FIELDS OF LORENTZ TRANSFORMATIONS ON SPACE-TIME

Daniel Henry Gottlieb

Abstract. Fields of Lorentz transformations on a space–time M are related to tangent bundle self isometries. In other words, a gauge transformation with respect to the − + ++ Minkowski metric on each fibre. Any such isometry L : T (M) → T (M) can be expressed, at least locally, as L = e^{F} where F : T (M) → T (M) is antisymmetric with respect to the metric. We find there is a homotopy obstruction and a differential obstruction for a global F. We completely study the structure of the singularity which is the heart of the differential obstruction and we find it is generated by “null” F which are “orthogonal” to infinitesimal rotations F with specific eigenvalues. We find that the classical electromagnetic field of a moving charged particle is naturally expressed using these ideas. The methods of this paper involve complexifying the F bundle maps which leads to a interesting algebraic situation. We use this not only to state and prove the singularity theorems, but to investigate the interaction of the “generic” and “null” F, and we obtain, as a byproduct of our calculus, a interesting basis for the 4 × 4 complex matrices, and we also observe that there are two different kinds of two dimensional complex null subspaces.

1. Introduction

Let M be a space–time and let T (M) denote its tangent bundle equipped with the Minkowski inner product ⟨ , ⟩ of type −+++. A field of Lorentz transformations refers to a vector bundle map L: T (M) → T (M) inducing the identity 1_{M}: M → M on M which preserves the inner product. We assume that L preserves orientation and the future light cones.

Now suppose F: T (M) → T (M) is a bundle map over the identity 1: M → M which is skew symmetric with respect to ⟨ , ⟩. That is ⟨F u, v⟩ = −⟨u, F v⟩. Then the exponential e^{F}: T (M) → T (M) is a field of Lorentz transformations, since e^{F} = I + F + \frac{F^{2}}{2} + … + \frac{F^{n}}{n!} + … and ⟨e^{F} u, e^{F} v⟩ = ⟨e^{−F} e^{F} u, v⟩ = ⟨e^{−F +F} u, v⟩ = ⟨u, v⟩. An interesting question, brought to the author’s attention by Chris Phillips, is: Under what circumstances is L = e^{F} for some F?

There is a homotopy obstruction. More interestingly, there is a differential obstruction. We show that the exponential map has a singular point at F if and only if e^{F} = I (excluding F = 0). The kernel of ∇e^{F} at the singular point is shown to consist of the “orthogonal” component to the two dimensional space of skew symmetric maps which commutes with F.

1991 Mathematics Subject Classification. 57R45, 17B90, 15A63. Key words and phrases. exponential map, singularity, electro-magnetism, energy-momentum,

vector bundles, Clifford Algebras.

Typeset by AMS-TEX

These singularities of the exponential map from 𝔰𝔬(3, 1) → SO(3, 1) taking F ↦ e^{F} are very provocative physically. We may think of a Lorentz transformation as being characterized by an orthonormal frame {e_{0}, e_{1}, e_{2}, e_{3}}. As this frame changes under some process, it is difficult to imagine that there is not some ‘infinitesimal” process F (t) which yields the motion of the frame by e^{F (t)}. Yet if the frame passes through its original position, there is a possibility that there is no covering infinitesimal process.

We might avoid the singularities by introducing two infinitesimal processes F_{1}(t) and F_{2}(t) so that e^{F_{1}(t)}e^{F_{2}(t)} reproduces the motion of the frame. But, alas, the map 𝔰𝔬(3, 1)×𝔰𝔬(3, 1) → SO(3, 1) also has singular points. If we seek a nice submanifold of 𝔰𝔬(3, 1) × (𝔰𝔬(3, 1) so that the map (F_{1}, F_{2}) ↦ e^{F_{1}} ∘ e^{F_{2}} induces a proper onto map, there will still be singularities, even though the manifold itself avoids the original singularities. Even if we consider n–processes, we still get singularities.

This is very closely related to the robot arm map, which takes the position of the arm and assigns it to the orientation of the end of the arm. The existence of singularities for an n–linked robot arm follows from the same homotopy argument as the existence of singularities of the Generalized Robot Arm Map which is defined

by ∏^{n} 𝔰𝔬(3, 1) → SO(3, 1): (F_{1}, …, F_{n}) ↦ e^{F_{1}} … e^{F_{n}}.

When the robot arm approaches a singularity, the parts of the arm moves faster and faster to keep the orientation along its programmed path. At last they drag on each other and the arm departs from the planned path. It could remain on the path only with infinite acceleration.

What happens when a physical process can be described by a moving 4–frame {e_{0}, e_{1}, e_{2}, e_{3}}? Does it avoid the singular points somehow; or does it pass through them, in which case, how does it react to the “infinite accelerations” physically? If I could be allowed to speculate, I would say on the basis of the mathematical structure of the singularities, that their existence might give an argument of why spinning objects tend to remain spinning around a fixed or precessing axis, and when this does not obtain, there is an emission or absorption of radiation.

I base my speculation on the following observations. Any F ∈ 𝔰𝔬(3, 1), which is a skew symmetric linear transformation F: ℝ^{3,1} → ℝ^{3,1}, can be classified into three classes, according to its eigenvectors. F has 1, 2 or infinitely many null eigenvector spaces (where a null vector s means ⟨s, s⟩ = 0). The case of infinitely many eigenvectors only occurs for the trivial case, F = 0. The case of one null eigenvector subspace corresponds to the null transformations, which the physicists say corresponds to the radiative case in electro–magnetism. The generic case has two null eigenvector spaces.

The singular F are infinitesimal rotations around an axis a multiple of 2π times for some observer. The skew symmetric operators which are “orthogonal” to that F consists of linear combinations of two null operators, one of whose eigenvectors agrees with one of the two eigenvector directions of F, and the other null operator’s eigenvector points along the opposite eigenvector of F. Now F plus one of these null operators is still a rotation through an angle 2πn about an axis, but its observer is different.

Thus we see that a spinning process must maintain the same eigenvectors, hence spin axis, every 2π rotation. And the “directions” having no effect on the spin are given by two null operators.

This paper depends on [Gottlieb (1998)] which may be found on my home page. We keep the same notation. In §2 we will review this notation, but from a generalized complex point of view in which the main features will be stated more quickly. In §3, as an application of this notation, we will produce a basis for the complex 4 × 4 matrices M_{4}(ℂ) which are Hermitian, clearly respect the decomposition of 𝔰𝔬(3, 1) ⊗ ℂ into two complex quaternionic subalgebras, and in which each of the 16 matrices has a square equal to the identity. This basis easily gives a set of multiplicative generators for the Clifford algebra Cl(4) ≃ M_{4}(ℂ).

In §4 we will prove the singularity theorem for the exponential map 𝔰𝔬(3, 1) → SO(3, 1). We see that the singular points occur at those F such that e^{F} = I and F ≠ 0 and the image of the induced differential on the tangent spaces is in a “direction’ parallel to F. In §5 we study adjoint mappings, finding several equations relating generic F to their two orthogonal null F. In §6 we discuss the obstructions to representing a Lorentz field by L = e^{F}. In §7, we show that the classical electromagnetic field of a moving charged particle has a very nice description in terms of conjugation by certain exponentials of null F.

I would like to thank the organizers of the conference on The Theory of Fixed Points and its Applications for inviting me to speak last in the cleanup position. From that vantage point I was able to connect some of the material here to concepts arising from various talks which preceded mine. My talk was necessarily philosophical, whereas this paper is completely rigorous, so I welcomed the opportunity to actually quote a precise result or two. In the paragraphs below, I will repeat those connections of this material to fixed point theory.

Lorentz transformations are strongly influenced by their eigenvectors, and of course eigenvectors are particularly useful types of fixed points. In one talk, I believe it was Phil Heath’s, an example of a transformation on a two-sphere with only one fixed point was mentioned. A interesting example of those can be defined by a Lorentz transformation with only one eigenvector. Such transformations can be associated to radiation, as mentioned above. The Lorentz transformation takes any future pointing null ray into another future pointing null ray. The future pointing null rays can be characterized as the points on a two-sphere, that is as the direction of a light ray from a given observer.

Robert Brown mentioned a generalization of the Brouwer degree of a map from oriented to unoriented cases. I defined another generalization which works for any continuous map between any topological spaces. I pointed out that I had predicted in print that this generalized concept should appear widely in mathematics, and I mentioned that I had in fact found that this degree determined whether or not a given field of Lorentz transformations had a vector field of null eigenvectors. This can be seen in 7.1 to 7.8 in [Gottlieb (1998)].

2. Notation

We will follow the notation and conventions of [Gottlieb (1998)] as closely as possible. But we shall outline the theory from a different point of view. In [Gottlieb (1998)], we started with real bundle maps on the tangent space of a space–time M whose tangent bundle was furnished with a Minkowskian metric ⟨ , ⟩. There we proceeded as geometrically as possible, defining dot products and cross products with their geometric meanings for example. One main reason for this was to try to understand the role of choices of orientation. As we proceeded, we added more layers of notation, creating at least three different calculi: Level −2 without choice of basis, Level −10 with matrices, and Level −16, the usual (t, x, y, z) of Minkowski space. We found, actually against our wishes, that complexifying was a powerful aid to calculation and understanding.

In this paper, we will begin with complexified Minkowski space ℝ^{3,1} ⊗ ℂ, mostly use Level −10 notation and recover the body of [Gottlieb (1998)] by specializing to the real case and extending to fibre bundles.

Let ℝ^{3,1} be ℝ^{4} with an inner product ⟨ , ⟩ of type − + ++. The space of linear maps Hom(ℝ^{3,1}, ℝ^{3,1}) is a 16–dimensional real vector space. Let the 6–dimensional subspace of linear maps F which are skew–symmetric with respect to ⟨ , ⟩ be denoted as 𝔰𝔬(3, 1). Skew symmetric means ⟨F u, v⟩ = −⟨u, F v⟩. We say that u ∈ ℝ^{3,1} is an observer if ⟨u, u⟩ = −1 and u is future pointing.

We wish to study ℝ^{3,1} ⊗ ℂ. We define the inner product ⟨ , ⟩_{ℂ} on ℝ^{3,1} ⊗ ℂ by letting ⟨u, iw⟩_{ℂ} = ⟨iv, w⟩_{ℂ} = i⟨v, w⟩ where v, w ∈ ℝ^{3,1}. This extends linearly to an inner product ⟨ , ⟩_{ℂ} on ℝ^{3,1} ⊗ ℂ. We will usually suppress the ℂ and write ⟨ , ⟩.

Remark 2.1. The usual choice of inner product on ℝ^{3,1} ⊗ ℂ ≅ ℂ^{4} is the Hermitian inner product ⟨⟨ , ⟩⟩ which can be defined in terms of ⟨ , ⟩_{ℂ} by ⟨⟨v, w⟩⟩ = ⟨v, \overline{w}⟩_{ℂ}. The advantage of ⟨⟨ , ⟩⟩ is that it is positive definite on the complex rest space of an observer, there are no null vectors (s ≠ 0, so that ⟨⟨s, s⟩⟩ = 0). The advantage of ⟨ , ⟩_{ℂ} is that it is linear and there are null vectors! In ℝ^{3,1} ⊗ ℂ, the subspaces of null vectors are two–dimensional and they come in two distinct types. In ℝ^{3,1}, the null spaces are one–dimensional.

Scholium 2.2. In the real case the null spaces are one–dimensional. The propagation of light is along these one–dimensional null spaces. One wonders what the physical meaning of the two types of null spaces is in the complex case? After all, every one who wants to study the real physical world is forced into complexification. God would be malicious if these null spaces had no meaning! I have a suggestion.The two different subspaces correspond to the two different helicities that a photon can have. In other words, one subspace corresponds to a photon moving in the same direction as its Poynting vector, and the other corresponds to a photon moving in the opposite direction as its Poynting vector.

The subspace of Hom(ℂ^{4}) consisting of F which are skew symmetric with respect to ⟨ , ⟩_{ℂ} is denoted 𝔰𝔬(3, 1) ⊗ ℂ. Thus ⟨F v, w⟩ = −⟨v, F w⟩ where v, w ∈ ℂ^{4}.

The above style of notation comes from Level −2 of [Gottlieb (1998)]. To go to Level −10, we must choose an orthonormal basis {e_{0}, e_{1}, e_{2}, e_{3}} of ℝ^{3,1} so that ⟨e_{0}, e_{0}⟩ = −1 and ⟨e_{i}, e_{j}⟩ = δ_{ij} otherwise. These naturally give rise to an orthonormal basis on ℝ^{3,1} ⊗ ℂ ≅ ℂ^{4}.

This orthonormal basis allows us to write linear transformations F: ℂ^{4} → ℂ^{4} as 4 × 4 matrices, so Hom(ℂ^{4}) →^{≅} M_{4}(ℂ), the space of 4 × 4 complex matrices. We see

FIELDS OF LORENTZ TRANSFORMATIONS ON SPACE-TIME 5

that 𝔰𝔬(3, 1) ⊗ ℂ is isomorphic to the vector space consisting of all the matrices of the form

( \begin{matrix}0 & E_{1} & E_{2} & E_{3} \\ E_{1} & 0 & B_{3} & −B_{2} \\ E_{2} & −B_{3} & 0 & B_{1} \\ E_{3} & B_{2} & −B_{1} & 0 \end{matrix} )

We find it convenient to use the block matrix notation

( \begin{matrix}0 & E⃗^{T} \\ E⃗ & ×B \end{matrix} ) where E⃗ = ( \begin{matrix}E_{1} \\ E_{2} \\ E_{3} \end{matrix} ) and × B⃗ = ( \begin{matrix}0 & B_{3} & −B_{2} \\ −B_{3} & 0 & B_{1} \\ B_{2} & −B_{1} & 0 \end{matrix} ) .

Note that (×B)v⃗ = v⃗ × B⃗ if we assume that e⃗_{1} × e⃗_{2} = e⃗_{3}.

Let us call the space of these matrices S for skew symmetric. Note they are not skew symmetric in the sense of matrices.

Now we introduce the ∗ operation on S. If

F = ( \begin{matrix}0 & E^{T} \\ E & ×B \end{matrix} ) , then F^{∗} = ( \begin{matrix}0 & −B^{T} \\ −B & ×E \end{matrix} ) .

This ∗ gives rise to a mapping F → F^{∗} on S to itself which is a linear transformation and F^{∗∗} = −F, so the star mapping composed with itself is −I where I: S → S stands for the identity.

The ∗ operation is really a Level −2 concept. It arises from the Hodge dual on two–forms and depends on a choice of a volume form for ℝ^{3,1}. Thus on level −2 we can define E⃗ and B⃗ for each observer u. Namely E⃗_{u} = F u and B⃗_{u} = −F^{∗}u where E⃗_{u} and B⃗_{u} are in the rest space T^{u} of u, that is E⃗_{u} and B⃗_{u} are orthogonal to u. So E⃗ and B⃗ are freed of depending on the whole basis, just the observer.

Now define linear maps c: S → S and \overline{c}: S → S given by cF: = F − iF^{∗} and \overline{c}F: = F + iF^{∗}.

Theorem 2.3. a) c and \overline{c} are linear maps 𝔰𝔬(3, 1) ⊗ ℂ → 𝔰𝔬(3, 1) ⊗ ℂ

b) c + \overline{c} = 2I

c) c ∘ \overline{c} = \overline{c} ∘ c = 0

d) c ∘ c = 2c, \overline{c} ∘ \overline{c} = 2\overline{c}

Proof.

a)  c(F + G) = (F + G) − i(F + G)^{∗}
= F − iF^{∗} + G − iG^{∗} = cF + cG

and c(aF) = aF − i(aF)^{∗} = aF − i(aF^{∗}) = a(F − iF^{∗}) = acF. Similarly for \overline{c}.

b) (c + \overline{c})(F) = F − iF^{∗} + F + iF^{∗} = 2F.

c)  c ∘ \overline{c}(F) = c(F + iF^{∗}) = cF + icF^{∗}
= F − iF^{∗} + i(F^{∗} − iF^{∗∗})
= F − iF^{∗} + iF^{∗} + F^{∗∗} = F + F^{∗∗} = 0.

Similarly for \overline{c} ∘ c = 0.

d)  c ∘ c(F) = c(F − iF^{∗}) = cF − icF^{∗} = cF − i(F^{∗} − iF^{∗∗})
= F − iF^{∗} − iF^{∗} − F^{∗∗} = 2F − 2iF^{∗} = 2cF.

Similarly for \overline{c} ∘ \overline{c}. □

The above result gives a decomposition of S = 𝔰𝔬(3, 1) ⊗ ℂ into the direct sum of two subspaces, cS ⊕ \overline{c}S. In Level −10 notation, c ( \begin{matrix}0 & E \\ E & ×B \end{matrix} ) = ( \begin{matrix}0 & A \\ A & −i(×A) \end{matrix} ) where A⃗ = E⃗ + iB⃗, and \overline{c} results in ( \begin{matrix}0 & A⃗ \\ A⃗ & i(×A⃗) \end{matrix} ) where A⃗ = E⃗ − iB⃗. Note that cS is the +i eigenspace of ∗: S ⊗ ℂ → S ⊗ ℂ and \overline{c}S is the −i eigenspace of ∗. For

( \begin{matrix}0 & A⃗ \\ A⃗ & ∓(i × A) \end{matrix} ) ↦^{∗} ( \begin{matrix}0 & ±iA⃗ \\ ±iA⃗ & ×A \end{matrix} ) = ±i ( \begin{matrix}0 & A \\ A & ∓i(×A) \end{matrix} ).

Now c and \overline{c} restricted to 𝔰𝔬(3, 1) are bijections c: 𝔰𝔬(3, 1) ⟶^{≅} cS and \overline{c}: 𝔰𝔬(3, 1) → \overline{c}S. So we have two different embedding of 𝔰𝔬(3, 1) into 𝔰𝔬(3, 1) ⊗ ℂ.

Now the beautiful algebra found in [Gottlieb (1998)] follows from the following two facts. Theorem 2.4. a) Any element of cS commutes with any element of \overline{c}S.

b) F ∈ S satisfies F^{2} = kI if and only if F is in either cS or \overline{c}S. Proof. Part a) is Theorem 4.8 of [Gottlieb (1998)] and part b) is Theorem 4.5 of the same paper. Corollary 2.5. a) cF \overline{c}G = \overline{c}GcF for F, G ∈ S.

b) cF cG + cGcF = kI for some k ∈ ℂ.

c) \overline{c}F \overline{c}G + \overline{c}G\overline{c}F = ℓI for some ℓ ∈ bC. Proof. Theorem 4.8 and Theorem 4.7 of [Gottlieb (1998)]. Remark 2.6. The c: 𝔰𝔬(3, 1) → S notation was used in [Gottlieb (1998)]. Now if F is real, then \overline{c}F = \overline{cF} and for eigenvalues, λ_{\overline{c}F} = \overline{λ}_{cF}. This is not true in general. This point should be understood in extending the results of §4, §5, §6 to 𝔰𝔬(3, 1)⊗ℂ. The complex conjugate should be restricted to \overline{c}. Then most of those results should generalize.

FIELDS OF LORENTZ TRANSFORMATIONS ON SPACE-TIME 7

Corollary 2.7. a) cF cG + cGcF = 2(A⃗_{1} · A⃗_{2})I where cF u = A⃗_{1} and cGu = A⃗_{2}.

We define ⟨cF, cG⟩: = ⟨cF u, cGu⟩_{ℂ}. This number is independent of the choice of the observer. This is an inner product on 𝔰𝔬(3, 1) ⊗ ℂ.

b) \overline{c}F \overline{c}G + \overline{c}G\overline{c}F = 2(A⃗_{3} · A⃗_{4})I where A⃗_{3} = \overline{c}F u and A⃗_{4} = \overline{c}Gu.

This defines another inner product ⟨F, G⟩_{−}: = ⟨\overline{c}F u, \overline{c}Gu⟩ on 𝔰𝔬(3, 1) ⊗ ℂ.

We now see from [Gottlieb (1998)] Theorem 4.3 and Corollary 4.4, that c[F_{1}, F_{2}]u = i(cF_{1}u) × (cF_{2}u) and \overline{c}[F_{1}, F_{2}]u = −i(\overline{c}F_{1}u) × (\overline{c}F_{2}u) and [cF_{1}, cF_{2}]u = 2i(cF_{1}u) × (cF_{2}u) with [\overline{c}F_{1}, \overline{c}F_{2}]u = −2i(\overline{c}F_{1}u) × (\overline{c}F_{2}u) .

In general, any F ∈ 𝔰𝔬(3, 1) ⊗ ℂ can be written F = \frac{1}{2}cF + \frac{1}{2} \overline{c}F.

So [F, G]u = \frac{1}{4}[(cF + \overline{c}F), (cG + \overline{c}G)]u

= \frac{1}{4}\{[cF, cG] + [\overline{c}F, \overline{c}G]\}u = \frac{1}{4}\{2icF u × cGu + 2(−i)\overline{c}F u × \overline{c}Gu\}

= \frac{i}{2}\{cF u × cGu − \overline{c}F u × \overline{c}Gu\}.

This proves:

Theorem 2.8. [F, G]u = \frac{i}{2}\{cF u × cGu − \overline{c}F u × \overline{c}Gu\}.

3. An Interesting Basis

We will use our notation to produce an interesting basis for M_{4}(ℂ), the vector space of complex 4 × 4 matrices.

We choose an orthogonal coordinate system (t, x, y, z) for Minkowski space ℝ^{3,1}. Thus we are in Level −16 notation of [Gottlieb (1998)]. Let e_{t}, e⃗_{x}, e⃗_{y}, e⃗_{z} be the unit vectors (with respect to the Minkowski metric). We express e⃗_{x} = (1, 0, 0)^{T} , e⃗_{y} = (0, 1, 0)^{T} , e⃗_{z} = (0, 0, 1)^{T}.

Let

E_{x}: = ( \begin{matrix}0 & e_{x}^{T} \\ e⃗_{x} & 0 \end{matrix} )

and define E_{y} and E_{z} by replacing e⃗_{x} by e⃗_{y} and e⃗_{z} respectively.

Similarly we define

B_{x}: = ( \begin{matrix}0 & 0^{T} \\ 0 & ×(e_{x}) \end{matrix} )

with B_{y} and B_{z} defined by replacing e⃗_{x} by e⃗_{y} and e⃗_{z} respectively.

Lemma 3.1. {E_{x}, E_{y}, E_{z}, B_{x}, B_{y}, B_{z}} form a basis for 𝔰𝔬(3, 1) as a vector space.

Now consider cE_{x}, cE_{y}, cE_{z}, and \overline{c}E_{x}, \overline{c}E_{y}, \overline{c}E_{z}. Note that \overline{c}E_{x} = \overline{cE_{x}} since E_{x} is real. Similarly for y and z.

Lemma 3.2. {cE_{x}, cE_{y}, cE_{z}} forms a basis of the complex vector space c(𝔰𝔬(3, 1)) in 𝔰𝔬(3, 1) ⊗ ℂ. Similarly \{\overline{cE_{x}}, \overline{cE_{y}}, \overline{cE_{z}}\} forms a basis for the image of \overline{c}.

Theorem 3.3. The set of sixteen matrices

a)  \begin{matrix}I, & cE_{x}c\overline{E_{x}}, & cE_{y}\overline{cE_{y}}, & cE_{z}\overline{cE_{z}}, \\ cE_{x}, & \overline{cE_{x}}, & cE_{y}\overline{cE_{z}}, & cE_{z}\overline{cE_{y}}, \\ cE_{y}, & \overline{cE_{y}}, & cE_{x}\overline{cE_{z}}, & cE_{z}\overline{cE_{x}}, \\ cE_{z}, & \overline{cE_{z}}, & cE_{x}\overline{cE_{y}}, & cE_{y}\overline{cE_{x}} \end{matrix}

forms a basis for M_{4}(ℂ), the vector space of 4 × 4 complex matrices.

b) The square of each of the matrices in the basis is I.

c) Each matrix is Hermitian.

Proof. a) Let e_{ij} represent the 4 × 4 matrix consisting of a 1 in the i, j position and zeros everywhere else. Then the sixteen e_{ij} form a basis for M_{4}(ℂ). Express the 16 matrices in a) as linear combinations of the e_{ij} using the order given in a). Then we must show that the coefficient matrix must have linearly independent rows. Although the matrix is 16 × 16 with 256 entries, most of the entries are zero. In fact, the only rows which can possibly be linearly dependent form four sets. Thus the first four rows are obviously independent of the remaining rows and similarly for the next four rows, and so on. It is easy to show each of the four sets of four rows is linearly independent.

b) Recall that cF^{2} = (A⃗·A⃗)I where A⃗ = cF e_{t} = E⃗+iB⃗. So cE_{x}cE_{x} = (e⃗_{x}·e⃗_{x})I = I. Similarly for cE_{y}, cE_{z} and c\overline{E}_{x}, c\overline{E}_{y}, c\overline{E}_{z}. Next recall that cF \overline{c}G = \overline{c}GcF, so (c\overline{E}_{x}c\overline{E}_{y})^{2} = cE_{x}^{2}c\overline{E}_{y}^{2} = I · I = I. Similar arguments will work for the remaining matrices.

c) Note that the transpose of ( \begin{matrix}0 & A⃗^{T} \\ A⃗ & ∓i(×A⃗) \end{matrix} ) is ( \begin{matrix}0 & A⃗^{T} \\ A⃗ & ±i(×A⃗) \end{matrix} ). If A⃗ is

a real vector, then the transpose equals the complex conjugate. Hence

( \begin{matrix}0 & A⃗^{T} \\ A⃗ & ±i(×A⃗) \end{matrix} ) is Hermitian if A⃗ is real.

Thus cE_{x}, cE_{y}, cE_{z}, \overline{cE_{x}}, \overline{cE_{y}}, \overline{cE_{z}} are all Hermitian, so is I. Now cE_{x}\overline{cE_{y}} is Hermitian since

[(cE_{x})(\overline{cE_{y}})]^{†} = [(\overline{cE_{x}})(cE_{y})]^{T} = (cE_{y})^{T} (\overline{cE_{x}})^{T}
= (\overline{cE_{y}})(cE_{x}) = cE_{x}\overline{cE_{y}}.

The remaining matrices are Hermitian by similar arguments. □

We can understand the multiplication properties of these matrices by the following theorem.

Theorem 3.4. a) cE_{x}cE_{y} = icE_{z} = cB_{z}

b) \overline{cE_{x}} \overline{cE_{y}} = −i\overline{cE_{z}} c) cE_{x}cE_{y} = −cE_{y}cE_{x} and \overline{cE_{x}} \overline{cE_{y}} = −\overline{cE_{y}} \overline{cE_{x}}

d) cF \overline{cG} = \overline{cG}cF

Proof. c) cE_{x}cE_{y} + cE_{y}cE_{x} = 2(e⃗_{x} · e⃗_{y})I = 0 and apply complex conjugate to this equation.

a) We know that [cE_{x}, cE_{y}]e_{t} = 2ie⃗_{x} × e⃗_{y} = 2ie⃗_{z} = 2icE_{z}. But since cE_{x}cE_{y} = −cE_{y}cE_{x} we have [cE_{x}, cE_{y}] = 2cE_{x}cE_{y}. So (cE_{x}cE_{y})e_{t} = icE_{z}e_{t}. Now cF e_{t} completely determines cF. So cE_{x}cE_{y} = cE_{z}.

b) Same argument or take complex conjugate of a). □

Thus we see that cE_{x}, cE_{y}, \overline{cE_{x}}, \overline{cE_{y}} generates multiplicitivity M_{4}(ℂ), and cE_{x} and cE_{y} generates the Pauli Algebra, that is, the quaternions complexified.

Now since M_{4}(ℂ) is the complex Clifford algebra Cℓ(4), there must be generators α_{0}, α_{1}, α_{2}, α_{3} so that α_{i}α_{j} + α_{j}α_{i} = δ_{ij}I. One such set of α’s is given by

α_{0} = cE_{x}
α_{1} = cE_{y}
α_{2} = cE_{z}\overline{cE_{x}}
α_{3} = cE_{z}\overline{cE_{y}}.

Of course, to obtain γ_{i} such that γ_{i}γ_{j} +γ_{j}γ_{i} = 2⟨e_{i}, e_{j}⟩I we let γ_{0} = iα_{0}, γ_{i} = α_{i} for i = 1, 2, 3 or

γ_{0} = icE_{x}
γ_{1} = cE_{y}
γ_{2} = cE_{z}\overline{cE_{x}}
γ_{3} = cE_{z}\overline{cE_{y}}.

4. The singularity theorem

Let exp: 𝔰𝔬(3, 1) → SO(3, 1) be the exponential map F ↦ e^{F}: = I + F + \frac{F^{2}}{2!} + \frac{F^{3}}{3!} + . . .. Our goal is to classify the singularities of exp.

Singularity Theorem. The only non–regular point of exp is the identity I ∈ SO(3, 1). Any F ≠ 0 ∈ 𝔰𝔬(3, 1) satisfies e^{F} = I if and only if λ_{cF} = 2πni where n is a non zero integer. The kernel of exp_{∗}: G ↦ ∇_{G}e^{F}: = \frac{d}{dt}(e^{F +tG})|_{t=0} consists of those G ∈ 𝔰𝔬(3, 1) such that ⟨cF, cG⟩ = 0. That is ∇_{G}e^{F} = 0 if and only if λ_{cF} = 2πni ≠ 0 and ⟨cF, cG⟩ = 0.

A general theorem for the differential of the exponential map is the following result due to Helgason, see [Helgason (1978)], page 105, Theorem 1.7. This will help simplify our proof of the singularity theorem.

Theorem (Helgason). Let 𝔤 be the Lie Algebra for a Lie group G. Then ∇_{G}e^{F}: = \frac{d}{dt}(e^{F +tG})|_{t=0} = e^{F} · (g[adF](G)) where adF: 𝔤 → 𝔤 sends X ⟼ [F, X] and g[ξ] is

the power series of the function g(ξ) = \frac{e^{−ξ} − 1}{−ξ}.

Corollary 4.1. ∇e^{F}: G ↦ ∇_{G}e^{F} is singular if and only there is a non zero integer n so that 2πin is an eigenvalue of ad(F): 𝔤 → 𝔤.

Proof. g(ξ) = 0 occurs only when ξ ∈ 2πi(ℤ − {0}).

Lemma 4.2. 2c ∘ ad(F) = ad(cF) ∘ c.

Proof. We must show that 2c ∘ ad(F)(G) = ad(cF)(cG) for all G ∈ 𝔰𝔬(3, 1). That is that 2c[F, G] = [cF, cG]. But this last is just Theorem 4.3 of [Gottlieb (1998)].

Lemma 4.3. exp: 𝔰𝔬(3, 1) → SO(3, 1) is singular at F exactly when exp: S →

M_{4}(ℂ) is singular at \frac{1}{2}cF.

Proof. Suppose exp: 𝔰𝔬(3, 1) → SO(3, 1) is singular at F. Then ad(F): 𝔰𝔬(3, 1) → 𝔰𝔬(3, 1) has an eigenvalue 2πni ≠ 0. Thus there is an eigenvector cG ∈ 𝔰𝔬(3, 1) ⊗ ℂ so that ad(F)(cG) = 2πnicG. That is [F, cG] = 2πnicG. Hence 2πnicG = [F, cG] = [\frac{cF}{2} + \frac{\overline{cF}}{2} , cG] = [\frac{cF}{2} , cG]. Hence ad(\frac{cF}{2} ): S → S has the same eigenvalue 2πni and hence ∇e^{\frac{cF}{2}} is singular.

Now e^{cF} = cosh(λ_{cF})I + \frac{sinh(λ_{cF})}{λ_{cF}} cF according to Theorem 8.5 [Gottlieb (1998)].

Also e^{F} = e^{\frac{cF}{2}} e^{\frac{\overline{cF}}{2}} . So exp: F ↦ \frac{cF}{2} ↦ e^{\frac{cF}{2}} ↦ e^{\frac{\overline{cF}}{2}} e^{\frac{cF}{2}} = e^{F} is a composition of first an injective map, then an exponential map, and finally an “absolute value” map. Hence the differential ∇e^{F} factors through the differential ∇e^{cF/2}. Hence if cG is in the kernel of ∇e^{cF/2}, then G must be in the kernel of ∇e^{F}. That is ∇_{cG}e^{cF/2} = 0 implies ∇_{G}e^{F} = 0.

Now we shall show that the kernel of ∇e^{cF/2} will be 2 complex dimensional in a 3 dimensional complex space. Hence ∇e^{F} has a kernel of 4 real dimensions. But the subspace of all G ∈ 𝔰𝔬(3, 1) which commutes with F is two dimensional and also it cannot be in the kernel of ∇e^{F} by the following lemma. Hence the kernel of ∇e^{F} must be 4 dimensional and so ∇_{G}e^{F} = 0 if and only if ∇_{cG}e^{cF/2} = 0. □

Lemma 4.4. a) If F commutes with G ≠ 0 (in any Lie Algebra) then ∇_{G}e^{F} ≠ 0.

b) The two dimensional space generated by F and F^{∗} commutes with F.

Proof. a) Since F commutes with G we have e^{F +tG} = e^{F} e^{tG}. Thus

∇_{G}e^{F} = \frac{d}{dt}e^{F +tG}|_{t=0} = \frac{d}{dt}e^{F} e^{tG}|_{t=0} = e^{F} \frac{d}{dt}e^{tG}|_{t=0}
= e^{F} G ≠ 0 since e^{F} is an isomorphism.

b) F^{∗} commutes with F by Theorem 3.2 of [Gottlieb (1998)]. □

Thus we will have proved the singularity theorem when we prove the following complex singularity theorem.

Theorem 4.5. a) e^{cF} = I if and only if 2nπi = λ_{cF} and e^{cF} = −I if and only if (2n + 1)πi = λ_{cF} for any integer n.

b) If [cF, cG] = 0, then ∇_{cG}e^{cF} ≠ 0.

c) Assume cF ≠ 0 and λ_{cF} = nπi. Then ∇_{cG}e^{cF} = 0 if and only if ⟨cF, cG⟩ = 0.

Proof. a) From Theorem 8.5 of [Gottlieb (1998)] we know that e^{cF} = cosh(λ_{cF})I +

\frac{sinh(λ_{cF})}{λ_{cF}} cF. Now e^{cF} = I if and only if cosh(λ_{cF}) = 1 and \frac{sinh(λ_{cF})}{λ_{cF}} = 0. If

λ_{cF} = 0, then e^{cF} = I + cF, so we can exclude that case. Hence, sinh λ_{cF} = 0 if and only if λ_{F} = nπi. And then cosh(nπi) = cos(nπ) = (−1)^{n}.

b) This is Lemma 4.4 a).

c) Let A_{t}: = cF + tcG and let λ_{t}: = λ_{A_{t}} be the eigenvalue of A_{t}.

Then we want to show that \frac{d}{dt} λ_{t}|_{t=0} = \frac{⟨cF, cG⟩}{λ_{cF}}.

We know A_{t}^{2} = λ_{t}^{2}I = (λ_{cF}^{2} +2t⟨cF, cG⟩+t^{2}λ_{cG}^{2})I. So \frac{d}{dt}λ_{t}^{2} = 2⟨cF, cG⟩+2tλ_{cG}^{2}.

Now formally, \frac{d}{dt} λ_{t}^{2} = 2λ_{t} \frac{dλ_{t}}{dt}, but caution is needed here, since although λ_{t}^{2} is a well–defined function of t, note that λ_{t} has an ambiguity of sign and a consistent choice of sign must be made in order to make λ_{t} a well–defined function of t. Now if λ_{t} runs over a closed loop in ℂ around 0, it may not be possible to define λ_{t} globally. This is no problem in our present argument, but anyone who wants to apply this formula should be careful.

So assuming λ_{t} is made well–defined, we see that

\frac{d}{dt}λ_{t}^{2}|_{t=0} = 2λ_{0} \frac{d}{dt}λ_{t}|_{t=0} = 2⟨cF, cG⟩.

Hence

(*)  \frac{dλ_{t}}{dt} |_{t=0} = \frac{⟨cF, cG⟩}{λ_{cF}}.

Now carrying out the differention and using (*) we see that

∇_{cG}e^{cF} = \frac{d}{dt}(cosh λ_{t}I + \frac{sinh λ_{t}}{λ_{t}} (cF + tcG))|_{t=0}
= αI + βcF + γcG

where

α = sinh(λ_{cF})\frac{⟨cF, cG⟩}{λ_{cF}}
β = \frac{⟨cF, cG⟩}{λ_{cF}^{3}}(cosh(λ_{cF})λ_{cF} − sinh(λ_{cF}))
γ = \frac{sinh λ_{cF}}{λ_{cF}} .

So ∇_{cG}e^{cF} = 0 if and only if α = β = γ = 0. This follows since cF and cG and I are linearly independent since otherwise they would commute Now γ = 0 if and only if λ_{cF} = nπi ≠ 0 and then β = 0 if and only if ⟨cF, cG⟩ = 0. □

Corollary 4.6. ∇_{cG}e^{cF} =

= ⟨cF, cG⟩ \{(\frac{sinh(λ_{cF})}{λ_{cF}}) I + (\frac{cosh(λ_{cF})}{λ_{cF}^{2}} − \frac{sinh(λ_{cF})}{λ_{cF}^{3}}) cF \}

+ \frac{sinh(λ_{cF})}{λ_{cF}} cG if λ_{cF} ≠ 0; and

∇_{cG}e^{cF} = ⟨cF, cG⟩(I + \frac{1}{3}cF) + cG if λ_{cF} = 0.

Proof. The first equation comes from the above proof. The second equation is the limit of the first equation as cF is chosen so that λ_{cF} → 0. L’Hopital’s rule is applied to the coefficients.

Corollary 4.7. \frac{d}{dt}λ_{cF +tcG}^{2}|_{t=0} = 0 if and only if ⟨cF, cG⟩ = 0. \frac{d}{dt}λ_{cF +tcG}^{2} = 0 if

and only if ⟨cF, cG⟩ = 0 and cG is null.

5. Exponential equations

Lemma 5.1. Suppose F and G ∈ 𝔰𝔬(3, 1) share a null eigenvector s. Then ⟨cF, cG⟩ = λ_{cF} λ_{cG}. In addition, one of cG or cF is null if and only if cF and cG anticommute.

Proof. (cF cG + cGcF)s = 2⟨cF, cG⟩s , hence

2λ_{cF} λ_{cG} = 2⟨cF, cG⟩
λ_{cF} λ_{cG} = ⟨cF, cG⟩.

Now if cF is null, then λ_{cF} = 0 so ⟨cF, cG⟩ = 0. So cF and cG anticommute. This argument is reversible. □

Theorem 5.2. Suppose F ∈ 𝔰𝔬(3, 1) and suppose N is null and shares a null eigenvector s corresponding to λ_{cF}. a) Then [cF, cN] = 2λ_{cF} cN. b) If N^{†} is another null operator which shares the other null eigenvector of a generic cF, then

[cN, cN^{†}] = \frac{2⟨cN, cN^{†}⟩}{λ_{cF}} cF.

Proof. a) In the case where cF is generic, consider \frac{1}{λ_{cF}} cF. This has eigenvalues 1 and −1, and so \frac{1}{λ_{cF}} cF u is a unit vector. There is a choice of observer u for which this unit vector is a real vector k⃗ and that u + k⃗ is parallel to the real eigenvector s.

Now since s is an eigenvector of cN also, we have that i⃗×j⃗ = k⃗ where i⃗ and j⃗ are parallel to E⃗ and B⃗ for cN. That is, i⃗ and j⃗ are unit vectors and cNu = E(i⃗ + ij⃗) for some constant E. Now k⃗ × (i⃗ + ij⃗) = −i(i⃗ + ij⃗). So cF u × cNu = −iλ_{cF} cNu when we restore the constants to the above equation. Now [cF, cN]u = 2i(cF u × cNu) = 2i(−iλ_{cF} cNu) = 2λ_{cF} cNu by Theorem 4.3 and Corollary 4.4 of [Gottlieb (1998)]. Since the two operators agree on an observer u, they must be equal, so [cF, cN] = 2λ_{cN}cN.

This relation remains true when cF is null since two null operators are orthogonal if and only if they commute, and λ_{cF} = 0.

b) Let A and B represent two null elements in cS with null eigenvectors s_{+} and s_{−} respectively. (In order to simplify the equations, we have dropped the c from cA and cB). Let F be a generic element in cS with F s_{+} = λ_{F} s_{+} and F s_{−} = −λ_{F} s_{−}.

Now AB +BA = 2⟨A, B⟩I. Apply this equation to s_{+} and s_{−}, making use of the

facts that As_{+} = 0 and Bs_{−} = 0 and \frac{F s_{±}}{λ_{F}} = ±s_{±}. This gives along with Theorem

5.2 a)

[A, B]s_{+} = 2⟨A, B⟩\frac{F s_{+}}{λ_{F}}

and

[A, B]s_{−} = 2⟨A, B⟩\frac{F s_{−}}{λ_{F}} .

Adding these two equations gives

[A, B](s_{+} + s_{−}) = 2⟨A, B⟩\frac{F}{λ_{F}} (s_{+} + s_{−}).

Since s_{+} + s_{−} is a time–like vector, this implies that

[A, B] = 2⟨A, B⟩\frac{F}{λ_{F}} ,

since if two skew operators agree on the same observer they must be the same operator (see Theorem 6.9 of [Gottlieb (1998)] for a vast generalization of this last reason), which is what was desired. □

Corollary 5.3. cF cN = λ_{F} cN if cN is null and ⟨cF, cG⟩ = 0.

Lemma 5.4. cGcF cG = 2⟨cF, cG⟩cG − λ_{cG}^{2}cF.

Proof. Multiply cF cG + cGcF = 2⟨cF, cG⟩I on the right by cG. □

We will drop the c in cF for ease of notation and say instead that F ∈ cS.

Lemma 5.5. Let F, G ∈ cS. Then F G = ⟨F, G⟩I + \frac{1}{2}[F, G].

Proof. F G = \frac{1}{2}\{F, G\} + \frac{1}{2}[F, G].

14 DANIEL HENRY GOTTLIEB

Theorem 5.6. Let F, G ∈ cS. Then e^{F} e^{G} = e^{D} where

\frac{sinh λ_{D}}{λ_{D}} D = (bαF + aβG + \frac{bβ}{2} [F, G]) and cosh λ_{D} = (aα + bβ⟨F, G⟩) ,
a = cosh λ_{F} , b = \frac{sinh λ_{F}}{λ_{F}} , α = cosh λ_{G}, β = \frac{sinh λ_{G}}{λ_{G}} .

We of course assume that λ_{D} ≠ (2n + 1)πi Proof. e^{F} e^{G} = (aI+bF)(αI+βG), which follows from Theorem 8.5 in [Gottlieb(1998)]. Expand and use the lemma above to get the equations for D and cosh λ_{D} using again the expansion of e^{D} from Theorem 8.5 to read off the equations. □ Corollary 5.7. [e^{F} , e^{G}] = \frac{sinh λ_{F} sinh λ_{G}}{λ_{F} λ_{G}} [F, G] if F, G ∈ cS. Proof. [e^{F} , e^{G}] = [aI + bF, αI + βG] = bβ[F, G]. Theorem 5.8. Let F and G be operators in cS. If e^{F} = e^{G} ≠ ±I, then F = G + 2πnB̂ where B̂ is unit rotation, that is B̂ has eigenvalues i and −i. Also F and G commute. Proof. 0 = [e^{F} , e^{G}] = \frac{sinh λ_{F}}{λ_{F}} \frac{sinh λ_{G}}{λ_{G}} [F, G]. Hence one of λ_{F} or λ_{G} must equal πni for n ≠ 0 ∈ ℤ, in which case e^{F} = ±I, say; or else [F, G] = 0. Hence I = e^{F} e^{−G} = e^{F −G}. Hence F − G = 2πnB̂ where B̂ is a unit rotation. Corollary 5.9. If F, G ∈ 𝔰𝔬(3, 1) and e^{F} = e^{G} ≠ I, then F = G + 2πnB̂ where B̂ is unit rotation in 𝔰𝔬(3, 1). Proof. Now e^{F} = e^{G} have the same null eigenvectors, so do F and G. Thus [F, G] = 0. Hence I = e^{F} e^{−G} = e^{F −G} = e^{2πnB̂}. Theorem 5.10. Let A, B ∈ cS. Then λ_{[A,B]}^{2} = 4(⟨A, B⟩^{2} − λ_{A}^{2}λ_{B}^{2}). Proof. [A, B]^{2} = (AB − BA)(AB − BA)

= ABAB + BABA − AB^{2}A − BA^{2}B

= {ABA, B} − 2λ_{A}^{2}λ_{B}^{2}I.
Now ABA = −A^{2}B + 2⟨A, B⟩A = −λ_{A}^{2}B + 2⟨A, B⟩A by Lemma 5.4.
So [A, B]^{2} = {−λ_{A}^{2}B + 2⟨A, B⟩A, B} − 2λ_{A}^{2}λ_{B}^{2}I
= −2λ_{A}^{2}λ_{B}^{2}I + 4⟨A, B⟩^{2}I − 2λ_{A}^{2}λ_{B}^{2}I
= 4(⟨A, B⟩^{2} − λ_{A}^{2}λ_{B}^{2}⟩)I.

□ Corollary 5.11. If A, B ∈ 𝔰𝔬(3, 1), then

λ_{c[A,B]}^{2} = ⟨cA, cB⟩^{2} − λ_{cA}^{2}λ_{cB}^{2}.

proof. We know that 2c[F, G] = [cF, cG] by Theorem 4.3 of [Gottlieb (1998)]. Hence the eigenvalue squared of c[F, G] is 1/4 times the eigenvalue squared of [cF, cG]. Now apply the preceeding theorem.

Theorem 5.12. Let A, C be null operators in cS. Then

e^{D} = e^{A}e^{C} = (1 + ⟨A, C⟩)I + A + C + ⟨A, C⟩Ê

where Ê is a unit boost operator orthogonal to A and B, and where D is characterized by the vector

\frac{sinh λ_{D}}{λ_{D}} D⃗ := \frac{sinh λ_{D}}{λ_{D}} Du = A⃗ + C⃗ + iA⃗ × C⃗

Proof. e^{A}e^{C} = (I + A)(I + C) = I + A + C + AC

= I + A + C + \frac{1}{2}\{A, C\} + \frac{1}{2}[A, C]

= I + A + C + ⟨A, C⟩I + \frac{1}{2}(2⟨A, C⟩Ê)

= (1 + ⟨A, C⟩)I + ⟨A, C⟩Ê + A + C.

The second to last equation follows from Theorem 5.2 b). The characterisation of D follows because

[cF, cG]u = 2iF⃗ × G⃗

from [Gottlieb (1998)] Theorem 4.3 and Corollary 4.4 as quoted in the paragraph after Corollary 2.7 in this paper. □

Corollary 5.13. Let A and C be null operators in cS. Then e^{A} = e^{C} if and only if A = C.

Proof. This follows from Theorem 5.8 since A = C + 2πnB̂, and A and C must commute. Thus A − C is null and equal to a rotation whose eigenvalue is 2πni. This can only happen if n = 0.

Theorem 5.14. Let G, F ∈ cS. Then

e^{−G}F e^{G} = ((cosh λ_{G})^{2} + (sinh λ_{G})^{2})F

− 2⟨F, G⟩\frac{(sinh λ_{G})^{2}}{λ_{G}^{2}} G + \frac{(sinh λ_{G})(cosh λ_{G})}{λ_{G}} [F, G].

Proof. e^{−G}F e^{G} = (aI − bG)F (aI + bG). Expanding yields

e^{−G}F e^{G} = a^{2}F + abF G − abGF − b^{2}GF G.

But

GF G = −λ_{G}^{2}F + 2⟨F, G⟩G.

So the result follows.

Corollary 5.15. Let F ∈ cS and N ∈ cS be null and let ⟨F, N⟩ = 0. Then

e^{−F} Ne^{F} = e^{−2λ_{F}} N

and

e^{−N}F e^{N} = F + 2λ_{F} N.

Proof. Apply Theorem 5.2 a) and the fact that λ_{N} = 0 to Theorem 5.14 above and simplify using the definition of the hyperbolic trigonometric functions.

Theorem 5.16. Let F ∈ 𝔰𝔬(3, 1). Let λ_{cF} be imaginary. Then the real exponential satisfies

e^{F} = cos^{2} (\frac{λ_{F^{∗}}}{2} ) I + \frac{2}{λ_{F^{∗}}^{2}} sin^{2} (\frac{λ_{F^{∗}}}{2} ) T_{F} + \frac{sin λ_{F^{∗}}}{λ_{F^{∗}}} F.

If λ_{cF} is real, then the real exponential is given by

e^{F} = cosh^{2} (\frac{λ_{F}}{2} ) I + \frac{2}{λ_{F}^{2}} sinh^{2} (\frac{λ_{F}}{2} ) T_{F} + \frac{sinh λ_{F}}{λ_{F}} F.

If λ_{cF} = 0, then the real exponential is

e^{F} = I + F + \frac{1}{2}F^{2}

Proof. In the imaginary case we may take λ_{cF} = iλ_{F^{∗}}.

So e^{\frac{cF}{2}} = cosh (i\frac{λ_{F^{∗}}}{2} ) I + \frac{sinh(i\frac{λ_{F^{∗}}}{2} )}{iλ_{F^{∗}}} cF

= cos (\frac{λ_{F^{∗}}}{2} ) I + \frac{i sin(\frac{λ_{F^{∗}}}{2} )}{iλ_{F^{∗}}} cF

So e^{\frac{cF}{2}} e^{\frac{\overline{c}F}{2}} = (cos (\frac{λ_{F^{∗}}}{2} ) I + \frac{sin(\frac{λ_{F^{∗}}}{2} )}{λ_{F^{∗}}} cF ) (cos (\frac{λ_{F^{∗}}}{2} ) I + \frac{sin (\frac{λ_{F^{∗}}}{2} )}{λ_{F^{∗}}} \overline{cF} )

e^{F} = cos^{2} (\frac{λ_{F^{∗}}}{2} ) I + \frac{sin (\frac{λ_{F^{∗}}}{2} ) cos (\frac{λ_{F^{∗}}}{2} )}{λ_{F^{∗}}} (cF + \overline{cF})

+ \frac{sin^{2} (\frac{λ_{F^{∗}}}{2} )}{λ_{F^{∗}}^{2}} cF \overline{cF}

= cos^{2} (\frac{λ_{F^{∗}}}{2} ) I + \frac{2 sin (\frac{λ_{F^{∗}}}{2} ) cos (\frac{λ_{F^{∗}}}{2} )}{λ_{F^{∗}}} F + \frac{2 sin^{2} (\frac{λ_{F^{∗}}}{2} )}{λ_{F^{∗}}^{2}} T_{F} .

Now since 2 sin α cos α = sin(2α), we obtain the desired result.

The real case and the null case follow similarly. □

Note in the above theorem that 2T_{F} = F^{2} for null F. Also note that (\frac{T_{F}}{λ_{T}} )^{2} = I,

so \frac{T_{F}}{λ_{T}} is an isometry since it is symmetric with respect to the inner product. So it must be the exponential of something. The following results tells us what that something is.

Theorem 5.17. Let F be generic in 𝔰𝔬(3, 1). Then T_{F} = λ_{T} e^{(2n+1)πB} where B = −\frac{i}{λ_{cF}} · F is a rotation. So T_{F} /λ_{T_{F}} is a rotation about 180^{∘}.

Proof. Note that λ_{cB} = −i. Then

e^{(2n+1)πB} = e^{(n+ \frac{1}{2} )πcB}e^{(n+ \frac{1}{2} )π\overline{c}B}.

Now e^{(n+ \frac{1}{2} )πcB} = (−1)^{n}cB. Hence e^{(2n+1)πB} = ((−1)^{n}cB)(−1)^{n}\overline{c}B) = cB\overline{cB}. Now cB = −\frac{i}{λ_{cF}} F. So cB\overline{cB} = cF \overline{cF} /λ_{cF} λ_{\overline{c}F} = 2T_{F} /2λ_{T_{F}} = T_{F} /λ_{T_{F}}. □

6. Topology and Physics

We look at the lifting problem. Suppose we have a Gauge transformation, that is a field of Lorentz transforms, that is a bundle isometry L

T (M) ⟶^{L} T (M)
↓^{p}  ↓^{p}
M ⟶^{1} M

Then we can regard L as a cross–section to the associated bundle SO(3, 1) → E ⟶^{p} M where each point e of E is an isometry on the tangent space based on the point p(e) ∈ M. Now suppose L = e^{F} where F: M → E′ is a cross–section where E′ → M is the bundle of tangents along the fibres of E. It has fibres 𝔰𝔬(3, 1). Since F can be homotopied to the zero section, it follows that its image e^{F} is homotopic to the identity I.

A more concrete picture arises when we consider M with trivial tangent bundle. Then the above cross–section L can be represented by a map f: M → SO(3, 1). Then there exists an F so that e^{F} = L only if there is a g which lifts f in the diagram

M ⟶^{g} 𝔰𝔬(3, 1)
↓^{1}  ↓^{exp}
M ⟶^{f} SO(3, 1).

This implies that f is homotopic to a constant map since 𝔰𝔬(3, 1) is contractible. Thus it is easy to find non homotopy trivial maps which will not give rise to e^{F}. But if M is contractible, say, this homotopy obstruction does not exist.

If I is in the image of f, then since I is not regular under exp, it is possible that the maps induced on the tangent bundle are not equal,

exp_{∗} ∘ g_{∗} ≠ f_{∗}: T_{m}(M) → T_{I}(𝔰𝔬(3, 1)),

since exp_{∗} ∘ g_{∗} is not onto at a critical point, whereas f_{∗} could be onto, for example.

However, there are a few interesting observations to be made:

First, f may be approximated by a smooth mapping f′ so that f′ avoids I. Since M is 4–dimensional and SO(3, 1) is six dimensional, that is easy to do.

Second, if we consider S ⊂ M a 3 dimensional space–like slice, then f: S → SO(3, 1) maps a 3 dimensional space into a 6 dimensional space. We can define self intersection numbers which propagate in space–time.

Now we give an result which is similar to the singularity theorem for robot arms [Gottlieb (1986, 1988)]. The singularities of the robot arm can physically be seen as infinite accelerations of the joints in the arm. The argument below follows that of the robot arm since the map R defined below is closely related to the robot arm map.

Theorem 6.1. Let R: 𝔰𝔬(3, 1)×…×𝔰𝔬(3, 1) → SO(3, 1) be the map (F_{1}, …, F_{n}) ↦ e^{F_{1}} … e^{F_{n}}. Suppose N ⊂ 𝔰𝔬(3, 1) × … × 𝔰𝔬(3, 1) is a submanifold so that R: N → SO(3, 1) is onto and proper, i.e. the pre-image of every compact set is compact. Then R must have a singularity.

Proof. Since R is proper, by a theorem of Ehresman, if R has no singularities then R is a fibre bundle F → N → SO(3, 1) where the fibre F is a compact closed manifold. Now R is homotopy trivial since R factors through ∏_{n} 𝔰𝔬(3, 1), a contractible space.

Hence the fibre F is homotopy equivalent to ΩSO(3, 1) × N. But ΩSO(3, 1) has non zero cohomology groups in infinitely many dimensions since the connected components are homotopy equivalent of the following series of loopspaces: ΩSO(3, 1) ≅ ΩSO(3) ≃ Ω(S^{3}). Since Ω(S^{3}) has cohomology in infinitely many dimensions F cannot be compact. □

7. A physical example

A moving particle of charge q moving along a time–like path in space–time has a classical electromagnetic field given as follows. (See [Parrott (1987)], p. 134 for details, for example.)

Suppose the particle p is situated at a point x in Minkowski space–time M. Suppose p has 4–velocity u ∈ T_{x}(M), so ⟨u, u⟩ = −1. Suppose p undergoes an acceleration a⃗. Now a⃗ must be orthogonal to u, that is a⃗ ∈ T_{x}^{u}, the rest space of p at x, or equivalently, the subspace of T_{x} orthogonal to u. Then the electro–magnetic field of p is defined along the future light cone C at x as follows. Any point z on C is given uniquely by z = ru+rw⃗ where r is a positive number and w⃗ is a unit vector in the rest space of p at x, namely w⃗ ∈ T_{x}^{u}. Now at z the electric field for observer

u is given by E⃗ = q (\frac{w⃗}{r^{2}} − \frac{1}{r} a⃗_{⊥}), whereas B⃗ = qa⃗_{⊥} × w⃗ where a⃗_{⊥}: = a⃗ − (a⃗ · w⃗)w⃗

is the orthogonal part of a⃗ with respect to w⃗, Hence a⃗_{⊥} · w⃗ = 0.

So if we let Fa⃗ stand for the skew symmetric field on M due to the particle p, we have that Fa⃗u = \frac{q}{r^{2}} Ew⃗ + \frac{q}{r} Na⃗ where Ew⃗ is the skew symmetric operator so that cEw⃗u = w⃗, or equivalently Ew⃗ has an eigenvector along u + w⃗ associated with eigenvalue λ = 1 and B = 0. Hence λ_{cE_{w}} = 1 is the eigenvalue associated with the eigenvector u+w⃗. And Na⃗ is the null operator with E⃗ = −a⃗_{⊥} and B⃗ = a⃗_{⊥} ×w⃗ and eigenvector s = a_{⊥}^{2}u − a_{⊥} × (a⃗_{⊥} × w⃗) = a_{⊥}^{2}(u + w⃗). So Na⃗ shares the eigenvector (u + w⃗) with Ew⃗.

Let E_{coul} = \frac{q}{r^{2}} Ew⃗ represent the boost operator corresponding to the Coulomb field of the particle at rest in its rest frame. Then with the above notation, the following Theorem states the relationship between the Coulomb field at rest, Na⃗, and the electro–magnetic field of the particle under an acceleration a⃗ denoted by Fa⃗.

Theorem 7.1. Fa⃗ is the image of E_{coul} under conjugation by Exp(rNa⃗). That is

Fa⃗ = e^{−rNa⃗}E_{coul}e^{rNa⃗}

Proof. Since Na⃗ and E_{coul} share an eigenvector, < Na⃗, E_{coul} >= 0. Hence we may apply Corollary 5.15 to cFa⃗ = cE_{coul} + \frac{q}{r} cNa⃗. That gives us e^{−cN}cF e^{cN} = cF +2λ_{cF} cN for null cN orthogonal to cF. So setting cN to \frac{r}{2}cNa⃗ and cF to E_{coul}, we see that λ_{E_{coul}} = q/r^{2} and so

e^{− \frac{r}{2} cNa⃗}cE_{coul}e^{\frac{r}{2} cNa⃗} = cE_{coul} + \frac{q}{r} cNa⃗ = cFa⃗

Conjugating the above equation with e^{\frac{r}{2} \overline{c}Na⃗} yields

e^{−rNa⃗}cE_{coul}e^{rNa⃗} = cFa⃗

and the real part of this equation is the result to be proved. □

References

Daniel H. Gottlieb (1998), Skew Symmetric Bundle Maps, Contemporary Mathematics . 220, pp. 117 - 141.

Daniel H. Gottlieb (1986), Robots and fibre bundles, Bulletin de la Societe Mathematique de Belgique 38, 219–223.

Daniel H. Gottlieb (1988), Topology and the Robot Arm, Applicandae Mathematicae 11, 117–121.

Helgason (1978), Sigurdur, Differentiable Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York.

Stephen Parrott (1987), Relativistic Electrodynamics and Differential Geometry, Springer-Verlag, New York.