Seminar Sophus Lie 2 (1992) 205–212

Distinguishability Condition and the Future Subsemigroup

A. Levichev and V. Levicheva^{∗}

Dedicated to Karl H. Hofmann on the occasion of his 60^{th} birthday

Abstract

The paper deals with two simply connected solvable four-dimensional Lie groups M_{1} and M_{2} . The first group is a direct product of the nilpotent Heisenberg Lie group and the one-dimensional Lie group. The second one is a direct product of the two-dimensional non-abelian Lie group and the two-dimensional abelian Lie group. Applying Methods of [4, 6] we investigate the causal structure of left-invariant Lorentzian metrics on M_{1} [8] and M_{2} [7]. Here we focus our attention on one concrete metric on M_{1} and on a certain one-parameter family g_{q}, q > 0 of metrics on M_{2} . We have proved in [7, 8] these Lorentzian spaces to be geodesically complete, satisfying the causality condition with a violation of uniform stable causality. In the present paper, we prove these spaces to be future distinguishing (that involves, because of their homogeneity, also the conditions of past distinguishing, strong causality, stable causality and continuity of causality). This result is of interest in causality theory since in accordance with [9], respectively, [5] the chronological (respectively, causal) structure of such spaces codes their conformal structure. It also characterizes the structure of the subsemigroup I^{+} , respectively, J^{+} which defines the chronological, respectively, causal structure of the considered Lorentzian Lie group.

For all unfamiliar definitions, the reader is referred to [1, 6].

1. General method to prove future distinguishability of a Lorentzian Lie group

Assume M to be a solvable connected Lie group and fix a symmetric nondegenerated form of Lorentzian signature +, …, +, − in the Lie algebra L of M. After the choice of future cone K^{+} in L the group M becomes a Lorentzian Lie group, or LLG for short. If an ideal [L, L] is lightlike, i.e., its intersection with K^{+} is a single ray lying in ∂K^{+} , then such an LLG M satisfies the causality condition [6, Theorem 4.2]. If the intersection of K^{+} and [L, L] is {0}, then M is uniformly stably causal [6, Theorem 4.1], hence distinguishing. We may, therefore, restrict our attention to the case K^{+} ∩ [L, L] = ℓ where ℓ is a light ray in L.

∗ This paper was presented by title at the Oberwolfach Conference on the analytical and topological theory of semigroups at Oberwolfach, January 30, 1989

Suppose that the hyperplane N contains ℓ and is a support hyperplane of K^{+} . Introduce in M a canonical coordinate system (x_{1}, …, x_{n}) of the second type associated with N. Then the Lie subgroup corresponding to N is characterized by the equation x_{n} = 0.

A Lorentzian manifold M with a prescribed time orientation is said to be future distinguishing (see [1], p. 24) if for any x, y ∈ M the assumption I_{x}^{+} = I_{y}^{+} implies x = y, where I_{x}^{+} , respectively, I_{x}^{−}, as usual, denotes the chronological future, respectively, past of x. If x = 1 then we shall simply write I^{+} instead of I_{1}^{+} etc. Also the causal future, respectively, past, of x will be denoted by J_{x}^{+} , respectively, J_{x}^{−} (and J^{+} , respectively, J^{−} if x = 1.). We want to make use of a result due to R. Penrose: If M fails to be future distinguishing, then Condition (e) of his Theorem 4.31 from [10] is valid. The latter condition deals with a certain light geodesic γ. Suppose now that our LLG M fails to be future distinguishing. It follows from the proof of Theorem 4.2 of [6] that the γ above corresponds to ℓ.

We recall Condition (e) itself:

For any u, v ∈ γ with u ≤ v, if u ≪ x and y ≪ v, then y ≪ x.

We may assume without loss of generality that 1 ∈ γ.

Lemma . γ is entirely contained in \overline{I^{+}} ∩ \overline{I^{−}}. Proof. In Condition (e) we take u = 1, v ∈ γ ∩ J^{+} . Let 1 ≪ x, i.e., x ∈ I^{+} and y ≪ v, i.e., y ∈ I_{v}^{−}. Let y tend to v in I_{v}^{−} and let x tend to 1 in I^{+} . This choice is possible, since v ∈ I_{v}^{−} ⊂ \overline{I_{v}^{−}}, 1 ∈ J^{+} ⊂ \overline{I^{+}} . Taking into consideration the continuous dependence of I_{y}^{±} on the point y itself, we deduce v ∈ \overline{I^{−}}.

We return to M and the canonical coordinates (x_{1}, …, x_{n}). Let x = x(t) denote a future timelike curve λ issuing from 1 = (0, …, 0). The subsemigroup I^{+} , the chronological future of 1, consists of the points on all such λ. Observe that the component z_{n} of the product z = x·y equals x_{n} +y_{n}. Thus the coordinate x_{n} of the point x(t) is increasing while we move along λ from 1 to the future. But the above lemma states that it must somehow reach the vicinity of γ^{−} = J^{−} ∩ γ.

Let exp denote the exponential map (in the geometrical sense) defined on some neighborhood U of 1. When t > 0 is sufficiently small, the points x(t) “concentrate near” exp K^{+} . They can return to γ^{−} only above such a “level” x_{n}, at or below which there are conjugate points to 1 along null geodesics issuing from 1. Therefore the we have following result.

Theorem 1. If, under the assumptions above, in a certain slice U =^{def} {x : 0 < x_{n} < ε} there are no points conjugate to 1 along future null geodesics issuing from 1, and if the set of all points on all lightlike future geodesics from 1 in U divides U into two components, then the Lorentzian Lie group M is future distinguishing.

We note that similar arguments have been used in [5] in the course of proving the future distinguishability of a certain class of Lorentzian symmetric spaces.

Note added in 1992. In [2] the authors introduced the notion of strict causality. For homogeneous Lorentzian spaces this concept agrees with that of distinguishability (see e.g. [4, 6]). In particular, the Lorentzian in the present article as well as the symmetric spaces in [5] are strictly causal.

We also avail ourselves the opportunity of pointing out that, in the English translation [5] of our article “Prescribing the conformal geometry…” the formula labelled (3) was inadvertantly omitted. It should read

(3)  ds^{2} = ∑_{i=1}^{n−2} dx_{i}^{2} − 2dx_{n−1}dx_{n} − (∑_{i=1}^{n−2} λ_{i}x_{i}^{2}) dx_{n}^{2}.

2. Future distinguishability of the space M_{1}

The Lie algebra L_{1} = L(M_{1}) can be defined by the commutation rule

[e_{4}, e_{2}] = e_{1}.  (1)

We can realize M_{1} as ℝ^{4} with multiplication z = x · y given by

z = (x_{1} + y_{1} − x_{2}y_{4}, x_{2} + y_{2}, x_{3} + y_{3}, x_{4} + y_{4}) .  (2)

We fix a Lorentzian form g̃ = (g̃_{ij}) with g̃_{13} = g̃_{31} = g̃_{22} = g̃_{44} = 1 in L_{1} and extend it to M_{1} via left translation to get a homogeneous Lorentzian manifold. We also fix the future cone K^{+} = {a ∈ L_{1} : a^{2} < 0, a_{3} > 0} and use for this LLG the same notation M_{1} as for the Lie group itself.

It was proved in [8] that M_{1} fails to be uniformly stably causal whence it is causal with I^{+} ⊂ {x : x_{3} > 0}.

To apply Theorem 1 we find the points conjugate to 1 along future light geodesics issuing from 1.

In the system (2) the equations for a geodesic x = x(t) passing through 1 with the initial tangent vector a ∈ L_{1} are as follows:

\begin{aligned}ẋ_{1} & = a_{1} + a_{3}x_{2}^{2} − a_{4}x_{2}, \\ ẋ_{2} & = a_{2} + a_{3}x_{4}, \\ ẋ_{3} & = a_{3}, \\ ẋ_{4} & = −a_{3}x_{2} + a_{4}, \end{aligned}  (3)

with the initial conditions x(0) = 1. The geodesic (3) is lightlike iff its initial tangent vector a is lightlike, i.e. satisfies a^{2} = 0.

We need ds^{2} = g_{ij}dx^{i}dx^{j} in the coordinate basis. The matrix g = (g_{ij}) is equal to B^{T} g̃B, where B = A^{−1} and A = \frac{∂z_{i}}{∂y_{k}} is the derivative of the left translation L_{x} at 1. We now compute the Christoffel symbols

Γ_{jk}^{i} = \frac{g^{im}}{2} (\frac{∂g_{mj}}{∂x_{k}} + \frac{∂g_{mk}}{∂x_{j}} − \frac{∂g_{jk}}{∂x_{m}} ) ,  (4)

where (g^{im}) = g^{−1} . The non-zero components are:

Γ_{23}^{1} = Γ_{32}^{1} = −\frac{x_{2}}{2} , Γ_{24}^{1} = Γ_{42}^{1} = \frac{1}{2},
Γ_{34}^{2} = Γ_{43}^{2} = −\frac{1}{2}, Γ_{23}^{4} = Γ_{32}^{4} = \frac{1}{2}.  (5)

The non-zero connection one-forms Γ_{j}^{i} = Γ_{jk}^{i} dx^{k} are:

\begin{aligned}Γ_{2}^{1} & = −\frac{x_{2}}{2} dx^{3} + \frac{1}{2}dx^{4}, Γ_{3}^{1} = −\frac{x_{2}}{2} dx^{2}, Γ_{4}^{1} = \frac{1}{2}dx^{2} , \\ Γ_{3}^{2} & = −\frac{1}{2}dx^{4}, Γ_{4}^{2} = −\frac{1}{2}dx^{3}, Γ_{2}^{4} = \frac{1}{2}dx^{3}, Γ_{3}^{4} = \frac{1}{2}dx^{2} . \end{aligned}

The components R_{jkm}^{i} of the curvature tensor R may be found from the

equality θ_{j}^{i} = \frac{R_{jmk}^{i}}{2} dx^{m} ∧ dx^{k} where curvature two-forms θ_{j}^{i} = dΓ_{j}^{i} + Γ_{s}^{i} ∧ Γ_{j}^{s}. Non-zero of them are:

θ_{2}^{1} = −\frac{dx^{2} ∧ dx^{3}}{4} , θ_{3}^{1} = \frac{x_{2}dx^{3} ∧ dx^{4}}{4} , θ_{4}^{1} = \frac{dx^{3} ∧ dx^{4}}{4} ,
θ_{3}^{2} = \frac{dx^{2} ∧ dx^{3}}{4} , θ_{3}^{4} = −\frac{dx^{3} ∧ dx^{4}}{4} .  (6)

The Jacobi field y = y(t) along the geodesic γ given by x = x(t) is found as the solution of the system

\frac{D^{2}y}{dt^{2}} + R(y, ẋ)ẋ = 0 ,  (7)

where D/dt is the covariant derivative along γ.

From the 3rd equation of (7) we extract y_{3} ≡ 0 since we are searching for only those Jacobi fields which become zero at 1 (i.e. when t = 0), and at least at one additional point of γ with t > 0. Let us also consider the three other equations. The second and third of them form the subsystem

ÿ_{2} − a_{3}ẏ_{4} = 0 ,
ÿ_{4} + a_{3}ẏ_{2} = 0 ,  (8)

which can be easily integrated to yield y_{2} = \frac{λ_{2}(1−C)−λ_{1}S}{a_{3}} , y_{4} = \frac{λ_{1}(1−C)−λ_{2}S}{a_{3}} , where we write S, C for sin a_{3}t and cos a_{3}t, respectively, and where λ_{1}, λ_{2} are integration constants. These solutions also fulfil the initial conditions y_{2}(0) = y_{4}(0) = 0.

Such a Jacobi field is orthogonal to the tangent vector of the geodesic γ [1, p. 294]. The component y_{1}(t), therefore, may be found from the equation

0 = ⟨y, ẋ⟩ = y_{1}a_{3} + ẋ_{2}y_{2} + 2x_{2}y_{4}a_{3} + ẋ_{4}y_{4} .

Note that there is only one null-geodesic through 1 with a_{3} = 0. It coincides with a one-parameter subgroup, is contained in T = {x: x_{3} = 0}, and has no points conjugate to 1 along itself. That is why our Jacobi field becomes zero iff x_{3} = πk with k ∈ ℤ. Therefore, the slice 0 < x_{3} < π has no points conjugate to 1 along null geodesics. Thus the first hypothesis of Theorem 1 is satisfied. In order to verify the second we note that the equations (3) are readily integrated and have the following solutions for a_{3} ≠ 0:

We set C(t) = cos a_{3}t, S(t) = sin a_{3}t, S_{2}(t) = sin(2a_{3}t) and C_{2}(t) = cos(2a_{3}t). Then we have

x(t; a_{1}, …, a_{4}) \{ \begin{matrix}x_{1}(t; a_{1}, …, a_{4}) = \frac{a_{2}a_{4}}{2a_{3}^{2}} + \frac{a_{2}a_{4}}{2a_{3}^{2}} (C_{2}(t) − 2C(t))+ \\ (a_{1} + \frac{a_{2}^{2}+a_{4}^{2}}{2a_{3}} )t + \frac{a_{4}^{2}−a_{2}^{2}}{4a_{3}^{2}} S_{2}(t) − \frac{a_{4}^{2}}{a_{3}^{2}} S(t), \\ x_{2}(t; a_{1}, …, a_{4}) = \frac{a_{4}}{a_{3}} (1 − C(t)) + \frac{a_{2}}{a_{3}} S(t), \\ x_{3}(t; a_{1}, …, a_{4}) = a_{3}t, \\ x_{4}(t; a_{1}, …, a_{4}) = \frac{a_{2}}{a_{3}} (C(t) − 1) + \frac{a_{4}}{a_{3}} S(t). \end{matrix}

The function f: L_{1} → M_{1} , f(a_{1}, a_{2}, a_{3}, a_{4}) = x(a_{3}; a_{1}, a_{2}, 1, a_{4}) maps a suitable open subset V in L_{1} diffeomorphically onto the slice U in M_{1} . Let γ^{+} denote the future geodesic ray from the identity in the direction of e_{1} . The f maps the portion ∂K^{+} \ γ^{+} of the boundary of the light cone onto the set ∂I^{+} ∩ U of all points on all light like geodesics in U. Since ∂K^{+} \ γ^{+} divides V into two components, then ∂I^{+} ∩ U separates U into two components. Thus the hypotheses of Theorem 1 are satisfied and we obtain the following result:

Theorem 2. A Lorentzian Lie group M_{1} is future distinguishing (hence also strongly causal, stably causal, causally continuous in view of homogeneity).

3. Future distinguishability of the spaces M_{2} = M_{2}(q).

The Lie algebra L_{2} = L(M_{2}) is defined via

[e_{4}, e_{1}] = e_{1} .  (9)

The Lie group M_{2} itself is R^{4} with z = x · y given by

z = (x_{1} + y_{1}e^{x_{4}}, x_{2} + y_{2}, x_{3} + y_{3}, x_{4} + y_{4}) .  (10)

The commutation rule for its Lie algebra in this coordinate system is exactly (9). We fix a Lorentzian form

g̃ = ( \begin{matrix}0 & 1 & 0 & 0 \\ 1 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q \end{matrix} )

at 1 = (0, 0, 0, 0) and extend it to M_{2} via left translations. We thus get for every q > 0 a homogeneous Lorentzian space, also denoted by M_{2} . It is proved in [7] that M_{2} is not uniformly stably causal. If a future cone in L_{2} is fixed by K^{+} = {a ∈ L_{2} : a^{2} < 0, a_{2} > 0}, then I^{+} is entirely contained in the halfspace x_{2} > 0 and M_{2} possesses no causal cycles.

Here are the equations for a geodesic γ : x = x(t) passing through 1 with initial tangent vector a :

ẋ_{1} = a_{1}e^{x_{4}} + a_{2}(e^{2x_{4}} − e^{x_{4}}), ẋ_{2} = a_{2}e^{x_{4}} ,
ẋ_{3} = a_{3}, ẋ_{4} = −\frac{a_{2}x_{1}}{q} + a_{4} .  (11)

They are more complicated than (3), but we shall solve our problems without integrating them.

We find ds^{2} = g_{ij}dx^{i}dx^{j} in the coordinates (10) as in Section 2:

ds^{2} = 2e^{−x_{4}}dx_{1}dx_{2} − dx_{2}^{2} + dx_{3}^{2} + qdx_{4}^{2} .

The non-zero connection one-forms are

\begin{aligned}Γ_{1}^{1} & = \frac{dx^{4}}{2} , Γ_{2}^{1} = −\frac{e^{x_{4}}dx^{4}}{2} , Γ_{4}^{1} = −\frac{dx^{1}}{2} − \frac{e^{x_{4}}dx^{2}}{2} , \\ Γ_{2}^{2} & = −\frac{dx^{4}}{2} , Γ_{4}^{2} = −\frac{dx^{2}}{2} , Γ_{1}^{4} = \frac{e^{−x_{4}}dx^{2}}{2q} , Γ_{2}^{4} = \frac{e^{−x_{4}}dx^{1}}{2q} . \end{aligned}

Similarly to (6), the expressions for the curvature two-forms are

\begin{aligned}θ_{1}^{1} & = −\frac{e^{−x_{4}}dx^{1} ∧ dx^{2}}{4q} , θ_{2}^{1} = \frac{dx^{1} ∧ dx^{2}}{4q} , θ_{4}^{1} = −\frac{dx^{1} ∧ dx^{4}}{2} , \\ θ_{2}^{2} & = \frac{e^{−x_{4}}dx^{1} ∧ dx^{2}}{4q} , θ_{4}^{2} = −\frac{dx^{2} ∧ dx^{4}}{4} , θ_{1}^{4} = \frac{e^{−x_{4}}dx^{2} ∧ dx^{4}}{4q} , \\ θ_{2}^{4} & = \frac{e^{−x_{4}}dx^{1} ∧ dx^{4}}{4q} − \frac{dx^{2} ∧ dx^{4}}{4q} . \end{aligned}

As in Section 2, we are searching for the solution y(t) of the system (7) with y(0) = y(t) = 0 for some t > 0. That is why y_{3} ≡ 0. The other equations form a system for y_{1}, y_{2}, y_{4} as follows:

ÿ_{1} − ẏ_{1}ẋ_{4} − ẏ_{2}ẋ_{2}e^{x_{4}} − ẏ_{4}(ẋ_{1} + ẋ_{2}) − y_{4}ẋ_{2}ẋ_{4} = 0,

ÿ_{2} − ẏ_{2}ẋ_{4} − ẏ_{4}ẋ_{2} = 0 , (12)

ÿ_{4} + \frac{a_{2}ẏ_{1} + ẋ_{1}e^{−x_{4}} ẏ_{2} − a_{2}ẋ_{1}y_{4}}{q} = 0.

We find y_{1} from 0 = ⟨y, ẋ⟩ = g_{ij}y^{i}ẋ^{j} and get a linear first-order system for ẏ_{2} , ẏ_{4} :

ÿ_{2} − ẋ_{4}ẏ_{2} − ẋ_{2}ẏ_{4} = 0 ,
ÿ_{4} + \frac{ẋ_{2}ẏ_{2}}{q} − ẋ_{4}ẏ_{4} = 0 .  (13)

It can be writen as ż = A(t)z with z = (ẏ_{2}, ẏ_{4}) and

A = ( \begin{matrix}ẋ_{4} & ẋ_{2} \\ −ẋ_{2}/q & ẋ_{4} \end{matrix} ) .

The matrix A of this system commutes with its integral

( \begin{matrix}x_{4} & x_{2} \\ −x_{2}/q & x_{4} \end{matrix} ) .

The latter makes it possible to find the fundamental solutions

ẏ_{2} = e^{x_{4}}C(x_{2}), ẏ_{4} = −\frac{e^{x_{4}}S(x_{2})}{\sqrt{q}},

and

ẏ_{2} = \sqrt{q}e^{x_{4}}S(x_{2}), ẏ_{4} = e^{x_{4}}C(x_{2}),

where C(x_{2}) = cos \frac{x_{2}}{\sqrt{q}}, S(x_{2}) = sin \frac{x_{2}}{\sqrt{q}}.

In order to integrate these equations, we will use (11):

y_{2}(τ) = ∫_{0}^{τ} e^{x_{4}(t)}C(x_{2}(t)) dt = ∫_{0}^{τ} \frac{ẋ_{2}(t)}{a_{2}}C(x_{2}(t)) dt = \frac{\sqrt{q}}{a_{2}} ∫_{0}^{x_{2}(τ)} dS = \frac{\sqrt{q}S(x_{2}(τ))}{a_{2}}.

Therefore,

y_{2} = \frac{λ_{1}\sqrt{q}S(x_{2}) + λ_{2}q(1 − C(x_{2}))}{a_{2}}, y_{4} = \frac{λ_{1}(C(x_{2}) − 1) + λ_{2}\sqrt{q}S(x_{2})}{a_{2}} (14)

is the general solution of (13). It is not difficult now to find y_{1}(t), but for our purposes this will not be necessary.

There is only one null geodesic with a_{2} = 0. It coincides with a oneparameter subgroup, is contained in T = {x : x_{2} = 0} and has no points conjugate to 1 along itself. We deduce, as in Section 2, that the points cojugate to 1 along null future geodesics closest with respect to x_{2} lie in the hypersurface x_{2} = π\sqrt{q}. So the first hypothesis of Theorem 1 is satisfied.

In order to verify the second, we integrate the equations (11): We set b = 1 + a_{3}^{2} + qa_{4}^{2} , Δ = 4a_{3}^{2} − b^{2} . Note that b > 0 and thus Δ < 0. Further set S(t) = sin \frac{a_{3}(t+λ)}{\sqrt{q}} and C(t) = cos \frac{a_{3}(t+λ)}{\sqrt{q}} with λ solving the equation

\sqrt{−Δ} sin \frac{a_{3}λ}{\sqrt{q}} = b^{2} − 2a_{3}^{2}.

Now we obtain

x(t; a_{1}, …, a_{4}) = \{ \begin{matrix}x_{1}(t; a_{1}, …, a_{4}) = q (a_{4} + \frac{(a_{3}/\sqrt{q})\sqrt{−Δ}C(t)}{\sqrt{−Δ}S(t)−b}) , \\ x_{2}(t; a_{1}, …, a_{4}) = h + 2\sqrt{q} arctan ^{\frac{b tan \frac{a_{3}(t+λ)}{2\sqrt{q}} −\sqrt{−Δ}}{2a_{3}}}, \\ x_{3}(t; a_{1}, …, a_{4}) = a_{3}t, \\ x_{4}(t; a_{1}, …, a_{4}) = log(2a_{3}^{2}) − log(b − \sqrt{−Δ}S(t)), \end{matrix} (15)

where h is a constant chosen so that x_{2}(0) = 0.

In the equation for x_{2} in (15) it is understood that the values of the function are defined for all t ∈ ℝ by continuous extension. By an argument similar to that used in the proof of Theorem 2 we now conclude that also the second hypothesis of Theorem 1 is satisfied. Therefore we obtain:

Theorem 3. The Lorentzian Lie group M_{2}(q) is future distinguishing for all q > 0. Hence is also strongly causal, stably causal and causally continuous in view of its homogeneity.

Acknowledgement. The authors would like to thank Karl-Hermann Neeb for his comments on the subject matter of this article which have contributed the clarification of certain aspects.

References

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[3] Hofmann, K. H., and J. D. Lawson, The local theory of semigroups in nilpotent Lie groups, Semigroup Forum 23 (1981), 343–357.

[4] Levichev, A., Some methods for investigation of causal structure of homogeneous Lorentzian manifolds, USSR Academy of Sciences, Siberian Branch, Institute for Mathematics, Novosibirsk, 1986, preprint (in Russian).

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[8] Levicheva, V., Left-invariant Lorentzian metrics on the group M_{4} , II, manuscript nr. 1583–B89, deposited at VINITI, 1989; abstract in: Siberian Math. J. 31 (1990), nr. 3, 209 (in Russian).

[9] Malament, D. B., The class of continuous time-like curves determines the topology of space-time, J. Math. Phys. 18 (1977), 1399–1404.

[10] Penrose, R., “Techniques of Differential Topology in Relativity”, Philadelphia, SIAM Publications, 1972.

Novosibirsk, Mathematical Institute of the Siberian Branch of the Russian Academy of Sciences

Received April 13, 1989 and in final form October 1, 1992