p2

Appendix B – THεµ Methodology
In the formalism of the THεµ methodology, the functions T and H are introduced by requiring that the
Lagrangian for the motion of particles (with charge e a and mass m 0a for the a th particle), under the joint
action of gravity and the electromagnetic field A α (α ≡ spacetime vector components), be expressed in the
canonical form
α
−1
−
( 0a a a α a ) ( 8π) ( ε µ )
∑∫ ∫ E B (B.1);
2 2 1 2 3
L = −m T − Hv + e A v dt + + d x dt
a
where the arbitrary functions T, H, ε, and µ are functions of the metric (a.k.a. gravitation field), v a α is the
a th particle four-vector velocity, and A α is the electromagnetic field four-vector potential, E and B are the
electric and magnetic field strengths, and (B.1) is in geometrodynamic natural units (ħ = c 0 = G = ε 0 = µ 0
= 1). The Lagrangian characterizes the motion of charged particles in an external gravitational field by
the two functions T and H, and characterizes the response of the electromagnetic fields to the external
gravitational field by the two functions ε and µ. For all standard (metric) theories of gravity, the four
functions are related by
H
ε = µ = (B.2);
T
and every metric theory of gravity satisfies this relation, such that the Einstein Equivalence Principle is
satisfied.
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