53

change as they move along geodesics we

12:55

saw in the previous video that volumes

12:57

can in fact change along geodesics what

13:01

this does mean is that the rules for

13:03

measuring volumes don't change along

13:05

geodesics so to prove this we start with

13:08

a geodesic path and by taking a set of

13:11

vectors that form a box we then parallel

13:15

transport these vectors along the path

13:17

we've

13:19

in previous videos that the levy chavita

13:21

connection has the property of metric

13:23

compatibility this means that when we

13:26

parallel transport vectors along a path

13:28

using the levy chibita connection the

13:31

lengths of the vectors stay the same and

13:33

the angles between the vectors stay the

13:35

same this means that when we parallel

13:38

transport these vectors the volume

13:41

created by these vectors will not change

13:43

in size therefore the covariant

13:46

derivative of this volume is zero when

13:49

we apply this covariant derivative we

13:51

are actually applying it to a product of

13:54

four terms the volume form and each of

13:57

these three vectors using product rule

14:00

we get four terms each with the

14:02

covariant derivative applied to a

14:04

different part of the formula now since

14:07

these vectors a B and C are all parallel

14:10

transported this means that their

14:13

covariant derivative is zero by

14:15

definition and since Omega is a tensor

14:18

and a multi linear map when one of the

14:21

inputs is zero that means that the

14:23

entire output is zero this can be proven

14:27

using the multi linearity properties of

14:29

a tensor so we're left with the

14:32

conclusion that the covariant derivative

14:34

of the volume form must be equal to zero

14:36

in other words the derivative of the

14:39

volume form components along this path

14:42

are zero so knowing this let's look at

14:46

what happens when we apply two covariant

14:48

derivatives to an arbitrary volume

14:50

spanned by some vectors we know that the

14:53

volume of this shape formed by the u W

14:56

and T vectors is just the summation of

14:59

the volume form components summed with

15:01

the vector components now recall

15:05

previously when thinking about curvature

15:07

we looked at the separation vector

15:09

between two neighboring geodesics now

15:13

when you look at these three vectors

15:15

that create the volume u W and T I want

15:19

you to think of these as separation

15:21

vectors each keeping track of the

15:23

separation distance with its own

15:25

neighboring geodesic path so I'm going

15:29

to rename u W and T

15:31

as s1 s2 and s3 since they are all

15:35

separation vectors for some geodesic I'm

15:39

also going to replace the summation

15:41

indexes with mu1 mu2 and mu3 I realize

15:47

this notation might be a bit confusing

15:49

but it basically just means that up here

15:51

I is the first summation index mu 1 J is

15:56

the second summation index mu 2 and K is

16:00

the third summation index finally I'm

16:03

going to rewrite the product of these

16:05

three separation vector components using

16:08

this product notation where we do the

16:11

product of D separation vector

16:13

components where D is the dimension of

16:16

the space and if we take two covariant

16:19

derivatives of the volume along this

16:21

geodesic since the volume is a scalar

16:24

this is the same thing as the ordinary

16:26

derivative along the geodesic path using

16:29

the path parameter lambda and recall

16:32

that since the volume form has a

16:34

derivative of zero we can take the

16:36

volume form components and put them

16:39

outside the derivative so let's apply

16:42

this first derivative to the product of

16:44

the S vector components you should know

16:47

by now that using product rule on a

16:49

product gives us a sum of terms where

16:52

every term has the derivative applied to

16:54

a different factor I'm denoting the

16:57

derivative here by an overhead dot we

17:01

can do the same thing here when we take

17:02

the derivative of this product by

17:05

pulling the S J components outside of

17:08

the product and applying a dot to take

17:10

the derivative there is a summation over

17:13

J here so this means that we get a sum

17:16

of terms where in every term a different

17:19

SJ component gets the derivative this is

17:23

very similar to what we've written out

17:25

here where in each term a different

17:27

component gets the derivative dot now

17:30

for the second derivative we can think

17:32

of it as doing product rule on this

17:34

factor and then this factor the first

17:37

factor is easy we just apply the

17:39

derivative to s J dot and get sjws

17:45

for the second factor we can use these

17:47

same reasoning as above and pull out

17:49

another index out in front

17:51

SK this time and apply a dot for the

17:54

derivative and removing the K index from

17:57

the product now recall that taking two

18:01

derivatives of a separation vector along

18:03

a geodesic just gives us this negative

18:06

Riemann tensor term this means that this

18:09

separation vector component with two

18:11

derivatives equals this summation with

18:14

the Riemann tensor components and the s

18:16

and V vector components this s and V are

18:21

given in the Y and Zed summations and

18:23

this V is given in the X summation so

18:28

now we can sub this expression in for

18:30

the S double dot now here notice in this

18:34

product that all the MU in Dex's are

18:37

used except form you J and recall in the

18:42

Levy chavita symbol that if there are

18:44

any repeated indexes it goes to zero now

18:48

this S term out here has a superscript y

18:52

that's involved in a summation so it can

18:55

take on many different values like mu 1

18:57

mu 2 and so on but if we use mu 1 or mu

19:02

2 and it's already in the leve chavita

19:04

symbol the term will go to 0 because

19:07

there's a repeated index this means that

19:10

the only term that will end up being

19:12

nonzero in the summation occurs when y

19:15

equals mu J because mu J does not

19:18

already occur in this product so we can

19:22

set y equals mu J because all other

19:25

versions of y in this summation go to 0

19:28

and now since this product doesn't

19:31

include the J index we can simply take

19:34

the S component with the J index and put

19:37

it back in the product so we get this

19:39

where the product goes from I equals 1

19:41

to D without skipping any indexes but

19:45

you'll notice that all of this right

19:47

here is really just the original formula

19:49

that defined the volume of the

19:51

parallelogram shape we were originally

19:53

talking about it has the components of

19:56

the volume form and the product of

19:58

all these separation vector components

20:00

so all of this is just equal to the

20:03

volume capital B and notice that since

20:07

the Riemann tensor components have a

20:09

summation on the upper index and the

20:11

lower middle index this is actually just

20:14

the definition of the Ricci tensor XZ

20:17

and of course this summation just gives

20:20

us the Ricci curvature in the direction

20:22

of the e vector so the conclusion after

20:26

all of this is that the Ricci tensor

20:28

tells us how volumes change in size as

20:31

we move around in space along geodesics

20:34

the second order derivative of the

20:37

volume has a term that is proportional

20:39

to the original volume with the Ricci

20:42

curvature being the constant of

20:43

proportionality now you might be

20:46

wondering about this other term here

20:48

that involves the first derivative of

20:50

the separation vector components recall

20:53

in the last video that I said geodesics

20:56

spread apart for two different reasons

20:58

when they accelerate apart this is due

21:01

to the curvature of the space but when

21:04

they move apart at a constant velocity

21:06

this can occur in flat space because of

21:09

the way the geodesics are angled so this

21:12

first term in the second derivative with

21:14

the Ricci tensor detects how volumes

21:16

change due to the curvature of the space

21:19

while the second term measures the

21:22

volume changes that can occur in flat

21:24

space because of the aligning of

21:26

geodesics so the conclusion is that the

21:29

Ricci tensor tells us how volumes change

21:32

specifically because of the curvature of

21:34

the space were in when we're traveling

21:36

along geodesics recall that for the

21:40

sphere the Ricci curvature is always

21:42

positive and with this negative sign

21:45

here this means that volume is moving

21:47

along geodesics on the sphere will

21:49

always shrink in volume because the

21:51

geodesics on the sphere are always

21:53

converging the last thing I'll talk

21:57

about is the Ricci scalar which is the

22:00

scalar we get when we do a summation on

22:02

the Ricci tensor with one upper index

22:05

and one lower index and recall that

22:08

raising an index involves doing a

22:10

summation with the inverse metric

22:12

in the last video I said that the Ricci

22:15

scalar tells us how much the size of a

22:17

ball in curved space deviates from the

22:20

standard volume of a ball in flat space

22:22

so in 2d the standard flat disk with the

22:26

given circumference will contain an area

22:29

equal to PI R squared and this area is

22:32

less than the surface area of an

22:34

upside-down bowl shape that sits on the

22:37

top of a sphere even though the bowl has

22:39

the exact same circumference so given

22:42

this circular circumference we don't

22:44

actually know how much area will fit

22:47

inside the answer could change depending

22:49

on the curvature of space there's the

22:52

completely flat space area but the area

22:55

will increase slightly if this circle

22:57

sits on a very large ball which causes

23:00

the area to bulge a little and as the

23:03

sphere gets smaller and smaller the

23:05

amount of curvature increases and the

23:08

amount of surface area contained within

23:10

the perimeter also increases in the edge

23:14

cases when we have flat space the

23:16

surface area is PI R squared and in the

23:20

case where the circle is the boundary on

23:22

the equator of a sphere the surface area

23:25

is 2 PI R squared so we can see that the

23:29

surface area inside the circle can

23:31

double if the curvature of the space is

23:34

large enough now when it comes to the

23:37

Ricci scalar were generally interested

23:40

in comparing ball sizes by their radius

23:43

not by their circumference and when I'm

23:46

talking about the radius I don't mean

23:48

this distance I mean the radius as seen

23:51

from the point of view of someone

23:53

traveling from the North Pole of the

23:55

ball out to the edge of the circle so

23:58

this radius is really an arc length

24:00

given by the radius of the ball times

24:03

this angle Phi so let's start by getting

24:07

the surface area of this upside-down

24:09

Bowl shape on a sphere of radius capital

24:13

script R we can compute this by taking a

24:16

collection of thin disks of radius Rho

24:19

and integrating over their thickness D s

24:23

we know from trigonometry

24:25

tree that row is equal to the sphere

24:28

radius times the sine of this angle

24:30

theta we also know that the thickness of

24:34

each disc is equal to this tiny arc

24:37

length D s which is equal to the spheres

24:40

radius times the small angle D theta so

24:44

subbing those formulas into our area

24:47

integral and taking the constants out in

24:49

front we get to PI capital R squared

24:53

times the integral of sine theta D theta

24:56

from zero to Phi and remember Phi is the

25:00

angle measured from the vertical to the

25:03

edge of the upside-down Bowl shape the

25:06

antiderivative of sine is negative

25:08

cosine and subbing in the limits of

25:11

integration we get cosine of 0 which is

25:14

1 and since these negative signs cancel

25:17

out we end up with this formula for the

25:21

surface area of the bowl and recall that

25:24

we said that the radius of this circle

25:26

as seen by someone walking from the

25:28

North Pole to the edge of the bowl is

25:30

given by this arc length lowercase R and

25:34

by the familiar arc length formula since

25:37

lowercase R equals capital R Phi and

25:41

this means that the angle Phi is given

25:43

by the radius of the bowl as seen by

25:46

someone walking from the North Pole

25:48

divided by the radius of the entire

25:50

sphere so this here is the formula for

25:54

the surface area of the upside-down Bowl

25:57

as a function of the bowls radius as

25:59

measured using the walking distance from

26:02

the North Pole now let's compare the

26:05

surface area of this bowl shape against

26:08

the standard flat space area of a circle

26:10

PI R squared we can immediately cancel

26:13

these pies and to handle this cosine I'm

26:17

going to expand it as a Taylor series

26:19

where we get a summation of all the even

26:23

powers of little R over big R with

26:25

alternating positive and negative signs

26:28

this positive one and this negative one

26:31

cancel and if we distribute this

26:33

negative sign into the brackets

26:35

containing the Taylor series all these

26:38

signs end up

26:39

next I'll distribute this term out in

26:42

front to all the terms inside the Taylor

26:45

series in the first term we can see that

26:48

all the numerators cancel with all the

26:50

denominators and were left with just one

26:53

in the second term some of the powers

26:56

cancel and we're left with 1 over 24

26:59

times 2 over big R squared times little

27:03

R squared and I'm not going to bother

27:06

with the higher-order terms because they

27:09

aren't as important you might be

27:11

wondering why I'm keeping this 2 here

27:13

instead of writing 1 over 12 that will

27:16

become clear soon so now given that we

27:20

know the metric tensor and the Ricci

27:22

tensor for the sphere of radius capital

27:24

R let's compute these spheres Ricci

27:27

scalar so we have a double summation

27:30

over i and j for the metric and Ricci

27:33

tensor x' that gives us four terms but

27:36

two of them go to zero in the remaining

27:39

terms these sign terms in the numerator

27:42

and denominator cancel and we're left

27:44

with two over big R squared so we can

27:48

see that the Ricci scalar does appear in

27:51

the second order term in the Taylor

27:53

series expansion between the area of the

27:57

bowl on the sphere and the area of a

27:59

circle in flat space because this ratio

28:03

is a little less than 1 it means that

28:06

the surface area of the bowl is slightly

28:08

less than the area of a flat disc given

28:12

the same radius little R so this might

28:16

be a bit confusing but just to review

28:18

the sphere has positive curvature and

28:21

that means that for a given boundary

28:23

circle the surface area on the sphere

28:26

given by this bowl shape is greater than

28:29

the area of the flat circle with the

28:31

same circumference but for a given

28:35

radius the surface area of the bowl

28:38

shape on the sphere is less than the

28:41

area of the circle in flat space with

28:43

the same radius so when we think of

28:46

positive curvature it can either mean

28:48

more area for the same circumference or

28:52

it can mean less area for the same

28:54

radius in a negatively curved space it

28:58

would be the opposite where we get less

29:00

area for the same circumference or more

29:03

area for the same radius so this is what

29:07

the Taylor series looks like for the

29:09

ratio of circle areas in the specific

29:12

case where our curved space is a sphere

29:14

for general curved spaces in d

29:18

dimensions the ratio of the volumes has

29:20

this Taylor expansion where the

29:22

second-order coefficient is the Ricci

29:25

scalar divided by 6 times the number of

29:28

dimensions plus 2 you might ask where

29:31

this formula comes from but

29:33

unfortunately I don't know

29:35

I tried searching for a proof online but

29:38

I wasn't able to find a proof that I

29:40

understood so unfortunately I can't

29:43

prove this for you if I do find a proof

29:46

that's understandable I may upload

29:48

another video about it but for now all I

29:50

can do is give you some links in the

29:52

description to the complex proofs that I

29:54

don't understand

29:56

so to summarize these past two videos

29:58

we've learned two interpretations of the

30:01

Ricci tensor and how it tracks volume

30:04

changes along geodesics one

30:06

interpretation involves sectional

30:08

curvature and the other interpretation

30:10

involves the derivative of the volume

30:13

form the sectional curvature tracks how

30:16

neighboring geodesics either converge

30:19

with positive curvature or diverge with

30:22

negative curvature by measuring the

30:24

second rate of change of a separation

30:26

vector when we look at all these

30:29

sectional curvatures along geodesics in

30:31

every direction in an orthonormal basis

30:34

we can add all these sectional

30:36

curvatures together to get the Ricci

30:38

curvature the Ricci curvature tells us

30:41

the overall volume change as a ball

30:43

travels along geodesics the Ricci

30:46

curvature does not tell us about how the

30:48

ball changes shape but it will tell us

30:51

how the ball changes volume the Ricci

30:53

curvature can be calculated using the

30:55

components of the Ricci tensor for the

30:59

second interpretation we learned that

31:01

the volume of a box formed by a set of

31:04

vectors is given by

31:06

the volume form acting on those vectors

31:08

and the components of the volume form

31:11

tensor is given by the square root of

31:13

the determinant of the metric tensor

31:15

matrix multiplied by the levy chibita

31:18

symbol we found that the second

31:20

derivative of the volume due to the

31:23

curvature of space is given by the Ricci

31:25

tensor term which computes the Ricci

31:28

curvature finally we discussed the Ricci

31:31

scalar which tells us how the volume of

31:34

a given radius and curved space deviates

31:37

from the volume of a ball with the same

31:39

radius and flat space if the Ricci

31:42

scalar is positive the volume will be

31:44

smaller than expected for a given radius

31:46

as seen on the sphere if the Ricci

31:50

scalar is negative the volume will be

31:52

bigger than expected for that radius in

31:55

the next video I'll talk about the

31:57

algebraic properties of the Ricci tensor

31:59

and Ricci scalar as always if you enjoy

32:03

my videos and would like to leave a tip

32:05

you can check the description for a link

32:07

to my donations page thanks