Minkowski space

(noun)

A four dimensional flat space-time. Because it is flat, it is devoid of matter.

Related Terms

Examples of Minkowski space in the following topics:

  • The Relativistic Universe

    • In this case, the set is the space-time and the elements are points in that space-time.
    • A space-time with the $\eta$ metric is called Minkowski space and $\eta$ is the Minkowski metric.
    • Four-dimensional Minkowski space-time is only one of many different possible space-times (geometries) which differ in their metric matrix.
    • Thus, energy and momentum curves space-time.
    • Minkowski space is the special space devoid of matter, and as a result, it is completely flat.

  • A Geometrical Picture

    • Any vector in the null space of a matrix, must be orthogonal to all the rows (since each component of the matrix dotted into the vector is zero).
    • Therefore all the elements in the null space are orthogonal to all the elements in the row space.
    • In mathematical terminology, the null space and the row space are orthogonal complements of one another.
    • Similarly, vectors in the left null space of a matrix are orthogonal to all the columns of this matrix.
    • This means that the left null space of a matrix is the orthogonal complement of the column $\mathbf{R}^{n}$ .

  • Problems

    • Use special relativity (the Minkowski metric) to figure this out.

  • Spaces Associated with a linear system Ax = y

    • Now the column space and the nullspace are generated by $A$ .
    • What about the column space and the null space of $A^T$ ?
    • These are, respectively, the row space and the left nullspace of $A$ .
    • The nullspace and row space are subspaces of $\mathbf{R}^{m}$ , while the column space and the left nullspace are subspaces of $\mathbf{R}^{n}$ .
    • We can summarize these spaces as follows:

  • B.4 Chapter 4

    • Use special relativity (the Minkowski metric) to figure this out.

  • Shape

    • The shape of an object is a description of space that the object takes up; the shape can change if the object is deformed.
    • The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as color, content, and material composition.
    • In geometry, two subsets of a Euclidean space have the same shape if one can be transformed to the other by a combination of translations, rotations (together also called rigid transformations), and uniform scalings.
    • Shapes of physical objects are equal if the subsets of space these objects occupy satisfy the definition above.
    • In particular, the shape does not depend on the size and placement in space of the object.

  • Four-Dimensional Space-Time

    • We live in four-dimensional space-time, in which the ordering of certain events can depend on the observer.
    • Observer A sets up a space-time coordinate system (t, x, y, z); similarly, A' sets up his own space-time coordinate system (t', x', y', z').
    • (See for an example. ) Therefore both observers live in a four-dimensional world with three space dimensions and one time dimension.
    • Back to our example: let us assume that at some point in space-time there is a beam of light that emerges.
    • In this situation, the space-time separation between the two events is space-like.

  • Linear Vector Spaces

    • Whereas our use of vector spaces is purely abstract.
    • The definition of such a space actually requires two sets of objects: a set of vectors $V$ and a one of scalars $F$ .
    • The simplest example of a vector space is $\mathbf{R}^n$ , whose vectors are n-tuples of real numbers.
    • In the case of $n=1$ the vector space $V$ and the field $F$$F$ are the same.
    • So trivially, $F$ is a vector space over $F$ .

  • Phase-Space Density

    • The phase-space density of particles gives the number of particles in an infinitesimal region of phase space,
    • If there is no dissipation, the phase-space density along the trajectory of a particular particle is given by
    • We would like to define some quantities that are integrals over momentum space that transform simply under Lorentz transformations.

  • Introduction to The Four Fundamental Spaces

    • A subspace of a vector space is a nonempty subset $S$ that satisfies
    • This subspace is called the column space of the matrix and is usually denoted by $R(A)$ , for "range".
    • The dimension of the column space is called the rank of the matrix.