# Can Gravity be described in terms of velocities and rotations of local Minkowski space?

In the paper The River Model of Black Holes:
Am.J.Phys.76:519-532,2008, Andrew J. S. Hamilton, Jason P. Lisle
http://arxiv.org/abs/gr-qc/0411060
The authors give a way of describing the action of a rotating (and or charged) black hole through a collection of local Minkowski frames, that is, as a sort of collection of preferred reference frames, or more accurately, by the tidal effects arising from the movement from one frame to another. Each frame is defined by a "river field" $\omega_{km}$ (See around equation 74). This field is composed as follows:
$\omega_{0m} = \beta_m$ is the "velocity" of the river, while
$\mu^i = 0.5 \epsilon^{ikm}\omega_{km}$ gives the "twist" of the river.

Then the motion of objects due to the black hole can be calculated from the tidal change $\delta\omega^k_m$ which is a local infinitesimal Lorentz transformation.

My question is this: Can this description of a black hole be used to describe general relativity? Note that there is an obvious limitation: since this is based on a flat background metric you can't get worm holes and the like, but I mean, subject to the requirement of trivial topology, can every GR situation be described by a "river field"?

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## 2 Answers

Daniel's answer may be a little bit generic; although I am not exactly sure what you are looking for one similar concept is that of a Tetrad; there are other names like Frame Field for the same concept.

Here is the introductory paragraph from Wikipedia to see if this is what you are interested in:

Frame fields 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.

The Tetrad formalism is closely related to 2-spinors and Tetrads can twist and turn as they move through space-time encoding lots of GR properties more directly than other formalisms.

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As i said in a response to a previous question of yours, a good way to think about these issues is in terms of the Frobenius theorem (in differential topology) — or, more generally, in terms of Foliations.

What is seems to me that was done in the work you cited is exactly this: a particular 'distribution' (in the sense of Fröbenius's theorem above) was chosen, i.e., a particular collection of flat approximations [to the curved spacetime] — or, if you will, a certain foliation was chosen.

(Note that there's a clear analogy between GR and Fluid Mechanics: both are so-called Classical Field Theories. Landau's "Course of Theoretical Physics", in 10 vols, comes in handy here. My point being that, effectively, it doesn't matter whether you're talking about GR or Fluid Mechanics — and this is the basis behind Unruh's "Sonic Holes". ;-)

So, in the end of the day, you can use the very same construction in GR, Fluid Mechanics, or any other 'Classical Field Theory'.

I think you can take a look at MacDowell-Mansouri gravity and Cartan geometry :: arXiv:gr-qc/0611154 and see how the tools of Cartan Geometry can be used in GR (and, more generally, in Differential Geometry) — in particular, you may be interested in the so-called repère mobile (moving frame).

PS: I should have said something similar to this in the answer to your previous question. Note how the Equivalence Principle fits nicely in this framework (moving frames, foliations, 'distributions', etc).

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