iSoul In the beginning is reality

# “Synchronizing” space

An isodistance map shows the contours of equal distances from a central point. These would be circles on a map if distance is measured “as the crow flies.” The shapes vary if distance depends on a road network:

But how do you tell if two distances are the same? Different observers have different distance measuring instruments. There must be a way to ensure consistency of the distance measuring instruments – e.g., rigid measuring rods.

The problem is analogous to the need to synchronize clocks for 1D time. The Einstein synchronization convention is the standard way to do this:

Position 1 has clock 1 and rod 1; position 2 has clock 2 and rod 2. A light signal is sent from position 1 at time τ1 (defined from a specified point in time) to position 2 and immediately back, e.g. by means of a mirror. Its arrival time back at position 1 is τ2. This synchronization convention sets clock 2 so that the time τ3 of signal arrival and reflection is defined as

τ3 = τ1 + ½ ( τ2 − τ1) = ½ ( τ1 + τ2).

Synchronization is also achieved by slowly transporting a third clock from position ρ1 to position ρ2.

Similarly, one needs to synmacronize measuring rods for 1D space. As synchronize is from transliterated Greek syn + chron + ize (“to occur at the same time”) so synmacronize is from syn + macron + ize (“to occur at the same length”). Follow the convention above and then:

Use the constant speed of light, c, to convert the times into corresponding distances, setting rod 2 so that the distance of signal arrival and reflection is defined to be

3 = 1 + ½ (21) = ½ (1 + 2), i.e.,

ρ3 = ρ1 + ½ (ρ2ρ1) = ½ (ρ1 + ρ2), where ρ1, ρ2, and ρ3 are the distances corresponding to τ1, τ2, and τ3.

Synmacronization can also be achieved by slowly transporting a third measuring rod from position 1 to position 2.

Both space and time require consistent measuring instruments.