The first problem in writing gravitation as a particle interaction is the fact that QFT works best on flat space, while general relativity is almost always written in arbitrary coordinates. One of the underlying principles of general relativity is that coordinates shouldn’t matter (background independence), so this problem appears to be a deep one. The point of view we will take here is that of the “new physics” .
That is, we will treat the equations of the old physics with more respect than we treat their theories. Consequently, instead of chasing after will-o-the-wisps like background independence, we will instead search for a method of writing general relativity using the mathematical tools of quantum field theory. Very fortunately for us, that method has already been found; it is the gauge theory of gravity discovered by the Cambridge Geometry Algebra Research Group. The purpose of this post is to introduce the theory to those who have not yet been exposed to it, and to note that this gravity theory (which is identical to GR so long as you restrict your attention to stuff that happens outside of the event horizons of black holes) picks out Painleve coordinates as a natural flat space (and therefore QFT compatible) coordinate system for a non rotating black hole.
Those with a graduate education in physics are already familiar with the Geometric Algebra (GA) in that it is equivalent to the Gamma matrices used throughout quantum field theory. So a gravitation theory that is equivalent to general relativity, but is written with gamma matrices, is a natural starting point for a search for a unified field theory.
The primary proponent for the use of GA in physics (outside of QFT) is David Hestenes, who applied it to classical and quantum mechanics. As the introduction to GA article at the Cambridge Geometry group’s website puts it:
We believe that there are two aspects of Hestenes’ work which physicists should take particularly seriously. The first is that the geometric algebra of spacetime is the best available mathematical tool for theoretical physics, classical or quantum. Related to this part of the programme is the claim that complex numbers arising in physical applications usually have a natural geometric interpretation that is hidden in conventional formulations. David’s second major idea is that the Dirac theory of the electron contains important geometric information, which is disguised in the conventional matrix based approaches.
Now that’s a pretty big claim: that geometric algebra is the best mathematical tool for all physics. I will spend the rest of this post exploring this claim in the case of general relativity, and then tracing the consequences for a unified field theory.
The very fact that general relativity, classical mechanics, and quantum mechanics can be efficiently described in the geometric algebra is a deep hint that this branch of mathematics is a natural one to explore a unification of the forces. But we will obtain slightly more than that. It turns out that when general relativity is written in geometric algebra, one ends up with a flat space gravity theory. And a flat space is just where the tools of quantum field theory are the easiest to use.
When I began studying geometric algebra some years ago, I eventually noticed that Dr. Hestenes had applied the theory to quantum mechanics but had not applied it to quantum field theory. I think it was the summer of 2005 that I found out the reason. I was in Phoenix installing and testing a surplus Boeing (megawatt) oven that my buddy had sold to a company making armor for the US military. Since Hestenes retired to a emeritus position at Arizona State University, I decided to drop by his office and say hello. We talked for about 90 minutes. I found that he was more inclined towards a Bohmian interpretation of quantum mechanics and thought that quantum field theory was bunk.
The effect of Hestenes’ tendencies towards looking for a realist’ interpretation of quantum mechanics has been to reduce the ability of geometric algebra to penetrate physics as a whole. As the above linked review article says, “David Hestenes’ willingness to ask the sort of question that Feynman specifically warned against, [i.e. "How can it be like that?" ] and to engage in varing degrees of speculation, has undoubtedly had the unfortunate effect of diminishing the impact of his first idea, that geometric algebra can provide a unified language for physics – a contention that we strongly believe.” Hestenes’ absence of interest in QFT has slowed the infiltration of geometric algebra techniques in QFT, but ironically it is in QFT that geometric algebra finds its easiest and most natural application; it amounts to treating the gamma matrices as fundamental geometric objects (vectors associated with directions in spacetime) instead of a fairly arbitrary set of matrices that were jiggered together to convert the 2nd order Klein Gordon equation into linear form.
But getting back to the application of geometric algebra to general relativity, there are several papers available on the web to introduce the reader to the gravity theory of the Cambridge geometry group:
Spacetime Geometry with Geometric Calculus, David Hestenes, [28pp, acrobat, 2006?, For the purposes of this post, note pages 22-23 and the Appendix. This paper gives a somewhat different interpretation of the mathematics from the Cambridge Geometry group's original paper.]
Geometric Calculus is developed for curved-space treatments of General Relativity and comparison with the flat-space gauge theory approach by Lasenby, Doran and Gull. Einstein’s Principle of Equivalence is generalized to a gauge principle that provides the foundation for a new formulation of General Relativity as a Gauge Theory of Gravity on a curved spacetime manifold. Geometric Calculus provides mathematical tools that streamline the formulation and simplify calculations. The formalism automatically includes spinors so the Dirac equation is incorporated in a geometrically natural way.
Gauge Theory Gravity with Geometric Calculus, David Hestenes, [58pp, Acrobat, 2005, This is Hestenes' description of Gauge Gravity. This paper includes an introduction to geometric algebra. For the purposes of this post, pages 35-36 solve the non rotating black hole problem. Equation (203) is the metric for Painleve coordinates though Hestenes has not marked it as such. To see this fact, compare equation (203) with equation (2.7) of this arXiv reference on Painleve coordinates.]
A new gauge theory of gravity on flat spacetime has recently been developed by Lasenby, Doran, and Gull. Einstein’s principles of equivalence and general relativity are replaced by gauge principles asserting, respectively, local rotation and global displacement gauge invariance. A new unitary formulation of Einstein’s tensor illuminates long-standing problems with energy-momentum conservation in general relativity. Geometric calculus provides many simplifications and fresh insights in theoretical formulation and physical applications of the theory.
Spacetime Calculus for Gravitation Theory, David Hestenes, [74pp, Acrobat, 1998, This is the first and most complete of Hestenes' papers on gauge gravity. It is a very complete introduction to geometric algebra and gauge gravity, except that it does not include his later ideas. Part II covers flat spacetime.]
A new gauge theory of gravitation on flat spacetime has recently been developed by Lasenby, Doran, and Gull in the language of Geometric Calculus. This paper provides a systematic account of the mathematical formalism to facilitate applications and extensions of the theory. It includes formulations of differential geometry, Lie derivatives and integrability theorems which are coordinate-free and gauge-covariant. Emphasis is on use of the language to express physical and geometrical concepts.
Gravity, Gauge Theories and Geometric Algebra Lasenby, Doran, Gull [112pp, Acrobat, 1998, This is the original paper. Quoting from the conclusion:]
In this paper we developed a theory of gravity consisting of gauge fields defined in a flat background spacetime. The theory is conceptually simple, and the role of the gauge fields is clearly understood — they ensure invariance under arbitrary displacements and rotations. While it is possible to maintain a classical picture of the rotation gauge group, a full understanding of its role is only achieved once the Dirac action is considered. The result is a theory which offers a radically different interpretation of gravitational interactions from that provided by general relativity. Despite this, the two theories agree in their predictions for a wide range of phenomena. Differences only begin to emerge over issues such as the role of topology, our insistence on the use of global solutions, and in the interaction with quantum theory. Furthermore, the separation of the gauge fields into one for displacements and one for local rotations is suggestive of physical effects being separated into an inertia field and a force field. …
Fermion absorption cross section of a Schwarzschild black hole Doran, Lasenby, Dolan, Hinder [11pp, acrobat, 2005, This is an application showing how gravity and the Dirac equation get combined in Painleve coordinates.]
We study the absorption of massive spin-half particles by a small Schwarzschild black hole by numerically solving the single-particle Dirac equation in Painleve-Gullstrand coordinates. …
For the purpose of this series of posts, I really don’t need to get into a detailed explanation of how the Cambridge Geometry Group’s “gauge gravity” works. Since my own preon theory is based on geometric calculus it is highly tempting, but I will stick to the plan at hand and the above references are sufficient.
The conclusion is that Painleve coordinates are special, and it is these coordinates that we should use when analyzing gravitation as a force. And it will be Painleve coordinates that we return to in the next post, which will be on how one computes the general relativistic acceleration of gravity for the case of the Schwarzschild black hole as written in Painleve coordinates. In other words, we will use Painleve coordinates to compute the Einstein’s gravitation in Newtonian form, as a simple differential equation with respect to coordinate time (rather than proper time as is usually done). This turns out to be a lot more trouble than one might suppose so we will devote the whole post to the derivation. [On second thought, I'll devote the entire post to the Painleve coordinates which are quite interesting.]
The resulting differential equation is the basis for the gravitation simulation applet I wrote last year, and this connection to general relativity and quantum field theory is the reason for my choice of that coordinate system.