Deviation equation

The deviation equation or geodetic deviation is an equation of Riemann's geometry or general relativity theory and describes the change in the distance between two neighboring geodesics with the help of the Riemann curvature tensor . This equation can be used to determine whether and in what way a space is curved by measuring the relative acceleration of two specimens on adjacent geodesics. If no relative acceleration is measured between two geodesics, the space is flat . The relative acceleration between the test specimens is only due to the curvature of the room, not from their mutual gravitational attraction, which would also have an additional effect in a real experiment.

Formulation of the equation

The mathematical formulation of the deviation equation is:

${\displaystyle {\frac {\mathrm {D} ^{2}V^{\alpha }}{\mathrm {D} \tau ^{2}}}={\frac {\partial x^{\beta }}{\partial \tau }}{\frac {\partial x^{\mu }}{\partial \tau }}V^{\nu }R_{\,\,\mu \beta \nu }^{\alpha }+{\frac {\mathrm {D} }{\mathrm {D} \tau }}\left(T_{\kappa \lambda }^{\alpha }{\frac {\partial x^{\kappa }}{\partial \tau }}V^{\lambda }\right)}$

and simplified in a torsion free space to

${\displaystyle {\frac {\mathrm {D} ^{2}V^{\alpha }}{\mathrm {D} \tau ^{2}}}={\frac {\partial x^{\beta }}{\partial \tau }}{\frac {\partial x^{\mu }}{\partial \tau }}V^{\nu }R_{\,\,\mu \beta \nu }^{\alpha }}$

The symbols in the equations mean the following:

• ${\displaystyle x^{\alpha }(\tau )}$denotes the geodesic and their tangential vector . ${\displaystyle \textstyle {\tfrac {\partial x^{\alpha }}{\partial \tau }}}$
• ${\displaystyle \textstyle V^{\alpha }{\mbox{d}}p={\tfrac {\partial x^{\alpha }}{\partial p}}\mathrm {d} p}$is the distance vector between two neighboring geodesics and thus the linear change in the distance between two infinitesimally neighboring geodesics. ${\displaystyle \textstyle {\tfrac {\partial x^{\alpha }}{\partial p}}}$
• ${\displaystyle T_{\kappa \lambda }^{\alpha }}$is the torsion tensor of space, in particular is the vector that closes the parallelogram spanned by and . This vector is zero in torsion-free spaces. ${\displaystyle T_{\kappa \lambda }^{\alpha }A^{\kappa }B^{\lambda }}$ ${\displaystyle A}$ ${\displaystyle B}$
• ${\displaystyle R_{\mu \beta \nu }^{\alpha }}$ is the Riemann curvature tensor.
• In addition, Einstein's sums convention is used, the Greek indices run from and and are first order tensors . ${\displaystyle 0\dots 3}$ ${\displaystyle x^{\alpha }}$ ${\displaystyle V^{\alpha }}$
• ${\displaystyle \textstyle {\tfrac {\mathrm {D} V^{\alpha }}{\mathrm {D} \tau }}}$denotes the covariant derivative .

In the flat space the distance intersecting geodesics growing two to and proportional to . If this is not the case, then this is a symptom of the curvature of space and corresponds to the above equation with a non-vanishing curvature tensor. ${\displaystyle x^{\alpha }(\tau ,p_{1})}$ ${\displaystyle x^{\alpha }(\tau ,p_{2})}$ ${\displaystyle \tau }$

literature

• Hans Stephani : General theory of relativity . 4th edition. Wiley-VCH, 1991, ISBN 3-326-00083-9

Individual evidence

1. ↑ Hendrik van Hees: Physics and the Trappings , Section Straight Lines ( Memento of the original from July 23, 2013 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
2. ↑ Hendrik van Hees: Physics and the Trappings , Section Torsion and Curvature ( Memento of the original from May 5, 2010 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
3. ^ Geometry of spacetime by Rainer Oloff, p. 141