Page 75 - Modern Spatiotemporal Geostatistics

56 Modern Spatiotemporal Geostatistics — Chapter 2

of ds, we can write the space/time metric as in Equation 2.28. Equation 2.28
attributes a physical meaning to the quantity \dp\, even if the points are so
chosen that the corresponding \dp\ does not vanish (Einstein, 1994 [1954]).
In particular, we say that the space/time distance is "space-like" when \dp\ is
real, "time-like" when \dp\ is imaginary, and "light-like" or "null" when \dp\ is
zero. Space-like distances occur between events with a spatial separation less
than the distance the light pulse travels between the times of their occurrence;
time-like distances occur between events with a spatial separation that exceeds
the distance that the light pulse can travel in the time between them; null
distances represent events in space/time that could just be connected by a
light pulse. Mathematically, the distance (Eq. 2.28) is the Minkowski metric
2
determined from Equation 2.27 by letting g 00 = —c , gu = 1 (i = 1,... ,n),
and

Restrictions on spatiotemporal geometry imposed
by physical laws
We have seen that we can learn about the nature of space/time by studying
the characteristics of the physical system. Nature does not, of course, allow
the natural processes to vary in an arbitrary manner, but imposes constraints
in the form of physical laws. Consider a physical law governing the distribution
of a natural field X(p) that is generally expressed as

where v = (v\,... ,Vk) are known physical coefficients, BC and 1C are given
boundary and initial conditions, and L[-\ is a known mathematical functional.
Typically, Equation 2.33 is regarded as determining the values of the field
X(p] from the known coefficients v, the BC and 1C, and the space/time
coordinates. When it comes to the spatiotemporal metrical structure, however,
there are several ways in which the physical law (Eq. 2.33) can help one's effort
to determine the appropriate metric form. Some of these ways are reviewed
next.
In certain applications, Equation 2.33 leads to the following explicit ex-
pression for the metric

where x ar| d x' are ^-values at points p and p', respectively; and g [•] has
a functional form associated with -£•[•]. As far as the metrical structure is
concerned, the interpretation of Equation 2.34 can be different from that of
Equation 2.33. Given any metric \dp\, Equation 2.34 will not be satisfied
automatically. In fact, Equation 2.34 will serve to cut down the number of
possible metric models. Equation 2.34 is thus regarded as determining the
metric \dp\ of the space/time geometry from the natural field values at p
and p', the coefficients v, the BC, and the 1C. Here, then, is a sense in
which Equation 2.34 restricts the space/time geometry. On imposing Equation