*Gravity and Inertia in General Relativity DOI: http://dx.doi.org/10.5772/intechopen.99760*

hyperbolic, open geometry. The metric for k = 0 FLRW cosmology is, aside from the scale factor which multiplies the spacelike part of the metric to produce cosmic expansion, time-independent. So, a universe that starts out spatially flat, stays spatially flat throughout its history. Brans' doctoral supervisor Robert Dicke, in the '70s, identified this as a paradox, for spatial flatness should be unstable against small perturbations that should drive cosmic spacetime curvature quickly into the spherical or hyperbolic state. Why, more than 10 billion years into cosmic expansion is the universe still spatially flat? Alan Guth created inflation to address this problem.

Curvature at cosmic scale is related to energetic considerations. This is especially clear in the Newtonian analog elaboration of cosmology. (See Bernstein's *An Introduction to Cosmology*, Prentice Hall, 1995, chapter 2 for example [9]). Spatial flatness is a consequence of the balancing of gravitational potential energy and "kinetic", that is, non-gravitational energy encompassed in the Eq. *E* = *mc*<sup>2</sup> . If gravitational energy exceeds non-gravitational energy, the universe is "closed" and will eventually contract after reaching some finite size. When non-gravitational energy exceeds gravitational energy, the universe is "open" and expands forever. Since gravitational and non-gravitational energies are balanced in a spatially flat universe, if we place a test particle of mass *m* anywhere/when in spacetime (which is the gravitational field according to Einstein), we will have:

$$m\_{\text{g}}\phi = m\_{i}\varepsilon^{2} \tag{4}$$

where the subscripts g and i identify the passive gravitational and inertial masses of the material particle. The Equivalence Principle identifies the passive gravitational and inertial masses as equal in magnitude, so:

$$
\phi = c^2 \tag{5}
$$

everywhere/when. The vacuum speed of light, a constant in special relativity, becomes a locally measured invariant in general relativity because, while local measurements always return the same number for *c*, non-local measurements may return different numbers. (Distant observers measure *c* in the vicinity of black holes to be much less than their locally measured value). It follows that *ϕ* too must be a locally measured invariant like *c* in spatially flat cosmology. Is spacetime spatially flat at cosmic scale? Observation answers this question in the affirmative.

*ϕ*/*c* <sup>2</sup> being a ratio of locally measured invariants in k = 0 cosmology does two things. First, it means that Eq. (4) can be interpreted as the assertion that inertial mass is induced by the action of gravity due to cosmic sources, as Einstein claimed to be the case – notwithstanding Brans'spectator matter argument. Indeed, since Brans' argument can be sidestepped by *ϕ* being a locally measured invariant (equal to *c* 2 ), his argument becomes a compelling argument **for** the gravitational induction of inertial mass. Second, *ϕ*/*c* <sup>2</sup> = 1 makes inertial reaction forces an inductive gravitational effect, again, as Einstein claimed should be the case. In this case, though, the claim is complicated by the tensorial nature of gravity.

In the vector approximation based on the analogy with electrodynamics, one writes for the "gravelectric" field equation:

$$E\_{grav} = -\nabla \mathcal{Q} - \frac{1}{c} \frac{\partial \mathcal{A}}{\partial t} \tag{6}$$

in Gaussian units. If one uses this equation for gravity, when one computes the equation of motion for a test particle, one gets *ϕ*/*c* <sup>2</sup> times *d***v**/*dt* = **a**, the acceleration, from the term in the time derivative of the vector potential, and *ϕ*/*c* <sup>2</sup> = 1 makes this term the inertial reaction force on the test particle. In tensor general relativity one must specify one's choice of coordinates.
