12
1 Congruences of World Lines
Fig. 1.2 Sketch of the two
arbitrarily parametrized world
lines Y (t) and X(˜ t), and the
geodesic Z connecting two
points on these world lines.
The squared length of the
interval between the two
points x and y is proportional
to the world function σ (x, y).
The (generalized) deviation
vector along the reference
world line Y is denoted by η y
As becomes clear by the definition, the world function is a two-point function,
or biscalar. One can also introduce generalizations of ordinary space-time tensor,
called bitensors, which depend on not one but two space-times points. Pioneering
references in this respect are [2, 9], and a comprehensive review can also be found
in [10]. Here we only review some of the essential bitensor concepts.
We start by the introduction of a condensed notation, in which the space-time
point to which an index of a bitensor belongs can be directly read from the index
itself, for example B x 0 x 1 y 0 y 1 y 2 (x, y) would denote a bitensor of second rank at x and
of third rank at y. Indices attached to the world function always denote covariant
derivatives, at the given point, i.e. σ y := ∇ y σ , hence we do not make explicit use of
the semicolon in case of the world function. For any bitensor, covariant derivatives at
different points commute with each other, for instance in case of the world function
σ y 0 y 1 x 0 y 2 x 1 = σ y 0 y 1 y 2 x 0 x 1 = σ x 0 x 1 y 0 y 1 y 2 .
In many calculations the limiting behaviour of a bitensor B ... (x, y) as x
approaches the reference point y is required. This so-called coincidence limit of
a bitensor B ... (x, y) is a tensor
[B ... ] = lim
x→y
B ... (x, y),
(1.61)
at y and will be denoted by square brackets. In particular, for a bitensor B with
arbitrary indices at different points (here just denoted by dots), we have the rule [2]
[B ... ] ;y =
B ...;y
+
B ...;x
.
(1.62)
1 Congruences of World Lines
Fig. 1.2 Sketch of the two
arbitrarily parametrized world
lines Y (t) and X(˜ t), and the
geodesic Z connecting two
points on these world lines.
The squared length of the
interval between the two
points x and y is proportional
to the world function σ (x, y).
The (generalized) deviation
vector along the reference
world line Y is denoted by η y
As becomes clear by the definition, the world function is a two-point function,
or biscalar. One can also introduce generalizations of ordinary space-time tensor,
called bitensors, which depend on not one but two space-times points. Pioneering
references in this respect are [2, 9], and a comprehensive review can also be found
in [10]. Here we only review some of the essential bitensor concepts.
We start by the introduction of a condensed notation, in which the space-time
point to which an index of a bitensor belongs can be directly read from the index
itself, for example B x 0 x 1 y 0 y 1 y 2 (x, y) would denote a bitensor of second rank at x and
of third rank at y. Indices attached to the world function always denote covariant
derivatives, at the given point, i.e. σ y := ∇ y σ , hence we do not make explicit use of
the semicolon in case of the world function. For any bitensor, covariant derivatives at
different points commute with each other, for instance in case of the world function
σ y 0 y 1 x 0 y 2 x 1 = σ y 0 y 1 y 2 x 0 x 1 = σ x 0 x 1 y 0 y 1 y 2 .
In many calculations the limiting behaviour of a bitensor B ... (x, y) as x
approaches the reference point y is required. This so-called coincidence limit of
a bitensor B ... (x, y) is a tensor
[B ... ] = lim
x→y
B ... (x, y),
(1.61)
at y and will be denoted by square brackets. In particular, for a bitensor B with
arbitrary indices at different points (here just denoted by dots), we have the rule [2]
[B ... ] ;y =
B ...;y
+
B ...;x
.
(1.62)