Randall–Sundrum model
Encyclopedia
In physics
, Randall–Sundrum models (also called 5-dimensional warped geometry theory) imagine that the real world is a higher-dimensional Universe described by warped geometry. More concretely, our Universe is a five-dimensional anti de Sitter space
and the elementary particles except for the graviton
are localized on a (3 + 1)-dimensional brane
or branes.
The models were proposed in 1999 by Lisa Randall
and Raman Sundrum
because they were dissatisfied with the universal extra dimensional
models then in vogue. Such models require two fine tunings; one for the value of the bulk
cosmological constant
and the other for the brane tensions. Later, while studying RS models in the context of the AdS/CFT correspondence
, they showed how it can be dual to technicolor model
s.
There are two popular models. The first, called RS1, has a finite size for the extra dimension with two branes, one at each end. The second, RS2, is similar to the first, but one brane has been placed infinitely far away, so that there is only one brane left in the model.
of the Standard Model
. It involves a finite five-dimensional bulk
that is extremely warped and contains two branes: the Planckbrane (where gravity is a relatively strong force; also called "Gravitybrane") and the Tevbrane (our home with the Standard Model particles; also called "Weakbrane"). In this model, the two branes are separated in the not-necessarily large fifth dimension by approximately 16 units (the units based on the brane and bulk energies). The Planckbrane has positive brane energy, and the Tevbrane has negative brane energy. These energies are the cause of the extremely warped spacetime
.
's probability function is extremely high at the Planckbrane, but it drops exponentially as it moves closer towards the Tevbrane. In this, gravity would be much weaker on the Tevbrane than on the Planckbrane.
. However, the change by 16 units causes the scales to change by 16 orders of magnitude. On the Planckbrane, strings would be 10−33 cm in size, but on the Tevbrane they'd be 10−17 cm. In fact, this makes the mass scale for the Weakbrane based on about a TeV. Therefore, you no longer have the strange range of masses and energies. If this is the case, we should see evidence when the Large Hadron Collider
begins working.
. The warping of the extra dimension is analogous to the warping of spacetime
in the vicinity of a massive object, such as a black hole
. This warping, or red-shifting, generates a large ratio of energy scales so that the natural energy scale at one end of the extra dimension is much larger than at the other end.
where k is some constant and η has "-+++" metric signature
. This space has boundaries
at y = 1/k and y = 1/Wk, with
where k is around the Planck scale
and W is the warp factor and Wk is around a TeV
. The boundary at y = 1/k is called the Planck brane and the boundary at y = 1/Wk is called the TeV brane. The particles of the standard model
reside on the TeV brane. The distance between both branes is only −ln(W)/k, though.
In another coordinate system
,
so that
and
a number of articles on a very similar theme. In those papers he showed that if the Universe is considered as a thin shell (a mathematical synonym
for "brane") expanding in 5-dimensional space then there is a possibility to obtain one scale for particle theory corresponding to the 5-dimensional cosmological constant
and Universe thickness, and thus to solve the hierarchy problem
. It was also shown that four-dimensionality of the Universe is the result of stability
requirement since the extra component of the Einstein equations giving the localized solution for matter
fields coincides with the one of the conditions of stability.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, Randall–Sundrum models (also called 5-dimensional warped geometry theory) imagine that the real world is a higher-dimensional Universe described by warped geometry. More concretely, our Universe is a five-dimensional anti de Sitter space
Anti de Sitter space
In mathematics and physics, n-dimensional anti de Sitter space, sometimes written AdS_n, is a maximally symmetric Lorentzian manifold with constant negative scalar curvature...
and the elementary particles except for the graviton
Graviton
In physics, the graviton is a hypothetical elementary particle that mediates the force of gravitation in the framework of quantum field theory. If it exists, the graviton must be massless and must have a spin of 2...
are localized on a (3 + 1)-dimensional brane
Membrane (M-Theory)
In theoretical physics, a membrane, brane, or p-brane is a spatially extended mathematical concept that appears in string theory and related theories...
or branes.
The models were proposed in 1999 by Lisa Randall
Lisa Randall
Lisa Randall is an American theoretical physicist and a leading expert on particle physics and cosmology. She works on several of the competing models of string theory in the quest to explain the fabric of the universe. Her most well known contribution to the field is the Randall-Sundrum model,...
and Raman Sundrum
Raman Sundrum
Raman Sundrum is a theoretical particle physicist. He is an Indian American of South Asian origin. His most famous contribution to the field is a class of models called the Randall-Sundrum models, first published in 1999 with Lisa Randall....
because they were dissatisfied with the universal extra dimensional
Large extra dimension
In particle physics, models with universal extra dimensions propose that there are one or more additional dimensions beyond the three spatial dimensions and one temporal dimension that are observed...
models then in vogue. Such models require two fine tunings; one for the value of the bulk
Bulk
-Industry:* Bulk cargo* Bulk liquids* Bulk mail* Bulk material handling* Bulk pack, packaged bulk materials/products* Bulk purchasing- Physics :*Bulk density*Bulk modulus...
cosmological constant
Cosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
and the other for the brane tensions. Later, while studying RS models in the context of the AdS/CFT correspondence
AdS/CFT correspondence
In physics, the AdS/CFT correspondence , sometimes called the Maldacena duality, is the conjectured equivalence between a string theory and gravity defined on one space, and a quantum field theory without gravity defined on the conformal boundary of this space, whose dimension is lower by one or more...
, they showed how it can be dual to technicolor model
Technicolor (physics)
Technicolor theories are models of physics beyond the standard model that address electroweak symmetry breaking, the mechanism through which elementary particles acquire masses...
s.
There are two popular models. The first, called RS1, has a finite size for the extra dimension with two branes, one at each end. The second, RS2, is similar to the first, but one brane has been placed infinitely far away, so that there is only one brane left in the model.
Overview
The model is a braneworld theory developed while trying to solve the hierarchy problemHierarchy problem
In theoretical physics, a hierarchy problem occurs when the fundamental parameters of some Lagrangian are vastly different than the parameters measured by experiment. This can happen because measured parameters are related to the fundamental parameters by a prescription known as renormalization...
of the Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
. It involves a finite five-dimensional bulk
Bulk
-Industry:* Bulk cargo* Bulk liquids* Bulk mail* Bulk material handling* Bulk pack, packaged bulk materials/products* Bulk purchasing- Physics :*Bulk density*Bulk modulus...
that is extremely warped and contains two branes: the Planckbrane (where gravity is a relatively strong force; also called "Gravitybrane") and the Tevbrane (our home with the Standard Model particles; also called "Weakbrane"). In this model, the two branes are separated in the not-necessarily large fifth dimension by approximately 16 units (the units based on the brane and bulk energies). The Planckbrane has positive brane energy, and the Tevbrane has negative brane energy. These energies are the cause of the extremely warped spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
.
Graviton Probability Function
In this warped spacetime that is only warped along the fifth dimension, the gravitonGraviton
In physics, the graviton is a hypothetical elementary particle that mediates the force of gravitation in the framework of quantum field theory. If it exists, the graviton must be massless and must have a spin of 2...
's probability function is extremely high at the Planckbrane, but it drops exponentially as it moves closer towards the Tevbrane. In this, gravity would be much weaker on the Tevbrane than on the Planckbrane.
Growing, Shrinking, and Changing Weight: Solving the Hierarchy Problem
Also in this model, any objects moving from the Planckbrane to the Tevbrane in the bulk would be growing, becoming lighter, and moving more slowly through time. Distance and time expand near the Tevbrane, and mass and energy shrink near it. This creates an alternate explanation of gravity's weakness in the Tevbrane: everything is lighter. An interesting part of this is that the hierarchy problem is automatically solved. The main scale on the Planckbrane would be the Planck scalePlanck scale
In particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 1019 GeV at which quantum effects of gravity become strong...
. However, the change by 16 units causes the scales to change by 16 orders of magnitude. On the Planckbrane, strings would be 10−33 cm in size, but on the Tevbrane they'd be 10−17 cm. In fact, this makes the mass scale for the Weakbrane based on about a TeV. Therefore, you no longer have the strange range of masses and energies. If this is the case, we should see evidence when the Large Hadron Collider
Large Hadron Collider
The Large Hadron Collider is the world's largest and highest-energy particle accelerator. It is expected to address some of the most fundamental questions of physics, advancing the understanding of the deepest laws of nature....
begins working.
The RS1 model
The RS1 model attempts to address the hierarchy problemHierarchy problem
In theoretical physics, a hierarchy problem occurs when the fundamental parameters of some Lagrangian are vastly different than the parameters measured by experiment. This can happen because measured parameters are related to the fundamental parameters by a prescription known as renormalization...
. The warping of the extra dimension is analogous to the warping of spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
in the vicinity of a massive object, such as a black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
. This warping, or red-shifting, generates a large ratio of energy scales so that the natural energy scale at one end of the extra dimension is much larger than at the other end.
where k is some constant and η has "-+++" metric signature
Metric signature
The signature of a metric tensor is the number of positive and negative eigenvalues of the metric. That is, the corresponding real symmetric matrix is diagonalised, and the diagonal entries of each sign counted...
. This space has boundaries
Boundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
at y = 1/k and y = 1/Wk, with
where k is around the Planck scalePlanck scale
In particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 1019 GeV at which quantum effects of gravity become strong...
and W is the warp factor and Wk is around a TeV
TEV
TEV may refer to:* TeV, or teraelectronvolt, a measure of energy* Total Enterprise Value, a financial measure* Total Economic Value, an economic measure* Tobacco etch virus, a plant pathogenic virus of the family Potyviridae....
. The boundary at y = 1/k is called the Planck brane and the boundary at y = 1/Wk is called the TeV brane. The particles of the standard model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
reside on the TeV brane. The distance between both branes is only −ln(W)/k, though.
In another coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
,
so that
and
The RS2 model
The RS2 model uses the same geometry as RS1, but there is no TeV brane. The particles of the standard model are presumed to be on the Planck brane. This model was originally of interest because it represented an infinite 5-dimensional model which, in many respects, behaved as a 4-dimensional model. This setup may also be of interest for studies of the AdS/CFT conjecture.Prior models
In 1998/99 Merab Gogberashvili published on ArxivArXiv
The arXiv |Chi]], χ) is an archive for electronic preprints of scientific papers in the fields of mathematics, physics, astronomy, computer science, quantitative biology, statistics, and quantitative finance which can be accessed online. In many fields of mathematics and physics, almost all...
a number of articles on a very similar theme. In those papers he showed that if the Universe is considered as a thin shell (a mathematical synonym
Synonym
Synonyms are different words with almost identical or similar meanings. Words that are synonyms are said to be synonymous, and the state of being a synonym is called synonymy. The word comes from Ancient Greek syn and onoma . The words car and automobile are synonyms...
for "brane") expanding in 5-dimensional space then there is a possibility to obtain one scale for particle theory corresponding to the 5-dimensional cosmological constant
Cosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
and Universe thickness, and thus to solve the hierarchy problem
Hierarchy problem
In theoretical physics, a hierarchy problem occurs when the fundamental parameters of some Lagrangian are vastly different than the parameters measured by experiment. This can happen because measured parameters are related to the fundamental parameters by a prescription known as renormalization...
. It was also shown that four-dimensionality of the Universe is the result of stability
Stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...
requirement since the extra component of the Einstein equations giving the localized solution for matter
Matter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...
fields coincides with the one of the conditions of stability.