Measuring space and time

Constant of proportionality

Kilometers are proportional to miles in measuring distance relative to a solid body of reference. The conversion factor from miles to kilometers is approximately (.6). We also say that (.6) is the constant of proportionality from miles to kilometers.

  1 kilometer = .6 miles

Measuring distance in space

We know that we can use miles or kilometers to measure distance. For a given distance we know that the particular measure of miles vs kilometers is not significant. If we know the distance in miles we can always find the distance in kilometers. Likewise, if we know the distance measured in kilometers, we can find the distance measured in miles. Question: what is the conversion factor from kilometers to miles? Distance in space is a concept we have in our minds that transcends a particular measuring method -- we don't care if we use kilometers or miles to measure distance.

Constant speed

Distance traveled is constant speed times time. We can think of this as a law of constant speed.

(1)   distance = speed * time

For example, traveling 60 miles per hour for 2 hours gives 120 miles traveled.

      120 miles traveled = 60 miles/hour * 2 hours

Now consider re-orgainzing the law of constant speed equation (1) above.

(2)   speed = distance / time

In the law of constant speed, if we fix the speed then we know that increasing the distance traveled requires that we increase the time of travel in order to keep the same speed. Similarly if we decrease the distance traveled, then we must decrease the time traveled. We say that for a given constant speed, distance traveled varies in direct proportion to travel time. For instance, at the constant speed 100 miles/hour traveling 100 miles requires 1 hour, traveling twice as far requires twice as much time, and traveling half as far requires half as much time.

      100 miles/hour = 100 miles / 1 hour
                     = 200 miles / 2 hours
                     = 50 miles / .5 hour

Now think of the law of speed expressed in equations (1) or (2), and imagine that we have a particular speed S that is somehow universal, that is, we can observe the speed S from everywhere in the universe. For the fixed speed S, distance is proportional to time and the fixed speed S is a conversion factor from time to distance.

      distance = S * time

For our fixed speed S, whenever the distance traveled increases, the time of travel must increase.

The constant speed of light c (= 300,000,000 meters per second)

Given what we have discussed above, for the universal constant speed of light c = 3 * 10^8 we can say

Distance of light travel varies in direct proportion to travel time of light.
The constant of proportionality of distance and time (of light travel) is the speed of light c.
Distance and travel time of light are always proportional
We can always convert time of light travel to distance of light travel using the conversion constant c

Question: What is the conversion factor from distance of light travel to time of light travel?

Question: What is one second of time measured as a distance (of light travel)?

The concept of spacetime

Because distance and time of light travel are proportional, we know that the particular measure of time or distance is not significant. Whenever we have a measure of distance, then we can convert that to a measure of time. Similarly, we can always convert a measure of time to a measure of distance. Consequently, we have a concept of spacetime that transcends a particular measuring method, just as we have a concept of distance in space that transcends a particular measure of distance.

For our concept of spacetime, it doesn't matter whether we use seconds or meters to measure spacetime. As a matter of convention, we use meters of light travel to measure time because the other dimensions of spacetime are measured in meters (length, height, width). The conversion factor from time to distance in space is the speed of light c = 300,000,000 meters per second. The idea of measuring time as distance (of light travel time) is very helpful in perceiving time (conceptually and mathematically) as a 4th dimension just like the other three dimensions.

Question: How old are you in meters?