Lorentz transformation under symmetric configuration

Lorentz transformation under symmetric configuration

In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other.

Assume there are two observers O_1 and O_2, each using their own Cartesian coordinate system to measure space and time intervals. O_1 uses (t_1, x_1, y_1, z_1) and O_2 uses (t_2, x_2, y_2, z_2). Assume further that the coordinate systems are oriented so that the x_1-axis and the x_2-axis overlap but in opposite directions. The y_1-axis is parallel to the y_2-axis but in opposite directions. The z_1-axis is parallel to the z_2-axis and in the same direction. The relative velocity between the two observers is v along the x_1 or x_2 axis. v is defined as a positive number when O_1 sees O_2 sliding in the direction of x_1. Also assume that the origins of both coordinate systems are the same. If all this holds, then the coordinate systems are said to be in symmetric configuration.

In this configuration, frame O_2 appears to O_1 in the identical way that frame O_1 appears to O_2. However, in the standard configuration, if O_2 sees O_1 going forward then O_1 sees O_2 going backward.

The Lorentz transformation for frames in symmetric configuration is:: t_1 = gamma (t_2 - frac{v x_2}{c^{2) ,: x_1 = gamma (v t_2 - x_2) ,: y_1 = - y_2 ,: z_1 = z_2 ,where gamma := 1/sqrt{1 - v^2/c^2} is the Lorentz factor.

The inverse transformation is:: t_2 = gamma (t_1 - frac{v x_1}{c^{2) ,: x_2 = gamma (v t_1 - x_1) ,: y_2 = - y_1 ,: z_2 = z_1 .

The above forward and inverse transformations are identical. This offers mathematical simplicity.

In matrix form the forward symmetric transformation is: :egin{bmatrix}c t_1 \ x_1 \ y_1 \ z_1end{bmatrix}=egin{bmatrix}gamma&-eta gamma&0&0\eta gamma&-gamma&0&0\0&0&-1&0\0&0&0&1\end{bmatrix}egin{bmatrix}c t_2 \ x_2 \ y_2 \ z_2end{bmatrix} .

where eta := frac{v}{c} .

The inverse symmetric transformation is::egin{bmatrix}c t_2 \ x_2 \ y_2 \ z_2end{bmatrix}=egin{bmatrix}gamma&-eta gamma&0&0\eta gamma&-gamma&0&0\0&0&-1&0\0&0&0&1\end{bmatrix}egin{bmatrix}c t_1 \ x_1 \ y_1 \ z_1end{bmatrix} .

A single transformation matrix is used for both the forward and the inverse operation.

As expected::egin{bmatrix}gamma&-eta gamma&0&0\eta gamma&-gamma&0&0\0&0&-1&0\0&0&0&1\end{bmatrix}egin{bmatrix}gamma&-eta gamma&0&0\eta gamma&-gamma&0&0\0&0&-1&0\0&0&0&1\end{bmatrix}=egin{bmatrix}1&0&0&0\0&1&0&0\0&0&1&0\0&0&0&1\end{bmatrix}.


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Lorentz transformation — A visualisation of the Lorentz transformation (full animation). Only one space coordinate is considered. The thin solid lines crossing at right angles depict the time and distance coordinates of an observer at rest with respect to that frame; the …   Wikipedia

  • Maxwell's equations — For thermodynamic relations, see Maxwell relations. Electromagnetism …   Wikipedia

  • Minkowski space — A diagram of Minkowski space, showing only two of the three spacelike dimensions. For spacetime graphics, see Minkowski diagram. In physics and mathematics, Minkowski space or Minkowski spacetime (named after the mathematician Hermann Minkowski)… …   Wikipedia

  • Spin-statistics theorem — Statistical mechanics Thermodynamics · …   Wikipedia

  • Magnetic monopole — It is impossible to make magnetic monopoles from a bar magnet. If a bar magnet is cut in half, it is not the case that one half has the north pole and the other half has the south pole. Inst …   Wikipedia

  • Supergravity — In theoretical physics, supergravity (supergravity theory) is a field theory that combines the principles of supersymmetry and general relativity. Together, these imply that, in supergravity, the supersymmetry is a local symmetry (in contrast to… …   Wikipedia

  • Spin (physics) — This article is about spin in quantum mechanics. For rotation in classical mechanics, see angular momentum. In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles… …   Wikipedia

  • Group (mathematics) — This article covers basic notions. For advanced topics, see Group theory. The possible manipulations of this Rubik s Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines …   Wikipedia

  • String theory — This article is about the branch of theoretical physics. For other uses, see String theory (disambiguation). String theory …   Wikipedia

  • Molecular Hamiltonian — In atomic, molecular, and optical physics as well as in quantum chemistry, molecular Hamiltonian is the name given to the Hamiltonian representing the energy of the electrons and nuclei in a molecule. This Hermitian operator and the associated… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”