Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists
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62
Chapter
3.
Differentiable Manifolds I
These
are
just
the
following two
empty-space
Maxwell equations:
~ ~
aE
\1xB=-
at
(3.64)
Thus
in
differential form language,
the
free Maxwell's
equations
are
sum-
marized
in:
dP
=0
d*P
=
0
(identity)
(dynamical equation)
(3.65)
In
realistic physical
situations
the
second
equation
acquires a
term
on
the
right
hand
side
representing
an
electrical charge
or
current
source.
Finally,
note
that
the
fundamental
concept
of
gauge invariance
in
Max-
well's
equations
has
a simple
interpretation
in
terms
of
differential forms.
If
we
change
the
vector
potential
by
A
-t
A'
=
A
+
dA (where A is a function,
or
O-form),
then
P
-t
p'
=
dA
'
=
d(A
+
dA)
=
dA
=
P
(3.66)
because
d
2
=
O.
This
is gauge invariance for free electrodynamics.
3.7
More
About
Vectors
and
Forms
Here we will display some
more
properties
of
the
structures
defined above.
In
particular,
this
will lead
to
a
binary
operation,
the
"Lie bracket" , which
maps
a
pair
of
vector
fields
to
a new vector field.
Choose a vector field
X
=
ai(x)
a/axi,
and
a 1-form
W
=
Wi(X) dxi
Recall
that
X
acts
on
a function
f
by
X : f
-t
X f
to
give a new function
a
i
(x)
g!i
.
Also,
the
inner
product
between
vector fields
and
I-forms
is a function:
(3.67)
Now given two vector fields,
we
can
act
with
them
successively
on
a func-
tion, in two different orders,
and
find
the
commutator
of
the
two actions.
Thus,
if
X
=
ai(x)
a/ax
i
, Y
=
bi(x)
a/ax
i
,
consider
f-tYf-tXYf=a'-.
bl-.
. a ( . a
f
)
ax'
ax]
(3.68)
and,
f
-t
X f
-t
Y X f
=
b
i
~
(a
j
a
f )
ax'
ax]
(3.69)
Computing
the
difference
of
these
two
operations:
XYf
-
YXf
=
(a
i
ab1
_
bi
aa
J
)
af
ax'
ax'
ax]
(3.70)
3.7. More
About
Vectors
and
Forms
63
This
defines a new vector field:
Definition:
The
Lie bracket
of
two vector fields
X
and
Y
is
the
vector
field
[X,
Y]:
j
-?
[X,
Ylf
=
XYj
-
YXj
(3.71)
[X,
Y]
is sometimes also
denoted
LxY,
the
Lie derivative
of
Y
along
X.
In
components,
(
.
8bi . 8a
j
) 8
[X,
Y]
=
a'
-8
. -
b'
-8·
-8·
x' x'
Xl
(3.72)
The
Lie
bracket
gives a Lie
algebra
structure
to
the
tangent
space.
This
arises from
the
the
following (easily derived) identities:
[X,Y]
=
-[Y,
X]
[Xl
+
X
2
,
Y]
=
[Xl,
Y]
+
[X2'
Y]
[eX,
Y]
=
c[X,
YJ,
c
E
IR
(3.73)
as
well as
the
Jacobi
identity:
[[X,
YJ,
Z]
+
[[Y,
ZJ,
X]
+
[Z,
[X,
Y]]
=
0
(3.74)
Next, we prove
an
interesting
identity
about
2-forms. Since a
I-form
maps
tangent
vector fields
to
real
functions,
W:
X
-?
(w,X),
a 2-form will do
the
same
for a
pair
of
vectors:
D
=
Dij(X) dx
i
1\
dx
j
Now
if
we
are
given a
I-form
wand
two
vector
fields
X
and
Y:
. 8
X=a'(x)-8·'
x'
then
dw
is a 2-form
and
we have
the
identity:
. 8
Y
=
b'(x)
-8
.
x'
(3.75)
(3.76)
(3.77)
I
(dw;X,
Y)
=
2{X(w,
Y)
-
Y(w,X)
-
(w,
[X, Y])}
(3.78)
The
proof
follows
by
expanding
both
sides:
I . .
LHS
=
2(8
i
wj
- 8jWi) a'
bl
(3.79)
64
Chapter
3.
Differentiable Manifolds I
The
Lie
bracket
of
vector fields
appears
in
the
last
term
in
this
identity,
and
indeed
the
identity
can
be
turned
around
to
provide
an
alternative
definition
of
the
Lie bracket.
Chapter
4
Differentiable Manifolds II
4.1
Riemannian
Geometry
So far we have
studied
differentiable manifolds
without
assigning
any
metric
to
them.
The
properties
of spaces
that
we have discussed so far - continuity for
topological spaces,
and
differentiability for manifolds - do
not
require
the
as-
signment of a metric. However, manifolds
equipped
with
a
notion
of
distance
are
fundamental
in physics, so we now
turn
to
the
study
of
Riemannian
manifolds,
namely manifolds
with
a metric.
Since a manifold is only locally like Euclidean space,
we
will have
to
start
by working locally. Moreover we
must
make
sure
that
whatever
distance we
define between two
points
will
not
depend
on
the
specific
coordinate
systems
around
the
points.
On
IR
n
with
Cartesian
coordinates
l
XM,
there
is a
natural
notion
of
distance
between two infinitesimally
separated
points
x
M
and
x
M
+
bx
M
given by
We
can
use
this
fact
to
define a distance
on
a general differentiable manifold
M.
Here,
in
a given
chart
with
coordinates
XM,
consider two
points
p, q
whose
coordinates differ infinitesimally:
Suppose
another
chart
with
coordinates
yM
overlaps
with
the
first one.
Then
on
the
overlap
of
charts,
the
displacement
bx
M
is given in
terms
of
byM
by
the
chain rule:
S:"
ox
M
s:
v
uX""
=
--uy
oyv
( 4.1)
1
In
this
section
we use
/-L
=
1,2,·
..
,
d
to
label
the
directions
of
a
d-dimensional
manifold.
This
notation
is
common
in
physics.
Summation
over
repeated
indices is
implied,
as
before.
66
Chapter
4.
Differentiable Manifolds II
Clearly
( 4.2)
so
the
Cartesian
definition
of
distance
is
not
coordinate-independent.
However,
the
above
equation
tells us how
to
modify
the
definition
such
that
it
becomes
coordinate
independent.
We need
to
"compensate"
the
transforma-
tion
of
coordinates
by
introducing
a
quantity
that
transforms
in
the
opposite
way.
Therefore
we define a rank-2 covariant
tensor
field
g!1-
v
(x):
(4.3)
where
x
is
the
coordinate
ofthe
point
p.
Given such a
tensor,
the
distance
d(p,
q)
between
points
p
and
q
(infinitesimally
separated
in
terms
of
their
coordinates
in
the
given
chart)
is defined by:
[d(p,
q)]2
==
ds
2
=
g!1-v(x)
Jx!1-Jx
v
(4.4)
This
is
coordinate
independent
by
virtue
of
the
transformation
law of a rank-2
covariant
tensor
field:
ox).,
ox
P
g~v(Y)
=
oy!1-
oyV
g).,p(x)
from which, using
the
chain
rule,
it
is easy
to
see
that:
g~v(y)Jy!1-JyV
=
g!1-V(x)Jx!1-Jx
v
(4.5)
(4.6)
The
tensor
g!1-V
must
be
chosen so
that
the
three
axioms defined
in
Chapter
1 for a
metric
on
a topological
space
are
satisfied.
g!1-V
is called
the
Riemann
metric
or
simply
metric tensor.
The
above definition only tells us
the
distance
between
infinitesimally sep-
arated
points. To
extend
it
to
compute
the
total
length
of
any
path
between
finitely
separated
points, simply
integrate
the
infinitesimal
ds
defined above
along
the
given
path.
We will define
the
distance
between
finitely-separated
points
p,
q
to
be
the
minimum
of all
path
lengths
between
the
points:
d(p,q)
=
min
l
q
ds
(4.7)
The
integral
is
evaluated
on
the
image
of
the
path
in some chosen
chart.
We
may
write
a
Riemannian
metric
as G
=
g!1-v(x)
dx!1-
0
dx
v
where
the
notation
0 highlights
the
fact
that
it
is similar
to
a differential form
except
that
it
is
not
antisymmetric.
Such
an
object
can
be
contracted
with
two vector fields
using
the
duality
of
tangent
and
cotangent
spaces
that
we have
already
noted:
o
(
dx!1-
-)
=
J!1-
(4.8)
'ox
v
v
Thus,
if
A
=
a!1-(x)
a~,"
and
B
=
b!1-(x)
a~,"
are
two
vector
fields,
then
(G;A,B)
=
(g!1-
v
(x)dx!1-0dx
V
;
a).,
(x)
(0).,
,
bP(x)oo)
x x
P
=
g!1-v(x)
a!1-(x)b
V
(x)
(4.9)
4.2.
Frames
67
Thus,
the
metric
defines
an
inner
product
on
the
space
of
vector fields:
(4.10)
This
provides new insight into
the
meaning
of
the
Riemannian
metric.
Indeed,
there
is a
relation
between
the
two roles
of
the
metric:
that
of
providing a
distance
on
the
manifold,
and
that
of
providing
an
inner
product
on
vector fields. To see this,
note
that:
(4.11)
where
we
have
parametrised
the
image
of
each curve joining
p
and
q
as
XIL(t)
with
0
::;
t
::;
1,
with
XIL(O),
xIL(l)
being
the
coordinates
of
p,
q
respectively.
Now
d::
are
just
the
components
of
the
tangent
vector:
dxIL
a
T=
- .
---
dt
axIL
(4.12)
to
the
curve. So
d(p,
q)
=
min
11
J
(T,
T)c
dt
(4.13)
This
expresses
the
distance between two
points
along a given curve as
an
integral
over
the
norm
of
the
tangent
vector
to
the
curve.
Definition:
A differentiable manifold
equipped
with
a
Riemannian
metric
as
defined above is said
to
be
a
Riemannian
manifold.
4.2
Frames
It
is useful
to
define a
basis
for
the
tangent
space
at
each
point
of
a manifold.
A continuously varying basis will
correspond
to
a collection
of
d
vector fields
on
M,
where
n
is
the
dimension
of
M.
We
can
choose
the
set
to
be
orthonormal
in
the
inner
product
on
Tp(JvI)
defined by
the
metric.
If
the
vector fields
are
denoted
Ea
(x)
=
E::
(x)
",a,
a
=
1,
...
,d
ux
lL
(4.14)
then
this
amounts
to
requiring
that:
(4.15)
68
Chapter
4.
Differentiable Manifolds II
Definition:
Vector fields
Ea(x)
satisfying
the
above requirements
are
called
orthonormal frames
or
vielbeins.
Typically
they
are
just
referred
to
as "frames".
We
may
also define
the
1-forms
dual
to
these
vector fields,
These
are
de-
noted:
(4.16)
and
are
defined
to
satisfy:
(e
a
,
Eb)
=
e~(x)
Et(x)
=
oa
b
(4.17)
For a d-dimensional manifold,
an
O(d)
rotation
of
the
frames gives a new
set
of
orthonormal
frames (here O(d) denotes
the
group
of
d x d
orthogonal
matrices).
The
duality
between
e
a
and
Ea
is preserved if
the
same
O(
d)
rotation
is
made
on
both.
The
frames
and
their
dual1-forms
in
turn
determine
the
metric
tensor
ofthe
manifold.
From
the
1-form
e
a
we
can
construct
the
O(d)-invariant
symmetric
second-rank tensor,
~~=1
e
a
(:9
ea.
As we have seen, such a
tensor
defines
an
inner
product
on
tangent
vectors.
Thus
we
may
compute:
d
(L
e
a
(:9
e
a
;
Eb,
Ec)
=
e~
e~
Et
E~
a=l
from which we conclude
that
n
G
==
g",vdx'"
(:9
dx
v
=
Lea
(:9
e
a
a=l
or
in
component
notation,
g",v(x)
=
e~(x)e~(x).
(the
repeated
index is
summed
over).
Similarly, if we define a rank-2
tensor
field
~:=1
Ea
(:9
E
a
,
write
LE
E
-
h"'v
[) [)
a(:9
a-
~(:9~
uX'"
ux
V
from which
it
follows easily
that
h"'V(x)gv;..(x)
=
0"';..
(4.18)
(4.19)
(4.20)
then
we
can
(4.21 )
( 4.22)
So
h"'v
is
the
matrix
inverse
of
g",v'
Henceforth we
write
h"'v
as
g"'v.
Exercise:
Check all
the
manipulations
above.
Once we
are
equipped
with
a
set
of
orthonormal
frames
and
their
dual
1-forms,
it
becomes convenient
to
study
vectors
and
tensors
by
referring
their
4.3.
Connections,
Curvature
and
Torsion
69
components
to
this
basis. For example, given
any
vector field
A(x)
=
Afl(X)
a~'"
we
can
take
its
inner
product
with
the
1-form
ea(x}:
(4.23)
The
n-component
objects
Aa(x)
are a collection of (scalar) functions on
the
manifold.
Under
changes of
the
coordinate
system,
both
e~
(x)
and
Afl(X)
change
such
that
their
inner
product
remains invariant.
Similarly, a I-form
B(x)
=
Bfl(X)dxfl
can
be converted
into
a collection of
(scalar) functions:
(4.24)
Although
coordinate
invariant,
the
functions
Aa(x)
and
Ba(x)
do
depend
on
the
choice
of
orthonormal
frames,
and
they
change
under
O(n)
rotations
of
these frames, for example:
(4.25)
So it may seem we have
not
gained
anything
by converting
the
form into
O(n)
vector-valued functions. Indeed, going from forms
to
O(n)
vectors is completely
reversible
and
this
is also
true
if we
extend
the
above relation
to
map
coordinate
tensors
and
O(n)
tensors
to
each
other
via:
Bala2"'an
=
E::,'
E::~
...
E::;:
B
fl1fl2
"'fln
Bfllfl2
"'fln
=
e~~
e~~
...
e~:
B
a1a2
,,·a
n
( 4.26)
This
suggests
that
it
is
merely a
matter
of convenience
to
work
with
quantities
that
transform
under
coordinate
transformations
(tensor fields
and
forms) as
against those
that
transform
under
O(n).
However
the
utility of
orthonormal
frames goes beyond convenience. O(
n)
is a semi-simple Lie
group
and
its
representations are well-understood.
In
partic-
ular (see
N.
Mukunda's
lectures in
this
volume)
it
admits
a family of spinor rep-
resentations. Fields
that
transform
in these representations
cannot
be
mapped
to
(or from)
coordinate
tensors or differential forms as above. Therefore by in-
troducing
frames
and
using fields
that
transform
in
O(n)
representations, we are
able
to
deal
with
a more general class of objects
on
manifolds.
This
possibility
is
of
crucial
importance
in
the
physical context of general relativity, since
this
is
precisely how fermions
are
introduced.
4.3
Connections,
Curvature
and
Torsion
Recall
that
we defined
the
exterior derivative on I-forms as
2
( 4.27)
2
As
usual
the
components
of
the
forms
depend
on
coordinates,
but
from
this
section
on-
wards
we
suppress
the
argument
to
simplify
our
notation.
70
Chapter
4.
Differentiable Manifolds
II
We have shown
that
dA
obtained
in
this way is indeed a 2-form.
Now suppose
we
first convert
the
1-form
A
into a zero-form using
the
frames,
as
in
the
previous section,
and
then
take
the
exterior derivative.
Thus
we
start
by defining:
( 4.28)
and
then
att
e
mpt
to
define:
( 4.29)
Is
the
result sensible?
It
is
certainly
a differential form.
But
it
is easy
to
see
that
it
fails
to
be
an
D(n)
vector.
In
fact,
under
a local
D(n)
rotation
of
the
frames,
( 4.30)
one finds
that:
(4.31)
Because
of
the
first
term,
which involves a derivative
of
Aa
b
,
this
is
not
how
an
D(n)
vector transforms.
Therefore we need
to
look for a new
type
of derivative
D,
called a
covariant
derivative,
on
D(n)
vectors.
It
is required
to
have
the
property
that,
when
acting
on
D(n)
vectors,
it
gives us
back
D(n)
vectors. To qualify as a derivative,
D
must
be
a linear
operation,
so
it
is
natural
to
try
a definition like:
( 4.32)
Here,
W b
=
W b
dx'"'
(4.33)
a
,",a
is a 1-form whose
D(n)
transformation
rules
remain
to
be
found,
and
will
be
determined
by requiring
that
the
result of differentiation is again
an
D(n)
vector.
For this, we simultaneously
carry
out
the
known
O(n)
transformation
on
the
1-form
and
an
arbitrary
transformation
on
w:
Aa
-+
A~
=
Aa
b
Ab
(4.34)
Then
(
DA')
=
dA'
+,
'bA'
a a
Wa
b
=
A
b(dA
+
(A-I)
cdA dA
+
(A-I)
c
w
'
dA
e
A
)
a b b
cd
b c
de
(4.35)
Requiring
(DA')a
terms,
we find:
Aa
b
DAb
as
expected
of
an
D(n)
vector,
and
comparing
(4.36)
4.3.
Connections,
Curvature
and
Torsion
71
where
matrix
multiplication
is intended. Hence
Wi
=
AwA-
1
-
dAA-
1
( 4.37)
Thus
we have shown
that
w
a
b
transforms
inhomogeneously
under
O(n).
It
is
called a
connection
(sometimes,
in
this
context,
the
spin connection).
We
can
extract
an
important
property
of
the
spin
connection from
the
above
transformation
law.
If
we
take
the
transpose
and
use
the
fact
that
AT
=
A-1,
we find:
WiT
=
AwTA-
1
+dAA-1
( 4.38)
Adding
and
subtracting
the
two
equations
above, we see
that
the
symmetric
part
of
w
transforms
as
a
tensor,
while
the
antisymmetric
part
has
the
characteristic
inhomogeneous
transformation
law
3
.
Antisymmetry
of
w,
as
demonstrated
above, allows us
to
check
that
the
following
identity
holds:
(4.39)
On
general
O(
n)
tensors
,
the
definition
of
the
covariant derivative
naturally
extends
to:
(4.40)
Exercise:
Check
that
the
O(
n)
transformation
Aab
...
c
--->
A,!
Ab
q
...
A/
A
pq
...
r
together
with
the
transformation
for w discovered above,
transforms
the
covari-
ant
derivative
(DA)ab
...
c
in
the
same
way
as
A.
The
spin
connection
is a differential
I-form
but
is
not
itself
an
O(n)
ten-
sor,
as
is
evident
from
the
inhomogeneous
transformation
law
written
above.
However, using
the
spin
connection
we
can
define two very basic
tensors
associ-
ated
to
a manifold. Applying
D
to
the
orthonormal
frames
e
a
(which
are
O(n)
vector-falued I-forms),
one
gets
an
O(n)
vector-valued 2-form.
Definition:
The
2-form:
(4.41 )
is called
the
torsion 2-form
on
the
manifold
M.
Another
tensor
arises
naturally
by
asking
the
following question. We have
seen
that
in
the
absence
of
a
spin
connection,
the
exterior
derivative d satisfies
3N
ote
that
it
makes
no
difference
whether
the
O(n)
indic
es
are
raised
or
lowered,
as
this
is
done
using
the
identity
metric
,sab.
In
applications
to
general
relativity
the
group
O(n)
that
acts
on
frames
will
be
replaced
by
the
Lorentz
group
O(n-l,
1)
in
which
case
the
metric
that
raises
and
lowers
these
indices is
the
Minkowski
metric
TJab
=
diag(
-1,1,
·
··
, 1).
Here
too
the
difference
between
raised
and
lowered
indices
is
trivial,
involving
at
most
a
change
of
sign.