Conformal gravity refers to gravity theories that are invariant under
conformal transformations in the
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...
sense; more accurately, they are invariant under
Weyl transformation :''See also Wigner–Weyl transform, for another definition of the Weyl transform.''
In theoretical physics, the Weyl transformation, named after Hermann Weyl, is a local rescaling of the metric tensor:
:g_\rightarrow e^g_
which produces anothe ...
s
where
is the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
and
is a function on
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
.
Weyl-squared theories
The simplest theory in this category has the square of the
Weyl tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal f ...
as the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
:
where
is the Weyl tensor. This is to be contrasted with the usual
Einstein–Hilbert action
The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the act ...
where the Lagrangian is just the
Ricci scalar
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry ...
. The equation of motion upon varying the metric is called the
Bach tensor,
:
where
is the
Ricci tensor
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
. Conformally flat metrics are solutions of this equation.
Since these theories lead to
fourth-order equations for the fluctuations around a fixed background, they are not manifestly unitary. It has therefore been generally believed that they could not be consistently quantized. This is now disputed.
Four-derivative theories
Conformal gravity is an example of a 4-
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
theory. This means that each term in the
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
can contain up to four derivatives. There are pros and cons of 4-derivative theories. The pros are that the quantized version of the theory is more convergent and
renormalisable. The cons are that there may be issues with
causality
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
. A simpler example of a 4-derivative wave equation is the scalar 4-derivative wave equation:
:
The solution for this in a central field of force is:
:
The first two terms are the same as a normal wave equation. Because this equation is a simpler approximation to conformal gravity, m corresponds to the mass of the central source. The last two terms are unique to 4-derivative wave equations. It has been suggested that small values be assigned to them to account for the
galactic acceleration constant (also known as
dark matter
Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not ab ...
) and the
dark energy
In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univer ...
constant.
The solution equivalent to the
Schwarzschild solution
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an
exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumpti ...
in
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
for a spherical source for conformal gravity has a metric with:
:
to show the difference between general relativity. 6bc is very small, and so can be ignored. The problem is that now c is the total
mass-energy of the source, and b is the
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
of density, times the distance to source, squared. So this is a completely different potential from
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
and not just a small modification.
The main issue with conformal gravity theories, as well as any theory with higher derivatives, is the typical presence of
ghosts
A ghost is the soul or spirit of a dead person or animal that is believed to be able to appear to the living. In ghostlore, descriptions of ghosts vary widely from an invisible presence to translucent or barely visible wispy shapes, to rea ...
, which point to instabilities of the
quantum
In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
version of the theory, although there might be a solution to the ghost problem.
An alternative approach is to consider the gravitational constant as a
symmetry broken scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
, in which case you would consider a small correction to
Newtonian gravity
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
like this (where we consider
to be a small correction):
:
in which case the general solution is the same as the Newtonian case except there can be an additional term:
:
where there is an additional component varying
sinusoidally
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ma ...
over space. The wavelength of this variation could be quite large, such as an atomic width. Thus there appear to be several stable potentials around a gravitational force in this model.
Conformal unification to the Standard Model
By adding a suitable gravitational term to the
Standard Model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
action in
curved
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
, the theory develops a local conformal (Weyl) invariance. The conformal gauge is fixed by choosing a reference mass scale based on the gravitational constant. This approach generates the masses for the
vector boson
In particle physics, a vector boson is a boson whose spin equals one. The vector bosons that are regarded as elementary particles in the Standard Model are the gauge bosons, the force carriers of fundamental interactions: the photon of electromag ...
s and matter fields similar to the
Higgs mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other bein ...
without traditional spontaneous symmetry breaking.
See also
*
Conformal supergravity
*
Hoyle–Narlikar theory of gravity
The Hoyle–Narlikar theory of gravity is a Machian and conformal theory of gravity proposed by Fred Hoyle and Jayant Narlikar that originally fits into the quasi steady state model of the universe.
Description
The gravitational constant ''G'' ...
References
Further reading
*
Falsification of Mannheim's conformal gravityat CERN
Mannheim's rebuttal of aboveat arXiv.
{{theories of gravitation, state=collapsed
Conformal geometry
Lagrangian mechanics
Spacetime
Theoretical physics
Theories of gravity