Prof. Dr. Lust
Summer2006
Assignment # 3
(Due May 22,2006)
1) Dierential Geometry for General Relativity
Consider the metri of a2-sphere of radiusa:
ds 2
=g
dx
dx
=a 2
[d 2
+sin 2
d 2
℄:
Themetrienodesallinformationonthe geometryofamanifold. InAssignment#2,
you already alulated the surfae area of the two-sphere. In the present problem,we
willdetermine allthose geometri quantities that are relevant forgeneral relativity:
a)The metri
Choosing x 1
= and x 2
=, read o the matrix g
.
b)The Christoel symbols
The Christoelsymbolsare dened as
= 1
2 g
g
x
+ g
x
g
x
!
:
They enter ovariantderivatives suh asr
V
=
V
+
V
, where the orretion
term with the Christoel symbols ensures that the ovariant derivative indeed trans-
forms\ovariantly"under arbitrary oordinatetransformationsx
!x 0
(x
), i.e.,
r
V
!(r
V
) 0
= x
x 0
x 0
x
r
V
;
withoutseond derivativesinthe oordinates.
Compute the non-vanishing Christoel symbols for the two-sphere. (Hint:
=
, soonly afew omponents have to be omputed expliitly.)
)The Riemann tensor
The Riemannurvature tensorhas the form
R
=
+
:
Calulate the non-vanishing omponents of R
for the two-sphere (Hint: Use the
antisymmetryin and to avoid redundant omputations).
quantifyingthe non-ommutativity of the ovariantderivatives:
[r
;r
℄V
=R
V
:
A spae with vanishing R
is at, i.e., the metri an be brought to the standard
Minkowskian (or Eulidean) formby meansof a oordinatetransformation.
d)The Riitensor
The Riitensor isdened as
Ri
=R
:
Calulate Ri
forS
2
.
e)The salar urvature
The salarurvature is given as
R =g
Ri
:
CalulateRforS 2
. Howdoesthesalarurvaturebehaveinthelimita!1? Interpret
this behaviour.
f) The Einstein tensor
The Einstein equation is the eld equation of general relativity, and it relates the
urvature of spaetime to the matterdistribution:
G
=8GT
;
where G denotes Newton's onstant, T
is the energy momentum tensor, and G
denotes the Einstein tensor:
G
=Ri
1
2 g
R:
Calulate G
forS
2
.
2) The Polyakov ation: I) The eld equations
Consider the Polyakov ation,
S
P
= T
2 Z
d 2
p
hh
X
X
:
a)Rememberingdet (expA) =exp (Tr A),show that
Æh= h
(Æh
)h;
whereh= det(h
)
inthe metri:
ÆS = T Z
d 2
p
hT
Æh
() T
= 1
T p
h ÆS
Æh
:
Compute T
forthe Polyakov ation.
) Find the equations of motion for h
and show that, after some manipulation and
re-insertionintoS
P
, one re-obtains the Nambu-Gotoation.
d)Showthat adding a \osmologialonstant term",
S
1
= Z
d 2
p
h
to the Polyakov ation leads to inonsistent eld equations for h
in the ombined
system S
P +S
1
when 6=0.
3) The Polyakov ation: II) The symmetries
a)Show in one line that the Weyl invariane S
P [e
2
h
;X
℄ =S
P [h
;X
℄ automati-
allyimplies h
T
=0 withoutthe use of the equationsof motion.
b)Verify the traelessnessof T
diretlybyusing your resultforT
fromProblem2)
b).
) How does h
have to transform under arbitrary reparameterizations (;) !
(~(;);(~ ;)) for S
P
tobeinvariant?
4) Lightone oordinates
In onformal gauge and after a Weyl resaling, the metri h
an be brought to the
standard Minkowskian form:
ds 2
=
d
d
= d
2
+d 2
:
a)Rewrite the metri interms of lightone oordinates
:=;
and read o the omponents
++
,
+
and .
b) Determine the omponents ++
, and +
of the inverse metri and use it to
derivethe relationbetween the omponents (V +
;V ) and the omponents (V
+
;V )of
a2D vetor by raising the indies.
)Determinethe derivatives
interms of
and
. (Use
=1and
=0):
