Causal automorphisms on two-dimensional Minkowski space
Do-Hyung Kim
Department of Applied Mathematics, College of Advanced Science, Dankook University, Cheonan-si, Chungnam, 330-714, Korea e-mail : mathph@dankook.ac.kr
(2010 Mathematics Subject Classification : 53B20, 53C15.)
Abstract. The general form of causal automorphism on two-dimensional Minkowski space- time is given and its group structure is analyzed.
1 Introduction
It is a well-known fact that causal automorphism on Minkowski space-timeRn1, for n ≥ 3, can be represented by a composite of orthochronous transformation, translation and dilatation.(Ref. [1]). In other words, for any causal automorphism F :Rn1 →Rn1, ifn≥3, then there exist a unique orthochronous matrixA, a unique real number a, and unique b ∈ Rn1 such that F(x) = a·Ax+b. From Zeeman’s theorem, we can see that the structure of the group of all causal automorphisms on Rn1 has a semi-direct product structure.
However, as Zeeman remarked, his theorem does not hold in two-dimensional Minkowski spacetime and it was an open problem for a long time.
Recently, it was shown that the group of causal automorphisms onR21 is of infi- nite dimensional and contains the group of all homeomorphisms onRas a subgroup.
In this paper, the answer to the long-lasting open problem is given and details of the proof can be found in [4] and [5]. In fact, this is the abbreviation of my previous papers [4] and [5].
2 Preliminaries
By spacetime, we mean a time-oriented Lorentzian manifold. We denote by x≪y if there exists a piecewise smooth, future-directed timelike curve from xto y and by x ≤ y if there exists a piecewise smooth, future-directed causal curve from x to y. We define a bijection f : M → M′ between two space-times to be chronological isomorphism ifx≪y ⇔ f(x)≪f(y) and to be causal isomorphism Key words and phrases: Lorentzian geometry, general relativity, causality, Cauchy surface, space-time, global hyperbolicity, causal automorphism, Minkowski space-time.
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ifx≤y ⇔ f(x)≤f(y). Whenf :M →M is a causal isomorphism, then we call f a causal automorphism. It is a well-known fact that if space-times are strongly causal, then chronological isomorphisms and causal isomorphisms are equivalent.
(See Ref. [3], [6].)
In connection with this, we have a famous theorem, called the Hawking’s theo- rem which is the following.
Theorem 2.1. If M andM′ are strongly causal, then any chronological isomor- phism (or causal isomorphism) betweenM andM′ is a smooth conformal isometry if the dimension of manifolds are greater than two.
Proof. See Ref. [6], [7], [8].
Note that the Hawking’s theorem does not hold if the dimension of given mani- fold is two. One of simple examples is given in Ref. [3]. However, if the space-times are strongly causal, since their topologies coincide with their Alexandrov topologies which are defined in terms of only chronological relations, any causal isomorphism or chronological isomorphism becomes a homeomorphism between the space-times.
Therefore, for the proof of our assertion, we need to replace the differentiable terms by topological terms. We define an open neighborhood U of pin M to be simple region if its topological closure U is compact and is contained in a totally convex, normal neighborhood. We also define a continuous curveγ: [0.1]→M to be future-directed timelike if we haveγ(s)≪γ(t) for anys < t.
Proposition 2.1. Forxandy in M, we havex≪y if and only if there exists a continuous, future-directed timelike curveγ: [0,1]→M fromxtoy.
Proof. The ‘only if’ part is trivial and we show the ‘if’ part. We coverγ([0,1]) by simple regions. Then, sinceγ([0,1]) is compact we can choose a finite subcoverU1, U2,· · ·, Un. Choose n1 such thatγ(0)∈Un1. Ifγ([0,1]) is contained inUn1 then we can connectγ(0) andγ(1) by a future-directed timelike geodesic and the proof is completed. If not, by definition of simple region, we can choose an open setV1such that Un1 ⊂V1 and we can choose a pointγ(t1) such that γ leavesUn1. Then we can connectγ(0) andγ(t1) by a timelike geodesicη1. We now choosen2 such that γ(t1)∈Un2. Ifγ([t1,1]) is contained inUn2, then we can connectγ(t1) andγ(1) by a timelike geodesicη2andη1∪η2gives a piecewise smooth, future-directed timelike curve. If we proceed inductively in this way, the step must ends at finite number of steps since the open subcoverU1, · · ·,Un is finite and the proof is completed.
In 1970, Geroch has shown that a space-time M is globally hyperbolic if and only if there is a topological hypersurface Σ such that every inextedible (continu- ous) timelike curve meets Σ exactly once. We call such a hypersurface a Cauchy surface.(See Ref. [9].)
Proposition 2.2. Let f : M →M′ be a causal isomorphism and Σ be a Cauchy surface of M. Then,Σ′=f(Σ) is a Cauchy surface ofM′.
Proof. If the dimension of M is greater than 2, then the assertion follows easily from the Hawking’s theorem and it remains to prove the case when the dimension ofM is two.
Let γ be an inextendible timelike curve in M′. Then, since f−1 is a homeo- morphism, f−1◦γ is a continuous, inextendible timelike curve inM. Since Σ is a Cauchy surface ofM, f−1◦γ must meet Σ exactly once at, say,x. Thenγ meets Σ′ =f(Σ) exactly once atf(x). Furthermore, since f is a homeomorphism, Σ′ is also a topological hypersurface and thus Σ′ is a Cauchy surface ofM′.
3 Causally Admissible System
In this section, we introduce causally admissible system and state some of its properties, which is developed in Ref. [2].
LetM be a globally hyperbolic space-time with its Cauchy surface Σ. For any point pin J+(Σ), we let Sp+ = J−(p)∩Σ. In general, it can be that Sp+ = S+q with p ̸=q as is easily seen in Einstein’s static universe. In Ref. [2], it is shown that if Σ is non-compact, then Sp+=Sq+only if p=q. (See Proposition 3.2 in Ref.
[2]). In other words, we can use the setC+={Sp+| p∈J+(Σ)} to represent points in J+(Σ) uniquely if Σ is non-compact. Likewise, if we let Sp− = J+(p)∩Σ for p∈J−(Σ), then we can use the setC−={Sp− |p∈J−(Σ)}to represent the points in J−(Σ) uniquely if Σ is non-compact. Therefore, when Σ is non-compact, if we form their pairC= (C+,C−), then the set Cuniquely determines all the points in M where we identify each singleton element inC+andC−. We call the elements of C+ and C− as future admissible subsets and past admissible subsets, respectively.
We also call the set Ca causally admissible system with respect to Σ
As is easily seen, ifp≤qforpandqin J+(Σ), then we have Sp+⊂Sq+. It can be shown that the converse also holds if Σ is non-compact.(See Theorem 4.1 in Ref.
[2]). Likewise, when Σ is non-compact, we have Sp− ⊂Sq− if and only if q≤pfor pand qin J−(Σ). Also, it is almost trivial to see thatS+q ∩S−p ̸=∅ if and only if p≤qforpin J−(Σ) andqin J+(Σ).
What we really have done is that we have the setCwhich representsM and we have replaced the causal relation ofM by the simple relation of inclusions between the elements of C. C is more tractable than M itself since Cis a set of subsets of Cauchy surface Σ as the following shows.
Let M and M′ be globally hyperbolic with their Cauchy surfaces Σ and Σ′ non-compact. We define a bijection f : Σ → Σ′ to be causally admissible if f induces a bijections from C+ to C′+ and from C− to C′−. Then, since the causal relation is encoded intoCthrough the relation of inclusion, it is not difficult to see the following theorem.
Theorem 3.1. Two space-times M and M′ with non-compact Cauchy surfaces are causally isomorphic if and only if there exists a causally admissible function f : Σ→Σ′ between their non-compact Cauchy surfaces.
The proof of the above theorem can be seen in Ref. [2]. Though the Hawking’s theorem was used implicitly in the proof, by a slight modification, it is easy to see that the above theorem also holds for two-dimensional case since f is a causal isomorphism and f(Σ) is a Cauchy surface whenever Σ is a Cauchy surface by Proposition 2.2.
SinceR21is globally hyperbolic with non-compact Cauchy surface, we can apply the theory of causally admissible system to analyze causal automorphisms onR21. This is the main tool of this paper.
4 Causal automorphisms of R21
For convenience, we use both tandy as time coordinate ofR21interchangeably in this section. In Ref. [3], a new method for imbedding a Lorentzian manifold is presented by use of causally admissible system and this also can be used to show the following.
Theorem 4.1. Let Rt0 ={(x, t)∈R21| t=t0} be a Cauchy surface ofR21. Then, for any homeomorphism f : R→ R can be extended to a causal automorphism f fromR21 ontoR21.
Proof. The key idea of this theorem is that we identify R and Rt0 so that we can view f as the homeomorphism from Rt0 onto Rt0. Since any compact and connected subset of Rt0 is a causally admissible subset, f naturally becomes a causally admissible function. If we extend this f, we get a corresponding causal automorphism. For details, see Theorem 5.1 and Theorem 5.2 in Ref. [3].
Let G be the group of all homeomorphisms of R and H be the group of all causal automorphisms ofR21. Then if we define a functionϕ:G→H byϕ(f) =f
defined in the above theorem, then it is easy to see that ϕ is an injective group homomorphism and thus the group of all homeomorphisms of Ris a subgroup of the group of all causal automorphisms ofR21. This is quite different from the case n≥3.
As commented in the proof of theorem 4.1, the characteristic property ofR21is that any connected and compact subset of Cauchy surfaceRt0 is causally admissible.
We next show that this also holds for its image Σ′=F(Σ) of Σ =Rt0 under causal automorphismF.
Proposition 4.1. Let F be a causal automorphism from R21 onto R21. For given Cauchy surface Σ = Rt0 of R21, any compact and connected subset of the Cauchy surfaceΣ′=F(Σ) is both future and past admissible.
Proof. LetI be a compact and connected subset of Σ′. Then, since F is a homeo- morphism,F−1(I) is also a compact and connected subset ofRt0, which is homeo- morphic to real line R. Since the only compact and connected subset of real line is a closed interval, we can letF−1(I) ={(x, t0)|a≤x≤b}for some real numbersa andb. Then there exist unique pointspandqinJ−(Rt0) andJ+(Rt0), respectively, such that Sp− =F−1(I) andSq+ =F−1(I). SinceF is a causal automorphism, we have I = F(Sq+) = F(J−(q)∩Rt0) = J−(F(q))∩Σ′. In other words, I is fu- ture admissible. Likewise, we can show thatI =J+(F(p))∩Σ′ andI is also past admissible.
The above theorem and its proof tell us that any causal automorphism F : R21→R21 is completely determined by its restriction to the Cauchy surfaceRt0. In other words, if two causal automorphismsF andGfromR21ontoR21coincide when we restrictF andGonRt0, thenF =G. We state this as a theorem.
Theorem 4.2. Let F, G : R21 → R21 be causal automorphisms. If, for some real number t0,F andGsatisfiesF(x, t0) =G(x, t0)for all xinR, thenF =G.
Proof. Choose any point pin J+(Rt0) and letSp+ =J−(p)∩Rt0. Then, since F is a causal automorphism, we have F(Sp+) =J−(F(p))∩Σ′ where Σ′ =F(Rt0) = G(Rt0). Likewise, we have G(Sp+) = J−(G(p))∩Σ′. Since F(Sp+) =G(Sp+), we have J−(F(p))∩Σ′ =J−(G(p))∩Σ′ and thus F(p) =G(p) for any p∈J+(Rt0).
By the similar method, we can show thatF(p) =G(p) for anyp∈J−(Rt0).
By the above theorem, to classify all automorphism onR21, we only need to see howRt0 is mapped by given causal automorphism.
Lemma 4.2. Let F : R21 → R21 be a causal automorphism and be given by (x, t) → (f(x, t), g(x, t)). Then, for any t0 ∈ R, the function x → f(x, t0) is a homeomorphism fromRontoR.
Proof. See [4].
From the above lemma, it is easy to see that for any given t0, the function f(·, t0) is either a monotonically increasing function or a monotonically decreasing function.
Lemma 4.3. Let F : R21 → R21 be a causal automorphism and be given by (x, t) → (f(x, t), g(x, t)). Then, for any x0 ∈ R, the function t → g(x0, t) is a homeomorphism fromRontoR.
Proof. See [4].
Since F is a causal automorphism, it must preserve time-orientation and thus the functiong(x0,·) is an increasing function. We now investigate the properties of the functionx→g(x, t0).
Lemma 4.4. Let F :R21→R21 be a causal automorphism and be given by(x, t)→ (f(x, t), g(x, t)). Then, we must havesup{g(x, t0)±f(x, t0)}=∞andinf{g(x, t0)± f(x, t0)}=−∞for any t0.
Proof. See [4].
As commented earlier, to classify causal automorphism F :R21 →R21, we need to study its restriction toRt0. If we letγ(x) = F(x, t0) = (f(x, t0), g(x, t0)), then γ(R) becomes an acausal Cauchy surface of R21 sinceF is a causal automorphism and Rt0 is an acausal Cauchy surface. Therefore, we need to study an imbedded curve inR21 of which the image is an acausal Cauchy surface. Also, note that, for anyt0, the curveγ(x) =F(x, t0) is an injective continuous curve since the function x→f(x, t0) is a homeomorphism by lemm 4.2.
Proposition 4.5. Letγ:R→R21 given byt→(f(t), g(t))be an injective, acausal, continuous curve in R21. Then, every inextendible causal curve inR21 meets γ(R)if and only ifsup(g±f) =∞andinf(g±f) =−∞.
Proof. See [4].
In the above proposition, since we assumedγ to be acausal, every inextendible causal curve must meet γ at an unique point and so γ(R) is a Cauchy surface.
Therefore, actually, we have the following theorem.
Theorem 4.3. Let γ:R→R21 given byt→(f(t), g(t))be an injective, continuous curve inR21. Then, γ(R) is an acausal Cauchy surface if and only if f is a home- omorphism,sup(g±f) =∞,inf(g±f) =−∞and|f(t+δt)g(t+δt)−−g(t)f(t)|<1 for all t and δt̸= 0.
Proof. Ifγ(R) is a Cauchy surface, then, the proof of lemma 4.4 shows thatγmust satisfy sup(g±f) =∞and inf(g±f) =−∞. SinceR21is globally hyperbolic,p≤q if and only if the straight line from pto q is a causal curve. This shows that, for γ(R) to be acausal, the above inequality must hold. The injectiveness ofγ shows that f is a homeomorphism by similar method as in lemma 4.2
Conversely, if a given curve γ satisfies given conditions, then proposition 4.5 shows that every inextendible timelike curve meets γ(R) and given inequality en- sures thatγ(R) is acausal. Therefore every inextendible timelike curve meetsγ(R) exactly once andγ(R) is an acausal Cauchy surface.
If Σ is a Cauchy surface of R21, then Σ is homeomorphic to R and since any causally admissible subset ofRis a closed interval [a, b] ofR, any causally admissible subset can be determined by its boundary points aand b. Therefore, we have the following.
Lemma 4.6. For given two points (x1, y1) and (x2, y2) with x1 < x2 on a Cauchy surface of R21, their future and past realizing points are given by (x1+x2 2 +
y2−y1
2 ,x2−2x1 +y1+y2 2)and(x1+x2 2 +y1−2y2,x1−2x2 +y1+y2 2), respectively.
Proof. The future realizing point can be obtained by solving simple equationsy− y1 = x−x1 and y −y2 = −(x−x2). Likewise, the past realizing point can be obtained by solving equations y−y1=−(x−x1) andy−y2=x−x2.
We now state the main theorem.
Theorem 4.4. Let F : R21 →R21 be a causal automorphism. Then, there exist a continuous functions g : R → R and a homeomorphism f : R → R which satisfy
sup(g±f) =∞,inf(g±f) =−∞and|f(t+δt)g(t+δt)−−g(t)f(t)|<1 for alltandδt, such that if f is increasing, thenF is given by
F(x, y) = (f(x−y)+f(x+y)2 +g(x+y)−2g(x−y),f(x+y)−2f(x−y)+g(x−y)+g(x+y)2 ) and if f is decreasing, then we have
F(x, y) = (f(x+y)+f(x2 −y)+g(x−y)−2g(x+y),f(x−y)−2f(x+y)+g(x+y)+g(x2 −y)).
Conversely, for any functionsf andg which satisfy the above conditions, the func- tionF :R21→R21 defined as above, is a causal automorphism.
Proof. See [4].
In the above theorem, we must note that, even if there exist two curves γ1 and γ2 such thatγ1(R) =γ2(R), the induced causal automorphisms do not necessarily coincide since the induced causal automorphisms depend on the parameterizations of γ1 and γ2. In a sense, Zeeman’s result tells us that the group of all causal automorphisms on Rn1 is finite dimensional for n ≥ 3 whereas our result tells us that the group of all causal automorphisms is infinite dimensional forR21.
Theorem 4.5. Let F : R21 → R21 be a causal automorphism on R21. Then, there exist unique homeomorphismsφandψofR, which are either both increasing or both decreasing, such that if φand ψ are increasing, then we have F(x, y) = 12(φ(x+ y) +ψ(x−y), φ(x+y)−ψ(x−y)), or if φ and ψ are decreasing, then we have F(x, y) =12(φ(x−y) +ψ(x+y), φ(x−y)−ψ(x+y)).
Conversely, for any given homeomorphismsφandψofR, which are either both increasing or both decreasing, the functionF defined as above is a causal automor- phism ofR21.
Proof. For any given causal automorphismF :R21→R21, by the previous theorem, we can get unique homeomorphism f and unique continuous functiong. Letφ= f+g andψ=f−g. Then, clearly,φandψ are continuous.
The conditions sup(g±f) = ∞, inf(g±f) = −∞ imply that φ and ψ are surjective. If φ(t) = φ(t′) with t ̸= t′, then this implies that f(tg(t′′))−−g(t)f(t) = −1, which contradicts to the condition|f(t+δt)g(t+δt)−−g(t)f(t)|<1 for alltandδt. Therefore,φis injective and likewise, we can show thatψis injective. Sinceφandψare continuous
bijections fromRto R, the topological domain of invariance implies thatφandψ are homeomorphisms.
The equationφ(t2)−φ(t1) = (f(t2)−f(t1))[1 +f(tg(t2)−g(t1)
2)−f(t1)] tells us that, by the condition|f(t+δt)g(t+δt)−−g(t)f(t)|<1,φis increasing if and only iff is increasing. Likewise, we can show thatψis increasing if and only if f is increasing.
By simple calculation, we can show thatF has the desired form when expressed in terms ofφandψ. This completes the proof of the first part.
To prove the converse, letφ andψ be increasing homeomorphisms on R. Let f = 12(φ+ψ) and g = 12(φ−ψ). Then g is continuous and f is an increasing homeomorphism. Sinceφandψare homeomorphisms, we have sup(f+g) = supφ=
∞, sup(f−g) = supψ=∞, inf(f+g) = infφ=−∞and inf(f−g) = infψ=−∞. We now show thatf and g satisfy the inequality |f(t+δt)g(t+δt)−−g(t)f(t)|<1. Let ∆ =
g(t)−g(t0)
f(t)−f(t0) = φ(t)φ(t)−−φ(tφ(t0)−(ψ(t)−ψ(t0))
0)+ψ(t)−ψ(t0) . Without loss of generality, we can assumet > t0
and it is easy to see that−1<∆<1, sinceφandψare increasing. Therefore, by the theorem ??, the functionF defined as in the statement, is a causal automorphism.
By the exactly same argument, we can show that the assertion also holds when φ andψ are both decreasing homeomorphisms.
5 The group of causal automorphisms on R21
In this section, we denote the group of all causal automorphisms on R21
by G, and we analyze its group structure. For this we let H(R) be the group of all homeomorphisms on R and let H = H+ ∪ H− where H+ = {(φ, ψ) ∈ H(R)×H(R)| φ, ψare increasing} and H− = {(φ, ψ) ∈ H(R) × H(R)| φ, ψare decreasing}. Then H is a subgroup of H(R)×H(R) under the operation induced fromH(R)×H(R).
From the Theorem 4.5, we can see that any causal automorphism F on R21
corresponds to a unique element inH and, conversely, each elements inH uniquely determines a causal automorphism on R21. Thus, there exists a one-to-one cor- respondence between G and H as a set. It might seem that G is isomorphic to H. However, we cannot obtain an isomorphism in this way and we define a new operation onH as follows.
We define aZ2-action onH bya·(φ, ψ) = (φ, ψ) ifa= 0 anda·(φ, ψ) = (ψ, φ) if a = 1. If we define a map π : H → Z2 by π(x) = 0 if x ∈H+ and π(x) = 1 ifx∈H−, thenπis a group homomorphism withH equipped with the operation induced from H(R)×H(R). Note that π(φ, ψ) = π(φ−1, ψ−1) = π(ψ, φ) and π(π(a, b)·(φ, ψ)) =π(φ, ψ) for any (a, b)∈H.
To get an isomorphism from G to H, we define a new operation ∗ on H by (α, β)∗(φ, ψ) = (α, β)◦π(α, β)·(φ, ψ) where·is theZ2-action defined above and
◦is the operation induced fromH(R)×H(R).
Theorem 5.1. The setH under ∗is a group and is isomorphic to G.
Proof. See [5].
In Ref. [3], it is shown thatH(R) is a subgroup ofGand this can also be seen in the above theorem as follows. If we define a map Ω :H(R)→H by Ω(f) = (f, f), then it is easy to see that Ω is an injective homomorphism and thus, H(R) is a subgroup ofGthrough an injective homomorphism Π−1◦Ω. Zeeman’s result tells us that the group of causal automorphisms onRn1 is finite dimensional whenn≥3 and our result tells us that the group is infinite dimensional whenn= 2.
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