47. Lorentz Transformations in Special Relativity

Introduction¶

Before we examine how the Dirac equation and Dirac wave function transform under Lorentz transformations we present some material on the Lorentz transformations themselves. In these notes we will work at the level of classical special relativity, without reference to quantum mechanics, but the presentation is tailored to our needs in the next set of notes when we examine the transformation properties of the Dirac equation. The discussion in these notes is similar to that in 11. Rotations in 3-Dimensional Space, which presented the theory of spatial rotations. Indeed, since spatial rotations are examples of Lorentz transformations, these notes constitute a generalization of 11. Rotations in 3-Dimensional Space in many respects. You may wish to review 11. Rotations in 3-Dimensional Space at this point to orient yourself. You may also wish to look over Appendix E. Introduction to Tensor Analysis for some general information on tensor analysis in special relativity, especially Secs. Lorentz Transformations--The Sign of the Metric, and a Notational Problem. In the following we call on some basic knowledge of special relativity and tensor analysis.

Lorentz Transformations and the Lorentz Group¶

The usual approach to Lorentz transformations in introductory treatments of special relativity is to work out the transformations that represent boosts, usually following Einstein’s postulate that the speed of light is the same in all reference frames, and then to show that these preserve the Minkowski scalar product shown in (3) below. It is assumed that you are familiar with this approach. Along the way it is sometimes mentioned that ordinary spatial rotations are also Lorentz transformations. In the following we will reverse this sequence of concepts, and define a Lorentz transformation as any linear transformation that satisfies (3). This has the advantage of unifying parity and time reversal with ordinary (proper) Lorentz transformations, which consist of boosts and rotations. In our approach, a general Lorentz transformation is a product of operations that can include rotations, boosts, parity and time reversal.

We begin with a space-time diagram, Figure 1, which shows the coordinate axes $txyz$ of a Lorentz frame as well as a light cone. The light cone has the equation,

It consists of two parts, the forward and backward light cones, the parts $t>0$ and $t<0$, respectively. The forward light cone is the locus of events generated by a light flash at $x=y=z=t=0$, that is, by the spherical shell of the light pulse created by the flash, expanding at velocity $c$. The backwards light cone consists of another light pulse in the form of a spherical shell, this one collapsing inward at velocity $c$ in such a way that the light reaches the spatial origin $x=y=z=0$ at time $t=0$. As usual in relativity theory, the interior and surface of the forward light cone are the set of events that can be reached from $x=y=z=t=0$ by a signal traveling with velocity $v\le c$, that is, events that lie in the causal future of the event $x=y=z=t=0$. The interior and surface of the backwards light cone consists of events that can communicate with the event $x=y=z=t=0$ by means of a signal traveling at velocity $v\le c$, that is, events that lie in the causal past of the event $x=y=z=t=0$.

Also shown in the figure is an event with coordinates $x^\mu =(ct,\xvec)$. We consider a mapping of space-time onto itself, taking the (old) event $x^\mu$ into a (new) event $x^{\prime\,\mu}$, as shown. We suppose the mapping is linear and specified by a matrix $\Lambda^\mu{}_\nu$,

Only linear transformations map free particle orbits (straight lines at constant velocity) into other free particle orbits. We define the mapping to be a Lorentz transformation if the Minkowski scalar product is preserved, that is, if

for all $x^\mu$, where $g_{\mu\nu}$ is the Minkowski metric with components

By this definition, the nature of the vector $x^\mu$ (space-like, time-light or light-like) is preserved by the transformation (3). In particular, the surface of the light cone (the set of light-like events $x^\mu$) is mapped into itself by the transformation, as is the interior (the set of time-like events) and the exterior (the set of space-like events). As we shall see, the interior and surface of the forward light cone may be mapped into itself by the Lorentz transformation, in which case the interior and surface of the backwards light cone is also mapped into itself, or the forward and backward light cones may be mapped into each other, depending on the Lorentz transformation.

Substituting (2) into (3) and juggling indices slightly, we obtain

or, since $x^\mu$ is arbitrary,

A Lorentz transformation, that is, a matrix $\Lambda^\mu{}_\nu$ specifying a Lorentz transformation, is one that satisfies this equation.

Let us put this into matrix form. Let

in which the first (second) index is understood as the row (column) index. As matrices, $\gmat$ and $\gmat^{-1}$ are actually the same [see (76) and (83)], but we shall keep them notationally distinct. Then the condition (6) defining a Lorentz transformation can be written,

where $t$ indicates the transpose.

Compare this to the definition of a $3\times3$ orthogonal matrix, which can be written

where all matrices are understood to be $3\times3$ and where $\Imat$ is the identity matrix, representing the Euclidean metric $\delta_{ij}$ on three-dimensional space. A matrix $\Rmat$ satisfying this equation is said to belong to $O(3)$, the group of $3\times3$ orthogonal matrices. The properties of $3\times3$ orthogonal matrices are discussed in 11. Rotations in 3-Dimensional Space. Similarly, (8) shows that a Lorentz transformation can be regarded as a matrix that is orthogonal with respect to the Minkowski metric $g_{\mu\nu}$. A matrix $\Lambdamat$ satisfying (8) is said to belong to the group $O(3,1)$, which indicates the group of matrices orthogonal with respect to a metric with three space-like and one time-like directions. That is, by our definition, $O(3,1)$ is the group of Lorentz transformations.

The proof of the group property works like this. First, note that $\Lambdamat=\Imat$ (now the $4\times4$ identity) satisfies (8). Next, by taking determinants of (8), we obtain

since the determinant of the transpose of a matrix is the same as the determinant of the matrix, and since $\det\gmat$ cancels from both sides. This shows that

and, in particular, $\det\Lambdamat\ne0$, so $\Lambdamat^{-1}$ exists. To obtain an expression for $\Lambdamat^{-1}$ we multiply (8) by $\gmat^{-1}$,

which shows that

This is not quite as simple as the rule $\Rmat^{-1} = \Rmat^t$ that applies to ordinary orthogonal matrices, but it does make it easy to invert a Lorentz transformation. If (13) is translated into index language, it becomes

Having shown that $\Lambdamat^{-1}$ exists, we now show that it is a Lorentz transformation. We do this by multiplying (8) from the left by $(\Lambdamat^{-1})^t$, and from the right by $\Lambdamat^{-1}$. This gives

which shows that $\Lambdamat^{-1}$ is a Lorentz transformation. Finally, if $\Lambdamat_1$ and $\Lambdamat_2$ are two Lorentz transformations, then

which shows that the product $\Lambdamat_1\Lambdamat_2$ is also a Lorentz transformation.

In this discussion and in the rest of these notes on the Dirac equation we view Lorentz transformations as mappings of space-time onto itself, that is, we will take the active point of view. See Active versus Passive Point of View for a discussion of the active and passive points of view. Most introductions to relativity theory take the passive point of view, in which coordinates or tensors are transformed from one coordinate system to another. But we used the active point of view in our development of rotations in quantum mechanics in earlier sets of notes, and for consistency we must continue it with Lorentz transformations. We will give some examples presently of how the active point of view works out in practical examples, as well as some comments about how to think about it.

The Group Manifold of $O(3,1)$; Proper and Improper Lorentz Transformations¶

We now discuss the distinction between proper and improper Lorentz transformations, which is explained geometrically in terms of the group manifold of $O(3,1)$. You may wish to review Group Manifolds for $O(3)$ and $SO(3)$, which explains the analogous situation for the group $O(3)$ of orthogonal transformations on 3-dimensional space.

The space of Lorentz transformations is 6-dimensional, that is, it takes six parameters to specify a Lorentz transformation. As we shall see, those parameters can be identified with the Euler angles of a rotation and the velocity of a boost. Recall that the space of rotations alone is 3-dimensional, and that it can be parameterized by the three Euler angles. The fact that we need six parameters to specify a Lorentz transformation follows from the definition (8). This equation can be regarded as a constraint that the matrix $\Lambdamat$ must satisfy in order to be a Lorentz transformation. In terms of components, there are 10 independent constraints, because both sides of (8) are symmetric, $4\times4$ matrices, which have 10 independent components. But a general $4\times4$ matrix has 16 components, so the number of independent parameters in a Lorentz transformation is $16-10=6$.

The group manifold of $O(3,1)$ can be thought of as the 6-dimensional surface in 16-dimensional matrix space (the space of $4\times4$, real matrices) on which (8) is satisfied. Recall that the group manifold $O(3)$ is the 3-dimensional surface in the 9-dimensional space of $3\times3$ matrices such that $\Rmat^t \Rmat=\Imat$, and that it consists of two disconnected components containing the proper and improper rotations. Recall also that the improper rotations contain the spatial inversion operation represented by the matrix $\Pmat$ [see (16)].

It turns out that the group manifold $O(3,1)$ consists of four disconnected components, which are distinguished by $\det\Lambdamat$, which can only take on the values $\pm1$ [see (11)], and by the sign of the time-time component of $\Lambdamat$, that is, $\Lambda^0{}_0$. The meaning of this component is the following. First of all, it is easy to show from the definition (7) that

Next, in a Lorentz transformation $x^{\prime\,\mu} = \Lambda^\mu{}_\nu \, x^\nu$, the 0-th component can be written,

that is, $\Lambda^0{}_0$ is the coefficient connecting the old time $t$ with the new time $t'$. This coefficient is the partial derivative,

where it is understood that $\xvec$ (the old spatial point) is held fixed in the differentiation. To within a sign, $\Lambda^0{}_0$ is the relativistic factor of time dilation, given in terms of the velocity $\vvec$ of the boost contained in $\Lambdamat$ by

If $\Lambda^0_{0} > 0$, then as the old time increases, so does the new time, and $\Lambda^0{}_0 = \gamma \ge 1$. In this case the Lorentz transformation maps the forward light cone into the forward light cone, and the backward into the backward. But if $\Lambda^0{}_0<0$, then as the old time increases, the new time decreases. This means that the Lorentz transformation contains a time-reversal operation, and that $\Lambda^0{}_0 = -\gamma \le -1$. In this case the Lorentz transformation maps the forward light cone into the backward light cone and vice versa.

We can exhibit some matrices taken from the four components of the Lorentz group $O(3,1)$. In the following $\Pmat$ is the matrix that brings about a spatial inversion but which does nothing to the time:

It is a generalization of the matrix $\Pmat$ discussed in 11. Rotations in 3-Dimensional Space\ and \parity\ [see (16)]. Also, $\Tmat$ is the matrix that inverts time but does nothing to the spatial components:

Finally, the matrix $\Pmat\Tmat$ inverts all the space-time coordinates:

These matrices appear in Table 1, which gives an example of a matrix taken from each of the four components of $O(3,1)$. It is easily checked that all four matrices in Table 1 are Lorentz transformations, according to the definition (8).

We can picture the group manifold of $O(3,1)$ geometrically as illustrated in Figure 2. The four components are each connected sets that are disconnected from one another. See Fig.~\figr\classrot.4 for the analogous diagram in the case of the group $O(3)$.

The component containing the identity $\Imat$ is called the identity component. It consists of Lorentz transformations that can be continuously connected with the identity, that is, transformations that can be built up out of products of infinitesimal Lorentz transformations. We will refer to the Lorentz transformations belonging to the identity component as proper, while all others will be considered improper. The idea is that proper Lorentz transformations consist of proper rotations or boosts, or products of these, but do not contain either a spatial inversion or a time reversal operation. The proper Lorentz transformations constitute a group by themselves, a subgroup of the full Lorentz group $O(3,1)$.

For the time being we will concentrate on proper Lorentz transformations, and defer a discussion of parity and time-reversal to a subsequent set of notes.

Pure Rotations¶

Pure rotations are examples of proper Lorentz transformations. They are specified by rotation matrices, which are exactly the same as those presented in 11. Rotations in 3-Dimensional Space, except that now we append a row and column corresponding to the time coordinate, which is not affected by the rotation. As in 11. Rotations in 3-Dimensional Space, we will denote a rotation by angle $\theta$ about axis ${\hat{\mathbf{n}}}$ by $\Rmat({\hat{\mathbf{n}}},\theta)$. For example, the rotations about the three coordinate axes are

These may be compared to (18). These matrices are interpreted in the active sense, as in 11. Rotations in 3-Dimensional Space, that is, they map an (old) unrotated point to a (new) rotated one.

Pure Boosts¶

Pure boosts are another example of proper Lorentz transformations. Boosts can be specified by the velocity $\vvec$ of the boost, but in the following we shall use a different parameterization, which is convenient for bringing out the analogy with rotations more clearly. First, we define the usual relativistic parameters $\beta$ and $\gamma$ in terms of the magnitude $v=|\vvec|$ of the velocity (a la Jackson, Classical Electrodynamics),

By these definitions $0\le v<c$ and $0\le\beta<1$, but sometimes it is convenient to let $\beta$ and $v$ take on negative values, so that $\vvec/c =(v/c){\hat{\mathbf{b}}} =\beta{\hat{\mathbf{b}}}$, where ${\hat{\mathbf{b}}}$ is a unit vector such that $\vvec$ is in the same direction as ${\hat{\mathbf{b}}}$ (when $\beta>0$) or opposite the direction of ${\hat{\mathbf{b}}}$ (when $\beta<0$). We call ${\hat{\mathbf{b}}}$ the axis of the boost. With these understandings, $-c < v < +c$, $-1 <\beta < +1$ and $1\le \gamma < \infty$. Next, it is convenient to introduce the rapidity parameter $\lambda$, defined by

Then the range $-\infty < \lambda < + \infty$ corresponds to $-1 < \beta < +1$. The rapidity $\lambda$ is a kind of hyperbolic angle, which plays the same role for boosts as does $\theta$ for rotations. We will write $\Bmat({\hat{\mathbf{b}}},\lambda)$ for the boost along axis ${\hat{\mathbf{b}}}$ with rapidity $\lambda$. The velocity of the boost is given by

We will now work out the explicit form of the matrices $\Bmat({\hat{\mathbf{b}}},\lambda)$. Here we call on some experience with Lorentz transformations in special relativity.

In any boost along a direction ${\hat{\mathbf{b}}}$, the two spatial coordinates orthogonal to ${\hat{\mathbf{b}}}$ are not affected, while the coordinate $x^0=ct$ and the spatial coordinate parallel to ${\hat{\mathbf{b}}}$ transform according to the $2\times2$ matrix,

We interpret this matrix in an active sense. An example will illustrate the meaning of active boosts, as well as showing how the signs in the matrix (28) were determined. Suppose we have a particle of mass $m$ at rest. Then its energy-momentum 4-vector [see (19)] is given by

where the 0-subscript means “at rest.” If we multiply the $x^0=ct$ and $x^1=x$ coordinates of this 4-vector by the matrix (28), keeping the $x^2=y$ and $x^3=z$ components fixed, we obtain

which corresponds to the correct energy $E=mc^2\gamma$ and momentum $p_x = mv\gamma$ for a particle moving down the $x$-axis with velocity $v$. The particle has been boosted with velocity $v$, which is just what we want in the active sense. This allows us to write down the matrix $\Bmat({\hat{\mathbf{x}}},\lambda)$.

Carrying out the same procedure for boosts down the three coordinate axes, we obtain explicit matrices $\Bmat({\hat{\mathbf{b}}},\lambda)$ when ${\hat{\mathbf{b}}}$ is one of the coordinate directions. These are

These boosts may be compared to the rotations (24).

We can also obtain an explicit expression for the boost $\Bmat({\hat{\mathbf{b}}},\lambda)$ along an arbitrary direction ${\hat{\mathbf{b}}}$. Let $X^\mu=(X^0,\Xvec)$ be an arbitrary 4-vector, and let

The boost transforms the time component $X^0$ and the component of the spatial vector $\Xvec$ along ${\hat{\mathbf{b}}}$, that is, ${\hat{\mathbf{b}}}\cdot\Xvec$, according to the matrix (28), while leaving the component of $\Xvec$ orthogonal to ${\hat{\mathbf{b}}}$ invariant. That is, (32) is equivalent to

Combining the last two equations, we write the overall transformation as

This may be compared to the analogous formula for pure rotations, (44). It may also be compared to Jackson, Classical Electrodynamics, 3rd ed, Eq.~(11.19). The formula in Jackson has minus signs compared to our (34), because he uses the passive interpretation.

A final remark about rapidities is that boosts along the same axis commute with one another, and the rapidities just add when boosts are composed. That is, we have

This may compared with (17), which shows the analogous property for rotations. We note that boosts along different axes do not commute, in general.

Property (35) may be proved by composing two transformations of the form (34), but since the coordinates orthogonal to the direction of a boost are not affected by the boost, the essence of the proof is just the multiplication of $2\times 2$ matrices of the form (28). In fact, calling that matrix $M(\lambda)$, it is easily verified that

Rapidities and velocities are related by

so the velocity that results on composing two boosts with rapidities $\lambda_1$ and $\lambda_2$ along the same axis is

where we use the addition formula for the $\tanh$ function. But this is equivalent to the usual rule in special relativity for composing velocities,

This rule has the property that if $|v_1| < c$ and $|v_2| < c$, then $|v|<c$. It is equivalent to the statement $|\tanh\lambda|<1$ for all $\lambda$.

A Useful Theorem, and Parameters of Proper Lorentz Transformations¶

We have presented boosts and rotations as examples of proper rotations, but there is a theorem that states that every proper Lorentz transformation can be factored in a unique way into the product of a rotation times a boost,

where $\Rmat$ has some axis ${\hat{\mathbf{n}}}$ and angle $\theta$, and where $\Bmat$ has some axis ${\hat{\mathbf{b}}}$ and rapidity $\lambda$. We will not prove this theorem (the proof is accomplished most easily in terms of the spinor representations of the proper Lorentz group), but it is a useful fact. It shows in particular that the six parameters specifying an arbitrary proper Lorentz transformation can be taken to be the three inherent in $({\hat{\mathbf{n}}},\theta)$ (or, equivalently, the three Euler angles), plus the three inherent in $({\hat{\mathbf{b}}},\lambda)$ (or, equivalently, the velocity $\vvec$).

Exponential Notation for Rotations and Boosts¶

It was shown in 11. Rotations in 3-Dimensional Space\ that proper rotations can be written in exponential form,

[see (40)]. Here $\Jmatvec$ is a “vector” of matrices, given by

These are the same as the matrices (22), except that we have appended a row and column of zeroes to take care of the time component. That is, these matrices have a $1+3$ block diagonal structure, where the $3\times3$ lower block is the same as in (22).

Boosts can also be represented in exponential form. First we consider an infinitesimal boost, in which $\lambda\ll 1$. Notice that when $\lambda$ is small, then $\lambda=v/c$, which gives a physical interpretation to the rapidity in this case. Specializing (34) to the case of small $\lambda$ and neglecting terms of order $\lambda^2$, we have

where $(b_1,b_2,b_3)$ are the spatial components of ${\hat{\mathbf{b}}}$. This can also be written,

where $\Kmatvec$ is another “vector” of $4\times4$ matrices, given explicitly by

(44) is the form of an infinitesimal boost, which may be compared to (32), which applies to infinitesimal rotations.

Now following the logic of Exponential Form of Finite Rotations, starting with (42), we can obtain an exponential form for boosts $\Bmat({\hat{\mathbf{b}}},\lambda)$ when $\lambda$ is not small. If $\lambda$ is not small then we can use (35) to write

where we can choose $N$ large enough to make $\lambda/N$ as small as we like. This makes it plausible that

which by the limit discussed in Exponential Form of Finite Rotations gives

This is the exponential form of boosts, which may be compared to (41) for rotations.

Infinitesimal Lorentz Transformations¶

Infinitesimal rotations were studied in 11. Rotations in 3-Dimensional Space; when the angle $\theta$ is small, then

[see (32)]. This can be regarded as the first two terms of the exponential series in (41). Similarly, (44) gives the form of infinitesimal boosts, which can be regarded as the first two terms of the exponential series in (48). Now multiplying the infinitesimal rotation (49) by the infinitesimal boost (44) and using the theorem (40), we obtain a general, proper, infinitesimal Lorentz transformation,

where we have neglected terms that are second order in the small corrections. We see that an infinitesimal Lorentz transformation is the identity plus a correction matrix that is a linear combination of the six matrices $\Jmat_i$, $\Kmat_i$, $i=1,2,3$.

These six matrices constitute the Lie algebra of the Lorentz group. They satisfy commutation relations that can be worked out directly from the definitions (42) and (45),

The first of these commutators was already given in (34), and we have seen that it is a classical version of the standard angular momentum commutation relations in quantum mechanics, as explained in 13. Representations of the Angular Momentum Operators and Rotations. The second of these equations is a classical statement of the fact that $\Kmatvec$ transforms as a vector under rotations [see (21) for the definition of a vector operator in quantum mechanics]. The third of these commutation relations shows that rotations can be generated as products of pure boosts. For example, when $\lambda$ is small, the Lorentz transformation

is a pure rotation about the $z$-axis. See the discussion in The Noncommutativity of Rotations. The fact that pure rotations can be generated by products of noncommuting pure boosts lies behind Thomas precession, an effective rotation that is experienced by accelerated frames. Thomas precession is responsible for canceling one half of the spin precession expected due to spin-orbit coupling. See the discussion in The Hamiltonian.

A Covariant Approach to Infinitesimal Lorentz Transformations¶

Infinitesimal Lorentz transformations are important because an arbitrary, proper Lorentz transformation can be built up as products of them. But the decomposition of them into rotations and boosts, as in (50), is not entirely covariant. That is, what is purely a boost in one frame is partly a boost, partly a rotation in another. We now develop a completely covariant approach to infinitesimal Lorentz transformations (that is, to the Lie algebra of the group), which will be useful to us in 48. Covariance of the Dirac Equation{} for establishing the covariance of the Dirac equation. (Note that the words “covariance” and “covariant” have two distinct meanings, as explained in Introduction.)

Let a Lorentz transformation have the form,

where $\Cmat$ is the correction matrix and $\epsilon$ is a small scale factor used to remind us that the correction is small. This Lorentz transformation is infinitesimal. Plugging this into the definition (8) of a Lorentz transformation, we have

or, through first order in $\epsilon$,

This shows that $\gmat\Cmat$ is an antisymmetric, $4\times4$ matrix. Compare this to what we did with ordinary rotations in Infinitesimal Rotations.

As in Infinitesimal Rotations, it is convenient to introduce a basis of antisymmetric matrices. The difference is that now we are dealing with $4\times4$ matrices, so the basis consists of six matrices altogether. Let $\Amat^{\mu\nu}$ be the $4\times4$ matrix with a 1 in component $\mu\nu$ and a -1 in component $\nu\mu$ (the $\Amat$ just stands for “antisymmetric”). The indices $\mu\nu$ attached to $\Amat^{\mu\nu}$ are labels of the matrix, not component indices. If we wish to specify the components of the matrix $\Amat^{\mu\nu}$, we will write $(A^{\mu\nu})_{\alpha\beta}$. According to the definition, these components are

where the two terms put a 1 in slot $(\alpha\beta)=(\mu\nu)$, and a -1 in slot $(\alpha\beta)=(\nu\mu)$. The matrices $\Amat^{\mu\nu}$ satisfy

in particular, they are antisymmetric. If we let $\mu$ and $\nu$ run over $0,1,2,3$, we get 16 matrices altogether. We can think of these matrices as constituting an antisymmetric, $4\times4$ “tensor” of antisymmetric matrices, much as $\Jmat_i$ constitutes a “vector” of matrices. Since this “tensor” is antisymmetric, that is, $\Amat^{\mu\nu}=-\Amat^{\nu\mu}$, four of these matrices vanish and the remaining twelve consist of six pairs that are equal and opposite. We see that there are only six linearly independent matrices $\Amat^{\mu\nu}$, which we get by restricting $\mu$ and $\nu$ to the range $0\le\mu<\nu\le3$.

Thus a given antisymmetric matrix, such as $\gmat\Cmat$ in (55), can be written as a linear combination of the six independent matrices $\Amat^{\mu\nu}$ with coefficients that we will call $\theta_{\mu\nu}$:

In this equation, you must remember that $\theta_{\mu\nu}$ are numbers and $\Amat^{\mu\nu}$ are matrices. Given $\gmat\Cmat$, the numbers $\theta_{\mu\nu}$ are uniquely determined for $\mu<\nu$. We now define $\theta_{\mu\nu}$ for other values of $\mu$, $\nu$ by requiring,

so that $\theta_{\mu\nu}$ can be thought of as an antisymmetric, $4\times4$ matrix or space-time tensor. Then using (57), the expansion (58) can be written,

where we drop the sum and use the summation convention, since now we have an unrestricted sum over $\mu$ and $\nu$.

Finally, we multiply (60) by $\gmat^{-1}$ and write the result as

where

Again, the indices $\mu$, $\nu$ on $\Jmat^{\mu\nu}$ are labels of the matrices, not component indices. You may compare (61), the expansion of the correction matrix for an infinitesimal Lorentz transformation in terms of a set of basis matrices, with (21), the analogous expression for pure rotations.

Combining (53) and (61), we can write the infinitesimal Lorentz transformation itself as

where we have absorbed the $\epsilon$ into the parameters $\theta_{\mu\nu}$, which, as noted, are small. Since both this expression and (50) represent general, infinitesimal Lorentz transformations, they must be equal to each other.

To establish the connection between these two expressions, we first work out the components of $\Jmat^{\mu\nu}$, which, according to (7) and (62), are the same as the components of $\Amat^{\mu\nu}$ with the first index raised. The components of $\Amat^{\mu\nu}$ are given by (56). Raising the index $\alpha$ in that equation, we obtain

If we set $(\mu\nu)=(0i)$ or $(\mu\nu)=(ij)$ in this expression and work out the components of the matrices, we find

where $\Jmat_i$ and $\Kmat_i$ are given by (42) and (45). Thus, the antisymmetric “tensor” of matrices $\Jmat^{\mu\nu}$ is equivalent to the two “vectors” of matrices $\Jmatvec$ and $\Kmatvec$, and it contains the Lie algebra of the Lorentz group. The commutation relations (51), expressed in terms of the matrices $\Jmatvec$, $\Kmatvec$, can be expressed in terms of the matrices $\Jmat^{\mu\nu}$. Alternatively, the commutation relations among the $\Jmat^{\mu\nu}$ can be worked out directly from (64). Although we will not use the result we note it for reference:

Breaking the sum on $\mu$, $\nu$ in (63) into its space and time components and using (65), we can write the infinitesimal Lorentz transformation as

Comparing this to (50), we can read off the relations between the six parameters $\theta$, ${\hat{\mathbf{n}}}$, $\lambda$ and ${\hat{\mathbf{b}}}$ of an infinitesimal Lorentz transformation, and the six independent parameters contained in $\theta_{\mu\nu}$. To do this it is convenient to define two vectors,

and

which are equivalent to

and

Then these vectors are related to the coefficients $\theta_{\mu\nu}$ by

and

If the infinitesimal Lorentz transformation is a pure rotation, then

and

If the infinitesimal Lorentz transformation is a pure boost, then

and

Active and Passive Lorentz Transformations¶

In Pure Boosts we made a few comments about how to interpret Lorentz transformations in an active sense. Here we will elaborate somewhat on this question. We mainly think in terms of boosts, since the active interpretation of rotations was covered in 11. Rotations in 3-Dimensional Space.

Let the world line of a particle be given by $x^\mu(\tau)$, where $\tau$ is the proper time. Then under an active, proper Lorentz transformation specified by matrix $\Lambda^\mu{}_\nu$, the state of motion of the particle specified by $x^\mu(\tau)$ mapped into a new state of motion $x^{\prime\,\mu}(\tau)$, given by

In Pure Boosts we considered the special case in which the original motion $x^\mu(\tau)$ was that of a particle at rest, but here $x^\mu(\tau)$ can represent any state of motion, including accelerated motion. The momentum $p^\mu$ transforms by the same matrix $\Lambda^\mu{}_\nu$ as the space-time vector $x^\mu$, since

See (49). Notice that $\tau$ is invariant under proper Lorentz transformations.

In the passive point of view which is used in most introductions to special relativity one speaks of two frames, the “lab” and “moving” frames, and one studies the relation between the space-time coordinates of the particle as seen in the two frames. To connect the present, active interpretation with the passive interpretation, first consider boosting a particle with space time coordinates $x^\mu$ in the active sense. Let the Lorentz transformation be parameterized by the velocity $\vvec_a$ of the boost, so that the space-time coordinates of the particle after the boost are

where we put an $a$ subscript on $\vvec$ to indicate that it is the velocity of the boost in the active interpretation. The meaning of $\vvec_a$ is that if we boost a particle at rest, then $\vvec_a$ is the velocity of the particle after the boost.

To obtain the passive interpretation we need a second frame. To obtain it we boost not only the particle but also the original Lorentz frame, both by the same Lorentz transformation $\Lambda (\vvec_a) ^\mu{} _\nu$. This produces a boosted frame, the velocity of which with respect to the original frame is $\vvec_a$. Also, the space-time coordinates of the particle with respect to the boosted frame are the same as the space-time coordinates of the original particle with respect to the original frame, that is, they are $x^\mu$ in (80). Now we interpret the boosted frame as the “lab” frame and the original frame as the “moving” frame. Then (80) can be written,

However, in the passive interpretation we usually parameterize the boost by the velocity of the moving frame with respect to the lab frame, which is $-\vvec_a$ in the present notation.

In summary, the active equation (80) is put into passive form by interpreting $x^{\prime\,\mu}$ as $x^\mu_moving$ and $x^\mu$ as $x^\mu_lab$, and changing the sign on the velocity parameter of the boost. You may compare our (34) with Jackson’s Eq.~(11.19) with this rule in mind.