How does the definition of a second affect timekeeping on Earth and Mars?

[sydneybself]

According to "Physics for Engineers and Scientists by Ohanian and Markert, a second is the time needed for 9,192,631,770 vibrations of a cesium atom.

In 1971, Hafele and Keating put four cesium-beam atomic clocks aboard commercial air liners which were flown around the world - two eastward, two westward and compared their times with two clocks which remained at the airport. After the test, the three pairs of clocks disagreed with one another. These results are supported by the experiments done with muons.

How long is a second, really? (if the question has a real[?] meaning)?
Wouldn't clocks on Mars keep different times from those on earth?
What about the length of a meter, which is based on the measurement of the speed of light?

 

[Student100]

To quote wiki

During the 1970s it was realized that gravitational time dilation caused the second produced by each atomic clock to differ depending on its altitude. A uniform second was produced by correcting the output of each atomic clock to mean sea level (the rotating geoid), lengthening the second by about 1×10−10. This correction was applied at the beginning of 1977 and formalized in 1980. In relativistic terms, the SI second is defined as the proper time on the rotating geoid.[24]

The definition of the second was later refined at the 1997 meeting of the BIPM to include the statement

This definition refers to a caesium atom at rest at a temperature of 0 K.

The revised definition seems to imply that the ideal atomic clock contains a single caesium atom at rest emitting a single frequency. In practice, however, the definition means that high-precision realizations of the second should compensate for the effects of the ambient temperature (black-body radiation) within which atomic clocks operate, and extrapolate accordingly to the value of the second at a temperature of absolute zero.

The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom—pretty much verbatim from various texts.

 

[Nugatory]

In 1971, Hafele and Keating put four cesium-beam atomic clocks aboard commercial air liners which were flown around the world - two eastward, two westward and compared their times with two clocks which remained at the airport. After the test, the three pairs of clocks disagreed with one another.

Yes. That's because a different number of seconds elapsed on the various journeys, and the scientists traveling along with the clocks aged by (imperceptibly) different amounts one the journey. This despite the fact that they started at the same place at the same time and ended at the same place at the same time.

So...

How long is a second, really? (if the question has a real[?] meaning)?

Just as you said... long enough for a cesium atom under your nose to undergo 9,192,631,770 oscillations. And therefore:

Wouldn't clocks on Mars keep different times from those on earth?

Yes. Not very different, but occasional corrections would be needed if you wanted to keep a clock on Mars and on Earth synchronized. For that matter, we have to worry about this effect with the GPS satellites in orbit around earth.

What about the length of a meter, which is based on the measurement of the speed of light?

The definition of the meter is subtly different from what you've said: it is defined as the distance that light travels in a particular time (which is of course given in seconds so is rooted in the cesium atom definition of the second). Thus, I can construct a one-meter measuring rod without knowing the speed of light, just by seeing how far light travels in a given time.