Frame fields in general relativity

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Physical interpretation

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Specifying a frame

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Specifying the metric using a coframe

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Comparison with coordinate basis

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Nonspinning and inertial frames

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Example: Static observers in Schwarzschild vacuum

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Example: Lemaître observers in the Schwarzschild vacuum

To find an inertial frame, we can boost our static frame in the ${\vec {e}}_{1}$ 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 can be written

${\vec {f}}_{0}={\frac {1}{1-2m/r}}\,\partial _{t}-{\sqrt {2m/r}}\,\partial _{r}$

${\vec {f}}_{1}=\partial _{r}-{\frac {\sqrt {2m/r}}{1-2m/r}}\,\partial _{t}$

${\vec {f}}_{3}={\frac {1}{r}}\,\partial _{\theta }$

${\vec {f}}_{3}={\frac {1}{r\sin(\theta )}}\,\partial _{\phi }$

Note that ${\vec {e}}_{0}\neq {\vec {f}}_{0},\;{\vec {e}}_{1}\neq {\vec {f}}_{1}$, and that ${\vec {e}}_{0}$ "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 ${\vec {e}}_{0}$) 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 at $r=2m$. Since the static polar spherical coordinates have a coordinate singularity 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

$T(t,r)=t-\int {\frac {\sqrt {2m/r}}{1-2m/r}}\,dr=t+2{\sqrt {2mr}}+2m\log \left({\frac {{\sqrt {r}}-{\sqrt {2m}}}{{\sqrt {r}}+{\sqrt {2m}}}}\right)$

This gives the Painleve chart. The new line element is

$ds^{2}=-dT^{2}+\left(dr+{\sqrt {2m/r}}\,dT\right)^{2}+r^{2}\left(d\theta ^{2}+\sin(\theta )^{2}\,d\phi ^{2}\right)$

$-\infty <T<\infty ,\;0<r<\infty ,\;0<\theta <\pi ,\;-\pi <\phi <\pi$

With respect to the Painleve chart, the Lemaître frame is

${\vec {f}}_{0}=\partial _{T}-{\sqrt {2m/r}}\,\partial _{r}$

${\vec {f}}_{1}=\partial _{r}$

${\vec {f}}_{3}={\frac {1}{r}}\,\partial _{\theta }$

${\vec {f}}_{3}={\frac {1}{r\sin(\theta )}}\,\partial _{\phi }$

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 hyperslices $T=T_{0}$ 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

$E[Y]_{ab}=R_{ambn}\,Y^{m}\,Y^{n}$

where we write $Y={\vec {f}}_{0}$ 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: $E[Y]_{11}=-2m/r^{3},\,E[Y]_{22}=E[Y]_{33}=m/r^{3}$. (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 ${\vec {e}}_{3}$ 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 distinguished (and mathematically gifted) astronomer Yusuke Hagihara.

In the static polar spherical chart, the Hagihara frame is

${\vec {h}}_{0}={\frac {1}{\sqrt {1-3m/r}}}\,\partial _{t}+{\frac {\sqrt {m/r^{3}}}{{\sqrt {1-3m/r}}\,\sin(\theta )}}\,\partial _{\phi }$

${\vec {h}}_{1}={\sqrt {1-2m/r}}\,\partial _{r}$

${\vec {h}}_{2}={\frac {1}{r}}\,\partial _{\theta }$

${\vec {h}}_{3}={\frac {\sqrt {1-2m/r}}{{\sqrt {1-3m/r}}\,\sin(\theta )}}\,\partial _{\phi }-{\frac {\sqrt {m/r^{3}}}{{\sqrt {1-2m/r}}\,{\sqrt {1-3m/r}}}}\,\partial _{t}$

which in the equatorial plane becomes

${\vec {h}}_{0}={\frac {1}{\sqrt {1-3m/r}}}\,\partial _{t}+{\frac {\sqrt {m/r^{3}}}{\sqrt {1-3m/r}}}\,\partial _{\phi }$

${\vec {h}}_{1}={\sqrt {1-2m/r}}\,\partial _{r}$

${\vec {h}}_{2}={\frac {1}{r}}\,\partial _{\theta }$

${\vec {h}}_{3}={\frac {\sqrt {1-2m/r}}{\sqrt {1-3m/r}}}\,\partial _{\phi }-{\frac {\sqrt {m/r^{3}}}{{\sqrt {1-2m/r}}\,{\sqrt {1-3m/r}}}}\,\partial _{t}$

The tidal tensor $E[Z]_{ab}$ where ${\vec {Z}}={\vec {h}}_{0}$ turns out to be given (in the equatorial plane) by

$E[Z]_{11}=-{\frac {m}{r^{3}}}\,{\frac {2-3m/r}{1-2m/r}}=-{\frac {2m}{r^{3}}}-{\frac {m^{2}}{r^{4}}}+O(1/r^{5})$

$E[Z]_{22}={\frac {m}{r^{3}}}\,{\frac {1}{1-3m/r}}=-{\frac {m}{r^{3}}}+{\frac {3m^{2}}{r^{4}}}+O(1/r^{5})$

$E[Z]_{33}={\frac {m}{r^{3}}}$

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 principle reason why is easy to spot: in this frame, each Hagihara observer keeps his spatial vectors radially aligned, so ${\vec {h}}_{1},\;{\vec {h}}_{3}$ rotate about ${\vec {h}}_{2}$ 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

In this article, we have focused on the application of frames to general relativity, and particularly on their physical interpretation. Here we very briefly discuss the general concept. In an n-dimensional Riemannian manifold or pseudo-Riemannian manifold, a frame field is an orthonormal set of vector fields which forms a basis for the tangent space at each point in the manifold. Once again, 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 $V$, we can define a new vector space consisting of all tuples of orthonormal bases for $V$. A frame field is just the analogue of the tangent bundle for this new vector field. More generally still, we can consider arbitrary principle fiber bundles over our Lorentzian manifold. 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
• Method of moving frames
• Paul Painleve
• Vierbein
• Yusuke Hagihara

References

• Flanders, Harley (1989). Differential Forms with Applications to the Physical Sciences. New York: Dover. ISBN 0-486-66169-5. See Chapter IV for frames in E³, 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.
• Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0-7167-0344-0.{{cite book}}: CS1 maint: multiple names: authors list (link) 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.
• Landau, L. D. & Lifschitz, E. F. (1980). The Classical Theory of Fields (4th ed.). London: Butterworth-Heinemann. ISBN 0-7506-2768-9.{{cite book}}: CS1 maint: multiple names: authors list (link) 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.
• De Felice, F.; and Clarke, C. J. (1992). Relativity on Curved Manifolds. Cambridge: Cambridge University Press. ISBN 0-521-42908-0.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: publisher location (link) See Chapter 4 for frames and coframes. If you ever need more information about frame fields, this might be a good place to look!