Terminology and notation

Notation and sign conventions adopted for the rest of the book are explained. The book employs index notation, but not abstract index notation. The metric signature for GR is taken as (-1,1,1,1). Terminology such as “local inertial frame” and “Rieman normal coordinates” is explained.

Further useful ideas

The chapter discusses several further aspects of the physics and mathematics that prove very useful in practice. First we define 4-velocity, 4-momentum and 4-acceleration. Then we introduce the tetrad and show how it can be used to relate a given 4-momentum to the energy and momentum observed in a LIF (local inertial frame). Then we define covariant version of the vector operators div, grad, curl, and obtain simplified expressions for the divergence of a vector and an antisymmetric tensor. The generalized Gauss divergence theorem is then presented.

Gravity as the Hidden Variable of Quantum Mechanics: Information-Matter Equation from Gravity

I localize gravity to match its measurements with the local inertial frame of special relativity. I find a geometric interpretation of the speed of light and mass. I find also the relation between every mass measured and the black hole entropy which introduce information-matter equation from gravity. Through localization of gravity, a timeless state of the universe emerges and the uncertainty principle does not hold since the velocity concept is replaced by distance. This would resolve the problem of time because timeless state of the universe emerges naturally and mathematically consistent. This would suggest that gravity form the hidden one variable of quantum mechanics which would complete the relation between quantum mechanics and gravity. We introduce also a principle of least computation which is achieved when the ratio equal to the difference in the process of local gravitational measurements.

Is the Fulling–Davies–Unruh effect valid for the case of an atom coupled to quantum electromagnetic field?

We study, from the viewpoint of a co-accelerated observer, the average rate of change of atomic energy for an atom in uniform acceleration and coupled to quantum electromagnetic field at a thermal state with an arbitrary temperature T. We show that only when the temperature of the thermal state in the co-accelerated frame is assumed to be the Fulling–Davies–Unruh (FDU) temperature, T = a/2[Formula: see text], can the average rate of change of atomic energy in a local inertial frame be recovered, which exemplifies the equivalence between the Minkowski vacuum and a thermal bath of Rindler particles. This conclusion is verified to be valid not only in a free spacetime, but also in a spacetime with a boundary.

IS GRAVITY AN INTRINSICALLY QUANTUM PHENOMENON? DYNAMICS OF GRAVITY FROM THE ENTROPY OF SPACE–TIME AND THE PRINCIPLE OF EQUIVALENCE

The two surprising features of gravity are (a) the principle of equivalence and (b) the connection between gravity and thermodynamics. Using principle of equivalence and special relativity in the local inertial frame, one could obtain the insight that gravity must possess a geometrical description. We show that, using the same principle of equivalence, special relativity and quantum theory in the local Rindler frame one can obtain the Einstein–Hilbert action functional for gravity and thus the dynamics of the space–time. This approach, which essentially involves postulating that the horizon area must be proportional to the entropy, uses the local Rindler frame as a natural extension of the local inertial frame and leads to the interpretation that the gravitational action represents the free energy of the space–time geometry. As an aside, one also obtains a natural explanation as to: (i) why the covariant action for gravity contains second derivatives of the metric tensor and (ii) why the gravitational coupling constant is positive. The analysis suggests that gravity is intrinsically holographic and even intrinsically quantum mechanical.

Realization of the Local Inertial Geocentric Frame in Relativity

There are two kinds of geocentric frames: local inertial and non-inertial geocentric frames. Ashby et al successfully constructed a local inertial geocentric frame in the neighborhood of the gravitating Earth. In the frame with origin at the Earth's center, the gravitational effects of the sun and of planets other than the Earth are basically reduced to their tidal forces, with very small relativistic corrections.However, the spatial base vectors of the local inertial frame essentially experience the geodesic (or deSitter) precession with respect to the solar system barycentric frame. Hence the realization of the local inertial frame requires that the general precession should exclude the geodesic precession. This requirement is inconsistent with the convention that the amount of geodesic precession is included in that of the general precession given by Lieske et al.

Relativistic Reference Frames in Astrometry

Astrometry can be defined as the measurement of space-time coordinates of photon events. For example, in principle, in classical optical astrometry, we measure the components of velocity, and hence the direction, of an incoming photon with respect to an instrumental coordinate system, and the clock time, at the instant of detection. The observer’s coordinate system at any instant can be identified with a local inertial frame. In the case of interferometric observations, the measurements are of clock times of arrival of a wavefront at two detectors whose spatial coordinates are specified with respect to instantaneous inertial frames.

Mach’s principle and the creation of matter

Accurate experiments have shown that the local inertial frame is the one with respect to which the distant parts of the universe are non-rotating. This coincidence, first noticed by Newton, later led to the formulation of Mach’s principle. It is known that relativity theory by itself cannot explain this coincidence. The introduction of a scalar ‘creation field’ into the theory is likely to improve the situation. Calculation shows that the continuous creation of matter has the effect of smoothing out any irregularities in the universe as it expands, while rotation, if present, becomes less and less. This explains the observed remarkable degree of homogeneity and isotropy in the universe.