Gravity, Dbranes, and Large Extra Dimensions

1
Gravity, D-branes, and Large Extra Dimensions

• Chad A. Middleton
• Mesa State College
• September 21, 2006

2
Outline

• Newtonian Gravity
• Einsteins General Relativity
• Distance in 3D Euclidean Space
• Distance in 4D Minkowski Space
• Distance in 4D Curved Space
• Principle of Equivalence
• Einstein Field Equations
• Schwarzschild Solution
• DGP model of Brane-Induced-Gravity
• DGP Field Equations
• Schwarzschild-like Solution

3
Newtons Universal Law of Gravitation

• Successes
• Described the motion of massive bodies
• on earth
• in the heavens

4
Shortcomings

• Action at a
• distance
• Mercurys perihelion precession

5
3D Euclidean Space

• Line element in Euclidean space
• is the line element measuring distance
• is invariant under rotations

6
Line element in 3D Euclidean space
Where

• Repeated indicies
• sum over indicies

7
In 1905, Einstein submitted his Special Theory of
Relativity

• Lorentz Transformations

• Length depends on reference frame

8
4D Minkowski Space

• Line element in Minkowski space
• is the line element measuring length
• is invariant under rotations

9
Line element in 4D Minkowski space
Where

• is the Minkowski metric

10
Line element in 4D curved space

• is the metric tensor

• is the flat limit of

• defines the geometry of spacetime

• Know , Know Geometry

11
Inertial vs. Gravitational Mass

• Newtons Second Law
• Newtons Law of Gravitation
• Force due to Electric field

12
Inertial vs. Gravitational Mass

• Eötvös experiment verified
• that to 1 part in 1012 !

The Weak Equivalence Principle is

• All massive objects fall at the same rate
• Suggests a preferred class of trajectories!
• ? inertial trajectories

13
Principle of Equivalence

• The motion of freely falling particles is the
  same in a
• gravitational field and a uniformly
  accelerated frame, in small enough regions of
  spacetime

14
What about massless particles?

• Elevator observer sees light
• moving in a curved path

• Elevator observer is non-inertial
• ? Earth observer is non-inertial

• Einstein suggested that we treat
• gravitation as a inertial effect

15
In 1915, Einstein gives the world his General
Theory of Relativity

• is the Einstein tensor describing the
  curvature of space
• is the stress-energy tensor
  describing the matter

16
Space is not an empty void but rather a dynamical
structure whose shape is determined by the
presence of matter and energy.

• Matter tells space how to curve
• Space tells matter how to move

17
Einstein Field Equations

• Subscripts label elements of each matrix
• metric ( ) encodes the geometry of
  spacetime
• 10 second-order, non-linear differential
  equations

18
In 1916, Karl Schwarzschild presented an EXACT
solution!

• Choose a point mass

• Choose a metric ansatz

• Plug metric ansatz into the equations

19
The solution is
Define
Notice
Schwarzschild radius
Spacetime singularity
20
In 1929, Edwin Hubble discovered that the
universe is expanding

• What is the fate of the
• decelerating Universe?
• Big Crunch
• Big Freeze
• Neither?

21
Recent Data from Type Ia Supernovae, WMAP and
SDSS implies

• The expansion of the Universe is
  ACCELERATING!!

• The Universe is flat

• Seems to imply a new form of Energy

22

• Are we witnessing
• a mysterious Dark Energy which is
• driving the accelerated expansion?
• the breakdown of Einsteins
• Theory of General Relativity?

23
Brane-Induced-Gravity picture

• Our Universe is a 3-brane embedded
• in a flat, infinite-volume bulk
• Everything minus gravity is confined
• to the brane
• Gravitons reside on the brane and
• in the bulk

24
The DGP model describes a 3-brane at the boundary
of a 5D, infinite-volume bulk space.
where

• ( ) is the 5D (4D) Einstein
  tensor

• In 4D Einstein tensor

• In 5D Einstein tensor

25
Attractive because

• Offers an alternative explanation to
• Dark Energy

? Modification of Einsteins GR itself

• Successfully explains the weakness of gravity

? Gravitons live in larger space!
26
Schwarzschild-like solution

• Choose a point mass

• Choose a metric ansatz

• Plug metric ansatz into the DGP Field
• equations

• Expand the metric 1st order in B,C,D
• 2nd order in ?

27
The solution on the brane is given by
where defines a crossover
distance
28
In the far regime ( ),

• This solution in the far regime corresponds
• to the standard perturbative solution

29
In the near regime ( ),

• This solution in the near regime corresponds
• to the expected Schwarzschild solution

30
Gravitational Behavior vs. Distance
31
Conclusions

• The DGP model of Brane-Induced-Gravity offers
  
• an alternative explanation to Dark
  Energy
• Successfully explains the weakness of gravity
• This work fixes the perturbative breakdown of
• DGP model in the weak-field limit
• This work predicts an intermediate distance
• regime

• Using a Brane to Probe the Bulk
• by Chad A. Middleton, Mercury,
• Journal of the Astronomical Society of the
  Pacific
• March/April 2006

32
String Essentials

• Points of QFT
• ? 1D Strings
• 2 Types
• ? Closed Open
• Different Vibrational Modes
• ? Different particles

33
String Theory demands Extra Dimensions
? Two possible descriptions

• Compactified
• Extra Dimensions

• Non-Compactified Extra Dimensions

34
String Theory admits a variety of
non-perturbative excitations of extended objects

• D(irichlet)-branes
• Closed strings
• ? Spin-2 gravitons
• Open strings
• ? Spin-1, -1/2 SM particles