Tidal tensor

Last updated

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

General relativity Theory by Albert Einstein, covering gravitation in curved spacetime

General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.

Contents

  1. tidal accelerations of a cloud of (electrically neutral, nonspinning) test particles,
  2. 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.

Tidal tensor for a spherical body

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 be multiplied by G.)

Gravitational constant empirical physical constant

The gravitational constant, denoted by the letter G, is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general theory of relativity.

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: .

In mathematics, multilinear algebra extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of p-vectors and multivectors with Grassmann algebra.

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.

Hessian formulation

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:

Gravitational potential

In classical mechanics, the gravitational potential at a location is equal to the work per unit mass that would be needed to move the object from a fixed reference location to the location of the object. It is analogous to the electric potential with mass playing the role of charge. The reference location, where the potential is zero, is by convention infinitely far away from any mass, resulting in a negative potential at any finite distance.

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.

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. The Laplacian Δf(p) of a function f at a point p, is the rate at which the average value of f over spheres centered at p deviates from f(p) as the radius of the sphere grows. 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.

A vacuum solution is a solution of a field equation in which the sources of the field are taken to be identically zero. That is, such field equations are written without matter interaction.

Harmonic function function with vanishing Laplacian

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : UR where U is an open subset of Rn that satisfies Laplace's equation, i.e.

The tidal tensor is given by the traceless part [1]

of the Hessian

where we are using the standard Cartesian chart for E3, 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

Spherically symmetric field

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.

See also

Related Research Articles

Spherical coordinate system

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances.

Spherical harmonics special functions defined on the surface of a sphere

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 that commonly occur in science. The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions are used to represent functions on a circle via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).

Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of continuum mechanics. The fundamental "linearizing" assumptions of linear elasticity are: infinitesimal strains or "small" deformations and linear relationships between the components of stress and strain. In addition linear elasticity is valid only for stress states that do not produce yielding. These assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in structural analysis and engineering design, often with the aid of finite element analysis.

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

Vector fields in cylindrical and spherical coordinates

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, the Hamilton–Jacobi equation (HJE) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.

In gas dynamics, Chaplygin's equation, named after Sergei Alekseevich Chaplygin (1902), is a partial differential equation useful in the study of transonic flow. It is

In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form

In general relativity, a frame field is a set of four 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.

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 space-time, 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 asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The most often-used variables in the formalism are the Weyl scalars, derived from the Weyl tensor. 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 convective form it is written:

Stokes stream function

In fluid dynamics, the Stokes stream function is used to describe the streamlines and flow velocity in a three-dimensional incompressible flow with axisymmetry. A surface with a constant value of the Stokes stream function encloses a streamtube, everywhere tangential to the flow velocity vectors. Further, the volume flux within this streamtube is constant, and all the streamlines of the flow are located on this surface. The velocity field associated with the Stokes stream function is solenoidal—it has zero divergence. This stream function is named in honor of George Gabriel Stokes.

Gravitational lensing formalism


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

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 geophysics, a geopotential model is the theoretical analysis of measuring and calculating the effects of Earth's gravitational field.

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.

References

  1. 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.