Let
g
g
be a Lorentzian metric on the plane
R
2
\mathbb {R}^2
that agrees with the standard metric
g
0
=
−
d
x
2
+
d
y
2
g_0=-dx^2+dy^2
outside a compact set and so that there are no conjugate points along any time-like geodesic of
(
R
2
,
g
)
(\mathbb {R}^2,g)
. Then
(
R
2
,
g
)
(\mathbb {R}^2,g)
and
(
R
2
,
g
0
)
(\mathbb {R}^2,g_0)
are isometric. Further, if
(
M
,
g
)
(M,g)
and
(
M
∗
,
g
∗
)
(M^*,g^*)
are two dimensional compact time oriented Lorentzian manifolds with space–like boundaries and so that all time-like geodesics of
(
M
,
g
)
(M,g)
maximize the distances between their points and
(
M
,
g
)
(M,g)
and
(
M
∗
,
g
∗
)
(M^*,g^*)
are “boundary isometric”, then there is a conformal diffeomorphism between
(
M
,
g
)
(M,g)
and
(
M
∗
,
g
∗
)
(M^*,g^*)
and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.