Relativistic Doppler effect in the context of Time dilation

In physics and general relativity, gravitational redshift (known as Einstein shift in older literature) is the phenomenon that electromagnetic waves or photons travelling out of a gravitational well lose energy. This loss of energy corresponds to a decrease in the wave frequency and increase in the wavelength, known more generally as a redshift. The opposite effect, in which photons gain energy when travelling into a gravitational well, is known as a gravitational blueshift (a type of blueshift). The effect was first described by Einstein in 1907, eight years before his publication of the full theory of relativity. Observing the gravitational redshift in the Solar System is one of the classical tests of general relativity.

Gravitational redshift can be interpreted as a consequence of the equivalence principle (that gravitational effects are locally equivalent to inertial effects and the redshift is caused by the Doppler effect) or as a consequence of the mass–energy equivalence and conservation of energy ('falling' photons gain energy), though there are numerous subtleties that complicate a rigorous derivation. A gravitational redshift can also equivalently be interpreted as gravitational time dilation at the source of the radiation: if two oscillators (attached to transmitters producing electromagnetic radiation) are operating at different gravitational potentials, the oscillator at the higher gravitational potential (farther from the attracting body) will tick faster; that is, when observed from the same location, it will have a higher measured frequency than the oscillator at the lower gravitational potential (closer to the attracting body).

In relativity, proper velocity (also known as celerity) w of an object relative to an observer is the ratio between observer-measured displacement vector ${\displaystyle {\textbf {x}}}$ and proper time τ elapsed on the clocks of the traveling object:

It is an alternative to ordinary velocity, the distance per unit time where both distance and time are measured by the observer.

The annus mirabilis papers (from Latin: annus mirabilis, lit. 'miraculous year') are four papers that Albert Einstein published in the scientific journal Annalen der Physik (Annals of Physics) in 1905. As major contributions to the foundation of modern physics, these scientific publications were the ones for which he gained fame among physicists. They revolutionized science's understanding of the fundamental concepts of space, time, mass, and energy.

1. The first paper explained the photoelectric effect, which established the energy of the light quanta ${\displaystyle E=hf}$, and was the only specific discovery mentioned in the citation awarding Einstein the 1921 Nobel Prize in Physics.
2. The second paper explained Brownian motion, which established the Einstein relation ${\displaystyle D=\mu \,k_{\text{B}}T}$ and compelled physicists to accept the existence of atoms.
3. The third paper introduced Einstein's special theory of relativity, which proclaims the constancy of the speed of light ${\displaystyle c}$ and derives the Lorentz transformations. Einstein also examined relativistic aberration and the transverse Doppler effect.
4. The fourth, a consequence of special relativity, developed the principle of mass–energy equivalence, expressed in the equation ${\displaystyle E=mc^{2}}$ and which led to the discovery and use of nuclear power decades later.

These four papers, together with quantum mechanics and Einstein's later general theory of relativity, are the foundation of modern physics.