Abstract
Einstein equation of gravity has on one side the momentum energy density tensor and on the other, Einstein tensor which is derived from Ricci curvature tensor. A better theory of gravity will have both sides geometric. One way to achieve this goal is to develop a new measure of time that will be independent of the choice of coordinates. One natural nominee for such time is the upper limit of measurable time form an event back to the big bang singularity. This limit should exist despite the singularity, otherwise the cosmos age would be unbounded. By this, the author constructs a scalar field of time. Time, however, is measured by material clocks. What is the maximal time that can be measured by a small microscopic clock when our curve starts at near the “big bang” event and ends at an event within the nucleus of an atom? Will our tiny clock move along geodesic curves or will it move in a non geodesic curve within matter? It is almost paradoxical that a test particle in General Relativity will always move along geodesic curves but the motion of matter within the particle may not be geodesic at all. For example, the ground of the Earth does not move at geodesic velocity. Where there is no matter, we choose a curve from near “big bang” to an event such that the time measured is maximal. Without assuming force fields, the gravitational field which causes that two or more such curves intersect at events, would cause discontinuity of the gradient of the upper limit of measurable time scalar field. The discontinuity can be avoided only if we give up on measurement along geodesic curves where there is matter. In other words, our tiny test particle clock will experience force when it travels within matter or near matter.
Highlights
Square Curvature in Positive Definite Metric Spaces JMP E
What is the maximal time that can be measured by a small microscopic clock when our curve starts at near the “big bang” event and ends at an event within the nucleus of an atom? Will our tiny clock move along geodesic curves or will it move in a non geodesic curve within matter? It is almost paradoxical that a test particle in General Relativity will always move along geodesic curves but the motion of matter within the particle may not be geodesic at all
The direction in space time of the particles measuring the maximum proper time forms a geodesic curve but near matter, without the existence of forces, not necessarily the gradient of the field would be parallel to a geodesic curve because: 1) more than one curve could reach the same event; 2) at that event near matter, even known forces would cause any test particle clock to move along non geodesic curves; and 3) one idea, even without explicitly assuming forces near matter, is that in a quantum model the intersection of such curves does manifest singularity of the gradient but by coupling by multiplication of the time field and a wave function of the entire system of particles, the 4-loaction of the singularity can be blurred by uncertainty
Summary
The Gradient Need Not to Be Parallel to any Geodesic Curve, Resolving Singularities. Our strategy will be to see what would happen to the gradient of the upper limit of measurable time from any event backwards, if no forces act on our test-clock-particles. Gravitational lenses are formed and events in the hollowed part of the ball are accessible by more than one curve as depicted by Figure 1(b) These singularities can be resolved too if any real world particle-like clock will not move along geodesic curves in the microscopic vicinity of matter, e.g. due to Casimir/Casimir Lifshitz [3]. We would like to describe the curvature of the gradient of the absolute maximum proper time from near “big bang” as a scalar field that is measured by our microscopic particle-like clocks and show its possible relation to Ricci curvature and to Einstein’s tensor. Given two infinitesimally close points in Rn , q1 and q2 q1 hV for some infinitesimal h , we would like that V q2 V q1 will be as parallel as possible to the field V q1
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