El viaje a la Luna y un pequeño ejemplo sobre Relatividad...

The state of rest and motionlessness is unknown in nature.

The whole of nature, from the smallest particles up to the most massive bodies,

is in a state of eternal creation and annihilation, in a perpetual flux, 

in unceasing motion and change. 

In the final analysis, every natural science studies some aspect of motion. 

Mathematical analysis is that branch of mathematics that provides methods 

for the quantitative investigation of various processes of change, motion, and 

dependence of one magnitude on another.

M.A. Lavrent'ev, s. M. Nikol'skil - Analysis - Mathematics: its content, methods and meanin, 1964

Hace un año, estuve conversando con Eldon Hall (matemático, físico y líder del proyecto del computador Apollo) sobre el concepto de límite y una curiosidad suya sobre la relatividad del tiempo/espacio aplicada en el viaje a la luna del Apollo.

JP: Following our talk about relativity and the journey to the moon, How you define "Limit"?

EH: A variable v which approaches 0 as a "limit" is called an "infinitesimal". That is very similar to your statements about the edge of your bed. The distance to the edge goes to 0 in the limit as you move toward the edge. There are special theorems that apply to infinitesimals and limits. These theorems have never meant much to me. I guess that means in the limit I am not a mathematician.

On Einstein's relativity I can use an example from Apollo to explain Einstein's thinking which led to his development of the special theory of relativity. I can ask the question "what time did the LM touch down?' The AGC has a clock and could record the time the touch down signal is received. This time can be sent down to the ground on the telemetry. The touch down signal that was sent to the computer could be sent down to the ground also and the ground could record the time the signal is received using a clock on the ground. If both of these records are sent, they will be delayed by the same transmission time but the time the computer recorded will be approximately 1.5 seconds earlier than the time recorded on the ground. Of course Einstein did not have examples like this but he did recognize that there would be differences in the time of events as recorded by observers separated in space. Another example is the Apollo spacecraft on the way to the moon. If time as measured in the AGC is sent to the ground and compared with a clock on the ground, the observer on the ground will see that the AGC clock is running slower than the clock on the ground. This results from the fact that the transmission time between the spacecraft and ground will be increasing as the spacecraft moves away from the earth. Here we can use the word limit by asking what happens to the ground's measurement of the AGC time when in the limit the spacecraft approaches the speed of light. The basic results of the special theory of relativity is an equation, called time dilation equation, for computing the time dilation as a function of the velocity of the spacecraft. In the limit as the spacecraft's speed approaches the speed of light this function approaches zero. That is in the limit the AGC clock will stop.

It is this equation that perplexes me. Using that equation at the velocities of the Apollo Spacecraft, there will be no difference between the ground and AGC time at the moon. This fact is the reason for the question I had for you about the landing. I do not know how the LM AGC was initialized. How was the time to fire the descent rockets determined? With my limited knowledge of orbital dynamics it seems to me if the rockets were fired approximately 1.5 sec. late, then the touch down point would be 3 or 4 miles long. If MIT experts made that kind of error, it would be quite embarrassing. I do not know how to find out. I ask Dr Battin once some time ago whether relativity was considered in the Apollo guidance and his answer was typical. The velocity was to low. But at that point he was considering position of stars. I did not have a chance to question him further.