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Mathematics for Physics

102 Frame

A coordinate system for measuring events that is used to assign positions and times to physical occurrences.

Note: Also called a reference frame. Also called a frame of reference. In geometry, also called an \(n\)-frame when it means a basis of vectors.

definition [d] (Frame = Reference Frame = Frame of Reference) From Knight: a coordinate system in which an experimenter makes position and time measurements of physical events is called a reference frame. Equivalently, a reference frame is a coordinate system in which experimenters equipped with meter sticks, stopwatches, and any other needed equipment make position and time measurements on moving objects.

where

  • the coordinate system provides spatial and temporal labels for events.
  • measurements are those made by observers associated with that system.

definition [d] (Frame = \(n\)-Frame) From Kosinski: an \(n\)-frame in an \(n\)-dimensional vector space \(E\) is an ordered set of \(n\) linearly independent vectors in \(E\), that is, a basis of \(E\).

where

  • \(E\) is an \(n\)-dimensional vector space.
  • the ordered set is a basis of \(E\).

definition [d] (Frame of Vector Fields) From Frankel: a frame of vector fields in a region \(U\) consists of \(n\) linearly independent smooth vector fields

  • \(e = (e_{1},\ldots,e_{n})\)

in \(U\).

where

  • \(U\) is a region of an \(n\)-dimensional manifold.
  • \(e_{1},\ldots,e_{n}\) are smooth vector fields on \(U\).

102.1 Elementary Example

102.1.1 Simple

The laboratory axes \((x,y,z)\) with synchronized clocks form a reference frame for measuring a ball’s position at successive times.

\[ (x,y,z,t) = (1\,\mathrm{m},\, 0,\, 0,\, 0),\quad (2\,\mathrm{m},\, 0,\, 0,\, 1\,\mathrm{s}) \]

where

  • each tuple is an event measured in that frame.

102.1.2 General

At a point of \(\mathbb{R}^{3}\), the standard basis is a \(3\)-frame.

\[ e_{1} = (1,0,0),\quad e_{2} = (0,1,0),\quad e_{3} = (0,0,1) \]

where

  • \(\{e_{1},e_{2},e_{3}\}\) is an ordered basis of \(\mathbb{R}^{3}\).

102.2 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — source for the heading explanation.
  2. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — reference frame as a coordinate system for event measurements.
  3. Kosinski, A. A. Differential Manifolds. 2008. — \(n\)-frame as an ordered basis.
  4. Frankel, T. The Geometry of Physics: An Introduction. Cambridge University Press, 2012. — frame of vector fields.
  5. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — frame of reference as another word for coordinate system.