Author:
Marín-Salvador Adrià,Rubio Roberto
Abstract
AbstractWe compute the contact manifold of null geodesics of the family of spacetimes $$\left\{ \left( \mathbb {S}^2\times \mathbb {S}^1, g_\circ -\frac{d^2}{c^2}dt^2\right) \right\} _{d,c\in \mathbb {N}^+\text { coprime}}$$
S
2
×
S
1
,
g
∘
-
d
2
c
2
d
t
2
d
,
c
∈
N
+
coprime
, with $$g_\circ $$
g
∘
the round metric on $$\mathbb {S}^2$$
S
2
and t the $$\mathbb {S}^1$$
S
1
-coordinate. We find that these are the lens spaces L(2c, 1) together with the pushforward of the canonical contact structure on $$ST\mathbb {S}^2\cong L(2,1)$$
S
T
S
2
≅
L
(
2
,
1
)
under the natural projection $$L(2,1)\rightarrow L(2c,1)$$
L
(
2
,
1
)
→
L
(
2
c
,
1
)
. We extend this computation to $$Z\times \mathbb {S}^1$$
Z
×
S
1
for Z a Zoll manifold. On the other hand, motivated by these examples, we show how Engel geometry can be used to describe the manifold of null geodesics of a certain class of three-dimensional spacetimes, by considering the Cartan deprolongation of their Lorentz prolongation. We characterize the three-dimensional contact manifolds that are contactomorphic to the space of null geodesics of a spacetime. The characterization consists in the existence of an overlying Engel manifold with a certain foliation and, in this case, we also retrieve the spacetime.
Funder
Universitat Autònoma de Barcelona
Publisher
Springer Science and Business Media LLC
Reference24 articles.
1. Adachi, J.: Engel structures with trivial characteristic foliations. Algebr. Geom. Topol. 2, 239–255 (2002)
2. Alfredo B.: Geometric structures and causality in the space of light rays of a spacetime. PhD thesis, Universidad Complutense de Madrid (2008)
3. Alfredo, B., Alberto, I., Javier, L.: On the space of light rays of a spacetime and a reconstruction theorem by Low. Class. Quant. Gravity 31(7), 075020 (2014)
4. Bernal, A.N., Sánchez, M.: On smooth Cauchy hypersurfaces and Geroch’s splitting theorem. Comm. Math. Phys. 243(3), 461–470 (2003)
5. Results in Mathematics and Related Areas;AL Besse,1978