Quasars have long been known to be variable sources at all wavelengths. Assuming fair sampling for all detected photons, and that each stellar photon’s color was set at emission, we observe statistically significant ≳7.31σ and ≳11.93σ violations of Bell’s inequality with estimated p values of ≲1.8×10−13 and ≲4.0×10−33, respectively, thereby pushing back by ∼600 years the most recent time by which any local-realist influences could have engineered the observed Bell violation. Here we report on a new experimental test of Bell’s inequality that, for the first time, uses distant astronomical sources as “cosmic setting generators.” In our tests with polarization-entangled photons, measurement settings were chosen using real-time observations of Milky Way stars while simultaneously ensuring locality. This, however, left open the possibility that an unknown cause affected both the setting choices and measurement outcomes as recently as mere microseconds before each experimental trial. In previous experiments, this “freedom of choice” was addressed by ensuring that selection of measurement settings via conventional “quantum random number generators” was spacelike separated from the entangled particle creation. That conflict is expressed by Bell’s inequality, which is usually derived under the assumption that there are no statistical correlations between the choices of measurement settings and anything else that can causally affect the measurement outcomes. In the same way, the impact and diverse applications of these geometries in other sciences are exposed in a general way, in particular cosmology, where space is conjugated with time, generating another type of space-time metrics such as the Riemannian one, which allows explanation and support to theories such as the general relativity of Einstein, and other physico-theoretical models related to quantum physics, giving way to new approaches on the characteristics and incidence of matter and energy in the macro and micro context of the universe.īell’s theorem states that some predictions of quantum mechanics cannot be reproduced by a local-realist theory. As a research methodology, we proceeded to carry out a study on the mathematical and geometric modeling that characterizes this type of geometry, establishing its differences framed in the theories formulated by its discoverers, simulating some of them, in order to show the spatial representation of the so-called geodesic shapes and curves. However, in the past two centuries, assorted non-Euclidean geometries have been derived based on using the first four Euclidean postulates together with various negations of the fifth.This article shows the results of the study conducted on Euclidean geometry, in particular the fifth postulate, which led to the emergence of non-Euclidean geometries. As a result, many mathematicians over the centuries have tried to prove the results of the Elements without using the Parallel Postulate, but to no avail. Mathematicians, and really most of us, value simplicity arising from simplicity, with the long complicated proofs, equations, and calculations needed for rigorous certainty done behind the scenes, and to have such a long sentence amidst such other straightforward, intuitive statements seems awkward. Postulate 5, the so-called Parallel Postulate was the source of much annoyance, probably even to Euclid, as it is not a simple, concise statement, as are the other four. If a straight line intersects two other straight lines, and so makes the two interior angles on one side of it together less than two right angles, then the other straight lines will meet at a point if extended far enough on the side on which the angles are less than two right angles.A circle may be described with any given point as its center and any distance as its radius.A straight line may be extended to any finite length. A straight line segment may be drawn from any given point to any other.Together with the five axioms (or "common notions") and twenty-three definitions at the beginning of Euclid's Elements, they form the basis for the extensive proofs given in this masterful compilation of ancient Greek geometric knowledge. The five postulates of Euclidean Geometry define the basic rules governing the creation and extension of geometric figures with ruler and compass. Postulates in geometry is very similar to axioms, self-evident truths, and beliefs in logic, political philosophy, and personal decision-making.
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