40.1 Minkowski space

40.1 Minkowski space

Concept links · terms present in this machine draft; source roles are unverified: vector space · dual space

Special relativity is based on the principle that one should consider space and time together, and take them to be a four dimensional space ${\mathbf{R}}^4$ with an indefinite inner product:

Definition (Minkowski space)

Minkowski space $M^4$ is the vector space ${\mathbf{R}}^4$ with an indefinite inner product given by

$$(x,y)\equiv x\cdot y=-x_0{y}_0+x_1{y}_1+x_2{y}_2+x_3{y}_3$$

where $(x_0,x_1,x_2,x_3)$ are the coordinates of $x\in {\mathbf{R}}^4,(y_0,y_1,y_2,y_3)$ the coordinates of $y\in {\mathbf{R}}^4$

Digression. We have chosen to use the − + ++ instead of $the+--$ sign convention for the following reasons:

• Analytically continuing the time variable $x_0$ to ix₀ gives a positive definite inner product.

• Restricting to spatial components, there is no change from our previous formulas for the symmetries of Euclidean space $E(3)$

• Only for this choice will we have a real (as opposed to complex) spinor representation (since $\mathrm{Cliff}(3,1)=M(4,\mathbf{R})\ne \mathrm{Cliff}(1,3))$ .

• Weinberg’s quantum field theory textbook $[100]$ uses this convention (although, unlike him, we’ll put the 0 component first).

This inner product will also sometimes be written using the matrix

$${\eta }_{\mu \nu }=\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$

as

$$x\cdot y=\sum _{\mu ,\nu =0}^3{\eta }_{\mu \nu }{x}_{\mu }{y}_{\nu }$$

Digression (Upper and lower indices). In many physics texts it is conventional in discussions of special relativity to write formulas using both upper and lower indices, related by

$$x_{\mu }=\sum _{\nu =0}^3{\eta }_{\mu \nu }{x}^{\nu }={\eta }_{\mu \nu }{x}^{\nu }$$

with the last form of this using the Einstein summation convention.

One motivation for introducing both upper and lower indices is that special relativity is a limiting case of general relativity, which is a fully geometrical theory based on taking space-time to be a manifold $M$ with a metric g that varies from point to point. In such a theory it is important to distinguish between elements of the tangent space $T_x(M)$ at a point $x\in M$ and elements of its dual, the co-tangent space $T_x^{\ast }(M)$ , while using the fact that the metric g provides an inner product on $T_x(M)$ and thus an isomorphism $T_x(M)\simeq T_x^{\ast }(M)$ . In the special relativity case, this distinction between $T_x(M)$ and $T_x^{\ast }(M)$ just comes down to an issue of signs, but the upper and lower index notation is useful for keeping track of those.

A second motivation is that position and momenta naturally live in dual vector spaces, so one would like to distinguish between the vector space $M^4$ of positions and the dual vector space of momenta. In the case though of a vector space like $M^4$ which comes with a fixed inner product ${\eta }_{\mu \nu }$ , this inner product gives a fixed identification of $M^4$ and its dual, an identification that is also an identification as representations of the Lorentz group. Given this fixed identification, we will not here try and distinguish by notation whether a vector is in $M^4$ or its dual, so will just use lower indices, not both upper and lower indices.

The coordinates $x_1,x_2,x_3$ are interpreted as spatial coordinates, and the coordinate $x_0$ is a time coordinate, related to the conventional time coordinate $t$ with respect to chosen units of time and distance by $x_0=ct$ where $c$ is the speed of light. Mostly we will assume units of time and distance have been chosen so that $c=1$

Vectors $v\in M^4$ such that $|\mathbf{v}{|}^2=\mathbf{v}\cdot \mathbf{v}>0$ are called “space-like”, those with $|v{|}^2<0\cdot {\mathrm{time}-\mathrm{like}}^{,}$ and those with $|v{|}^2=0$ are said to lie on the “light cone”. Suppressing one space dimension, the picture to keep in mind of Minkowski space looks like this:

Figure 40.1: Light cone structure of Minkowski spacetime.

Chapter contents · Next section: 40.2 The Lorentz group and its Lie algebra · 中文版本

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原书 PDF · 印刷页 425、426、427、428、429、430、431、432、433、434

来源版本：2025-10-20

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