Measuring time and other spatio-temporal quantities

Hartmut Traunmüller
Apeiron 5 (1998): 213-218.

ABSTRACT. Ordinary clocks do not measure time in the common and Newtonian sense, and there is a similar problem for spatial measurements due to effects of motion and gravitation. Einstein's theories of relativity are based on the denial of the possibility of the 'absolute' measurements that would be required. In this paper it is, nevertheless, shown how space-time can be surveyed in agreement with common sense. For this purpose, a light-clock (or equivalent) is combined with a space-time odometer. While a light-clock counts the number of reflections of a pulse of light propagating between mirrors, a space-time odometer counts the zero crossings in the field of the cosmic microwave background radiation. The readings of these two devices allow to calculate the time interval and the path length followed by the devices in Euclidean space even in the presence of local variations in gravitational potential.

When local gravitational fields can be neglected, the light clock shows proper time to, with

to = ( t2 - x2 - y2 - z2 )1/2,

while the space-time odometer shows path time ts, with

ts = ( t2 + x2 + y2 + z2 )1/2.

In all four dimensions, the (average) zero-crossing period is the unit of measurement.

When the devices are neither in motion with respect to the background radiation nor affected by a local gravitational field, both function as true clocks - in other cases neither of them can be said to measure common time t as such.

When at rest in a gravitational field, the readings of both devices are affected as if they were moving at the local velocity of escape.

When the motion as well as the gravitational field can be considered uniform, the travel time t and the spatial path length s (common sense notions!) can be calculated very simply and without any need of distinguishing between effects of velocity and gravitation as:

t = [ ( ts2 + to2 ) / 2 ]1/2

and
s = [ ( ts2 - to2 ) / 2 ]1/2.

The devices are here assumed to run in synchrony (average) when they are outside any strong local gravitational fields and at rest with respect to the background radiation.


Abstract posted in August 1996, additional information in February 1997. | pdf _ (20 kB)