A frame field 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 ...
(also called a tetrad or vierbein) is a set of four
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
-
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
vector fields, one
timelike
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 ...
and three
spacelike
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 ...
, defined on a
Lorentzian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
that is physically interpreted as a model of
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 differ ...
. The timelike unit vector field is often denoted by
and the three spacelike unit vector fields by
. All
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
ial quantities defined on the
manifold can be expressed using the frame field and its
dual coframe field.
Frame were introduced into general relativity by
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
in 1928 and by
Hermann Weyl in 1929.
[ Hermann Weyl "Elektron und Gravitation I", ''Zeitschrift Physik'', 56, p330–352, 1929.]
The index notation for tetrads is explained in
tetrad (index notation).
Physical interpretation
Frame fields of a
Lorentzian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
always correspond to a family of ideal observers immersed in the given spacetime; the
integral curve
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.
Name
Integral curves are known by various other names, depending on the nature and interpret ...
s of the timelike unit vector field are the
worldline
The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics.
The concept of a "world line" is distinguished from con ...
s of these observers, and at each event along a given worldline, the three spacelike unit vector fields specify the spatial triad carried by the observer. The triad may be thought of as defining the spatial coordinate axes of a local ''laboratory frame'', which is valid very near the observer's worldline.
In general, the worldlines of these observers need not be timelike
geodesics. If any of the worldlines bends away from a geodesic path in some region, we can think of the observers as
test particles In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insuf ...
that
accelerate
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
by using ideal rocket engines with a thrust equal to the magnitude of their
acceleration vector. Alternatively, if our observer is attached to a bit of matter in a ball of
fluid in
hydrostatic equilibrium
In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary ...
, this bit of matter will in general be accelerated outward by the net effect of
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
holding up the fluid ball against the attraction of its own gravity. Other possibilities include an observer attached to a free charged test particle in an
electrovacuum solution In general relativity, an electrovacuum solution (electrovacuum) is an exact solution of the Einstein field equation in which the only nongravitational mass–energy present is the field energy of an electromagnetic field, which must satisfy the (c ...
, which will of course be accelerated by the
Lorentz force, or an observer attached to a ''spinning'' test particle, which may be accelerated by a spin–spin force.
It is important to recognize that frames are ''geometric objects''. That is, vector fields make sense (in a smooth manifold) independently of choice of a
coordinate chart In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
, and (in a Lorentzian manifold), so do the notions of orthogonality and length. Thus, just like vector fields and other geometric quantities, frame fields can be represented in various coordinate charts. Computations of the components of tensorial quantities, with respect to a given frame, will always yield the ''same'' result, whichever coordinate chart is used to represent the frame.
These fields are required to write the
Dirac equation in curved spacetime
In mathematical physics, the Dirac equation in curved spacetime is a generalization of the Dirac equation from flat spacetime (Minkowski space) to curved spacetime, a general Lorentzian manifold.
Mathematical formulation
Spacetime
In full ...
.
Specifying a frame
To write down a frame, a
coordinate chart In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
on the Lorentzian manifold needs to be chosen. Then, every vector field on the manifold can be written down as a linear combination of the four
coordinate basis In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold is a set of basis vector fields defined at every point of a region of the manifold as
:\mathbf_ = \lim_ \frac ,
where is the displacem ...
vector fields:
:
Here, the
Einstein summation convention is used, and the vector fields are thought of as
first order linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
differential operators, and the components
are often called
contravariant components. This follows the standard notational conventions for
sections of a
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
. Alternative notations for the coordinate basis vector fields in common use are
In particular, the vector fields in the frame can be expressed this way:
:
In "designing" a frame, one naturally needs to ensure, using the given
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
, that the four vector fields are everywhere orthonormal.
More modern texts adopt the notation
for
and
or
for
. This permits the visually clever trick of writing the spacetime metric as the outer product of the coordinate tangent vectors:
:
and the flat-space Minkowski metric as the product of the gammas:
:
The choice of
for the notation is an intentional conflation with the notation used for the
Dirac matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
; it allows the
to be taken not only as vectors, but as elements of an algebra, the
spacetime algebra
In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra . According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of speci ...
. Appropriately used, this can simplify some of the notation used in writing a
spin connection
In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz tr ...
.
Once a signature is adopted, by
duality every ''vector'' of a basis has a dual ''covector'' in the cobasis and conversely. Thus, every ''frame field'' is associated with a unique ''coframe field'', and vice versa; a coframe fields is a set of four orthogonal sections of the
cotangent bundle.
Specifying the metric using a coframe
Alternatively, the
metric tensor can be specified by writing down a coframe in terms of a coordinate basis and stipulating that the metric tensor is given by
:
where
denotes
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
.
This is just a fancy way of saying that the coframe is ''orthonormal''. Whether this is used to obtain the metric tensor after writing down the frame (and passing to the dual coframe), or starting with the metric tensor and using it to verify that a frame has been obtained by other means, it must always hold true.
Relationship with metric tensor, in a coordinate basis
The vierbein field,
, has two kinds of indices:
labels the general spacetime coordinate and
labels the local Lorentz spacetime or local laboratory coordinates.
The vierbein field or frame fields can be regarded as the “matrix square root” of the
metric tensor,
, since in a coordinate basis,
:
where
is the
Lorentz metric
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which t ...
.
Local Lorentz indices are raised and lowered with the Lorentz metric in the same way as general spacetime coordinates are raised and lowered with the metric tensor. For example:
:
The vierbein field enables conversion between spacetime and local Lorentz indices. For example:
:
The vierbein field itself can be manipulated in the same fashion:
:
, since
And these can combine.
:
A few more examples: Spacetime and local Lorentz coordinates can be mixed together:
:
The local Lorentz coordinates transform differently from the general spacetime coordinates. Under a general coordinate transformation we have:
:
whilst under a local Lorentz transformation we have:
:
Comparison with coordinate basis
Coordinate basis vectors have the special property that their pairwise
Lie brackets vanish. Except in locally flat regions, at least some Lie brackets of vector fields from a frame will ''not'' vanish. The resulting baggage needed to compute with them is acceptable, as components of tensorial objects with respect to a frame (but not with respect to a coordinate basis) have a direct interpretation in terms of measurements made by the family of ideal observers corresponding to the frame.
Coordinate basis vectors can be
null
Null may refer to:
Science, technology, and mathematics Computing
* Null (SQL) (or NULL), a special marker and keyword in SQL indicating that something has no value
* Null character, the zero-valued ASCII character, also designated by , often use ...
, which, by definition, cannot happen for frame vectors.
Nonspinning and inertial frames
Some frames are nicer than others. Particularly in
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...
or
electrovacuum solution In general relativity, an electrovacuum solution (electrovacuum) is an exact solution of the Einstein field equation in which the only nongravitational mass–energy present is the field energy of an electromagnetic field, which must satisfy the (c ...
s, the physical experience of inertial observers (who feel no forces) may be of particular interest. The mathematical characterization of an inertial frame is very simple: the
integral curve
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.
Name
Integral curves are known by various other names, depending on the nature and interpret ...
s of the timelike unit
vector field must define a
geodesic congruence, or in other words, its acceleration vector must vanish:
:
It is also often desirable to ensure that the spatial triad carried by each observer does not
rotate. In this case, the triad can be viewed as being
gyrostabilized. The criterion for a nonspinning inertial (NSI) frame is again very simple:
:
This says that as we move along the worldline of each observer, their spatial triad is
parallel-transported. Nonspinning inertial frames hold a special place in general relativity, because they are as close as we can get in a curved Lorentzian manifold to the Lorentz frames used in
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
(these are special nonspinning inertial frames in the
Minkowski vacuum).
More generally, if the acceleration of our observers is nonzero,
, we can replace the
covariant derivatives
:
with the (spatially projected)
Fermi–Walker derivatives to define a nonspinning frame.
Given a Lorentzian manifold, we can find infinitely many frame fields, even if we require additional properties such as inertial motion. However, a given frame field might very well be defined on only part of the manifold.
Example: Static observers in Schwarzschild vacuum
It will be instructive to consider in some detail a few simple examples. Consider the famous
Schwarzschild vacuum that models spacetime outside an isolated nonspinning spherically symmetric massive object, such as a star. In most textbooks one finds the metric tensor written in terms of a static polar spherical chart, as follows:
:
:
More formally, the metric tensor can be expanded with respect to the coordinate cobasis as
:
A coframe can be read off from this expression:
:
To see that this coframe really does correspond to the Schwarzschild metric tensor, just plug this coframe into
:
The frame dual is the coframe inverse as below: (frame dual is also transposed to keep local index in same position.)
:
(The plus sign on
ensures that
is ''future pointing''.) This is the frame that models the experience of static observers who use rocket engines to ''"hover" over the massive object''.
The thrust they require to maintain their position is given by the magnitude of the acceleration vector
:
This is radially inward pointing, since the observers need to accelerate ''away'' from the object to avoid falling toward it. On the other hand, the spatially projected Fermi derivatives of the spatial basis vectors (with respect to
) vanish, so this is a nonspinning frame.
The components of various tensorial quantities with respect to our frame and its dual coframe can now be computed.
For example, the
tidal tensor
Tidal is the adjectival form of tide.
Tidal may also refer to:
* ''Tidal'' (album), a 1996 album by Fiona Apple
* Tidal (king), a king involved in the Battle of the Vale of Siddim
* TidalCycles, a live coding environment for music
* Tidal (servic ...
for our static observers is defined using tensor notation (for a coordinate basis) as
:
where we write
to avoid cluttering the notation. Its only non-zero components with respect to our coframe turn out to be
:
The corresponding coordinate basis components are
:
(A quick note concerning notation: many authors put
caret
Caret is the name used familiarly for the character , provided on most QWERTY keyboards by typing . The symbol has a variety of uses in programming and mathematics. The name "caret" arose from its visual similarity to the original proofreade ...
s over ''abstract'' indices referring to a frame. When writing down ''specific components'', it is convenient to denote frame components by 0,1,2,3 and coordinate components by
. Since an expression like
doesn't make sense as a
tensor equation, there should be no possibility of confusion.)
Compare the
tidal tensor
Tidal is the adjectival form of tide.
Tidal may also refer to:
* ''Tidal'' (album), a 1996 album by Fiona Apple
* Tidal (king), a king involved in the Battle of the Vale of Siddim
* TidalCycles, a live coding environment for music
* Tidal (servic ...
of Newtonian gravity, which is the
traceless
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix ().
It can be proved that the trace o ...
part of the
Hessian of the gravitational potential
. Using tensor notation for a tensor field defined on three-dimensional euclidean space, this can be written
:
The reader may wish to crank this through (notice that the trace term actually vanishes identically when U is harmonic) and compare results with the following elementary approach:
we can compare the gravitational forces on two nearby observers lying on the same radial line:
:
Because in discussing tensors we are dealing with
multilinear algebra
Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of ''p' ...
, we retain only first order terms, so
. Similarly, we can compare the gravitational force on two nearby observers lying on the same sphere
. Using some elementary trigonometry and the small angle approximation, we find that the force vectors differ by a vector tangent to the sphere which has magnitude
:
By using the small angle approximation, we have ignored all terms of order
, so the tangential components are
. Here, we are referring to the obvious frame obtained from the polar spherical chart for our three-dimensional euclidean space:
:
Plainly, the coordinate components
computed above don't even scale the right way, so they clearly cannot correspond to what an observer will measure even approximately. (By coincidence, the Newtonian tidal tensor components agree exactly with the relativistic tidal tensor components we wrote out above.)
Example: Lemaître observers in the Schwarzschild vacuum
To find an inertial frame, we can boost our static frame in the
direction by an undetermined boost parameter (depending on the radial coordinate), compute the acceleration vector of the new undetermined frame, set this equal to zero, and solve for the unknown boost parameter. The result will be a frame which we can use to study the physical experience of observers who fall freely and radially toward the massive object. By appropriately choosing an integration constant, we obtain the frame of Lemaître observers, who fall in ''from rest at spatial infinity''. (This phrase doesn't make sense, but the reader will no doubt have no difficulty in understanding our meaning.) In the static polar spherical chart, this frame is obtained from
Lemaître coordinates and can be written as
:
:
:
:
Note that
, and that
"leans inwards", as it should, since its integral curves are timelike geodesics representing the world lines of ''infalling'' observers. Indeed, since the covariant derivatives of all four basis vectors (taken with respect to
) vanish identically, our new frame is a ''nonspinning inertial frame''.
If our massive object is in fact a (nonrotating)
black hole, we probably wish to follow the experience of the Lemaître observers as they fall through the
event horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s.
In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact ob ...
at
. Since the static polar spherical coordinates have a
coordinate singularity A coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate frame that can be removed by choosing a different frame.
An example is the apparent (longitudinal) singularity at the 90 degree latitude in sph ...
at the horizon, we'll need to switch to a more appropriate coordinate chart. The simplest possible choice is to define a new time coordinate by
:
This gives the
Painlevé chart. The new line element is
:
:
With respect to the Painlevé chart, the Lemaître frame is
:
:
:
:
Notice that their spatial triad looks exactly like the frame for three-dimensional euclidean space which we mentioned above (when we computed the Newtonian tidal tensor). Indeed, the
spatial hyperslices
turn out to be
locally isometric to flat three-dimensional euclidean space! (This is a remarkable and rather special property of the Schwarzschild vacuum; most spacetimes do not admit a slicing into flat spatial sections.)
The tidal tensor taken with respect to the Lemaître observers is
:
where we write
to avoid cluttering the notation. This is a ''different tensor'' from the one we obtained above, because it is defined using a ''different family of observers''. Nonetheless, its nonvanishing components look familiar:
. (This is again a rather special property of the Schwarzschild vacuum.)
Notice that there is simply no way of defining static observers on or inside the event horizon. On the other hand, the Lemaître observers are not defined on the entire ''exterior region'' covered by the static polar spherical chart either, so in these examples, neither the Lemaître frame nor the static frame are defined on the entire manifold.
Example: Hagihara observers in the Schwarzschild vacuum
In the same way that we found the Lemaître observers, we can boost our static frame in the
direction by an undetermined parameter (depending on the radial coordinate), compute the acceleration vector, and require that this vanish ''in the equatorial plane''
. The new Hagihara frame describes the physical experience of observers in ''stable circular orbits'' around our massive object. It was apparently first discussed by the astronomer
Yusuke Hagihara.
In the static polar spherical chart, the Hagihara frame is
:
:
:
:
which in the equatorial plane becomes
:
:
:
:
The tidal tensor
where
turns out to be given (in the equatorial plane) by
:
:
:
Thus, compared to a static observer hovering at a given coordinate radius,
a Hagihara observer in a stable circular orbit with the same coordinate radius will measure ''radial'' tidal forces which are slightly ''larger'' in magnitude, and ''transverse'' tidal forces which are no longer isotropic (but slightly larger orthogonal to the direction of motion).
Note that the Hagihara frame is only defined on the region
. Indeed, stable circular orbits only exist on
, so the frame should not be used inside this locus.
Computing
Fermi derivatives shows that the frame field just given is in fact ''spinning'' with respect to a gyrostabilized frame. The principal reason why is easy to spot: in this frame, each Hagihara observer keeps his spatial vectors ''radially aligned'', so
rotate about
as the observer orbits around the central massive object. However, after correcting for this observation, a small precession of the spin axis of a gyroscope carried by a Hagihara observer still remains; this is the ''de Sitter precession'' effect (also called the ''geodetic precession'' effect).
Generalizations
This article has focused on the application of frames to general relativity, and particularly on their physical interpretation. Here we very briefly outline the general concept. In an ''n''-dimensional
Riemannian manifold or
pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
, a frame field is a set of
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
vector fields which forms a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
for the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
at each point in the manifold. This is possible globally in a continuous fashion if and only if the manifold is
parallelizable
In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist smooth vector fields
\
on the manifold, such that at every point p of M the tangent vectors
\
provide a basis of the tangent space at p. Equi ...
. As before, frames can be specified in terms of a given coordinate basis, and in a non-flat region, some of their pairwise Lie brackets will fail to vanish.
In fact, given any
inner-product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, we can define a new space consisting of all tuples of orthonormal bases for
. Applying this construction to each tangent space yields the orthonormal
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
of a (pseudo-)Riemannian manifold and a frame field is a section of this bundle. More generally still, we can consider frame bundles associated to any
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
, or even arbitrary
principal fiber bundles. The notation becomes a bit more involved because it is harder to avoid distinguishing between indices referring to the base, and indices referring to the fiber. Many authors speak of internal components when referring to components indexed by the fiber.
See also
*
Exact solutions in general relativity
In general relativity, an exact solution is a solution of the Einstein field equations whose derivation does not invoke simplifying assumptions, though the starting point for that derivation may be an idealized case like a perfectly spherical sh ...
*
Georges Lemaître
Georges Henri Joseph Édouard Lemaître ( ; ; 17 July 1894 – 20 June 1966) was a Belgian Catholic priest, theoretical physicist, mathematician, astronomer, and professor of physics at the Catholic University of Louvain. He was the first to t ...
*
Karl Schwarzschild
Karl Schwarzschild (; 9 October 1873 – 11 May 1916) was a German physicist and astronomer.
Schwarzschild provided the first exact solution to the Einstein field equations of general relativity, for the limited case of a single spherical non-r ...
*
Method of moving frames
In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.
Introduction
In lay ...
*
Paul Painlevé
Paul Painlevé (; 5 December 1863 – 29 October 1933) was a French mathematician and statesman. He served twice as Prime Minister of the Third Republic: 12 September – 13 November 1917 and 17 April – 22 November 1925. His entry into politic ...
*
Vierbein
The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independent ...
*
Yusuke Hagihara
References
*
* See ''Chapter IV'' for frames in E
3, then see ''Chapter VIII'' for frame fields in
Riemannian manifolds. This book doesn't really cover Lorentzian manifolds, but with this background in hand the reader is well prepared for the next citation.
* In this book, a frame field (coframe field) is called an ''anholonomic basis of vectors (covectors)''. Essential information is widely scattered about, but can be easily found using the extensive index.
* In this book, a frame field is called a ''tetrad'' (not to be confused with the now standard term ''NP tetrad'' used in 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 ...
). See ''Section 98''.
*{{cite book , author1=De Felice, F. , author2=Clarke, C. J. , title=Relativity on Curved Manifolds , publisher=Cambridge: Cambridge University Press, year=1992 , isbn=0-521-42908-0 See ''Chapter 4'' for frames and coframes. If you ever need more information about frame fields, this might be a good place to look!
Frames of reference
Mathematical methods in general relativity