PhD- thesis proposal

1. Explorations in Higher
Dimensional Gravity Theory
PhD- thesis proposal
Prepared by: Safinaz Ramadan
March 2019
Al-Azhar University
Faculty of Science
Physics Department
1

2. Highlights
Introduction: ( the way for higher dimensions )
- 1915: General Theory of Relativity.
- 1970: Standard Model of Elementary particles.
- 1973: Supersymmetry ( SUSY).
- 1976: Supergravity (SUGRA).
- 1984: Manifestation of SUGRA as an effective theory of
string theory.
D = 5 N = 2 supergravity
- Dimensional reduction from D=11 supergravity.
- Model the universe as a 3-brane embedded in 5-D
spacetime N=2 supergravity.
- Studying the time evolution of the universe.
- Studying the cosmological constant problem.
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3. Unification to Higher
Dimensions
3

4. From Special Relativity (1905) to General
Relativity
The line interval between A and
B can be defined by the metric
ds2
= − c2
dt2
+ dx2
+ dy2
+ dz2
( Minkowski flat space- time)
The speed of light in a vacuum is
constant and nothing can exceed
light’s speed.
But what if
the sun
disappeared
??
4

5. Einstein Field equations
Riemann curvature tensor
Rμν −
1
2
gμνR = κ Tμν
Rμν = Rρ
μρν
Minkowski s-t
Rρ
μνσ = 0 →
Tμν = (ρ + p)uμ
uν
− pδμν
.
The Energy-Momentum tensor
In IRF for perfect
fluid
Ricci scalar
R = gμνRμν
Curvature in s-t
Matter
Ricci tensor
Rμν −
1
2
gμνR = κ Tμν
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6. By demanding we get EFE in the absence of matter.
The Einstein–Hilbert action
F(R) gravity
SEH =
∫R
R −g d4
x
δSEH = 0
S =
∫
d4
x −g(R + R2
+ 3R3
+ . . . . ),
Modified Gravity Theories
S =
∫
d4
x −g F(R) .
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7. dF
dR
= FR, and make conformal transformation ̂
gμν =
1
FR
gμν ,
S =
∫
d4
x −g[ ̂
R −
1
2
(∂α
ϕ)(∂αϕ) − V(ϕ)] .
The ordinary E.H.A term plus extra scalar field terms
( the inflaton).
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8. The Standard Model
Internal symmetry group
SU(3)C × SU(2)L × U(1)Y
ψ′

(x) = eiϵa(x) ta ψ(x),
a = 1,2,3, SU(2), 1,..,8, SU(3) .
Drawbacks 🙁
- it is not a truly unified theory because the
gluons and the photons governed by totally
different rules.
- Too many unanswered questions ( masses,
and charges of particles) , and too many
constants ( with brute force values).
- Dark matter 23% and dark energy 73% of
our universe !! Couldn’t been explained.
- Unification? No gravity !!
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9. Supersymmetry
- Space-time
4D QFT.
-Poincaré
symmetry
δψ = − iσμ
ζ̄∂ϕ
δϕ = ζ . ψ
Lfree WZ = ∂μϕ*∂μ
ϕ + ψ̄γμ
∂μψ .
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10. 10

11. Supergravity
Introduce vector spinor field with spin 3/2 ( the
gravitino) with Noether coupling
LWZ is not invarinat under ϵ → ϵ(x)
δL = ∂μϵα
Kμ
α + h . c .
Kα
μ = − ∂μϕ*ψα
−
i
2
ψβ
(σμσ̄ν
)α
β∂μϕ*,
Ψμ
α → Ψμ
α +
1
k
∂μ
ϵα,
LN = kKα
μΨμ
α, Transforming as
However δ(L + LN) = kψ̄μγνϵTμν
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12. Lg = − gμνTμν
This contribution can only be canceled adding a new term
δgμν = kΨ̄μγνϵ
In the presence of local supersymmetry we must
include also the gravity supermultiplet the
graviton and gravitino respectively (N=1) SUGRA.
(gμν, Ψα
μ)
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13. SUGRA as an effective theory of superstring
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14. D=5 N=2
Supergravity
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15. Dimensional reduction over Calabi-Yau
Manifold
S11 =
1
2κ2
11
∫
d11
x −G (R −
1
48
F2
) −
1
12κ2
11
∫
A ∧ F ∧ F,
S5 =
1
2κ2
5
∫
d5
x −g [R −
1
2
(∂μσ)(∂μ
σ) − Gi¯
j (∂μzi
)(∂μ
z
¯
j
)
−
1
48
e−2σ
FμνρσFμνρσ
−
1
24
ϵ̄μνρσαFμνρσ
Kα
(ζ, ζ̄) + eσ
Lμ
μ(ζ, ζ̄)],
Calabi-Yau
Manifold
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16. Where the z's are the CY's complex structure (M) moduli,
h2,1 is the dimension of M.
Two gravitini and a set of hyperini; the superpartenrs
of the hypermultiples bosons.
The particle content at 5 D N=2
(a, σ, ζ0
, ¯
ζ0
) the universal hypermultiplet
(zi
, z̄i
, ζi
, ζ̄i
: i = 1,...,h2,1)
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17. The universe as a 3-brane embedded
in 5 D
5 D space-time.
The Bulk:
The moduli
Live here
We live
here
4 spatial D.
3 spatial D.
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18. Time evolution of the universe
More understanding for the inflation epoch where the early
universe expanded exponential rate for 10^-36 seconds after the
Big Bang , then the universe continued to expand in less rapid
rates until nowadays
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19. The cosmological constant
Einstein’s Greatest Blunder
In 1916 Rμν −
1
2
gμνR = κTμν − Λgμν .
1929 Edwin Hubble : the expansion of the universe
It gives negative pressure, and thus acts as a repulsive
force counteracting the attractive gravitational effect ->
static universe
In 1998 : from observing type IA supernova indicated that
the universe is also accelerating in its expansion
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20. So the cosmological constant term should be considered
to explain the vacuum energy or “ Dark Energy” that
produces repulsion of the universe
However when measuring the quantum vacuum energy
On the other hand the observed value of the darkenergy
density required for the current rate of acceleration of
the universe is roughly
The cosmological constant problem !!
ρ ∼ 10112
erg/cm3
ρ ∼ 10−8
erg/cm3
catastrophy !!
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