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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 and three spacelike, defined on a Lorentzian manifold 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 differen ...
. The timelike unit vector field is often denoted by \vec_0 and the three spacelike unit vector fields by \vec_1, \vec_2, \, \vec_3. 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 tenso ...
ial quantities defined on the
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
can be expressed using the frame field and its
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
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 Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
in 1929.
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
"Elektron und Gravitation I", ''Zeitschrift Physik'', 56, p330–352, 1929.
The index notation for tetrads is explained in
tetrad (index notation) 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 ...
.


Physical interpretation

Frame fields of a Lorentzian manifold always correspond to a family of ideal observers immersed in the given spacetime; the integral curves 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
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s. 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 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 physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
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, which will of course be accelerated by the
Lorentz force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
, 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, 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.


Specifying a frame

To write down a frame, a coordinate chart 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 vector fields: : \vec = X^\mu \, \partial_. Here, the
Einstein summation convention In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...
is used, and the vector fields are thought of as
first order In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
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 r ...
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
s, and the components X^\mu are often called contravariant components. This follows the standard notational conventions for
sections Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...
of a tangent bundle. Alternative notations for the coordinate basis vector fields in common use are \partial / \partial x^\mu\equiv\partial_\equiv\partial_\mu. In particular, the vector fields in the frame can be expressed this way: : \vec_a = ^\mu \, \partial_. 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 mathema ...
, that the four vector fields are everywhere orthonormal. More modern texts adopt the notation \mathbf_\mu for \partial_ and \gamma_a or \sigma_a for \vec_a. This permits the visually clever trick of writing the spacetime metric as the outer product of the coordinate tangent vectors: :g_ = \mathbf_\mu\cdot\mathbf_\nu and the flat-space Minkowski metric as the product of the gammas: :\eta_=\gamma_a\cdot\gamma_b The choice of \gamma_a 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 \gamma_a 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 special ...
. Appropriately used, this can simplify some of the notation used in writing a spin connection. Once a signature is adopted, by
duality Duality may refer to: Mathematics * Duality (mathematics), a mathematical concept ** Dual (category theory), a formalization of mathematical duality ** Duality (optimization) ** Duality (order theory), a concept regarding binary relations ** Dual ...
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 In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...
.


Specifying the metric using a coframe

Alternatively, 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 ...
can be specified by writing down a coframe in terms of a coordinate basis and stipulating that the metric tensor is given by :g = -\sigma^0 \otimes \sigma^0 +\sum_^3 \sigma^i \otimes \sigma^i, where \otimes 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 \otimes W ...
. 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, e^_, has two kinds of indices: \mu \, labels the general spacetime coordinate and a \, 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 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 ...
, g^ \,, since in a coordinate basis, :g^= e^_ e^_ \eta^ \, where \eta^ \, 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 the r ...
. 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: :T^a = \eta^ T_b. The vierbein field enables conversion between spacetime and local Lorentz indices. For example: :T_a = e^\mu_ T_\mu. The vierbein field itself can be manipulated in the same fashion: :e^\nu_ = e^\mu_ e^\nu_ \,, since e^\nu_ = \delta^\nu_\mu. And these can combine. :T^a = e_\mu^ T^\mu. A few more examples: Spacetime and local Lorentz coordinates can be mixed together: :T^=e_\nu^ T^. The local Lorentz coordinates transform differently from the general spacetime coordinates. Under a general coordinate transformation we have: :T'^ = \fracT^ whilst under a local Lorentz transformation we have: :T'^ = \Lambda(x)^a_ T^.


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, 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 dis ...
or electrovacuum solutions, 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 curves of the timelike unit vector field must define a
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
congruence Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
, or in other words, its acceleration vector must vanish: : \nabla_ \, \vec_0 = 0 It is also often desirable to ensure that the spatial triad carried by each observer does not
rotate Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
. In this case, the triad can be viewed as being gyrostabilized. The criterion for a nonspinning inertial (NSI) frame is again very simple: : \nabla_ \, \vec_j = 0, \; \; j = 0 \dots 3 This says that as we move along the worldline of each observer, their spatial triad is
parallel-transport In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bu ...
ed. 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 o ...
(these are special nonspinning inertial frames in the Minkowski vacuum). More generally, if the acceleration of our observers is nonzero, \nabla_\,\vec_0 \neq 0, we can replace the covariant derivatives : \nabla_ \, \vec_j, \; j = 1 \dots 3 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: :ds^2 = -(1-2m/r) \, dt^2 + \frac + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right) : -\infty < t < \infty, \; 2 m < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi More formally, the metric tensor can be expanded with respect to the coordinate cobasis as :g = -(1-2m/r) \, dt \otimes dt + \frac \, dr \otimes dr + r^2 \, d\theta \otimes d\theta + r^2 \sin(\theta)^2 \, d\phi \otimes d\phi A coframe can be read off from this expression: : \sigma^0 = \sqrt \, dt, \; \sigma^1 = \frac, \; \sigma^2 = r d\theta, \; \sigma^3 = r \sin(\theta) d\phi To see that this coframe really does correspond to the Schwarzschild metric tensor, just plug this coframe into :g = -\sigma^0 \otimes \sigma^0 + \sigma^1 \otimes \sigma^1 + \sigma^2 \otimes \sigma^2 + \sigma^3 \otimes \sigma^3 The frame dual is the coframe inverse as below: (frame dual is also transposed to keep local index in same position.) : \vec_0 = \frac \partial_t, \; \vec_1 = \sqrt \partial_r, \; \vec_2 = \frac \partial_\theta, \; \vec_3 = \frac \partial_\phi (The plus sign on \sigma^0 ensures that \vec_0 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 : \nabla_ \vec_0 = -\frac \, \vec_1 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 \vec_0) 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 : E = R_ \, X^m \, X^n where we write \vec = \vec_0 to avoid cluttering the notation. Its only non-zero components with respect to our coframe turn out to be : E = -2m/r^3, \; E = E = m/r^3 The corresponding coordinate basis components are : E = -2m/r^3/(1-2m/r), \; E = m/r, \; E = m \sin(\theta)^2/r (A quick note concerning notation: many authors put carets 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 t,r,\theta,\phi. Since an expression like S_ = 36 m/r doesn't make sense as a
tensor equation 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 tens ...
, 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 ...
\Phi 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 A Hessian is an inhabitant of the German state of Hesse. Hessian may also refer to: Named from the toponym *Hessian (soldier), eighteenth-century German regiments in service with the British Empire **Hessian (boot), a style of boot **Hessian f ...
of the gravitational potential U. Using tensor notation for a tensor field defined on three-dimensional euclidean space, this can be written :\Phi_ = U_ - \frac _ \, \eta_ 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: : m/(r+h)^2 - m/r^2 = -2mh/r^3 + 3mh^2/r^4 + O(h^3) Because in discussing tensors we are dealing with multilinear algebra, we retain only first order terms, so \Phi_ = -2m/r^3. Similarly, we can compare the gravitational force on two nearby observers lying on the same sphere r = r_0. 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 : \frac \, \sin(\theta) \approx \frac \, \frac = \frac \, h By using the small angle approximation, we have ignored all terms of order O(h^2), so the tangential components are \Phi_ = \Phi_ = m/r^3. Here, we are referring to the obvious frame obtained from the polar spherical chart for our three-dimensional euclidean space: : \vec_1 = \partial_r, \; \vec_2 = \frac \, \partial_\theta, \; \vec_3 = \frac \, \partial_\phi Plainly, the coordinate components E , \, E 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 \vec_1 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 Lemaître coordinates are a particular set of coordinates for the Schwarzschild metric—a spherically symmetric solution to the Einstein field equations in vacuum—introduced by Georges Lemaître in 1932. English translation: See also: & ...
and can be written as :\vec_0 = \frac \, \partial_t - \sqrt \, \partial_r :\vec_1 = \partial_r - \frac \, \partial_t :\vec_2 = \frac \, \partial_\theta :\vec_3 = \frac \, \partial_\phi Note that \vec_0 \neq \vec_0, \; \vec_1 \neq \vec_1, and that \vec_0 "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 \vec_0) vanish identically, our new frame is a ''nonspinning inertial frame''. If our massive object is in fact a (nonrotating)
black hole A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
, we probably wish to follow the experience of the Lemaître observers as they fall through the event horizon at r = 2m. 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 : T(t,r) = t - \int \frac \, dr = t + 2 \sqrt + 2m \log \left( \frac \right) This gives the Painlevé chart. The new line element is : ds^2 = -dT^2 + \left( dr + \sqrt \, dT \right)^2 + r^2 \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right) : -\infty < T < \infty, \; 0 < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi With respect to the Painlevé chart, the Lemaître frame is :\vec_0 = \partial_T - \sqrt \, \partial_r :\vec_1 = \partial_r :\vec_2 = \frac \, \partial_\theta :\vec_3 = \frac \, \partial_\phi 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 hyperslice Spatial may refer to: *Dimension *Space *Three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determ ...
s T=T_0 turn out to be
locally isometric In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighbourhood (mathematics), n ...
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 : E = R_ \, Y^m \, Y^n where we write Y = \vec_0 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: E = -2m/r^3, \, E = E = m/r^3. (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 \vec_3 direction by an undetermined parameter (depending on the radial coordinate), compute the acceleration vector, and require that this vanish ''in the equatorial plane'' \theta=\pi/2. 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 was a Japanese astronomer noted for his contributions to celestial mechanics. Life and work Hagihara graduated from Tokyo Imperial University with a degree in astronomy in 1921 and became an assistant professor of astronomy there two years ...
. In the static polar spherical chart, the Hagihara frame is :\vec_0 = \frac \, \partial_t + \frac \, \partial_\phi :\vec_1 = \sqrt \, \partial_r :\vec_2 = \frac \, \partial_\theta :\vec_3 = \frac \, \partial_\phi + \frac \, \partial_t which in the equatorial plane becomes :\vec_0 = \frac \, \partial_t + \frac \, \partial_\phi :\vec_1 = \sqrt \, \partial_r :\vec_2 = \frac \, \partial_\theta :\vec_3 = \frac \, \partial_\phi + \frac \, \partial_t The tidal tensor E where \vec = \vec_0 turns out to be given (in the equatorial plane) by :E = -\frac \, \frac = -\frac - \frac + O(1/r^5) :E = \frac \, \frac = -\frac + \frac + O(1/r^5) :E = \frac 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 r > 3m. Indeed, stable circular orbits only exist on r > 6m, 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 \vec_1, \; \vec_3 rotate about \vec_2 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 In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
or pseudo-Riemannian manifold, 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 of ...
for the tangent space at each point in the manifold. This is possible globally in a continuous fashion if and only if the manifold is parallelizable. 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 space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
V, we can define a new space consisting of all tuples of orthonormal bases for V. Applying this construction to each tangent space yields the orthonormal frame bundle 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 po ...
, or even arbitrary principal
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
s. 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 * Georges Lemaître *
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 ...
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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 ...
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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 ...
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Yusuke Hagihara was a Japanese astronomer noted for his contributions to celestial mechanics. Life and work Hagihara graduated from Tokyo Imperial University with a degree in astronomy in 1921 and became an assistant professor of astronomy there two years ...


References

* * See ''Chapter IV'' for frames in E3, then see ''Chapter VIII'' for frame fields in
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
s. 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). 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