Coordinate singularity

Coordinate singularities at the north and south poles of a sphere when using geographic coordinates

In physics this is called a coordinate singularity if in a coordinate system by its characteristics for a particular point no clear coordinates can be specified. For example, at the north and south poles of the earth, unambiguous information on the geographical longitude is neither possible nor necessary, as all longitudes intersect at this point . Unlike a physical singularity , a coordinate singularity is inconspicuous to an observer, since it only appears on the basis of the properties of the coordinate system. It disappears when a more suitable coordinate system is used.

definition

A coordinate singularity is present at the points at which a variable leaves its permissible value range or is not unambiguous, but this can be remedied by choosing a different coordinate system.

description

Coordinate singularities can occur in different situations. For example, a coordinate singularity arises at a point of a - dimensional submanifold of Euclidean space with or an (abstract) manifold of this dimension, if this point has no unambiguous coordinates in the chosen coordinate system . The nature of such a coordinate singularity can be seen if one looks at another coordinate system in which the point has unique coordinates . In the case of Euclidean space, these can be Cartesian coordinates ; in the case of manifolds, this can be done with a map . Then there is a coordinate transformation of the shape ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle m\geq n}$ ${\displaystyle (v_{1},\ldots ,v_{n})}$ ${\displaystyle (x_{1},\ldots ,x_{n})}$ ${\displaystyle T}$

${\displaystyle (x_{1},\ldots ,x_{n})=T(v_{1},\ldots ,v_{n}),}$

which, however, cannot be inverted at a coordinate singularity . If the coordinate transformation can be differentiated component by component , which is the case with common coordinate systems, then the Jacobi matrix is ${\displaystyle T}$

${\displaystyle J(T)={\frac {\partial (x_{1},\ldots ,x_{n})}{\partial (v_{1},\ldots ,v_{n})}}}$

singular at a coordinate singularity , hence the term "coordinate singularity ".

Examples

Polar coordinates

Polar coordinates ${\displaystyle (r,\theta )}$

In the polar coordinate system , each point on the plane is described by a radial coordinate and an angular coordinate . The conversion of polar coordinates into Cartesian coordinates takes place through the coordinate transformation ${\displaystyle r\in \mathbb {R} _{+}}$ ${\displaystyle \theta \in (-\pi ,\pi ]}$ ${\displaystyle (x,y)\in \mathbb {R} ^{2}}$

${\displaystyle x=r\cdot \cos \theta }$

${\displaystyle y=r\cdot \sin \theta }$

A coordinate singularity is obtained at the zero point : is , the result of the transformation is independent of the angle coordinate . The zero point therefore has no clear representation in polar coordinates. If polar coordinates are extended by a height coordinate that specifies the distance from the plane of the polar coordinate system, ${\displaystyle (0,0)}$ ${\displaystyle r=0}$ ${\displaystyle \theta }$ ${\displaystyle h}$

${\displaystyle z=h}$

one obtains cylindrical coordinates of the space which are singular at all points . ${\displaystyle (0,0,z)}$

Spherical coordinates

Spherical coordinates ${\displaystyle (r,\phi ,\theta )}$

In the spherical coordinate system , each point in space is described by a radial coordinate and two angular coordinates and . The conversion of spherical coordinates into Cartesian coordinates is carried out by the coordinate transformation ${\displaystyle r\in \mathbb {R} _{+}}$ ${\displaystyle \phi \in (-\pi ,\pi ]}$ ${\displaystyle \theta \in [0,\pi ]}$ ${\displaystyle (x,y,z)\in \mathbb {R} ^{3}}$

${\displaystyle x=r\cdot \sin \theta \cdot \cos \varphi }$

${\displaystyle y=r\cdot \sin \theta \cdot \sin \varphi }$

${\displaystyle z=r\cdot \cos \theta }$

This transformation results in the following coordinate singularities:

• is , the result of the transformation is the point on the positive z -axis independent of the angular coordinate ${\displaystyle \theta =0}$ ${\displaystyle (0,0,r)}$ ${\displaystyle \varphi }$
• is , the result of the transformation is the point on the negative z -axis independent of the angular coordinate ${\displaystyle \theta =\pi }$ ${\displaystyle (0,0,-r)}$ ${\displaystyle \varphi }$
• is , the result of the transformation is the zero point independent of both angular coordinates and ${\displaystyle r=0}$ ${\displaystyle (0,0,0)}$ ${\displaystyle \varphi }$ ${\displaystyle \theta }$

In spherical coordinates, the entire z- axis has no clear representation. Enforcing obtained spherical coordinates ( geographical coordinates ) on the ball surface , only at the two poles and are singular. ${\displaystyle r=1}$ ${\displaystyle (0,0,1)}$ ${\displaystyle (0,0,-1)}$

See also

• Schwarzschild metric

literature

• Franz Embacher: Mathematical foundations for teaching physics . 2nd revised edition. Vieweg + Teubner, Wiesbaden 2011, ISBN 978-3-8348-0948-3 , pp. 167 ( online ).
• Hans Jörg Dirschmid: Tensors and fields . 1st edition. Springer, Vienna 1996, ISBN 3-211-82754-4 , p. 492 ( online ).
• Thomas Filk, Domenico Giulini: In the beginning there was eternity: in search of the origin of time . 1st edition. Beck, Munich 2004, ISBN 3-406-52187-8 , pp. 243 ( online ).

Individual evidence

1. ↑ Hans-Jürgen Schmidt: Einstein's work in relation to modern cosmology . 2005, p. 2 ( Online (PDF; 146 kB)).
2. ↑ Hilmar W. Duerbeck , Wolfgang R. Dick (ed.): Einstein's cosmos: investigations into the history of cosmology . 2005, ISBN 3-8171-1770-1 , pp. 110 .