A frame field in general relativity (also called a tetrad or vierbein) is a set of four pointwise-orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime. The timelike unit vector field is often denoted by
\vec{e}0
\vec{e}1,\vec{e}2,\vec{e}3
Frame fields were introduced into general relativity by Albert Einstein in 1928[1] and by Hermann Weyl in 1929.[2]
The index notation for tetrads is explained in tetrad (index notation).
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 worldlines 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 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 hydrostatic equilibrium, this bit of matter will in general be accelerated outward by the net effect of pressure 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, 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.
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}=X\mu
\partial | |
x\mu |
.
X\mu
\partial/\partial
\mu\equiv\partial | |
x | |
x\mu |
\equiv\partial\mu.
In particular, the vector fields in the frame can be expressed this way:
\vec{e}a=
\mu | |
{e | |
a} |
\partial | |
x\mu |
.
More modern texts adopt the notation
g\mu
\partial | |
x\mu |
\gammaa
\sigmaa
\vec{e}a
g\mu\nu=g\mu ⋅ g\nu
ηab=\gammaa ⋅ \gammab
\gammaa
\gammaa
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 field is a set of four orthogonal sections of the cotangent bundle.
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
g=-\sigma0 ⊗ \sigma0
3 | |
+\sum | |
i=1 |
\sigmai ⊗ \sigmai,
⊗
The vierbein field,
\mu | |
e | |
a |
\mu
a
The vierbein field or frame fields can be regarded as the "matrix square root" of the metric tensor,
g\mu
g\mu=
\mu | |
e | |
a |
\nu | |
e | |
b |
ηab
ηab
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:
Ta=ηabTb.
The vierbein field enables conversion between spacetime and local Lorentz indices. For example:
Ta=
\mu | |
e | |
a |
T\mu.
The vierbein field itself can be manipulated in the same fashion:
\nu | |
e | |
a |
=
\mu | |
e | |
a |
\nu | |
e | |
\mu |
\nu | |
e | |
\mu |
=
\nu | |
\delta | |
\mu. |
And these can combine.
Ta=
a | |
e | |
\mu |
T\mu.
A few more examples: Spacetime and local Lorentz coordinates can be mixed together:
T\mu
a | |
=e | |
\nu |
T\mu.
The local Lorentz coordinates transform differently from the general spacetime coordinates. Under a general coordinate transformation we have:
T'\mu=
\partialx'\mu | |
\partialx\nu |
T\nu
whilst under a local Lorentz transformation we have:
T'\mu=
a | |
Λ(x) | |
b |
T\mu.
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.
Some frames are nicer than others. Particularly in vacuum 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 congruence, or in other words, its acceleration vector must vanish:
\nabla\vec{e0
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:
\nabla\vec{e0}\vec{e}j=0, j=0...3
More generally, if the acceleration of our observers is nonzero,
\nabla\vec{e0}\vec{e}j, j=1...3
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.
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:
ds2=-(1-2m/r)dt2+
dr2 | |
1-2m/r |
+r2\left(d\theta2+\sin(\theta)2d\phi2\right)
-infty<t<infty, 2m<r<infty, 0<\theta<\pi, -\pi<\phi<\pi
g=-(1-2m/r)dt ⊗ dt+
1 | |
1-2m/r |
dr ⊗ dr+r2d\theta ⊗ d\theta+r2\sin(\theta)2d\phi ⊗ d\phi
\sigma0=\sqrt{1-2m/r}dt, \sigma1=
dr | |
\sqrt{1-2m/r |
g=-\sigma0 ⊗ \sigma0+\sigma1 ⊗ \sigma1+\sigma2 ⊗ \sigma2+\sigma3 ⊗ \sigma3
The frame dual is the coframe inverse as below: (frame dual is also transposed to keep local index in same position.)
\vec{e}0=
1 | |
\sqrt{1-2m/r |
\sigma0
\vec{e}0
\nabla\vec{e0}\vec{e}0=-
m/r2 | |
\sqrt{1-2m/r |
\vec{e}0
The components of various tensorial quantities with respect to our frame and its dual coframe can now be computed.
For example, the tidal tensor for our static observers is defined using tensor notation (for a coordinate basis) as
E[X]ab=RambnXmXn
\vec{X}=\vec{e}0
E[X]11=-2m/r3, E[X]22=E[X]33=m/r3
E[X]rr=-2m/r3/(1-2m/r), E[X]\theta=m/r, E[X]\phi=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
Sab=36m/r
\Phi
U
\Phiij=U,i-
1 | |
3 |
{U,k
m/(r+h)2-m/r2=-2mh/r3+3mh2/r4+O(h3)
\Phi11=-2m/r3
r=r0
m | ||||||
|
\sin(\theta) ≈
m | ||||||
|
h | |
r0 |
=
m | ||||||
|
h
O(h2)
\Phi22=\Phi33=m/r3
\vec{\epsilon}1=\partialr, \vec{\epsilon}2=
1 | |
r |
\partial\theta, \vec{\epsilon}3=
1 | |
r\sin\theta |
\partial\phi
Plainly, the coordinate components
E[X]\theta,E[X]\phi
To find an inertial frame, we can boost our static frame in the
\vec{e}1
\vec{f}0=
1 | |
1-2m/r |
\partialt-\sqrt{2m/r}\partialr
\vec{f}1=\partialr-
\sqrt{2m/r | |
\vec{f}2=
1 | |
r |
\partial\theta
\vec{f}3=
1 | |
r\sin(\theta) |
\partial\phi
\vec{e}0 ≠ \vec{f}0, \vec{e}1 ≠ \vec{f}1
\vec{f}0
\vec{e}0
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 at
r=2m
T(t,r)=t-\int
\sqrt{2m/r | |
ds2=-dT2+\left(dr+\sqrt{2m/r}dT\right)2+r2\left(d\theta2+\sin(\theta)2d\phi2\right)
-infty<T<infty, 0<r<infty, 0<\theta<\pi, -\pi<\phi<\pi
\vec{f}0=\partialT-\sqrt{2m/r}\partialr
\vec{f}1=\partialr
\vec{f}2=
1 | |
r |
\partial\theta
\vec{f}3=
1 | |
r\sin(\theta) |
\partial\phi
T=T0
The tidal tensor taken with respect to the Lemaître observers is
E[Y]ab=RambnYmYn
Y=\vec{f}0
E[Y]11=-2m/r3,E[Y]22=E[Y]33=m/r3
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.
In the same way that we found the Lemaître observers, we can boost our static frame in the
\vec{e}3
\theta=\pi/2
In the static polar spherical chart, the Hagihara frame is
\vec{h}0=
1 | |
\sqrt{1-3m/r |
\vec{h}1=\sqrt{1-2m/r}\partialr
\vec{h}2=
1 | |
r |
\partial\theta
\vec{h}3=
\sqrt{1-2m/r | |
\vec{h}0=
1 | |
\sqrt{1-3m/r |
\vec{h}1=\sqrt{1-2m/r}\partialr
\vec{h}2=
1 | |
r |
\partial\theta
\vec{h}3=
\sqrt{1-2m/r | |
E[Z]ab
\vec{Z}=\vec{h}0
E[Z]11=-
m | |
r3 |
2-3m/r | |
1-2m/r |
=-
2m | |
r3 |
-
m2 | |
r4 |
+O(1/r5)
E[Z]22=
m | |
r3 |
1 | |
1-3m/r |
=-
m | |
r3 |
+
3m2 | |
r4 |
+O(1/r5)
E[Z]33=
m | |
r3 |
Note that the Hagihara frame is only defined on the region
r>3m
r>6m
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{h}1, \vec{h}3
\vec{h}2
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, a frame field is a set of orthonormal vector fields which forms a basis 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.
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