Gauge theory gravity

The foundation of GTG comes from two principles. First, position-gauge invariance demands that arbitrary local displacements of fields not affect the physical content of the field equations. Second, rotation-gauge invariance demands that arbitrary local rotations of fields not affect the physical content of the field equations. These principles lead to the introduction of a new pair of linear functions, the position-gauge field and the rotation-gauge field. A displacement by some arbitrary function f

${\displaystyle x\mapsto x'=f(x)}$

gives rise to the position-gauge field defined by the mapping on its adjoint,

${\displaystyle {\bar {\mathsf {h}}}(a,x)\mapsto {\bar {\mathsf {h}}}'(a,x)={\bar {\mathsf {h}}}(f^{-1}(a),f(x)),}$

which is linear in its first argument and a is a constant vector. Similarly, a rotation by some arbitrary rotor R gives rise to the rotation-gauge field

${\displaystyle {\bar {\mathsf {\Omega }}}(a,x)\mapsto {\bar {\mathsf {\Omega }}}'(a,x)=R{\bar {\mathsf {\Omega }}}(a,x)R^{\dagger }-2a\cdot \nabla RR^{\dagger }.}$

We can define two different covariant directional derivatives

${\displaystyle a\cdot D=a\cdot {\bar {\mathsf {h}}}(\nabla )+{\tfrac {1}{2}}{\mathsf {\Omega }}({\mathsf {h}}(a))}$

${\displaystyle a\cdot {\mathcal {D}}=a\cdot {\bar {\mathsf {h}}}(\nabla )+{\mathsf {\Omega }}({\mathsf {h}}(a))}$

or with the specification of a coordinate system

${\displaystyle D_{\mu }=\partial _{\mu }+{\tfrac {1}{2}}\Omega _{\mu }}$

${\displaystyle {\mathcal {D}}_{\mu }=\partial _{\mu }+\Omega _{\mu }\times ,}$

where × denotes the commutator product.

The first of these derivatives is better suited for dealing directly with spinors whereas the second is better suited for observables. The GTG analog of the Riemann tensor is built from the commutation rules of these derivatives.

${\displaystyle [D_{\mu },D_{\nu }]\psi ={\tfrac {1}{2}}{\mathsf {R}}_{\mu \nu }\psi }$

${\displaystyle {\mathcal {R}}(a\wedge b)={\mathsf {R}}({\mathsf {h}}(a\wedge b))}$

The field equations are derived by postulating the Einstein–Hilbert action governs the evolution of the gauge fields, i.e.

${\displaystyle S=\int \left[{1 \over 2\kappa }\left({\mathcal {R}}-2\Lambda \right)+{\mathcal {L}}_{\mathrm {M} }\right](\det {\mathsf {h}})^{-1}\,\mathrm {d} ^{4}x.}$

Minimizing variation of the action with respect to the two gauge fields results in the field equations

${\displaystyle {\mathcal {G}}(a)-\Lambda a=\kappa {\mathcal {T}}(a)}$

${\displaystyle {\mathcal {D}}\wedge {\bar {\mathsf {h}}}(a)=\kappa {\mathcal {S}}\cdot {\bar {\mathsf {h}}}(a),}$

where ${\displaystyle {\mathcal {T}}}$ is the covariant energy–momentum tensor and ${\displaystyle {\mathcal {S}}}$ is the covariant spin tensor. Importantly, these equations do not give an evolving curvature of spacetime but rather merely give the evolution of the gauge fields within the flat spacetime.

For those more familiar with general relativity, it is possible to define a metric tensor from the position-gauge field in a manner similar to tetrads. In the tetrad formalism, a set of four vectors ${\displaystyle \{{e_{(a)}}^{\mu }\}}$ are introduced. The Greek index μ is raised or lowered by multiplying and contracting with the spacetime's metric tensor. The parenthetical Latin index (a) is a label for each of the four tetrads, which is raised and lowered as if it were multiplied and contracted with a separate Minkowski metric tensor. GTG, roughly, reverses the roles of these indices. The metric is implicitly assumed to be Minkowski in the selection of the spacetime algebra. The information contained in the other set of indices gets subsumed by the behavior of the gauge fields.

We can make the associations

${\displaystyle g_{\mu }={\mathsf {h}}^{-1}(e_{\mu })}$

${\displaystyle g^{\mu }={\bar {\mathsf {h}}}(e^{\mu })}$

for a covariant vector and contravariant vector in a curved spacetime, where now the unit vectors ${\displaystyle \{e_{\mu }\}}$ are the chosen coordinate basis. These can define the metric using the rule

${\displaystyle g_{\mu \nu }=g_{\mu }\cdot g_{\nu }.}$

Following this procedure, it is possible to show that for the most part the observable predictions of GTG agree with Einstein–Cartan–Sciama–Kibble theory for non-vanishing spin and reduce to general relativity for vanishing spin. GTG does, however, make different predictions about global solutions. For example, in the study of a point mass, the choice of a "Newtonian gauge" yields a solution similar to the Schwarzschild metric in Gullstrand–Painlevé coordinates. General relativity permits an extension known as the Kruskal–Szekeres coordinates. GTG, on the other hand, forbids any such extension.

• David Hestenes: Spacetime calculus for gravitation theory – an account of the mathematical formalism explicitly directed to GTG