Gravity Dbranes and Large Extra Dimensions Chad A
Gravity, D-branes, and Large Extra Dimensions Chad A. Middleton Mesa State College September 21, 2006
Outline… § Newtonian Gravity § Einstein’s General Relativity § § § Distance in 3 D Euclidean Space Distance in 4 D Minkowski Space Distance in 4 D Curved Space Principle of Equivalence Einstein Field Equations Schwarzschild Solution § DGP model of Brane-Induced-Gravity § DGP Field Equations § Schwarzschild-like Solution
Newton’s Universal Law of Gravitation Successes Described the motion of massive bodies… …on earth …in the heavens
Shortcomings… n n “Action at a distance” Mercury’s perihelion precession
3 D Euclidean Space Line element in Euclidean space is the line element measuring distance n is invariant under rotations n
Line element in 3 D Euclidean space Where ; § Repeated indicies = sum over indicies
In 1905, Einstein submitted his Special Theory of Relativity § Lorentz Transformations § Length depends on reference frame
4 D Minkowski Space Line element in Minkowski space n is the line element measuring ‘length’ n is invariant under ‘rotations’
Line element in 4 D Minkowski space Where ; § is the Minkowski metric
Line element in 4 D curved space § is the metric tensor § is the flat limit of § defines the geometry of spacetime § Know , Know Geometry
Inertial vs. Gravitational Mass n n n Newton’s Second Law Newton’s Law of Gravitation Force due to Electric field
Inertial vs. Gravitational Mass n 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
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
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
In 1915, Einstein gives the world his General Theory of Relativity n is the Einstein tensor describing the curvature of space n is the stressenergy tensor describing the matter
Space is not an empty void but rather a dynamical structure whose shape is determined by the presence of matter and energy. n Matter tells space how to curve n Space tells matter how to move
Einstein Field Equations § Subscripts label elements of each matrix § metric ( ) encodes the geometry of spacetime § 10 second-order, non-linear differential equations
In 1916, Karl Schwarzschild presented an EXACT solution! § Choose a point mass § Choose a metric ansatz § Plug metric ansatz into the equations
The solution is Define: Notice: : Schwarzschild radius : Spacetime singularity
In 1929, Edwin Hubble discovered that the universe is expanding… What is the fate of the “decelerating” Universe? § “Big Crunch” § “Big Freeze” § Neither?
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
Are we witnessing… § a mysterious Dark Energy which is driving the accelerated expansion? § the breakdown of Einstein’s Theory of General Relativity?
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
The DGP model describes a 3 -brane at the boundary of a 5 D, infinite-volume bulk space. where § ( ) is the 5 D (4 D) Einstein tensor § In 4 D Einstein tensor § In 5 D Einstein tensor
Attractive because… § Offers an alternative explanation to Dark Energy Modification of Einstein’s GR itself § Successfully explains the weakness of gravity Graviton’s live in larger space!
Schwarzschild-like solution § Choose a point mass § Choose a metric ansatz § Plug metric ansatz into the DGP Field equations § Expand the metric: 1 st order in B, C, D 2 nd order in
The solution on the brane is given by… where defines a crossover distance
In the far regime ( ), § This solution in the far regime corresponds to the standard perturbative solution
In the near regime ( ), § This solution in the near regime corresponds to the expected Schwarzschild solution
Gravitational Behavior vs. Distance
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
String Essentials… n Points of QFT 1 D Strings n 2 Types Closed & Open n Different Vibrational Modes Different particles
String Theory demands Extra Dimensions Two possible descriptions n Compactified Extra Dimensions n Non-Compactified Extra Dimensions
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
- Slides: 34