Hilbert's nineteenth problem

In general relativity, geodesic deviation describes the tendency of objects to approach or recede from one another while moving under the influence of a spatially varying gravitational field. Put another way, if two objects are set in motion along two initially parallel trajectories, the presence of a tidal gravitational force will cause the trajectories to bend towards or away from each other, producing a relative acceleration between the objects.

Mathematically, the tidal force in general relativity is described by the Riemann curvature tensor, and the trajectory of an object solely under the influence of gravity is called a geodesic. The geodesic deviation equation relates the Riemann curvature tensor to the relative acceleration of two neighboring geodesics. In differential geometry, the geodesic deviation equation is more commonly known as the Jacobi equation.

Geodesic deviation equation

To quantify geodesic deviation, one begins by setting up a family closely spaced geodesics indexed by a continuous variable s and parametrized by an affine parameter t. That is, for each fixed s, the curve swept out by γ_s(t) as t varies is a geodesic with affine parameter. If x^μ(s, t) are the coordinates of the geodesic γ_s(t), then the tangent vector of this geodesic is

$T^{\mu }=\frac{\partial x^{\mu }(s,t)}{\partial t}.$

One can also define a deviation vector, which is the displacement of two objects travelling along two infinitesimally separated geodesics:

$X^{\mu }=\frac{\partial x^{\mu }(s,t)}{\partial s}.$

The relative acceleration A^μ of the two objects is defined, roughly, as the second derivative of the separation vector X^μ as the objects advance along their respective geodesics. Specifically, A^μ is found by taking the directional covariant derivative of X along T twice:

$A^{\mu }=T^{\alpha }{\mathrm{\nabla }}_{\alpha }(T^{\beta }{\mathrm{\nabla }}_{\beta }{X}^{\mu }).$

The geodesic deviation equation relates A^μ, T^μ, X^μ, and the Riemann tensor R^μ_νρσ:

$A^{\mu }={R^{\mu }}_{\nu \rho \sigma }{T}^{\nu }{T}^{\rho }{X}^{\sigma }.$

An alternate notation for the directional covariant derivative $T^{\alpha }{\mathrm{\nabla }}_{\alpha }$ is $D/dt$, so the geodesic deviation equation may also be written as

$\frac{D^2{X}^{\mu }}{d{t}^2}={R^{\mu }}_{\nu \rho \sigma }{T}^{\nu }{T}^{\rho }{X}^{\sigma }.$

The geodesic deviation equation can be derived from the second variation of the point particle Lagrangian along geodesics, or from the first variation of a combined Lagrangian.Template:Clarify The Lagrangian approach has two advantages. First it allows various formal approaches of quantization to be applied to the geodesic deviation system. Second it allows deviation to be formulated for much more general objects than geodesics (any dynamical system which has a one spacetime indexed momentum appears to have a corresponding generalization of geodesic deviation).Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

See also

• Bernhard Riemann
• Curvature
• Glossary of Riemannian and metric geometry

References

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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
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  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

External links

• General Relativity and Quantum Cosmology
• Tensors and Relativity: Geodesic deviation