filler beef

weasly and his alt army

SK

aye bro i kept the show going

unity

arxiv.org/pdf/hep-th/0106113

i was there

The production on Strictly 4 My N.I.G.G.A.Z.

boy if u think im clicking that

I don't understand it

Like if you made a joke that upset me, unless it was like something absolutely crazy, I would just ask you in a decent way to not joke like that

incredible post

u only pretending to understand to appear smart

projecting as usual

fAbstract
A holographic duality is proposed relating quantum gravity on dSD (D-dimensional de Sitter space) to conformal field theory on a single SD−1 ((D-1)-sphere), in which bulk de Sitter correlators with points on the boundary are related to CFT correlators on the sphere, and points on I+ (the future boundary of dSD) are mapped to the antipodal points on SD−1 relative to those on I−. For the case of dS3, which is a***yzed in some detail, the central charge of the CFT2 is computed in an a***ysis of the asymptotic symmetry group at I±. This dS/CFT proposal is supported by the computation of correlation functions of a massive scalar field. In general the dual CFT may be non-unitary and (if for example there are sufficently massive stable scalars) contain complex conformal weights. We also consider the physical region O− of dS3 corresponding to the causal past of a timelike observer, whose holographic dual lives on a plane rather than a sphere. O− can be foliated by asymptotically flat spacelike slices. Time evolution along these slices is generated by L0 + L¯0, and is dual to scale transformations in the boundary CFT2.

in fact it's me actually
bouta burn like some calories
she throw it back on me magically
drop that ass on me, gravity
drill her ass that's her cavity
big ol booty not average meat
i f*** her so fast and savagely
that i'm infatuated with rapid banging
i gradually masturbated in the back of the cabin ay
out in Zurich i'm Zuriching
off the percs she hallucinating
dramamine got me too sedated
worried bout hos like who u chasin

bout to type an essay attacking you

...as usual?

you have no idea what ANY of this means lmfao

im f***ing crying

i don't know s*** about physics
for the record neither does anyone itt so i'm still smartest

but posting that was so beautifully schizo

tldr

Well, I don't have a Physics-related degree.
Do you?

think for yourself bozo

dont care plus you are gay

2tU
′′∂z¯ +
1
2
U
′
∂t, (2.3)
where U = U(z) is holomorphic and the prime denotes differentiation. In order to obtain a
real vector field one must add the complex conjugate, however for notational simplicity we
suppress this addition in (2.3) and all subsequent formula. In general the metric transforms
under a diffeomorphism as the Lie derivative
δζ gmn = −Lζ gmn. (2.4)
For ζ parametrized by U as in (2.3), (2.4) becomes
δU gzz = −
ℓ
2
2
U
′′′
,
δU gzz¯ = δU gzt = δU gtt = 0. (2.5)
The change (2.5) in the metric satisfies (2.2) and so (2.3) generates an asymptotic symmetry
of de Sitter space on Iˆ−.
A special case of (2.3) is
U = α + βz + γz2
, (2.6)
5
where α, β, γ are complex constants. In this case U
′′′ vanishes, and the metric is therefore
invariant. These transformations generate the SL(2, C) global isometries of 2+1 de Sitter.
In conclusion the asymptotic symmetry group is the conformal group of the complex
plane, and the isometry group is SL(2, C) subgroup of the asymptotic symmetry group.
3. The Boundary Stress Tensor
Brown and York 25 have given a general prescription for defining a stress tensor
associated to a boundary of a spacetime. Our treatment parallels the discussion of AdS3
in 26. We will be interested in the boundary t → −∞ at Iˆ−. In the case at hand the
stress tensor is4
T
µν = −
4π
√γ
δS
δγµν
, (3.1)
where γ is the induced metric on the boundary. The action is
S =
1
16πG Z
M
d
3x
√
−g(R −
2
ℓ
2
) + 1
8πG Z
∂M
√
γK +
1
8πGℓ Z
∂M
√
γ + Smatter, (3.2)
K here is the trace of the extrinsic curvature defined by Kµν = −∇(µnν) = −
1
2Lnγµν with
n
µ
the outward pointing unit normal. The second integral in (3.2) is the usual gravitational
surface term. The third integral is a surface counterterm required for finiteness of T at
an asymptotic boundary, and uniquely fixed by locality and general covariance 26. The
matter action is assumed not to be relevant near the boundary and will henceforth be
suppressed.
The sign in the definition (3.1) leads to a positive mass5
for Schwarzschild-de Sitter
32. It would be interesting to see if (3.1) reproduces the canonical operator product
expansion for a 2D CFT stress tensor.
Using (3.2) to evaluate (3.1) we learn that, for solutions of the bulk equations of
motion,
T
µν =
1
4G

Kµν − (K +
1
ℓ
)γ
µν
. (3.3)
This vanishes for de Sitter space on Iˆ− in the coordinates (2.1). For more general asymptotically dS3 spacetimes (obeying (2.2)) (3.3) implies
Tzz =
1
4G

Kzz +
1
ℓ
γzz
. (3.4)
4 Our conventions in this section are those of 25,26 except for a factor of −2π in this equation,
in order to conform with the more standard conventions 31 for 2D CFT stress tensor.
5 More precisely, a positive value of the AD mass L0 + L¯0, as defined below in equation 7.4. 6
4. The Central Charge
A central charge can be associated to dS3 by a***yzing the behavior of the stress
tensor on I
−. We follow related discussions for AdS3 21,26 (which in turn followed the
earlier work 33). Under the conformal transformations (2.3), one finds
δU Tzz = −
ℓ
8G
U
′′′
. (4.1)
This transformation identifies the central charge as6
c =
3ℓ
2G
. (4.2)
It is presumably also possible to derive c by an alternate method 34,33,26 which
relates it to the trace anomaly by using spherical rather than planar spatial sections.
5. The Plane and Iˆ− Correlators
In the previous sections we have seen that the conformal group of euclidean R2 has
an action on Iˆ−. One therefore expects that appropriately rescaled gravity correlators
restricted to Iˆ− will be those of a euclidean 2D conformal field theory. In this section we
verify this expectation for the case of a massive scalar.
Consider a scalar field of mass m with wave equation
m2
ℓ
2φ = ℓ
2∇2φ = −∂
2
t φ + 2∂tφ + 4e
2t
∂z∂z¯φ. (5.1)
Near Iˆ− the last term in (5.1) is negligible and solutions behave as
φ ∼ e
h±t
, t → −∞, (5.2)
where
h± = 1 ±
p
1 − m2ℓ
2. (5.3)
We first consider the case 0 < m2
ℓ
2 < 1 so that h± are real and h− < 1 < h+. As a
boundary condition on Iˆ− we demand
lim
t→−∞
φ(z, z, t ¯ ) = e
h−tφ−(z, z¯), (5.4)
6 This equation was arrived at from a different perspective in 9, and a related equation
appears in 11.
7
with corrections suppressed by at least one power of e
2t
. The dS3 /CFT2 correspondence
proposes, in direct a***ogy with the AdS/CFT correspondence, that φ− is dual to an
operator Oφ of dimension h+ in the boundary CFT. The two point correlator of Oφ is (up
to normalization) the quadratic coefficient of φ− in the expression7
lim
t→−∞ Z
Iˆ−
d
2
zd2
v

e
−2(t+t
′
)φ(t, z, z¯)
↔∂ tG(t, z, z¯;t
′
, v, v¯)
↔∂ t
′φ(t
′
, v, v¯)

t=t
′
. (5.5)
G here is the de Sitter invariant Green function given in the appendix. Near Iˆ− it reduces
to
lim
t,t′→−∞
G(t, z, z¯;t
′
, v, v¯) = c+e
h+(t+t
′
)
|z − v|
2h+
+
c−e
h−(t+t
′
)
|z − v|
2h−
, (5.6)
where c± are constants. Inserting (5.4) and (5.6), (5.5) is proportional to8
Z
Iˆ−
d
2
zd2
vφ−(z, z¯)|z − v|
−2h+ φ−(v, v¯). (5.7)
We conclude that the dual operator Oφ obeys
hOφ(z, z¯)Oφ(v, v¯)i =
const.
|z − v|
2h+
, (5.8)
as is appropriate for an operator of dimension h+.
It should be noted that the boundary conditions (5.4) are not the most general. There
are also solutions with the subleading behavior φ+(z, z¯)e
h+t at Iˆ−. Including these would
lead to an additional term in (5.7) proportional to
Z
Iˆ−
d
2
zd2
vφ+(z, z¯)|z − v|
−2h− φ+(v, v¯), (5.9)
which might be associated with an operator of dimension h−. In the next section we will
find that this extra boundary condition can imposed on the second boundary at I
+, which
is not within the coordinate patch O− covered in the planar coordinates (2.1). Alternately
7 This is equivalent to the usual expression used in AdS/CFT (except of course with a different
boundary and G), as can be readily seen from the formula for the bulk Green function G in terms
of the bulk-to-boundary Green function. A clear discussion can be found in 35. We have avoided
use of the bulk-to-boundary Green function because in the de Sitter case we need to keep track
of both terms in (5.6).
8 Similar holographic expressions were derived for general coset spaces in 36, but the case of
de Sitter space was not explicitly considered.
8
this second set of independent fields might be eliminated by imposing a suitable boundary
condition at the future horizon t → ∞, which would lead to a different Green function.9
What happens if m2
ℓ
2 > 1? In that case the conformal weight h− is no longer real.
h± are complex conjugates with real part equal to unity. Of course the appearance of
imaginary conformal weights suggests that the dual CFT is not unitary. This might mean
that consistent theories of de Sitter quantum gravity have no stable scalars with m2
ℓ
2 > 1. On the other hand we know of no obvious reason that the dual CFT needs to be unitary.
The preceding discussion is reminiscent of Liouville theory and indeed suggests that
the boundary CFT has a Liouville-like form. In some discussions of Liouville theory
operators with complex dimensions are encountered. Further there are various kinds of
operators (called macroscopic and microscopic in 37) which may or may not be allowed
depending on the context.
6. The Sphere and I
± Correlators
An alternate form of the dS3 metric is
ds2
ℓ
2
= −dτ 2 + 4 cosh2
τ
dwdw¯
(1 + ww¯)
2
(6.1)
w = tan θ
2
e
iφ here is a complex coordinate on the round sphere. This metric describes dS3
as a contracting/expanding two-sphere. These coordinates cover the entire space which
has future and past S
2 boundaries I
±, as depicted in figure 2. In general we can consider
correlators with points on either or both of the boundaries. North Pole South Pole
I
I
Fig. 2: Lines of constant τ in global spherical coordinates are the spacelike
two-spheres indicated by dashed lines.
9
I thank J. Maldacena for this suggestion.
9
We begin with the two-point correlator with both points on I
−. We wish to compute
the spherical a***og of (5.5), which is
lim τ→−∞ Z
I−
d
2wd2
v
p
h(w)h(v)

e
−2(τ+τ
′
)φ(τ, w, w¯)
↔∂ τG(τ, w, w¯; τ
′
, v, v¯)
↔∂ τ
′φ(τ
′
, v, v¯)

τ=τ
′
,
(6.2)
where h(w) = 2(1 + ww¯)
−2
is the measure on the sphere. Near I
−, φ can be decomposed
as
lim τ→−∞
φ(τ, w, w¯) = φ
in

• (w, w¯)e
  h+τ + φ
  in
  −(w, w¯)e
  h−τ
  . (6.3)
  The superscripts ”in” (”out”) are used to denote quantities on I
  − (I
  +). As seen in the
  appendix, the propagator behaves as
  lim
  τ,τ′→−∞
  G(τ, w, w¯; τ
  ′
  , v, v¯) = c+e
  h+(τ+τ
  ′
  )
  (1 + ww¯)
  h+ (1 + vv¯)
  h+
  |w − v|
  2h+
• c−e
  h−(τ+τ
  ′
  )
  (1 + ww¯)
  h− (1 + vv¯)
  h−
  |w − v|
  2h−
  .
  (6.4)
  (6.2) is then proportional to
  Z
  I−
  d
  2
  vd2w
  p
  h(v)h(w)

c+φ
in
−(w, w¯)∆h+ (w, w¯; v, v¯)φ
in
−(v, v¯)

• c−φ
  in
• (w, w¯)∆h− (w, w¯; v, v¯)φ
  in
• (v, v¯)
  
  
  .
  (6.5)
  ∆h± here is the two point function for a conformal field of dimension h± on the sphere:
  ∆h± =
  (1 + ww¯)(1 + vv¯)
  |w − v|
  2
  h±
  , (6.6)
  including the normalization factor from the Weyl anomaly on a curved geometry. Hence
  we see that, as in the planar case, the two-point scalar correlators can be identified with
  correlators of conformal fields of dimension h±, except that now they are on the sphere
  rather than the plane. A similar expression holds for the boundary at I
  +.
  Life becomes more interesting when we put one point on I
  − and one on I
  +. Then we
  must compute
  lim τ→−∞ Z
  I−
  d
  2w
  Z
  I+
  d
  2
  v
  p
  h(w)h(v)

e
2(τ−τ
′
)φ(τ, w, w¯)
↔∂ τG(τ, w, w¯; τ
′
, v, v¯)
↔∂ τ
′φ(τ
′
, v, v¯)

τ=−τ
′
.
(6.7)
For this case, as shown in the appendix, the relevant limit of the Green function is
lim
τ→−∞,τ′→+∞
G(τ, w, w¯; τ
′
, v, v¯) = c+ cos(πh+)e
h+(τ−τ
′
)∆h+ (w, w¯; −
1
v¯
, −
1
v
) + (h+ ↔ h−).
(6.8)
1
It is convenient to define the inverted boundary field at I
+
φ˜out

• (v, v¯) = φ
  out
• (−
  1
  v¯
  , −
  1

That was merely something I'd found.