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The progress of modern cosmology took off in 1917 when A. Einstein published his paper on general theory of relativity extending his work of special theory of relativity (1905). In 1922 Alexander Friedmann constructed a mathematical model for expanding Universe that had a big bang in remote past. The experimental evidences could come in 1929 by the pioneering work on nebular red shifts by Edwin Hubble and Milton Humason. Doppler’s red shift for light also comes as a deduction of special theory of relativity which provides a fundamental formalism for observational astronomy. It can also be deduced from cosmological red shift arising from the curvature of space time warp of the Universe depending on the evolution model opted for the Universe. The gravitational red shift comes as a consequence of the principle of equivalence in presence of weak gravitational field which expresses the identity of the gravitational and inertial mass. The gravitation itself is a manifestation of curvature in space time. Massive objects show noticeable bending of light near it which explains well the apparent images formation when added with the gravitational fall of the photons towards the object. Special and general theory of relativity can be realized as the best mathematical narrations of the cosmic dance. This paper highlights the experimental evidences in favor of relativity and their applications in cosmology which could be possible with the invention of ultra high precision instruments in the latter half of the last century.