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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 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 vector calculus, **divergence** is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

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

In physics, the **Navier–Stokes equations** are a set of 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.

In mathematics, the **Laplace operator** or **Laplacian** is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇^{2} or Δ. 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*).

In mathematics and physical science, **spherical harmonics** are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

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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.

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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.

In general relativity, a **frame field** 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.

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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

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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). 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 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.

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