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To the second order in metric and the first order in equations of motion in the local coordinates of an accelerated rotating observer, the inertial effects and gravitational effects are simply additive. To look into the coupled inertial and gravitational effects, we derive the third-order expansion of the metric and the second-order expansion of the equations of motion in local coordinates. Besides purely gravitational (purely curvature) effects, the equations of motion contain, in this order, the following coupled inertial and gravitational effects: redshift corrections to electric, magnetic, and double-magnetic type curvature forces; velocity-induced special relativistic corrections; and electric, magnetic, and double-magnetic type coupled inertial and gravitational forces. An example is provided with a static observer in the Schwarzchild spacetime.

We present a bird's-eye survey on the development of fundamental ideas of quantum gravity, placing emphasis on perturbative approaches, string theory, loop quantum gravity (LQG) and black hole thermodynamics. The early ideas at the dawn of quantum gravity as well as the possible observations of quantum gravitational effects in the foreseeable future are also briefly discussed. Int. J. Mod. Phys. D Downloaded from www.worldscientific.com by CAMBRIDGE UNIVERSITY on 08/11/15. For personal use only. S. Carlip et al. 1530028-2 Int. J. Mod. Phys. D Downloaded from www.worldscientific.com by CAMBRIDGE UNIVERSITY on 08/11/15. For personal use only. Quantum gravity: A brief history of ideas and some prospectsNinety-nine years passed since the very first conception of Einstein. Today, quantum gravity has grown into a vast area of research in many different (both perturbative and nonperturbative) approaches. Many directions have led to significant advances with various appealing and ingenious ideas; these include causal sets, dynamical triangulation, emergent gravity, H-space theory, LQG, noncommutative geometry, string theory, supergravity, thermogravity, twistor theory and much more. (For surveys and references on various approaches of quantum gravity, see Refs. 22-25; for detailed accounts of the history and development, see Refs. 6 and 26; for a popular science account of quantum gravity, see the book Ref. 27.) Perturbative Quantum GravityThe quantization of general relativity on the perturbative level (with or without matter) was the life work of DeWitt, for which he was awarded the 1987 Dirac Medal and the 2005 Einstein Prize. 28-30 His program eventually succeeded -against the expectation of most expert opinion during the 1960s -in generalizing Feynman's covariant quantization of quantum electrodynamics so that it could be applied as well to theories such as general relativity and Yang-Mills that are based on non-Abelian gauge symmetries. There were three key steps:

A coordinatefree derivation of a generalized geodesic deviation equationFermi normal coordinates about a geodesic form a natural coordinate system for the nonrotating geodesic (freely falling) observer. Expansions of the affinity, metric, and geodesic equations in these coordinates in powers of proper distance normal to the geodesic are calculated here to third order, fourth order, and third order, respectively. An iteration scheme for calculation to higher orders is also given. For generality, we compute the affinity and the geodesic equations in an arbitrary affine manifold, and compute the metric in a Riemannian manifold with arbitrary signature.