In
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, 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
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
between the objects.
Mathematically, the tidal force in general relativity is described by the
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
,
and the trajectory of an object solely under the influence of gravity is called a ''
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
''. The geodesic deviation equation relates the Riemann curvature tensor to the relative acceleration of two neighboring geodesics. In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, the geodesic deviation equation is more commonly known as the
Jacobi equation.
Mathematical definition
To quantify geodesic deviation, one begins by setting up a family of closely spaced geodesics indexed by a continuous variable ''s'' and parametrized by an
affine parameter
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection ...
τ. That is, for each fixed ''s'', the curve swept out by γ
''s''(τ) as τ varies is a geodesic. When considering the geodesic of a massive object, it is often convenient to choose τ to be the object's
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval b ...
. If ''x''
μ(''s'', τ) are the coordinates of the geodesic γ
''s''(τ), then the
tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...
of this geodesic is
:
If τ is the proper time, then ''T''
μ is the
four-velocity
In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
of the object traveling along the geodesic.
One can also define a ''deviation vector'', which is the displacement of two objects travelling along two infinitesimally separated geodesics:
:
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:
:
The geodesic deviation equation relates ''A''
μ, ''T''
μ, ''X''
μ, and the
Riemann tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
''R''
μνρσ:
:
An alternate notation for the directional
covariant derivative is
, so the geodesic deviation equation may also be written as
:
The geodesic deviation equation can be derived from the
second variation
The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ...
of the point particle
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
along geodesics, or from the first variation of a combined Lagrangian. 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
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
which has a one
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
indexed momentum appears to have a corresponding generalization of geodesic deviation).
Weak-field limit
The connection between geodesic deviation and tidal acceleration can be seen more explicitly by examining geodesic deviation in the
weak-field limit, where the metric is approximately Minkowski, and the velocities of test particles are assumed to be much less than ''c''. Then the tangent vector ''T''
μ is approximately (1, 0, 0, 0); i.e., only the timelike component is nonzero.
The spatial components of the relative acceleration are then given by
:
where ''i'' and ''j'' run only over the spatial indices 1, 2, and 3.
In the particular case of a metric corresponding to the Newtonian potential Φ(''x'', ''y'', ''z'') of a massive object at ''x'' = ''y'' = ''z'' = 0, we have
:
which is the
tidal tensor
Tidal is the adjectival form of tide.
Tidal may also refer to:
* ''Tidal'' (album), a 1996 album by Fiona Apple
* Tidal (king), a king involved in the Battle of the Vale of Siddim
* TidalCycles, a live coding environment for music
* Tidal (servic ...
of the Newtonian potential.
See also
*
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
*
Curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
*
Glossary of Riemannian and metric geometry
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.
The following articles may also be useful; they either contain specialised vocabulary or provi ...
References
*.
*.
External links
General Relativity and Quantum Cosmology
{{DEFAULTSORT:Geodesic Deviation Equation
Geodesic (mathematics)
Riemannian geometry
Equations