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Lecture

Lorentz Transformations and Covariant Tensors

In course

PHYS-324: Classical electrodynamics

The goal of this course is the study of the physical and conceptual consequences of Maxwell equations. ...

Description

This lecture covers Lorentz transformations of currents and potentials, contravariant and covariant tensors, Lorentz boosts, addition of velocities, Lorentz invariance, and the covariant formulation of electrodynamics. It also discusses rotational invariance, linear transformations, and examples illustrating rotational invariance and vector spaces. The lecture delves into the concepts of contravariant and covariant tensors, emphasizing the transformation of components under linear transformations. It concludes with examples demonstrating the application of covariant tensors in rotational invariance and linear functions.

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Instructor

Victor Gorbenko

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Physics

Relativity: Special relativity

Related concepts (32)

Lorentz transformation

In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz.

Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.

Linear map

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a .

Lorentz covariance

In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same for all observers that are moving with respect to one another within an inertial frame. It has also been described as "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space".

Transformation matrix

In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then for some matrix , called the transformation matrix of . Note that has rows and columns, whereas the transformation is from to . There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors. Matrices allow arbitrary linear transformations to be displayed in a consistent format, suitable for computation.

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