This article needs additional citations for verification .(April 2018) |

In Newton's theory of gravitation and in various relativistic classical theories of gravitation, such as general relativity, the **tidal tensor** represents

- Tidal tensor for a spherical body
- Hessian formulation
- Spherically symmetric field
- In General Relativity
- See also
- References
- External links

*tidal accelerations*of a cloud of (electrically neutral, nonspinning) test particles,*tidal stresses*in a small object immersed in an ambient gravitational field.

The tidal tensor represents the relative acceleration due to gravity of two test masses separated by an infinitesimal distance. The component represents the relative acceleration in the direction produced by a displacement in the direction.

The most common example of tides is the tidal force around a spherical body (*e.g.*, a planet or a moon). Here we compute the tidal tensor for the gravitational field outside an isolated spherically symmetric massive object. According to Newton's gravitational law, the acceleration *a* at a distance *r* from a central mass *m* is

(to simplify the math, in the following derivations we use the convention of setting the gravitational constant G to one. To calculate the differential accelerations, the results are to be multiplied by G.)

Let us adopt the frame in polar coordinates for our three-dimensional Euclidean space, and consider infinitesimal displacements in the radial and azimuthal directions, and , which are given the subscripts 1, 2, and 3 respectively.

We will directly compute each component of the tidal tensor, expressed in this frame. First, compare the gravitational forces on two nearby objects lying on the same radial line at distances from the central body differing by a distance *h*:

Because in discussing tensors we are dealing with multilinear algebra, we retain only first order terms, so . Since there is no acceleration in the or direction due to a displacement in the radial direction, the other radial terms are zero: .

Similarly, we can compare the gravitational force on two nearby observers lying at the same radius but displaced by an (infinitesimal) distance *h* in the or direction. Using some elementary trigonometry and the small angle approximation, we find that the force vectors differ by a vector tangent to the sphere which has magnitude

By using the small angle approximation, we have ignored all terms of order , so the tangential components are . Again, since there is no acceleration in the radial direction due to displacements in either of the azimuthal directions, the other azimuthal terms are zero: .

Combining this information, we find that the tidal tensor is diagonal with frame components This is the *Coulomb form* characteristic of spherically symmetric central force fields in Newtonian physics.

In the more general case where the mass is not a single spherically symmetric central object, the tidal tensor can be derived from the gravitational potential , which obeys the Poisson equation:

where is the mass density of any matter present, and where is the Laplace operator. Note that this equation implies that in a vacuum solution, the potential is simply a harmonic function.

The *tidal tensor* is given by the *traceless part*^{ [1] }

of the Hessian

where we are using the standard *Cartesian chart* for E^{3}, with the Euclidean metric tensor

Using standard results in vector calculus, this is readily converted to expressions valid in other coordinate charts, such as the *polar spherical chart*

As an example, we can calculate the tidal tensor for a spherical body using the Hessian. Next, let us plug the gravitational potential into the Hessian. We can convert the expression above to one valid in polar spherical coordinates, or we can convert the potential to Cartesian coordinates before plugging in. Adopting the second course, we have , which gives

After a rotation of our frame, which is adapted to the polar spherical coordinates, this expression agrees with our previous result. The easiest way to see this is to set to zero so that the off-diagonal terms vanish and , and then invoke the spherical symmetry.

In general relativity, the tidal tensor is generalized by the Riemann curvature tensor. In the weak field limit, the tidal tensor is given by the components of the curvature tensor.

In mathematics and physics, **Laplace's equation** is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as

The **Navier–Stokes equations** are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842-1850 (Stokes).

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 , (where is the nabla operator), or . 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 Δ*f* (*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*).

**Linear elasticity** is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.

**Stellar dynamics** is the branch of astrophysics which describes in a statistical way the collective motions of stars subject to their mutual gravity. The essential difference from celestial mechanics is that the number of body

This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.

Note: This page uses common physics notation for spherical coordinates, in which is the angle between the *z* axis and the radius vector connecting the origin to the point in question, while is the angle between the projection of the radius vector onto the *x-y* plane and the *x* axis. Several other definitions are in use, and so care must be taken in comparing different sources.

In mathematics, a **Killing vector field**, named after Wilhelm Killing, is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the **Killing vector** will not distort distances on the object.

The **Kerr–Newman metric** is the most general asymptotically flat, stationary solution of the Einstein–Maxwell equations in general relativity that describes the spacetime geometry in the region surrounding an electrically charged, rotating mass. It generalizes the Kerr metric by taking into account the field energy of an electromagnetic field, in addition to describing rotation. It is one of a large number of various different electrovacuum solutions, that is, of solutions to the Einstein–Maxwell equations which account for the field energy of an electromagnetic field. Such solutions do not include any electric charges other than that associated with the gravitational field, and are thus termed vacuum solutions.

The Schwarzschild solution describes spacetime under the influence of a massive, non-rotating, spherically symmetric object. It is considered by some to be one of the simplest and most useful solutions to the Einstein field equations.

A **frame field** in general relativity is a set of four pointwise-orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime. The timelike unit vector field is often denoted by and the three spacelike unit vector fields by . All tensorial quantities defined on the manifold can be expressed using the frame field and its dual coframe field.

In theoretical physics, **Nordström's theory of gravitation** was a predecessor of general relativity. Strictly speaking, there were actually *two* distinct theories proposed by the Finnish theoretical physicist Gunnar Nordström, in 1912 and 1913 respectively. The first was quickly dismissed, but the second became the first known example of a metric theory of gravitation, in which the effects of gravitation are treated entirely in terms of the geometry of a curved spacetime.

In the mathematical description of general relativity, the **Boyer–Lindquist coordinates** are a generalization of the coordinates used for the metric of a Schwarzschild black hole that can be used to express the metric of a Kerr black hole.

The **Newman–Penrose** (**NP**) **formalism** is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members often asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.

The **Cauchy momentum equation** is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum.

In mathematics, **vector spherical harmonics** (**VSH**) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.

In general relativity, a point mass deflects a light ray with impact parameter by an angle approximately equal to

The **Jeans equations** are a set of partial differential equations that describe the motion of a collection of stars in a gravitational field. The Jeans equations relate the second-order velocity moments to the density and potential of a stellar system for systems without collision. They are analogous to the Euler equations for fluid flow and may be derived from the collisionless Boltzmann equation. The Jeans equations can come in a variety of different forms, depending on the structure of what is being modelled. Most utilization of these equations has been found in simulations with large number of gravitationally bound objects.

In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of *nested round spheres*. In such a spacetime, a particularly important kind of coordinate chart is the **Schwarzschild chart**, a kind of polar spherical coordinate chart on a static and spherically symmetric spacetime, which is *adapted* to these nested round spheres. The defining characteristic of Schwarzschild chart is that the radial coordinate possesses a natural geometric interpretation in terms of the surface area and Gaussian curvature of each sphere. However, radial distances and angles are not accurately represented.

In general relativity, the **Weyl metrics** are a class of *static* and *axisymmetric* solutions to Einstein's field equation. Three members in the renowned Kerr–Newman family solutions, namely the Schwarzschild, nonextremal Reissner–Nordström and extremal Reissner–Nordström metrics, can be identified as Weyl-type metrics.

- ↑ Baldauf, Tobias; Seljak, Uros; Desjacques, Vincent; McDonald, Patrick (13 January 2018). "Evidence for Quadratic Tidal Tensor Bias from the Halo Bispectrum".
*Physical Review D*.**86**(8): 083540. arXiv: 1201.4827 . Bibcode:2012PhRvD..86h3540B. doi:10.1103/PhysRevD.86.083540. S2CID 21681130.

- Sperhake, Ulrich. "Part II General Relativity Lecture Notes" (PDF): 19. Retrieved 13 January 2018.
`{{cite journal}}`

: Cite journal requires`|journal=`

(help) - Renaud, F.; Boily, C. M.; Naab, T.; Theis, Ch. (20 November 2009). "Fully Compressive Tides in Galaxy Mergers".
*The Astrophysical Journal*.**706**(1): 68. arXiv: 0910.0196 . Bibcode:2009ApJ...706...67R. doi:10.1088/0004-637X/706/1/67. S2CID 15831572. - Duc, Pierre-Alain; Renaud, Florent. "Gravitational potential and tidal tensor".
*ned.ipac.caltech.edu*. Caltech . Retrieved 13 January 2018.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.