Gravitational potential in the context of Newtonian potential

In physics, a redshift is an increase in the wavelength, or equivalently, a decrease in the frequency, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and increase in frequency and energy, is known as a blueshift.

Three forms of redshift occur in astronomy and cosmology: Doppler redshifts due to the relative motions of radiation sources, gravitational redshift as radiation escapes from gravitational potentials, and cosmological redshifts caused by the universe expanding. In astronomy, the value of a redshift is often denoted by the letter z, corresponding to the fractional change in wavelength (positive for redshifts, negative for blueshifts), and by the wavelength ratio 1 + z (which is greater than 1 for redshifts and less than 1 for blueshifts). Automated astronomical redshift surveys are an important tool for learning about the large-scale structure of the universe. Redshift and blueshift can also be related to photon energy and, via Planck's law, to a corresponding blackbody temperature.

Time dilation is the difference in elapsed time as measured by two clocks, either because of a relative velocity between them (special relativity), or a difference in gravitational potential between their locations (general relativity). When unspecified, "time dilation" usually refers to the effect due to velocity. The dilation compares "wristwatch" clock readings between events measured in different inertial frames and is not observed by visual comparison of clocks across moving frames.

These predictions of the theory of relativity have been repeatedly confirmed by experiment, and they are of practical concern, for instance in the operation of satellite navigation systems such as GPS and Galileo.

A satellite galaxy is a smaller companion galaxy that travels on bound orbits within the gravitational potential of a more massive and luminous host galaxy (also known as the primary galaxy). Satellite galaxies and their constituents are bound to their host galaxy, in the same way that planets within the Solar System are gravitationally bound to the Sun. While most satellite galaxies are dwarf galaxies, satellite galaxies of large galaxy clusters can be much more massive. The Milky Way is orbited by about fifty satellite galaxies, the largest of which is the Large Magellanic Cloud.

Moreover, satellite galaxies are not the only astronomical objects that are gravitationally bound to larger host galaxies (see globular clusters). For this reason, astronomers have defined galaxies as gravitationally bound collections of stars that exhibit properties that cannot be explained by a combination of baryonic matter (i.e. ordinary matter) and Newton's laws of gravity. For example, measurements of the orbital speed of stars and gas within spiral galaxies result in a velocity curve that deviates significantly from the theoretical prediction. This observation has motivated various explanations such as the theory of dark matter and modifications to Newtonian dynamics. Therefore, despite also being satellites of host galaxies, globular clusters should not be mistaken for satellite galaxies. Satellite galaxies are not only more extended and diffuse compared to globular clusters, but are also enshrouded in massive dark matter halos that are thought to have been endowed to them during the formation process.

The tidal force or tide-generating force is the difference in gravitational attraction between different points in a gravitational field, causing bodies to be pulled unevenly and as a result are being stretched towards the attraction. It is the differential force of gravity, the net between gravitational forces, the derivative of gravitational potential, the gradient of gravitational fields. Therefore tidal forces are a residual force, a secondary effect of gravity, highlighting its spatial elements, making the closer near-side more attracted than the more distant far-side.

This produces a range of tidal phenomena, such as ocean tides. Earth's tides are mainly produced by the relative close gravitational field of the Moon and to a lesser extent by the stronger, but further away gravitational field of the Sun. The ocean on the side of Earth facing the Moon is being pulled by the gravity of the Moon away from Earth's crust, while on the other side of Earth there the crust is being pulled away from the ocean, resulting in Earth being stretched, bulging on both sides, and having opposite high-tides. Tidal forces viewed from Earth, that is from a rotating reference frame, appear as centripetal and centrifugal forces, but are not caused by the rotation.

Gravitational energy or gravitational potential energy is the potential energy an object with mass has due to the gravitational potential of its position in a gravitational field. Mathematically, it is the minimum mechanical work that has to be done against the gravitational force to bring a mass from a chosen reference point (often an "infinite distance" from the mass generating the field) to some other point in the field, which is equal to the change in the kinetic energies of the objects as they fall towards each other. Gravitational potential energy increases when two objects are brought further apart and is converted to kinetic energy as they are allowed to fall towards each other.

Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events, as measured by observers situated at varying distances from a gravitating mass. The lower the gravitational potential (the closer the clock is to the source of gravitation), the slower time passes, speeding up as the gravitational potential increases (the clock moving away from the source of gravitation). Albert Einstein originally predicted this in his theory of relativity, and it has since been confirmed by tests of general relativity.

This effect has been demonstrated by noting that atomic clocks at differing altitudes (and thus different gravitational potential) will eventually show different times. The effects detected in such Earth-bound experiments are extremely small, with differences being measured in nanoseconds. Relative to Earth's age in billions of years, Earth's core is in effect 2.5 years younger than its surface. Demonstrating larger effects would require measurements at greater distances from the Earth, or a larger gravitational source.

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).

Bernoulli's principle is a key concept in fluid dynamics that relates pressure, speed and height. For example, for a fluid flowing horizontally, Bernoulli's principle states that an increase in the speed occurs simultaneously with a decrease in pressure. The principle is named after the Swiss mathematician and physicist Daniel Bernoulli, who published it in his book Hydrodynamica in 1738. Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler in 1752 who derived Bernoulli's equation in its usual form.

Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a steady flow, the sum of all forms of energy in a fluid is the same at all points that are free of viscous forces. This requires that the sum of kinetic energy, potential energy and internal energy remains constant. Thus an increase in the speed of the fluid—implying an increase in its kinetic energy—occurs with a simultaneous decrease in (the sum of) its potential energy (including the static pressure) and internal energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same because in a reservoir the energy per unit volume (the sum of pressure and gravitational potential ρ g h) is the same everywhere.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols ⁠${\displaystyle \nabla \cdot \nabla }$⁠, ${\displaystyle \nabla ^{2}}$ (where ${\displaystyle \nabla }$ is the nabla operator), or ⁠${\displaystyle \Delta }$⁠. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δf (p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f (p).

The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that density distribution. Solutions of Laplace's equation Δf = 0 are called harmonic functions and represent the possible gravitational potentials in regions of vacuum.

In mathematics and mathematical physics, potential theory is the study of harmonic functions.

The term "potential theory" dates from 19th-century physics when it was realized that the two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the gravitational potential and electrostatic potential, both of which satisfy Poisson's equation—or in the vacuum, Laplace's equation.

In geophysics and physical geodesy, a geopotential model is the theoretical analysis of measuring and calculating the effects of Earth's gravitational field (the geopotential).The Earth is not exactly spherical, mainly because of its rotation around the polar axis that makes its shape slightly oblate. However, a spherical harmonics series expansion captures the actual field with increasing fidelity.

If Earth's shape were perfectly known together with the exact mass density ρ = ρ(x, y, z), it could be integrated numerically (when combined with a reciprocal distance kernel) to find an accurate model for Earth's gravitational field. However, the situation is in fact the opposite: by observing the orbits of spacecraft and the Moon, Earth's gravitational field can be determined quite accurately. The best estimate of Earth's mass is obtained by dividing the product GM as determined from the analysis of spacecraft orbit with a value for the gravitational constant G, determined to a lower relative accuracy using other physical methods.

Gravitational energy or gravitational potential energy is the potential energy an object with mass has due to the gravitational potential of its position in a gravitational field. Mathematically, is a scalar quantity attached to the conservative gravitational field and equals the minimum mechanical work that has to be done against the gravitational force to bring a mass from a chosen reference point (often an "infinite distance" from the mass generating the field) to some other point in the field, which is equal to the change in the kinetic energies of the objects as they fall towards each other. Gravitational potential energy increases when two objects are brought further apart and is converted to kinetic energy as they are allowed to fall towards each other.

In astronomy, the Roche lobe is the region around a star in a binary system within which orbiting material is gravitationally bound to that star. It is an approximately teardrop-shaped region bounded by a critical gravitational equipotential, with the apex of the teardrop pointing towards the other star (the apex is at the L₁ Lagrangian point of the system).

The Roche lobe is different from the Roche sphere, which approximates the gravitational sphere of influence of one astronomical body in the face of perturbations from a more massive body around which it orbits. It is also different from the Roche limit, which is the distance at which an object held together only by gravity begins to break up due to tidal forces. The Roche lobe, Roche limit, and Roche sphere are named after the French astronomer Édouard Roche.

In physics, a redshift is an increase in the wavelength, or equivalently, a decrease in the frequency, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and increase in frequency and energy, is known as a blueshift.

Three forms of redshift occur in astronomy and cosmology: Doppler redshifts due to the relative motions of radiation sources, gravitational redshift as radiation escapes from gravitational potentials, and cosmological redshifts caused by the universe expanding. The value of a redshift is often denoted by the letter z, corresponding to the fractional change in wavelength (positive for redshifts, negative for blueshifts), and by the wavelength ratio 1 + z (which is greater than 1 for redshifts and less than 1 for blueshifts). Automated astronomical redshift surveys are an important tool for learning about the large-scale structure of the universe. Redshift and blueshift can also be related to photon energy and, via Planck's law, to a corresponding blackbody temperature.