9.1 Minkowski space-time

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In Newtonian mechanics an inertial frame is a one-to-one correspondence between physical events and points of ${\mathbb{R}}^4$ , each event being assigned coordinates $(x,y,z,t)$ such that the motion of any free particle is represented by a rectilinear path $\mathbf{r}=\mathbf{u}t+{\mathbf{r}}_0$ . This is Newton’s first law of motion. Coordinate transformations $(x,y,z,t)(x^{\prime },y^{\prime },z^{\prime },t^{\prime })$

between inertial frames are called Galilean transformations, shown in Example 2.29 to have the form

$$t^{\prime }=t+a,{\mathbf{r}}^{\prime }=\mathbf{Ar}-\mathbf{v}t+\mathbf{b}\text{\hspace{1em}(9.1)}$$

where $a$ is a real constant, $v$ and $b$ are constant vectors, and $A$ is a $3\times 3$ orthogonal matrix, ${\mathsf{A}}^T\mathsf{A}=\mathsf{I}.$

If there is no rotation, $A=I$ in Eq. (9.1), then a rectilinear motion $\mathbf{r}=\mathbf{u}t+{\mathbf{r}}_0$ is transformed to

$${\mathbf{r}}^{\prime }={\mathbf{u}}^{\prime }{t}^{\prime }+{\mathbf{r}}_0^{\prime }$$

where ${\mathbf{r}}_0^{\prime }={\mathbf{r}}_0+\mathbf{b}-a(\mathbf{u}-\mathbf{v})$ and

$${\mathbf{u}}^{\prime }=\mathbf{u}-\mathbf{v}.$$

This is known as the law of transformation of velocities and its inverse form,

$$\mathbf{u}={\mathbf{u}}^{\prime }+\mathbf{v}$$

is called the Newtonian law of addition of velocities.

In 1888 the famous Michelson–Morley experiment, using light beams oppositely directed at different points of the Earth’s orbit, failed to detect any motion of the Earth relative to an ‘aether’ postulated to be an absolute rest frame for the propagation of electromagnetic waves. The apparent interpretation that the speed of light be constant under transformations between inertial frames in relative motion is clearly at odds with Newton’s law of additior of velocities. Eventually the resolution of this problem came in the form of Einstein’s principle of relativity (1905). This is essentially an extension of Galileo’s and Newton’s ideas on invariance of mechanics, made to include electromagnetic fields (of which ligh is a particular manifestation). The geometrical interpretation due to Hermann Minkowski (1908) is the version we will discuss in this chapter.

Poincaré and Lorentz transformations

In classical mechanics we assume that events $(x,y,z,t)$ form a Galilean space-time, as described in Example 2.29. In relativity the structure is somewhat different. Instead of separate spatial and temporal intervals there is a single interval defined between pairs of events, written

$$\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2-c^2\Delta t^2$$

where $c$ is the velocity of light $(c\approx 3\times 10^8\mathrm{m}{\mathrm{s}}^{-1})$ . This singles out events connected by a light signal as satisfying $\Delta s^2=0$ . Setting $(x^1=x,x^2=y,x^3=z,x^4=ct)$ , the interval reads

$$\Delta s^2=g_{\mu \nu }\Delta x^{\mu }\Delta x^{\nu },\text{\hspace{1em}(9.2)}$$

where

$$\mathsf{G}=[g_{\mu \nu }]=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right).$$

Throughout this chapter, Greek indices $\mu$ , $\nu$, etc. will range from 1 to 4 while indices $i,j,k\dots$ . range from 1 to 3. The set ${\mathbb{R}}^4$ with this interval structure is called Minkowski space-time, or simply Minkowski space. The geometrical version of the principle of relativity says that the set of events forms a Minkowski space-time. The definition of Minkowski space as given here is not altogether satisfactory. We will give a more precise definition directly, in terms of an affine space.

The restricted class of coordinate systems for which the space-time interval has the form Eq. (9.2) will be called inertial frames. We will make the assumption, as in Newtonian mechanics, that free particles have rectilinear paths with respect to inertial frames in Minkowski space-time. As shown in Example 2.30, transformations preserving Eq. (9.2) are of the form

$$x^{\prime {\mu }^{\prime }}=L_{\nu }^{{\mu }^{\prime }}{x}^{\nu }+a^{{\mu }^{\prime }}\text{\hspace{1em}(9.3)}$$

where the coefficients $L_{\nu }^{{\mu }^{\prime }}$ satisfy

$$g_{\rho \sigma }=g_{{\mu }^{\prime }{\nu }^{\prime }}{L}_{\rho }^{{\mu }^{\prime }}{L}_{\sigma }^{{\nu }^{\prime }}.\text{\hspace{1em}(9.4)}$$

Equation (9.3) is known as a Poincaré transformation, while the linear transformations $x^{\prime {\mu }^{\prime }}=L_{\rho }^{{\mu }^{\prime }}{x}^{\rho }$ that arise on setting $a^{{\mu }^{\prime }}=0$ are called Lorentz transformations.

We define the light cone $C_p$ at an event $p=(x,y,z,t)$ to be the set of points connected to $p$ by light signals,

$$C_p=\{p^{\prime }=(x^{\prime },y^{\prime },z^{\prime },c{t}^{\prime })|\Delta x^2+\Delta y^2+\Delta z^2-c^2\Delta t^2=0\}$$

where $\Delta x=x^{\prime }-x,\Delta y=y^{\prime }-y$ , etc. Events $p^{\prime }$ on $C_p$ can be thought of either as a receiver of light signals from $p,$ or as a transmitter of signals that arrive at $p.$ . Poincaré transformations clearly preserve the light cone $C_p$ at any event $p$ .

As for Eq. (5.9), the matrix version of Eq. (9.4) is (see also Example 2.30)

$$\mathsf{G}={\mathsf{L}}^T\mathsf{GL},\text{\hspace{1em}(9.5)}$$

where $\mathsf{G}=[g_{\mu \nu }]$ and $\mathsf{L}=[L_{\nu }^{{\mu }^{\prime }}]$ . Taking determinants, we have $\det \mathsf{L}=\pm 1$ . It is further possible to subdivide Lorentz transformations into those having $L_4^{4^{\prime }}\ge 1$ and those having $L_4^{4^{\prime }}\le -1$ (see Problem 9.2). Those Lorentz tansformations for which both $\det \mathsf{L}=+1$ and $L_4^{4^{\prime }}\ge 1$ are called proper Lorentz transformations. They are analogous to rotations about the origin in Euclidean space. All other Lorentz transformations are called improper.

Affine geometry

There is an important distinction to be made between Minkowski space and a Minkowskian vector space as defined in Section 5.1. Most significantly, Minkowski space is not a vector space since events do not combine linearly in any natural sense. For example, consider two events $q$ and $p,$ , having coordinates $q^{\mu }$ and $p^{\mu }$ with respect to some inertial frame. If the linear combination $q+bp$ is defined in the obvious way as being the event having coordinates $(q+bp{)}^{\mu }=q^{\mu }+b{p}^{\mu }$ , then under a Poincaré transformation Eq. (9.3)

$$q^{\prime {\mu }^{\prime }}+b{p}^{\prime {\mu }^{\prime }}=L_{\nu }^{{\mu }^{\prime }}(q^{\nu }+b{p}^{\nu })+(1+b)a^{{\mu }^{\prime }}\ne L_{\nu }^{{\mu }^{\prime }}(q+bp{)}^{\nu }+a^{{\mu }^{\prime }}.$$

In particular, the origin $q^{\mu }=0$ of Minkowski space has no invariant meaning since it is transformed to a non-zero point under a general Poincaré transformation. The difference of any pair of points, $q^{\mu }-p^{\mu }$ , does however always undergo a linear transformation

$$q^{\prime {\mu }^{\prime }}-p^{\prime {\mu }^{\prime }}=L_{\nu }^{{\mu }^{\prime }}(q^{\nu }-p^{\nu })$$

and can be made to form a genuine vector space. Loosely speaking, a structure in which differences of points are defined and form a vector space is termed an affine space.

More precisely, we define an affine space to be a pair ($M$, $V$) consisting of a set $M$ and a vector space $V,$ , such that $V$ acts freely and transitively on $M$ as an abelian group of transformations. The operation of $V$ on $M$ is written $+:M\times VM$ , and is required to satisfy

$$p+(u+v)=(p+u)+v,p+0=p$$

for all $p\in M$ . There is then no ambiguity in writing expressions such as $p+u+v$ . Recall from Section 2.6 that afree action means that if $p+u=p(p\in M)$ then $u=0;$ , while the action is transitive if for any pair of points $p,q\in M$ there exists a vector $u\in V$ such that $q=p+u$ . The vector $u$ in this equation is necessarily unique, for if $q=p+u=p+u^{\prime }$ then $p+u-u^{\prime }=p$ , and since the action is free it follows tha $u=u^{\prime }$

Let $p_0$ be a fixed point of $M$. For any point $p\in M\mathrm{let}x(p)\in V$ be the unique vector such that $p=p_0+x(p)$ . This establishes a one-to-one correspondence between the underlying set $M$ of an affine space and the vector space $V$ acting on it. If $e_i$ is any basis for $V$ then the real functions $p\mapsto x^i(p)$ where $x(p)=x^i(p)e_i$ are said to be coordinates on $M$ determined by the basis $e_i$ and the origin $p_0$

As anticipated above, in an affine space it is always possible to define the difference of any pair of points $q-p$ . Given a fixed point $p_0\in M$ let $x(p)$ and $x(q)$ be the unique vectors in $V$ such that $p=p_0+x(p)$ and $q=p_0+x(q)$ , and define the difference of two points of $M$ to be the vector $q-p=x(q)-x(p)\in V$ . This definition is independent of the choice of fixed point $p_0$ , for if $p_0^{\prime }$ is a second fixed point such that $p_0=p_0^{\prime }+v$ then

$$p=p_0^{\prime }+v+x(p)=p_0^{\prime }+x^{\prime }(p),$$ $$q=p_0^{\prime }+v+x(q)=p_0^{\prime }+x^{\prime }(q),$$

and

$$x^{\prime }(q)-x^{\prime }(p)=v+x(q)-v-x(p)=x(q)-x(p)=q-p.$$

Minkowski space and 4-tensors

Minkowski space can now be defined as an affine space $(M,V)$ where $V$ is a four dimensional Minkowskian vector space having metric tensor $g,$ , acting freely and transitively on the set $M$. $\mathrm{If}\left\{e_1,e_2,e_3,e_4\right\}$ is an orthonormal basis of $V$ such that

$$g_{\mu \nu }=g(e_{\mu },e_{\nu })=\{\begin{array}{cl}1& \text{ if }\mu =\nu <4\\ -1& \text{ if }\mu =\nu =4\\ 0& \text{ if }\mu \ne \nu \end{array}$$

we say an inertial frame is a choice of fixed point $p_0\in M,$ , called the origin, together with the coordinates $x^{\mu }(p)$ on $M$ defined by

$$p=p_0+x^{\mu }(p)e_{\mu }(p\in M).$$

The interval between any two events $q$ and $p$ in $M$ is defined by

$$\Delta s^2=g(p-q,p-q).$$

This is independent of the choice offixed point $p_0$ or orthonormal frame $e_{\mu }:$ , since it depends only on the vector difference between $p$ and $q$ and the metric tensor $g.$ . In an inertial frame the interval may be expressed in terms of coordinates

$$\Delta s^2=g_{\mu \nu }\Delta x^{\mu }\Delta x^{\nu }\text{ where }\Delta x^{\mu }=x^{\mu }(q)-x^{\mu }(p)=x^{\mu }(q-p).\text{\hspace{1em}(9.6)}$$

Under a Lorentz transformation $e_{\nu }=L_{\nu }^{{\mu }^{\prime }}{e}_{\mu }^{\prime }$ and a change of origin $p_0=p_0^{\prime }+a^{{\mu }^{\prime }}{e}_{{\mu }^{\prime }}^{\prime }$ we have for an arbitrary point $p$

$$\begin{array}{rl}& p=p_0+x^{\nu }(p)e_{\nu }\\ & =p_0^{\prime }+a^{{\mu }^{\prime }}{e}_{{\mu }^{\prime }}^{\prime }+x^{\nu }(q)L_{\nu }^{{\mu }^{\prime }}{e}_{{\mu }^{\prime }}^{\prime }\\ & =p_0^{\prime }+x^{\prime {\mu }^{\prime }}(p)e_{{\mu }^{\prime }}^{\prime }\end{array}$$

where $x^{\prime {\mu }^{\prime }}(p)$ is given by the Poincaré transformation

$$x^{\prime {\mu }^{\prime }}(p)=L_{\nu }^{{\mu }^{\prime }}{x}^{\nu }(p)+a^{{\mu }^{\prime }}.\text{\hspace{1em}(9.7)}$$

It is a simple matter to verify that the coordinate expression Eq. (9.6) for $\Delta s^2$ is invariant with respect to Poincaré transformations Eq. (9.7).

Elements $v=v^{\mu }{e}_{\mu }$ of $V$ will be termed 4-vectors. With respect to a Poincaré transformation Eq. (9.7) the components $v^{\mu }$ transform as

$$v^{\prime {\mu }^{\prime }}=L_{\nu }^{{\mu }^{\prime }}{v}^{\nu },$$

where $L_{\nu }^{{\mu }^{\prime }}$ satisfy Eq. (9.4). The inverse transformations are

$$v^{\nu }=L_{{\rho }^{\prime }}^{\prime \nu }{v}^{\prime {\rho }^{\prime }}\text{ where }{L}_{{\rho }^{\prime }}^{\prime \nu }{L}_{\mu }^{{\rho }^{\prime }}={\delta }_{\mu }^{\nu }.$$

Elements of $V(r,s)$ , defined in Chapter 7, are termed 4-tensors of type $(r,s)$ . Since we restrict attention to orthonormal bases of $V$, the components $T^{\sigma \tau ...}{}_{\mu \nu ..}$ of a 4-tensor are only required to transform as a tensor with respect to the Lorentz transformations,

$$T_{{\mu }^{\prime }{\nu }^{\prime }\dots }^{\prime {\sigma }^{\prime }{\rho }^{\prime }\dots }=T_{\mu \nu \dots }^{\sigma \rho \dots }{L}_{\sigma }^{\prime {\sigma }^{\prime }}{L}_{\rho }^{\prime {\rho }^{\prime }}\dots L_{{\mu }^{\prime }}^{\mu }{L}_{{\nu }^{\prime }}^{\nu }\dots$$

4-tensors of type (0, 1) are called 4-covectors, and 4-tensors of type (0, 0) will be termed 4-scalars or simply scalars. The important thing about 4-tensors, as for general tensors, is that if a 4-tensor equation can be shown to hold in one particular frame it holds in all frames. This is an immediate consequence of the homogeneous transformation law of components.

By Eq. (9.4) $g$ is a covariant 4-tensor of rank 2 since its components $g_{\mu \nu }$ transform as

$$g_{{\mu }^{\prime }{\nu }^{\prime }}^{\prime }=g_{\mu \nu }{L}_{{\mu }^{\prime }}^{\prime \mu }{L}_{{\nu }^{\prime }}^{\prime \nu }$$

where $g_{{\mu }^{\prime }{\nu }^{\prime }}^{\prime }=g_{{\mu }^{\prime }{\nu }^{\prime }}$ . The inverse metric $g^{\mu \nu }$ , defined by

$$g^{\mu \rho }{g}_{\rho \nu }={\delta }_{\nu }^{\mu },$$

has identical components to $g_{\mu \nu }$ and is a contravariant tensor of rank 2,

$$g^{\prime {\mu }^{\prime }{\nu }^{\prime }}=g^{\mu \nu }{L}_{\mu }^{{\mu }^{\prime }}{L}_{\nu }^{{\nu }^{\prime }},g^{\mu \nu }=g^{\prime {\mu }^{\prime }{\nu }^{\prime }}{L}_{{\mu }^{\prime }}^{\prime \mu }{L}_{{\nu }^{\prime }}^{\prime \nu }.$$

We will use $g^{\mu \nu }$ and $g_{\mu \nu }$ to raise and lower indices of 4-tensors; for example,

$$U^{\mu }=g^{\mu \nu }{U}_{\nu },W_{\mu }=g_{\mu \nu }{W}^{\nu },T_{\mu \nu }^{\rho }=g_{\mu \alpha }{g}^{\rho \beta }{T}_{\nu \beta }^{\alpha }.$$

Given two 4-vectors $A=A^{\mu }{e}_{\mu },B=B^{\nu }{e}_{\nu }$ , define their inner product to be the scalar

$$g(A,B)=A^{\mu }{B}_{\mu }=g_{\mu \nu }{A}^{\mu }{B}^{\nu }=A_{\mu }{B}^{\mu }=g^{\mu \nu }{A}_{\mu }{B}_{\nu }.$$

We say the vectors are orthogonal if $A^{\mu }{B}_{\mu }=0$ . The magnitude ofa 4-vector $A^{\mu }$ is defined to be $g(A,A)=A^{\mu }{A}_{\mu }.$ . A non-zero 4-vector $A^{\mu }$ is called

spacelike if $g(A,A)=A^{\mu }{A}_{\mu }>0$

timelike if $g(A,A)=A^{\mu }{A}_{\mu }<0.$

null if $g(A,A)=A^{\mu }{A}_{\mu }=0.$

The set of all null 4-vectors is called the null cone. This is a subset of the vector space $V$ of 4-vectors. The concept of a light cone at $p\in M.$ , defined in Section 2.30, is the set of points of $M$ that are connected to $p$ by a null vector, $C_p=\{q|g(q-p,q-p)=0\}\subset M.$ Figure 9.1 shows how the null cone separates 4-vectors into the various classes. Timelike or null vectors falling within or on the upper half of the null cone are called future-pointing, while those in the lower half are past-pointing.

Spacelike vectors, however, lie outside the null cone and form a continuously connected region of $V$, making it impossible to define invariantly the concept of a future-pointing or past-pointing spacelike vector – see Problem 9.2.

Figure 9.1 The null cone in Minkowski space

Problems

Problem 9.1

Show that

$$\left[L_{\nu }^{{\mu }^{\prime }}\right]=\left(\begin{array}{cccc}1& 0& -\alpha & \alpha \\ 0& 1& -\beta & \beta \\ \alpha & \beta & 1-\gamma & \gamma \\ \alpha & \beta & -\gamma & 1+\gamma \end{array}\right)\text{ where }\gamma =\frac{1}{2}({\alpha }^2+{\beta }^2)$$

is a Lorentz transformation for all values of $\alpha$ and $\beta .$ . Find those 4-vectors $V^{\mu }$ whose components are unchanged by all Lorentz transformations of this form.

Problem 9.2

Show that for any Lorentz transformation $L_{\nu }^{{\mu }^{\prime }}$ one must have either

$$L_4^4\ge 1\text{ or }{L}_4^4\le -1.$$

(a) Show that those transformations having $L_4^4\ge 1$ have the property that they preserve the concept of ‘before’ and ‘after’ for timelike separated events by demonstrating that they preserve the sign of $\Delta x^4$

(b) What is the effect of a Lorentz transformation having $L_4^4\le -1$?

(c) Is there any meaning, independent of the inertial frame, to the concepts of ‘before’ and ‘after for spacelike separated events?

Problem 9.3

Show that (i) if $T^{\alpha }$ is a timelike 4-vector it is always possible to find a Lorentz transformation such that $T^{\prime {\alpha }^{\prime }}$ will have components $(0,0,0,a)$ and (ii) if $N^{\alpha }$ is a null vector then it is always possible to find a Lorentz transformation such that $N^{\prime {\alpha }^{\prime }}$ has components $(0,0,1,1)$.

Let $U^{\alpha }$ and $V^{\alpha }$ be 4-vectors. Show the following:

(a) If $U^{\alpha }{V}_{\alpha }=0$ and $U^{\alpha }$ is timelike, then $V^{\alpha }$ is spacelike.

(b) If $U^{\alpha }{V}_{\alpha }=0$ and $U^{\alpha }$ and $V^{\alpha }$ are both null vectors, then they are proportional to each other.

(c) $\mathrm{If}{U}^{\alpha }$ and $V^{\alpha }$ are both timelike future-pointing then $U_{\alpha }{V}^{\alpha }<0$ and $U^{\alpha }+V^{\alpha }$ is timelike.

(d) Find other statements similar to the previous assertions when $U^{\alpha }$ and $V^{\alpha }$ are taken to be various combinations of null, future-pointing null, timelike future-pointing, spacelike, etc.

Problem 9.4

If the 4-component of a 4-vector equation $A^4=B^4$ is shown to hold in all inertia frames, show that all components are equal in all frames, $A^{\mu }=B^{\mu }$