Wolfram Alpha:

```The Essential Mathematics of General Relativity
-----------------------------------------------

We will begin our discussion with a review of vectors
and dual vectors in flat spacetime and then transition
the discussion to curved spacetime.

Coordinate Transformations in Flat Spacetime
--------------------------------------------

Vectors exist independently of coordinate systems.
For example, a velocity is a velocity regardless
of the coordinate system we choose.  If we are
moving in a car at 50 MPH in a westerly direction
the speedometer reads 50 MPH and a compass will
read W without any notion of an coordinate system.
Therefore, it is a real physical quantity.  Note:
A position vector is different.  Its direction will
depend on the coordinate system because it is defined
with respect to one coordinate system.  However, a
displacement vector (the difference between 2 points)
is coordinate independent.  If a coordinate basis is
chosen, then the components of vector with respect to
that basis can be written.  We can see this easily.

V = v1e1 + v2e2 + v3e3

Where,

v1, v2, v3 are the components (numbers).
e1, e2, e3 is the basis (i.e. i, j, k).

In Einstein notation this can be written as V = Vμeμ

It should be obvious that as the coordinate basis
changes, the components of the vector must change
accordingly to maintain the invariance. We will now
look at this in more detail.

Vectors
-------

Vectors, V, live in vector space, V.

x2
|   e2
|   ^
|   |
|   |
|    -----> e1
---------------- x1

Consider a vector, V:

V = Vμeμ -> V' = Vμ'eμ'

= Λμ'μVμeμ'

= (Λμ'μVμ)(Λσμ'eσ)

= (component)(basis)

Now, Λμ'μΛσμ' = δσμ so the implication is that Λσμ'
has to be the inverse of Λμ'μ.  So if the components
change in one way, the basis has to change in the
opposite way for ds to be invariant.  Thus,

Vμ -> Vμ' = Λμ'νdxν

and,

eμ -> eμ' = Λσμ'eσ

Mathematically, components are said to transform
contravariantly and bases are said to transform
covariantly.

Dual Vectors
------------

Dual vectors, ω, live in dual vector space, V*.
Like vectors They are invariant and can be
expressed in terms of components and dual basis
vectors.  They linearly 'eat' vectors to produce
scalars (more on this later).

x2
|   θ2
|   ^
|   |
|   |
|    -----> θ1
---------------- x1

We will define 'eating' in terms of basis vectors and
dual basis vectors as follows:

θμeν = δμν

This is interpreted as θμ acting on eν to produce a
scalar.

The bases transform as:

θμeν = δμν -> θμ'eν' = δμ'ν'

Now eν' = Λνν'eν and θμ' = Λμ'μθμ

Therefore,

δμ'ν' = (Λμ'μθμ)(Λνν'eν)

= Λμ'μΛνν'θμeν

So, in order for this to hold true:

Λμ'μ ≡ Λ-1

and,

Λνν' ≡ Λ

In spacetime the simplest example of a dual
vector is the gradient of a scalar function,
φ, which is given by:

ω = (∂φ/∂xμ)θμ

Proof:

We can rewrite ∂φ/∂xμ using the Chain Rule as:

∂φ/∂xμ -> ∂φ/∂xμ' = (∂xμ/∂xμ')(∂φ/∂xμ)

= λμμ'(∂φ/∂xμ)

λμμ' is the transformation law for dual vector
components so we can conclude that ∂φ/∂xμ is a
dual vector.

The Dot Product
---------------

Consider a dual vector, ω, eating a vector, V.

ωV = ωμθμvνeν

= ωμvνδμν

= ω0v0 + w1v1 + w2v2 + w3v3

This looks like the dot product.  Therefore, the dot
product is 'secretely' the act of taking one of the
vectors and turning it into a dual vector by the
application of the metric tensor i.e.,

A.B = ημνAμeμBνeν

= AνθμBνeν

= AνBνδμν

The Lorentz Transformation
--------------------------

It is conventional to write the inverse Lorentz
transformation as:

(Λαβ)-1 ≡ Λαβ

Consider a boost in the x direction.  The partial
boost matrix is:

-        -
B = |  γ   -βγ | = Bμν
| -βγ   γ  |
-        -

With inverse,

-      -
B-1 = | γ   βγ | = Bμσ
| βγ  γ  |
-      -

-----------------------------------------------------

Digression:

Consider a boost in the x direction:

-        -  -  -     -         -
|  γ   -βγ || ct | = |  γt - βγx |
| -βγ   γ  ||  x |   | -βγt + γx |
-        -  -  -     -         -

And the inverse transformation:

-      -  -   -     -           -
| γ   βγ || ct' | = |  γt' + βγx' |
| βγ   γ ||  x' |   | βγt' + γx'  |
-      -  -   -     -           -

From these we get:

x' = γx - βγct

x = γx' + βγct'

t' = γct - γβx

t = γct' + βγx'

-----------------------------------------------------

The components transform as:

v0' = Bμνvν = B00v0 + B01v1

= γv0 - βγv1

v1' = Bμνvν = B10v0 + B11v1

= -βγv0 + γv1

The basis transforms as:

e0' = Bμσeσ = B00e0 + B01e1

= γe0 + βγe1

e1 = Bμσeσ = B10e0 + B11e1

= βγe0 + γe1

The transformed vector is identical to the vector
before the transformation.

Proof:

v0'e0' + v1'e1' = (γv0 - βγv1)(γe0 + βγe1)
+ (-βγv0 + γv1)(βγe0 + γe1)

= (γ2v0e0 + βγ2v0e1 - βγ2v1e0 - β2γ2v1e1)
+ (-β2γ2v0e0 - βγ2v0e1 + βγ2v1e0 + γ2v1e1)

= (γ2v0e0 - β2γ2v1e1) + (-β2γ2v0e0 + γ2v1e1)

= γ2(1 - β2)v0e0 + γ2(1 - β2)v1e1

= v0e0 + v1e1

Now consider a rotation about the z axis.  The
partial rotation matrix is:

-    -
R(90°) = |  0 1 | = Rμν
| -1 0 |
-    -

With inverse,

-    -
R(-90°) = | 0 -1 | = Rμσ
| 1  0 |
-    -

The components transform as:

v1' = Rμνvν = R11v1 + R12v2

= v2

v2' = Rμνvν = R21v1 + R22v2

= -v1

The basis transforms as:

e1' = Rμσeσ = R11e1 + R12e2

= e2

e2' = Rμσeσ = R21e1 + R22e2

= -e1

e2                              e1' (= e2)
|                                |
|                                |
|               ->               |
|                                |
--------- e1       e2' ---------
(= -e1)

The transformed vector is identical to the vector
before the transformation.

Proof:

v1'e1' + v2'e2' = v2e2 + (-v1)(-e1)

= v1e1 + v2e2

Lorentz Transform as a Coordinate Transformation
------------------------------------------------

The Lorentz transformation is just a special case
of a coordinate transformation.  Again, consider a
boost in the x direction:

Λμ'μ
|
v
-                 -  - -     -                -
|  coshζ -sinhζ 0 0 || t | = |  coshζt - sinhζx |
| -sinhζ  coshζ 0 0 || x |   | -sinhζt + coshζx |
|    0      0   1 0 || y |   |          y       |
|    0      0   0 1 || z |   |          z       |
-                 -  - -     -                -

-  -
| t' |
= | x' |
| y' |
| z' |
-  -

We can write this as a coordinate transformation
as a JACOBEAN MATRIX:

-                              -
| ∂t'/∂t  ∂t'/∂x  ∂t'/∂y  ∂t'/∂z |
Λμ'μ = | ∂x'/∂t  ∂x'/∂x  ∂x'/∂y  ∂x'/∂z |
| ∂y'/∂t  ∂y'/∂x  ∂y'/∂y  ∂y'/∂z |
| ∂z'/∂t  ∂z'/∂x  ∂z'/∂y  ∂z'/∂z |
-                              -

-                 -
|  coshζ -sinhζ 0 0 |
= | -sinhζ  coshζ 0 0 |
|    0      0   1 0 |
|    0      0   0 1 |
-                 -

Alternatively, we can see this in another way:

t' = γct - γβx
x' = γx - βγct
y' = y
z' = z

Therefore,

∂t'/∂t = γc  ∂t'/∂x = -γβ  ∂t'/∂y = ∂t'/∂z = 0

And,

∂x'/∂t = -γβc  ∂x'/∂x = γ

In general we can write:

Λμ'μ = ∂xμ'/∂xμ

This produces the matrix:

-          -
|  γ  -γβ 0 0 |
Λμ'μ = | -γβ  γ  0 0 |
|  0   0  1 0 |
|  0   0  0 1 |
-          -

Now consider the inverse transform given by:

t = γct' + βγx'
x = γx' + βγct'
y = y'
z = z'

In this case we get:

∂t/∂t' = γc  ∂t/∂x' = βγ  ∂t/∂y' = ∂t/∂z' = 0

And,

∂x/∂t' = βcγ  ∂x/∂x' = γ

In general we can write:

Λμ'μ = ∂xμ/∂xμ'

This produces the inverse transformation matrix we

-         -
| γ  γβ 0 0 |
Λμ'μ = | γβ γ  0 0 |
| 0  0  1 0 |
| 0  0  0 1 |
-         -

Summary of Transformation Laws
------------------------------

Vector Components:  Vμ -> Vμ' = Λμ'μVμ

Basis vectors:  eμ -> eμ' = Λσμ'eσ

Dual Vector Components:  ωμ -> ωμ' = Λμμ'ωμ

Dual Basis vectors:  θμ -> θμ' = Λμ'σθσ

Vectors:  V -> V' = Λμ'μVμΛσμ'eσ

= V

Dual Vectors:  ω -> ω' = Λμμ'ωμΛμ'σθσ

= ω

Λσμ'Λμ'μ = δσμ

The direction of the transformation is always from
the bottom index to the top index.

Coordinate Transformations in Curved Spacetime
----------------------------------------------

Manifolds
---------

A CP manifold, M,  is a collection of points in
spacetime with a maximal atlas.

We can break up M into PATCHES.  All of M needs to be
covered.  We have only shown a few patches to illustrate
the concept.

A COORDINATE CHART (aka chart) is a subset of M with a
1-to-1 map, φ:

φ: Uα -> ℝn where ℝn is n-dimensional Euclidean space.
(Note: R ≡ ℝ in diagrams).

Such that the image of φ is open.  Open means the
interior of an n - 1 dimensonal closed surface (i.e.
the edges of the image are not included).  Note that
charts must have the same dimensionality as the
manifold.

The fact that manifolds look locally like ℝn, which
is manifested by the construction of coordinate charts,
introduces the possibility of analysis on manifolds,
including operations such as differentiation and
integration.  In other words φ(p) converts a point,
p, in U on M to a point with coordinates xμ in ℝn.
Thus,

φ(p) = (x1(p), x2(p) ... xn(p))

or, in general:

φμ(p) = xμ(p)

Where xμ(p) are COORDINATE FUNCTIONS.  They are scalar
functions that return the μth coordinate of P as a
real number.  In a fixed coordinate system, each of
the 4 coordinates, xμ, can be thought of as a function
on spacetime.  Therefore,

xμ = xμ(p)

For example:

. .             ---------
. uα  .          |       ℝ2|
.  p--.-- φ(p) --|--->xμ   |
. .            |         |
---------

This takes a point, p, in Uα on a 2D manifold, M2 and
sends it to (x,y) ∈ ℝ2.

An important thing to recognize is that if a manifold
can be covered by a single chart, we can have a global
coordinate system on M.  If it cannot be covered by a
single chart then it is not possible to have a system
of global coordinates.

Atlases
-------

An ATLAS on M is a collection of charts {(Uα,φ)} such
that:

1.  The union of all charts Uα is equal to M

2.  The charts are sewn together with CP TRANSITION
FUNCTIONS.

CP Transition Functions
-----------------------

Suppose that 2 charts overlap.

Now,

P3 = φ3(p)

= φ3 o φ4-1(P4): ℝn -> ℝn

Now, the interesting thing is that P3 and P4 are both
in ℝn.  Therefore, we are taking the point P4 in one
set of coordinates and expressing it as the point P3
in another set of coordinates.  In other words, this
is just a ccoordinate transformation.

P4 with coordinates (x4, y4, z4 ...)
-> P3 with coordinates (x3, y3, z3 ...)

The transition function, φ43 is defined as:

φ43 = φ3 o φ4-1

Where φ3 and φ4-1 are coordinate and inverse coordinate
functions respectively.

Now that we are dealing with functions from ℝn to ℝn
we can talk about using calculus, in particular we
can consider the derivatives of φαβ.  If the pth
derivative of all φαβ exists and is continuous then
φαβ is Cp.  If P = ∞ (i.e. C∞) then we call φαβ
smooth.

Maximal Atlas
-------------

It is possible that a manifold may be covered by
different choices of atlases.  In this case each
atlas is an equally good representation of the
manifold so they belong to the maximal atlas.
Therefore, a Cp n-dimensional manifold is a set,
M, with a maximal atlas.

In summary, we can construct an atlas of charts
and use the maps φα: uα -> ℝn to coordinatize M
on each chart.  We can then sew them together
with transition functions which are equivalent
to coordinate transformations.

Manifolds in SR and GR
----------------------

With curved manifolds it is important to remove
the idea that a vector stretches from one point to
another, but instead is just an object associated
with a single point (the tangent space - see next)
that facilitates the use of multivariable calculus.

Manifolds can then be equipped with a metric at each
point that allows the existence of the inner product.
Consider the infinitesimally small displacement
vector, ds:

ds2 = gαβdxαdxβ

= g00dx0dx0 + g01dx0dx1 + g10dx1dx0 + g11dx1dx1

If ds2 is always positive the manifold is RIEMANNIAN

If ds2 can take any value it is PSEUDO-RIEMANNIAN.

If ds2 takes the form:

ds2 = -dt2 + dx2 + dy2 + dz2

it is LORENTZIAN.

Because ds2 can have different signs in GR, the
manifolds are pseudo-Riemannian.

Armed with this knowledge we can now proceed with
a discussion of tangent and cotangent spaces on
manifolds.

Tangent Space
-------------

Consider an n-dimensional mamifold, M.

Consider a SCALAR field, ψ(p), that lives in the manifold.
γ(λ) maps λp in ℝ1 to the 1 dimensional curve, γ(λp), that
passes through a particular point, p, on the manifold.  So
as λp changes, P will move along the curve γ.  f is a scalar
function that maps any point, P, on the manifold to αp in
ℝ1.  Therefore, f takes any point on the manifold and
assigns to it a real number (i.e. a scalar).

Note, λ and α are not charts since they do not have the
same dimensions as the manifold.

We want to find how the arbitrary function, f, changes
as we change λp.  Since f, gives us αp this is equivalent
to seeing how αp changes with λp.  Thus,

df/dλ = d(f o γ)/dλ

This is called the DIRECTIONAL DERIVATIVE OF f ALONG THE
CURVE γ

Note: F o G ≡ F(G(x)) so the order of execution is from
right to left.

This is equivalent to the tangent at P along the curve
γ.  If we repeat this for other curves through P we wlll
construct set of directional derivatives that forms a
vector space of tangents at P called the TANGENT SPACE, Tp,
at p.  This tangent space is an example of a vector spaces.
The set of all the tangent spaces of a manifold is called
the TANGENT BUNDLE.

Now that we have a tangent space we can now make use of
charts to setup a basis and, therefore, coordinate
representations of vectors in that tangent space.

λp -> P -> xμ -> P -> αp

f o γ = f o φ-1 o φ o γ: λp -> αp

df/dλ = d[(f o φ-1) o (φ o γ)]/dλ

Now φ o γ: ℝ1 -> ℝn so we can call it (φ o γ)μ

Using the chain rule:

df/dλ = [∂(f o φ-1)/∂(φ o γ)μ][d(φ o γ)μ)/dλ]

Since (φ o γ)μ: ℝ -> ℝn we can call it xμ(λ)

Since (f o φ-1): ℝn -> ℝ we can call it f(xμ)

Therefore;

df/dλ = (∂f/∂xμ)(dxμ/dλ)

Or,

d/dλ = (∂/∂xμ)(dxμ/dλ)

Where,

(dxμ/dλ) are the components of the TV.

and,

(∂/∂xμ) = ∂μ are the bases of the tangent space ≡ eμ

d/dλ is a tangent vector in tangent space.  Therefore,
we should be able to generalize this for any vector, V,
in that space.  Therefore, we can write:

V = Vμ∂μ

The basis vectors in the primed frame are given by
the chain rule.  Therefore,

∂μ -> ∂μ' = (∂xμ/∂xμ')∂μ

Since V = Vμ∂μ must be invariant under a transformation
this implies:

Vμ∂μ = Vμ'∂μ'

= Vμ'(∂xμ/∂xμ')∂μ

Summarizing:

Directional Derivative:

d/dλ = (∂/∂xμ)(dxμ/dλ)

Transformation law for basis vectors:

∂μ -> ∂μ' = (∂xμ/∂xμ')∂μ

Transformation law for vector components:

Vμ -> Vμ' = (∂xμ'/∂xμ)Vμ

So, as in the case of flat spacetime, the basis
transforms in the opposite direction to the components
to preserve the invariance of the vector.

Basis vectors are usually not written explicitly with
the understanding that a change of coordinates induces
a change of basis.

-----------------------------------------------------
A slightly different view:

Consider a curve on a manifold. We can define a
displacement (separation) vector Δp along the curve.
Δp has to short so that we can assume that the surface
local to P is flat and we can use basic calculus to
figure out the tangent vector, A, to the curve at p.

The tangent vector to the curve, A, is given by:

A = dp/dε ≡ lim [P(ε + Δε) - P(ε)]/Δε
Δε->0

Any point on the curve can be written in terms of
coordinates, xα.

Coordinate basis:  eα ≡ ∂p/∂xα

Components of vector:  Aα = dp/dε ∂xα/∂ε

We can write A = dp/dε as:

A  = (dxα/dε)(∂p/∂xα)  .. 1.

≡ Aαeα

Therefore, we can identify the components of the
vector as:

Aα = dxα/dε

Directional Derivative
----------------------

Consider a scalar field, φ(p), that lives in the
manifold at position, p, on the curve.  The
derivative of φ along A, ∂Aφ, is given by:

∂Aφ(P(ε)) = Aα∂[φ(P(ε))]/∂xα

Or,

∂A = Aα∂/∂xα

= (dxα/dε)(∂/∂xα)  ... 2.

= (d/dε) along the curve.

If we compare 1. with 2. we see that there is a
direct 1-to-1 correspondence between the tangent
vector and the directional derivative.  Thus,

A ≡ ∂A = ∂p/∂ε = ∂/∂ε

Cotangent (Dual) Space
----------------------

In addition to tangent space it is also possible to
construct COTANGENT SPACES, Tp* at each point on the
manifold.  Just as vectors are defined as elements of
the tangent space, dual vectors are defined as elements
of the cotangent space.  The set of all cotangent
spaces over a manifold is called the COTANGENT BUNDLE.

A dual space, V*, is a vector space of linear maps
from a vector space, V, to its field of scalars.  The
maps are also referred to as linear functionals,
linear forms, one-forms, covectors or dual vectors.
In layman's terms a linear functionals at P 'eat' the
tangent vector at P producing a scalar.  Therefore,

ω: V -> ℝ

Where ω ∈ V* (aka Tp*) and V ∈ V (aka Tp)

Note that dim V = dim V*.

In tangent space we used charts to set up coordinates
that enabled us to define the basis for the tangent
vector along λ.  With cotangent space we use the
gradients of the scalar coordinate functions, φμ(p)
(≡ xμ(p)), denoted by (df), to determine the basis.
Note that dxμ is not the same as dxμ which is an
infinitesimal displacement vector.  Therefore, we
can write:

(df): TpV -> ℝ

Where (df) = (∂f/∂xμ)

Dual spaces have to satisfy the operation of addition
and scalar multiplication.

(df)(V + V') = (df)(V) + (df)(V')

(df)(αV) = α(df)(V)

We now let (df) 'act' on the tangent vector d/dλ.
Note 'act' in this case does NOT mean differentiate.
Therefore,

(∂f/∂xμ)(d/dλ)

Now by the definition of linearity:

(df)V = V o f    V ∈ V, f ∈ V*

Therefore, we get:

(df)(d/dλ) = (d/dλ) o f

= df/dλ

We now write the tangent vector in terms of its
components and bases.

df(dxμ/dλ)(∂μ) = df/dλ

We are free to write this as:

(dxμ/dλ)(df)(∂μ) = df/dλ

Now the RHS can be rewritten using the chain rule.

(dxμ/dλ)(df)(∂μ) = (dxμ/dλ)(∂f/∂xμ)

By comparison:

(df)(∂μ) ≡ ∂f/∂xμ

But f = xν(P), the coordinate functions, so:

(dxν)(∂μ) = ∂xν/∂xμ

= δμν

In conclusion,

θμ = (dxμ)

Now, we had said previously that dual vectors
'eat' vectors to produce scalars, i.e.

θμeν = δμν

Or,

dxμ∂ν = δμν

Therefore, for invariance,

dxμ∂ν = dxμ'∂ν'

So,

dxμ' = (∂xμ'/∂xμ)dxμ

If this is the transformation law for dual basis
vectors then the transformation law for dual
vector components has to be:

ωμ -> ωμ' = (∂xμ/∂xμ')ωμ

Comparison with Quantum Mechanics
---------------------------------

Dual spaces play an important role in Quantum Mechanics.
QM uses complex functions to represent vectors and dual
vectors.  Since vectors live in complex Hilbert space,
the dual vectors have to live in a conjugate space to
ensure that the resulting wavefunctions are scalars.
The state |α> does not represent a wave function until
it is projected onto a basis in position or momentum
space.  For example,

ψ(p) = <p|α> or ψ(x) = <x|α>

ψ is a scalar and <p|α> just specifies the 'components'
of |α> in that basis.

We can also project <α| onto a basis:

ψ*(p) = <α|p>* or ψ*(x) = <α|x>

= <p|α>* and ψ*(x) = <x|α>*

Likewise, the overlap of 2 states <φ|ψ> is a scalar
since probabilities have to be postive numbers.

Dual Space                   Vector Space
----------                   ------------

Dual vector                  Vector
One-form                     Tangent vector
Covariant vector             Contravariant vector
Row vector                   Column vector

Complex conjugate function   Complex function  (QM)

Summary of Transformation Laws
------------------------------

SR:  Vμ -> Vμ' = Λμ'μVμ and eμ -> eμ' = Λμμ'eμ

ωμ -> ωμ' = Λμμ'ωμ and θμ -> θμ' = Λμ'μθμ

Where Λμ'μ and Λμμ' allows global transformations
of (t,x,y,z).

GR:  Vμ -> Vμ' = (∂xμ'/∂xμ)Vμ and ∂μ -> ∂μ' = (∂xμ/∂xμ')∂μ

ωμ -> ωμ' = (∂xμ/∂xμ')ωμ and dxμ -> dxμ' = (∂xμ'/∂xμ)dxμ

Where ∂xμ'/∂xμ and ∂xμ/∂xμ' allows local or
coordinate dependent transformations of
(t,x,y,z).

Vectors:  V -> V' = Λμ'μVμΛσμ'eσ

= V

Dual Vectors:  ω -> ω' = Λμμ'ωμΛμ'σθσ

= ω

Λσμ'Λμ'μ = δσμ

Pullbacks and Pushforwards
--------------------------

Consider 2 manifolds M and N.

The map:

φ: M -> N

φ is called a DIFFEOMORPHISM if it is an invertible
function that maps one differentiable manifold to
another such that both the function and its inverse
are smooth.

The PULLBACK of the function, f, from N to M is:

f o φ  (≡ f(φ))

Which is written as:

φ*f

While we can pull functions back there is no
construction that allows them to be pushed forward.
However, a tangent vector is defined as Aμ∂μ.
This allows us to define the PUSHFORWARD of a
tangent vector, V, from M to N, as:

φ o f  (≡ φ(f))

Which is written as:

φ*V

The action of the pushed forward vector on f is
defined to be:

(φ*V)f = V(φ*f)

What this means is that pushing V forward to N
and acting on f is equivalent to pulling f back
to M and acting on it with V.

To see how this works in practice, apply the
pushed forward vector Vα∂α to an arbitrary
function, f.  Therefore,

(φ*V)α∂αf = Vμ∂μ(φ*f)

= Vμ∂μ(f o φ)

= Vμ(∂φ/∂xμ)(∂f/∂yα)

= Vμ(∂yα/∂xμ)(∂f/∂yα)

= Vμ(∂yα/∂xμ)∂αf

We can write this as:

(φ*V)α∂α = (φ*)αμVμ∂α

Where (φ*)αμ = ∂yα/∂xμ is a matrix that represents
the vector transformation law under a change of
coordinates.  When M and N are the same manifold
the constructions are identical and ∂yα/∂xμ = 1.

Example:

Consider the map:

-     -
φ(x,y) = |   xy  | or (xy,x + y)
| x + y |
-     -

And an arbitrary curve, γ(t), on the manifold,
M, defined as:

γ(t) = (1 + 3t,2 - 2t)

Here we have replaced the parameter, λ, with the
parameter, t.

If we plot γ(t) for different values of t we get
a straight line with slope = -2/3.  Taking the
differential of γ(t) w.r.t. t gives the vector
(3,-2) which we can define at the point (1,2),
i.e.

The image curve of γ(t), β(t), is:

β(t) = φ o γ = ((1 + 3t)(2 - 2t),(1 + 3t)

+ (2 - 2t))

= ((2 + 4t - 6t2),(3 + t))

β(0) = (2,3)

dβ(t)/dt = ((4 - 12t),1)

The tangent vector to the image curve at (2,3)
when t = 0 is (4,1).  Again, we define dβ(t)/dt
at (2,3). i.e.

Let us now compute this in a different way. We
let:

x = a(t), y = b(t)

where a(t) and b(t) are coordinate functions.
Therefore,

-           -
β(t) = φ o γ = |   a(t)b(t)  |
| a(t) + b(t) |
-           -

and,

-                           -
dβ(t)/dt = | a(t)db(t)/dt + b(t)da(t)/dt |
|     da(t)/dt + db(t)/dt     |
-                           -

-          -  -        -
= | a(t)  b(t) || db(t)/dt |
|  1     1   || da(t)/dt |
-          -  -        -

The first matrix is just the Jacobean:

-             -
J = | ∂fx/dx ∂fx/dy |
| ∂fy/dx ∂fy/dy |
-             -

-   -
= | x y |
| 1 1 |
-   -

The vertical sum of each column gives the
coordinates of the pushed forward vector.

x:  a(t) + 1 = 1 + 1 = 2

y:  b(t) + 1 = 2 + 1 = 3

To get the vector at (2,3) we apply the Jacobean
to the vector at (1,2):

-   -  -  -     - -
| 1 2 || -2 | = | 4 |
| 1 1 ||  3 |   | 1 |
-   -  -  -     - -

This can also be found as:

(3∂x - 2∂y)(xy) = 3y - 2x = 6 - 2 = 4

(3∂x - 2∂y)(x + y) = 3 - 2 = 1

Vectors can be pushed forward from M to N but
there is no construction that allows them to
be pulled back.  On the other hand, cotangent
vectors, ω, behave in just the opposite manner.
They can be pulled back from N to M but there is
no construction to allow them to push forward
from N to M.

The action of the pulled back dual vector on V
is defined as:

(φ*ω)V = ω(φ*V)

What this means is that pulling ω back to M and
acting on V to produce a scalar is equivalent to
pushing forward V to N and acting on it with ω.

We can repeat the above process and write:

(φ*ω)μ = (φ*)αμωα

Where (φ*)αμ = ∂yα/∂xμ is the same matrix as before
except that we now contract on a different index.

We can extend the idea of pulling back dual vectors
to tensors with an arbitrary number of lower indices.

One common occurrence of a map between two manifolds
is when M is actually a smooth embedded submanifold
of N.  Under these circumstances there is a map from
M to N which just takes an element of M to the same
element of N.  Therefore, every embedded smooth
submanifold inherits a metric from being embedded in
a Riemannian manifold.  For example, in String Theory
there are 2 manifolds that are involved in string
propagation.

- The spacetime in which the string propagates.

- The worldsheet of the string itself.

The 2 dimensional world sheet is considered to be
embedded in the spacetime manifold such that points
(σ1,σ1) on the worldsheet correspond to points in
the manifold xμ(σ1,σ1).  The worldheet then inherits
its metric from the metric on the ambient spacetime
manifold.

Orthonormal and Coordinate Bases
--------------------------------

Up until now we have defined the bases of tangent
and cotangent space as:

eμ = ∂μ and θν = dxν

Where we set up the basis vectors to point along
the coordinate axes.

In 2D this corresponds to the following diagram.

We can define the commutator [∂μeμ,∂νeν] = 0

There is nothing to stop us, however, from setting
up any bases we like.  We will choose these basis
vectors to be orthonormal (orthogonal and of unit
length) and denote them as ea and eb.

In 2D this corresponds to the following diagram.

The left is a flat space and the commutator equals
0.  However, when we move to a curved surface the
commutator no longer equals 0 but represents a
difference vector (A).

In general we can write:

[eσ,eρ] = cσρλeλ

Where cσρλ is the commutation coefficient.

There is a theorem that says a basis is a coordinate
basis if and only if cσρλ = 0.

Calculations are easier in the coordinate basis.
On the other hand, the orthonormal basis is the
most familiar to people and is better for physical
interpretation.  For example, Quantum Mechanics,
SR and classical physics generally are developed
in terms of orthonormal bases.  From the above
diagram it should be clear that we cannot just
impose an orthonormal basis on curved spacetime.
What we need, therefore, is a mechanism that
enables us to move from one coordinate system to
the other.  The approach in GR that replaces the
coordinate basis with an orthonormal basis is
referred to as the TETRAD FORMALISM.  This was
proposed by by Albert Einstein in 1928.

-------

In 4 dimensions, the set of vector fields in an
orthonomal basis is called a TETRAD (aka VIERBEIN
or FRAME FIELD).

We can express the coordinate basis in terms of

Vectors:  eμ = eaμea or ea = eμaeμ

One-forms:  θμ = eμaθa or θa = eaμθμ

Components of a vector written in a coordinate
basis can be written in an orthonormal basis as:

Va = eaμVμ

alternatively,

Vμ = eμaVa

The components of eaμ etc. form n x n invertible
matrices where the Latin indeces represent the
orthonormal basis and the Greek indeces represent
the coordinate basis.  If the distinction between
a vector and its components is dropped, eaμ etc.
are regarded as the vierbeins.  Naturally, they
are functions of position on the manifold.  They
have the following properties:

eμa = (eaμ)-1

eμaeaν = δμν  eaμeμb = δab

g(ea,eb) = ea.eb = ηab

g(eμ,eν) = eμ.eν = gμν

Now,

ea.eb = eμaeνbeμ.eν

= eμaeνbgμν

Therefore,

ηab = eμaeνbgμν

or equivalently,

gμν = eaμebνηab

This last result expresses the fact that eaμ and
ebν are very much like √gμν.

Tetrads can be looked at as linear maps from
tangent space to Minkowski space that preserves
the inner product.  The fact that we can write
the Minkowski metric in terms of gμν gives us the
freedom to perform LOCAL LORENTZ TRANSFORMATIONS
at every point in space.  Therefore,

Λa'aΛbbηab = ηa'b'

ea -> ea' = Λa'aea

θa -> θa' = Λa'aθa

Where (Λ-1)ν'μ = Λν'μ

We also retain the freedom to make general
coordinates transformations at the same time.
Therefore, we can define a mixed tensor
transformation law as follows:

Ta'μ'b'ν' = Λa'a(∂xμ'/∂xμ)Λb'b(∂xν/∂xν')Taμbν

Spin Connection
---------------

The introduction of tetrads requires changing
the covariant derivative.  ∇μeaν must transform
like a (1,1) tensor and  a 4-vector.  In order
to account for this we need to modify the idea
of a connection.

Now ∇μηab ≡ ∂μηab = 0 so we can impose the same

∇μeaν = 0

∇μeaν = ∂μeaν - Γρμνeaρ + ωabμebν = 0

Multiply through by eνb to get:

ωabμ = eνbΓρμνeaρ - eνb∂μeaν

ωabμis called the SPIN CONNECTION.

We can define a curvature associated with this
connection:

Rμνab = ∂μωνab - ∂νωμab + ωeμaωνeb - ωeνaωμeb

Note that:

eaρebαRμνab = Rμνρα

Which is the Riemann tensor.

------------

Tetrads provide the ability to describe spinor
fields on spacetime and take their covariant
derivatives.  They are the most natural way to
represent a relativistic quantum field theory
in curved space.  It is also gauge field theory
for gravity with the tetrad playing the role of
a gauge field but not exactly like the vector
potential field does in Yang-Mills theory.

Covariant Derivative
--------------------

We have established the transformation laws for
vectors and dual vectors as well as scalars and
partial derivatives.  In summary,

Scalars: φ -> φ

Gradients of scalars:  ∂φ/∂xμ -> ∂φ/∂xμ'

= (∂φ/∂xμ)(∂xμ/∂xμ')

Partial derivatives:  ∂μ -> ∂μ' = (∂/∂xμ')∂μ

Vectors: Vμ -> Vμ' = (∂xμ'/∂xμ)Vμ

Dual vectors:  Vμ -> Vμ' = (∂xμ/∂xμ')Vμ

Tensors:  Tμν -> Tμ'ν' = (∂xμ/∂xμ')(∂xν/∂xν')Tμν

THe question is how do the derivatives of vectors
and tensors transform?.  Consider, a vector:

∂μVν = lim [Vν(xμ + εμ) - Vν(xμ)]/εμ
ε->0
The problem is that Vν(xμ + εμ) and Vν(xμ) live in
different tangent spaces so the coordinate system
is not constant.  This problem manifests itself
as follows:

∂μVν -> ∂μ'Vν' = (∂xμ/∂xμ')∂μ(∂xν'/∂xν)Vν

= (∂xμ/∂xμ')(∂xν'/∂xν)∂μVν
+ (∂xμ/∂xμ')Vν∂μ(∂xν'/∂xν)

The first term is a tensor but the second is not.
Therefore, since ∂μ'Vν' does not ransform like a
tensor it is not a tensor.  We can write the above
equation as follows:

∇μVν = ∂μVν + ΓνμλVλ

Where ∇μ is the COVARIANT DERIVATIVE and  Γνμλ
are the CHRISTOFFEL SYMBOLS.

The Christoffel symbols solves the problem by
taking Vν(xμ + εμ) and PARALLEL TRANSPORTING it
back to xμ before performing the subtraction.
Therefore,

∇μVν = lim [V//ν(xμ + εμ) - Vν(xμ)]/εμ
ε->0

Note.  ∇μ of a scalar is the same as ∂μ.   Thus,

∇μφ = ∂μφ

How do the symbols transform?

∇μVν -> ∇μ'Vν' = (∂xμ/∂xμ')(∂xν'/∂xν)∇μVν

Γν'μ'λ' = (∂xμ/∂xμ')(∂xν'/∂xν)(∂xλ/∂xλ')Γνμλ

+ (∂xμ/∂xμ')(∂xλ/∂xλ')(∂2xν'/∂xμxλ)

The first term is a tensor but the second term
is not.  It is this second term that counters
the second term in the covariant derivative to
produce a result that is a tensor.

Computing the Christoffel Symbols
---------------------------------

Before we do this we need to consider 2 things -
torsion and metric compatibility.

Torsion Tensor
--------------

The torsion tensor is defined as:

Tμνλ = Γλμν - Γλνμ

Torsion is a difficult concept to explain.  Loosely
speaking it characterizes how tangent spaces twist
about a curve when they are parallel transported
whereas curvature describes how the tangent spaces
roll along the curve.

Suffice to say that in GR we assume the torsion to
be 0 meaning that the Christoffel symbols are
symmetric under the interchange of their lower
indeces.  Likewise the metric is also symmetric
under the interchange its indeces.

Metric Compatibility
--------------------

A connection is metric compatible if the covariant
derivative of the metric with respect to that
connection is everywhere zero, i.e.,

∇ρgμν = 0

Now we're ready to compute the Christoffel symbols.
Lets expand out the equation of metric compatibility
for three different permutations of the indices:

1.  ∇ρgμν = ∂ρgμν - Γλρμgλν - Γλρνgμλ = 0

2.  ∇μgνρ = ∂μgνρ - Γλμνgλρ - Γλμρgνλ = 0
^
|
≡ Γλρμgλν

3.  ∇νgρμ = ∂νgρμ - Γλνρgλμ - Γλνμgρλ = 0
^
|
≡ Γλρνgμλ

1. - 2. - 3. gives:

∂ρgμν - ∂μgνρ - ∂νgρμ + 2Γλμνgλρ

Γλμν = (1/2)(-∂ρgμν + ∂μgνρ + ∂νgρμ)/gλρ

Noting that 1/gλρ = gσρ (i.e. gλρgσρ = δσλ)

Γσμν = (1/2)gσρ(∂μgνρ + ∂νgρμ - ∂ρgμν)

Note that Γσμν -> 0 as we get closer and closer to
the local coordinate frame where the derivative of
the metric = 0 and the space can be considered to
be flat (tangent space).  Γσμν = 0 does not by
itself imply that the overall space is flat.  This
determination is made by examining the 2 second
derivative of the metric as discussed above.

Covariant Directional Derivative
--------------------------------

Recall, for a scalar the directional derivative is:

d/dλ = (dxμ/dλ)∂μ

For a vector/tensor this becomes we can define the
DIRECTIONAL COVARIANT DERIVATIVE as:

D/dλ = (dxμ/dλ)∇μ

Therefore, for a vector, Vν,

DVν/dλ = (dxμ/dλ)∇μVν

= lim [V//ν(xμ(λ + δλ)) - Vν(xμ(λ))]/Δxμ
Δxμ->0

∇μVν picks a coordinate, μ, and shifts Vν along xμ.

DVν/dλ shifts Vν along the curve xμ(λ).

Parallel Transport
------------------

From the above we have the CDD:

DVν/dλ = (dxμ/dλ)∇μVν

To transport Vν(xμ + εμ) along xμ to P requires
keeping the vector as PARALLEL TO ITSELF AS POSSIBLE
as it is transported.  This condition implies that:

DVν/dλ = (dxμ/dλ)∇μVν = 0

= (dxμ/dλ)∂μVν + Γνμρ(dxμ/dλ)Vρ = 0

In other words, the components of the tangent vector
dxμ/dλ don't change as it is transported.

Now, (dxμ/dλ)∂μ is just the ordinary DD so we can
write:

dVν/dλ + ΓνμρVρ = 0

This is a differential equation that can be solved
for any point on xμ(λ).

Geodesics
---------

Geodesics are paths taken by objects under the
influence of gravity.  They are curves xμ(λ) that
parallel transport their tangent vectors.  In
other words, if xμ(λ) is a geodesic then dxμ/∂λ
(the components of the tangent vector) should be
constant along the path (i.e. should be parallel
to each other.

The 4-velocity, uμ, along the curve is given by the
directional derivative:

uμ = dxμ/dλ

The covariant directional derivative of uμ along
the curve is given by:

Duμ/dλ = D(dxμ/dλ)/dλ

The fact that the components of the tangent vector
should be constant implies D(dxμ/dλ)/dλ = 0.
Therefore,

D(dxμ/dλ)/dλ ≡ (dxν/dλ)∇ν(dxμ/dλ)

= (dxν/dλ)[∂ν(dxμ/dλ) + Γμνα(dxα/dλ)]

= (dxν/dλ)∂ν(dxμ/dλ) + Γμνα(dxν/dλ)(dxα/dλ)

= (dxν/dλ)(∂(dxμ/dλ)/∂xν)
+ Γμνα(dxν/dλ)(dxα/dλ)

= d2xμ/dλ2 + Γμνα(dxν/dλ)(dxα/dλ)

= 0
Therefore,

d2xμ/dλ2 = -Γμνα(dxν/dλ)(dxα/dλ)

If we now substitute the proper time, τ, for
λ to create a world line we get:

d2xμ/dτ2 = -Γμνα(dxν/dτ)(dxα/dτ)

Flatness and Curvature
----------------------

Consider the 2 sphere which represents the surface
of a sphere (not the interior).

The 2 sphere is not flat and has the metric:

-         -
gμν = | R2    0    |
| 0  R2sin2θ |
-         -

Which can be written as a line element as:

ds2 = R2dθ2 + R2sinθ2dφ2

This is derived using spherical coordinates as
follows:

x = Rsinθcosφ

y = Rsinθsinφ

z = Rcosθ

ds2 = dx2 + dy2 + dz2

ds2 = dR2 + R2dθ2 + R2sin2θdφ2

Consider a point, P, at the North pole where θ = 0.
If we do this we get a horrible metric.

-    -
gμν = | R2 0 |
| 0  0 |
-    -

Which is degenerate.  We need to find a 'good' set
of coordinates that gets rid of this degeneracy.
Such coordinates are referred to as RIEMANN NORMAL
COORDINATES.

The basic idea behind Riemann normal coordinates is
to use the geodesics through a given point to define
the coordinates at a nearby point that lies on the
same geodesic.

We choose a point, P, to be at xμ(0) in the tangent
space.  The components of the tangent vector at P
are given by:

Vμ = dxμ/dλ|λ= 0

We now push Vμ along the geodesic in n tiny increments
to the nearby point, Q.  We can write this as:

λQ = lim (1 + λ/n)n
n->∞

~ 1

So we have increased λ by 1 unit of distance.  The
change from xμ(0) to xμ(1) gives a map from the set
of tangent vectors at P to points in the manifold,
and locally this map is one-to-one.  In this way we
can use Vμ to define a local coordinate system on the
manifold near P.  Now consider the geodesic equation:

d2xμ/dλ2 = -Γμνα(dxν/dλ)(dxα/dλ)

If the space at P is flat Γμνα = 0 and d2xμ/dλ2 = 0

Geodesics that satisfy this differential equation
have the form:

xμ(λ) = λVμ

Therefore, any geodesic that passes through P allows
us to define a local coordinate system on the manifold.

Now lim (1 + λ/n)n is also the definition of the
n->∞
exponential function so the map from vectors at P
to points in the manifold is called the EXPONENTIAL
MAP.

exp: V in Tp -> M

In other words,

expP(Vμ) = xμ(1)

A 'better' set of coordinates can be found as
follows.  Consider a point very close to the
North pole where sinθ ~ θ and the z coordinate
can be regarded as constant.  By doing this we
are effectively emulating the exponential map
discussed above.  Therefore, we can approximate x
and y as:

x = Rθcosφ

y = Rθsinφ

So that:

x2 + y2 = R2θ2(sin2φ + cos2φ)

Therefore,

θ = √(x2 + y2)/R

and,

dθ = (1/2)(x2 + y2)-1/2[2xdx + 2ydy]/R

= (xdx + ydy)/R√(x2 + y2)

y/x = tanφ

Therefore,

φ = arctan(y/x)

d(arctan(y/x))= -ydx/(x2 + y2)

and

d(arctan(y/x)) =  xdy/(x2 + y2)

Therefore,

dφ = (xdy - ydx)/(x2 + y2)

The line element for the 2 Sphere is:

ds2 = R2 + R2dθ2 + R2sin2θdφ2

For constant R we get:

ds2 = R2dθ2 + R2sin2θdφ2

Taking this term by term and substituting gives:

R2dθ2 = R2(xdx + ydy)2/R2(x2 + y2)

= (x2dx2 + y2dy2 + 2xydydx)/(x2 + y2)

and,

R2sin2θdφ2 = R2sin2(√(x2 + y2)/R)(x2dy2 + y2dx2

- 2xdydx)/(x2 + y2)2

Therefore, R2dθ2 + R2sin2θdφ2 equals:

[x2/(x2 + y2) + R2sin2(√(x2 + y2)/R)(y2/(x2 + y2)2]dx2

+

[y2/(x2 + y2) + R2sin2(√(x2 + y2)/R)(x2/(x2 + y2)2]dy2

-

2[xy/(x2 + y2) - R2sin2(√(x2 + y2)/R)(xy/(x2 + y2)2]dydx

Now sin2a = a2 - a4/3 + .... (Taylor series)

Therefore,

sin2(√(x2 + y2)/R) = (√(x2 + y2))2/R2 - (√(x2 + y2))4/3R4 ...

For the dx2 term we get:

[y2/(x2 + y2) + R2{(√(x2 + y2))2/R2
- (√(x2 + y2))4/3R4)}(y2/(x2 + y2)2]dx2

= (1 - y2/3R2)dx2

Similarly, for the dy2 we get:

(1 - x2/3R2)dy2

and for the dxdy term we get:

(2xy/3R2)dxdy

Therefore, the line elements becomes:

ds2 = (1 - y2/3R2)dx2 + (1 - x2/3R2)dy2 + (2xy/3R2)dxdy

and the metric looks like:

-                     -
gμν = | 1 - y2/3R2   2xy/3R2   |
|  2xy/3R2    1 - x2/3R2 |
-                     -

At P, x = y = 0 the metric becomes:

-   -
gμν = | 1 0 |
| 0 1 |
-   -

and the first derivative vanishes.

What this tells us is that just looking at the metric
and its first derivative to see it vanishes, doesn't
tell us if the overall space we are working in is flat.
It is possible to be working in a local coordinate
system without knowing it ahead of time, and that the
space is in fact curved.  However, in general the second
derivative of the metric will not vanish if the space is
curved even if you happen to be in a local coordinate
system.  For example, ∂2(1 - x2/3R2)/∂x2 = 2/R3 ≠ 0 for
any x.  If the second derivative does vanish then it
means the the space is truly flat.  This is the reason
why the second derivative of the metric plays a key
role in the structure of the Riemann curvature tensor
that we will discuss next.

Riemann normal coordinates, with the associated basis
vectors constitute a local Lorentz frame.  They are
the best approximation to flat space that is available.
The fact that the metric at P looks like that of flat
space to first order support the idea that small enough
regions of spacetime look like Minkowski space.  This
implies that locally all physics looks the same, which
is the famous equivalence principle.

The Riemann Curvature Tensor
----------------------------

Parallel transport can detect curvature.  Previously
we have talked about curvature how this might be
measured using the second derivative of the metric.
We now look at this in more detail.  Consider the
following schematic representation of the parallel
transport of a tangent vector, VAλ, around a square
in flat space.

VAλ, VA'λ  -Δxν
A <-------------
|              ^
|              |
Δxμ |              | -Δxμ
|              |
v              |
-------------> B VBλ, VB'λ
Δxν

Let Δxμ, Δxν, -Δxμ and -Δxν act as infinitessimal
operators that do the shifting.  Moving counter-
clockwise we get:

(-Δxν)(-Δxμ)(Δxν)(Δxμ)VAλ = VA'λ

Therefore,

VA'λ - VAλ = δVλ

Alternatively, we can split this into 2 steps

Counter-clockwise from A to B:  (Δxν)(Δxμ)VAλ = VBλ

Clockwise from A to B:  (Δxμ)(Δxν)VAλ = VB'λ

Therefore,

VB'λ - VBλ = [(Δxμ),(Δxν)]Vλ = δVλ

We can do the same thing with the 2-Sphere which is
a curved surface.

So again the idea is to transport the tangent vector
in suvh a way that it remains as parallel as possible
from point to point (the vector has to lie in the
tangent plane and cannot protrude above it.).  If
this operation is perform on a manifold that is not
flat, the original vector and the transported vector
will not coincide if brought back to the same point
along a closed trajectory.

Now, we have previously seen that the covariant
derivative parallel transports a vector along a
curve.  Therefore, we can replace the Δs with ∇s.

[∇μ,∇ν]Vλ = δVλ

If we expand this out we get:

[∇μ,∇ν]Vλ = (∂μΓλνρ - ∂νΓλμρ + ΓλμαΓανρ - ΓλναΓαμρ)Vρ

= RλρμνVρ

Rλρμν is the RIEMANN CURVATURE TENSOR.  It is comprised
of Christoffel symbols and their partial derivatives.
In turn, the Christoffel symbols are comprised of the
metric tensor and its derivatives.  Therefore, the
Riemann tensor consists of 2nd derivativeS of metric
which, from the previous discussion, is a measure of
the curvature.  The Riemann tensor can be shown to
transform in the following way proving indeed that it
is a tensor.

Rλρμν -> Rλ'ρ'μ'ν' = (∂xλ'/∂xλ)(∂xρ/∂xρ')(∂xμ/∂xμ')(∂xν/∂xν')Rλρμν

Rλρμν

ρ corresponds to the transported vector

μ and μ correspond to the paths taken.

λ corresponds to the components of the difference vector.

If one is interested only in certain aspects of curvature,
it is possible to form other tensors from the Riemann
tensor.  The simplest of those is the Ricci tensor,
which is formed by contracting the upper index with
the lower middle index.  This the trace of the Riemann
tensor.

Symmetries
----------

The Riemann tensor has the following symmetries:

Rαρμν = gαλRλρμν

Rαρμν = -Rαρνμ

Rαρμν = -Rραμν

Rαρμν = Rμνρα

Rαρμν + Rανρμ + Rαμνρ = 0

The original Riemann tensor has 256 components but
after these symmetries have been applied, the number
of independent components reduces to 20..

Geodesic Deviation Equation
---------------------------

We start by looking at the geodesic deviation under
Newtonian gravity.  Consider the following diagram.

At the P the geodesics are parallel and the separation
vector, ξ is, orthogonal to both geodesics.
Therefore,

dξ/dt = 0 and ξ.t = 0

The acceleration can be written as:

d2ξ/dt2 = -𝕋(_,ξ)

𝕋(_,ξ) is referred to as the TIDAL TENSOR.

Alternatively, we can write this in component (index)
form as:

d2ξj/dt2 = -Tjkξk

The equation of motions of particles moving along
A and B are given by:

(d2xj/dt2)A = -(∂φ/∂xj)A  ... 1.

and,

(d2xj/dt2)B = -(∂φ/∂xj)B

Where φ is the gravitational potential.  Note that
φ = mgh ∴ F = ma = φ/d so the RHS is equivalent to
a force as one would expect.

We can expand (φ)A in terms of ξ using a Taylor
series as follows:

(φ)A = (φ)B + ξ(∂φ/∂xj)B

If we substitute this into 1 we get:.

(d2xj/dt2)A = -(∂φ/∂xj + ξ∂2φ/∂xj∂xj)B

Now, d2ξ/dt2 = (d2xj/dt2)A - (d2xj/dt2)B so

Therefore,

d2ξ/dt2 = -(∂φ/∂xj + ξ∂2φ/∂xj∂xj)B + (∂φ/∂xj)B

= -ξ∂2φ/∂xj∂xj

d2ξj/dt2 = -(∂2φ/∂xj∂xk)ξk

This is the geodesic deviation in 3D space.  Lets
now see how this looks in General Relativity. We
replace d with D which is the covariant directional
derivative of ξ along τ to get:

D2ξ/dτ2 = -R(_,u,ξ,u)

Where u is the 4-velocity = dx/dτ and the geodesic
is treated as a world line parametized by τ instead
of λ.

Note that R(_,ξ,u,u) doesn't work because of the
asymmetry in these indeces.

In component form this is:

D2ξα/dτ2 = -Rαβγδ(dxβ/dτ)ξγ(dxδ/dτ)

The Riemann tensor produces the rate of change of
separation between neighbouring geodesics over time.
This is a vector since it has both magnitude and
direction.  It is the link between curvature and
a force (≡ accelerration) that causes the geodesics
to converge. These are the tidal forces of GRAVITY.

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