The Lanczos tensor or Lanczos potential is a
rank 3 tensor 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 ...
that generates 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 ...
.
[Hyôitirô Takeno, "On the spintensor of Lanczos", ''Tensor'', 15 (1964) pp. 103–119.] It was first introduced by
Cornelius Lanczos
__NOTOC__
Cornelius (Cornel) Lanczos ( hu, Lánczos Kornél, ; born as Kornél Lőwy, until 1906: ''Löwy (Lőwy) Kornél''; February 2, 1893 – June 25, 1974) was a Hungarian-American and later Hungarian-Irish mathematician and physicist. Accor ...
in 1949.
The theoretical importance of the Lanczos tensor is that it serves as the
gauge field
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie group ...
for the
gravitational field in the same way that, by analogy, the
electromagnetic four-potential
An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. W ...
generates the
electromagnetic field.
[P. O’Donnell and H. Pye, "A Brief Historical Review of the Important Developments in Lanczos Potential Theory", ''EJTP'', 7 (2010) pp. 327–350. ]
Definition
The Lanczos tensor can be defined in a few different ways. The most common modern definition is through the Weyl–Lanczos equations, which demonstrate the generation of the Weyl tensor from the Lanczos tensor.
These equations, presented below, were given by Takeno in 1964.
The way that Lanczos introduced the tensor originally was as a
Lagrange multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied e ...
on constraint terms studied in the
variational approach to general relativity. Under any definition, the Lanczos tensor ''H'' exhibits the following symmetries:
:
:
The Lanczos tensor always exists in four dimensions
but does not generalize to higher dimensions. This highlights the
specialness of four dimensions.
Note further that the full
Riemann tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
cannot in general be derived from derivatives of the Lanczos potential alone.
The
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
must provide 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 measur ...
to complete the components of the
Ricci decomposition In the mathematical fields of Riemannian and pseudo-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a Riemannian or pseudo-Riemannian manifold into pieces with special algebraic properties. Th ...
.
The
Curtright field In theoretical physics, the Curtright field (named after Thomas Curtright) is a tensor quantum field of mixed symmetry, whose gauge-invariant dynamics are Hodge dual, dual to those of the general relativistic graviton in higher (''D''>4) spacetime ...
has a gauge-transformation dynamics similar to that of Lanczos tensor. But Curtright field exists in arbitrary dimensions > 4D.
Weyl–Lanczos equations
The Weyl–Lanczos equations express the Weyl tensor entirely as derivatives of the Lanczos tensor:
:
where
is the Weyl tensor, the semicolon denotes the
covariant derivative, and the subscripted parentheses indicate
symmetrization. Although the above equations can be used to define the Lanczos tensor, they also show that it is not unique but rather has
gauge freedom under an
affine group
In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself.
It is a Lie group if is the real or complex field or quaternions.
Rela ...
. If
is an arbitrary
vector field, then the Weyl–Lanczos equations are invariant under the gauge transformation
:
where the subscripted brackets indicate
antisymmetrization. An often convenient choice is the Lanczos algebraic gauge,
which sets
The gauge can be further restricted through the Lanczos differential gauge
. These gauge choices reduce the Weyl–Lanczos equations to the simpler form
:
Wave equation
The Lanczos potential tensor satisfies a wave equation
:
where
is the
d'Alembert operator
In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of Mi ...
and
:
is known as the
Cotton tensor. Since the Cotton tensor depends only on
covariant derivatives of 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 measur ...
, it can perhaps be interpreted as a kind of matter current.
The additional self-coupling terms have no direct electromagnetic equivalent. These self-coupling terms, however, do not affect the
vacuum solutions, where the Ricci tensor vanishes and the curvature is described entirely by the Weyl tensor. Thus in vacuum, the
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
are equivalent to the
homogeneous wave equation
in perfect analogy to the vacuum wave equation
of the electromagnetic four-potential. This shows a formal similarity between
gravitational waves and
electromagnetic wave
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
s, with the Lanczos tensor well-suited for studying gravitational waves.
In the weak field approximation where
, a convenient form for the Lanczos tensor in the Lanczos gauge is
:
Example
The most basic nontrivial case for expressing the Lanczos tensor is, of course, for the
Schwarzschild metric.
The simplest, explicit component representation in
natural units
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
for the Lanczos tensor in this case is
:
with all other components vanishing up to symmetries. This form, however, is not in the Lanczos gauge. The nonvanishing terms of the Lanczos tensor in the Lanczos gauge are
:
:
:
It is further possible to show, even in this simple case, that the Lanczos tensor cannot in general be reduced to a linear combination of the spin coefficients of the
Newman–Penrose formalism
The Newman–Penrose (NP) formalism The original paper by Newman and Penrose, which introduces the formalism, and uses it to derive example results.Ezra T Newman, Roger Penrose. ''Errata: An Approach to Gravitational Radiation by a Method of Sp ...
, which attests to the Lanczos tensor's fundamental nature.
Similar calculations have been used to construct arbitrary
Petrov type D solutions.
See also
*
Bach tensor
*
Ricci calculus
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be cal ...
*
Schouten tensor In Riemannian geometry the Schouten tensor is a second-order tensor introduced by Jan Arnoldus Schouten defined for by:
:P=\frac \left(\mathrm -\frac g\right)\, \Leftrightarrow \mathrm=(n-2) P + J g \, ,
where Ric is the Ricci tensor (defined by ...
*
tetradic Palatini action
The Einstein–Hilbert action for general relativity was first formulated purely in terms of the space-time metric. To take the metric and affine connection as independent variables in the action principle was first considered by Palatini. It is ...
*
Self-dual Palatini action
Ashtekar variables, which were a new canonical formalism of general relativity, raised new hopes for the canonical quantization of general relativity and eventually led to loop quantum gravity. Smolin and others independently discovered that the ...
References
{{Reflist, 30em
External links
*Peter O'Donnell, ''Introduction To 2-Spinors In General Relativity''
World Scientific 2003.
Gauge theories
Differential geometry
Tensors
Tensors in general relativity
1949 introductions