In general relativity, Synge's world function is an example of a bitensor, i.e. a tensorial function of pairs of points in the spacetime. Let
x
,
x
′
be two points in spacetime, and suppose
x
belongs to a convex normal neighborhood of
x
so that there exists a unique geodesic
γ
(
λ
)
from
x
to
x
′
, up to the affine parameter
λ
. Suppose
γ
(
λ
0
)
=
x
′
and
γ
(
λ
1
)
=
x
. Then Synge's world function is defined as:
σ
(
x
,
x
′
)
=
1
2
(
λ
1
−
λ
0
)
∫
γ
g
μ
ν
(
z
)
t
μ
t
ν
d
λ
where
t
μ
=
d
z
μ
d
λ
is the tangent vector to the affinely parametrized geodesic
γ
(
λ
)
. That is,
σ
(
x
,
x
′
)
is half the square of the geodesic length from
x
to
x
′
. Synge's world function is well-defined, since the integral above is invariant under reparametrization. In particular, for Minkowski spacetime, the Synge's world function simplifies to half the spacetime interval between the two points:
σ
(
x
,
x
′
)
=
1
2
η
α
β
(
x
−
x
′
)
α
(
x
−
x
′
)
β
