I've heard explanations for time dilation and relativity many times before, and yet whenever I read about it, I'm still very confused.
This NPR report isn't helping:
Clocks tick faster on Mars than they do on Earth, in part because Mars experiences less gravitational pull from the Sun. Now scientists have calculated just how much faster — 477 microseconds [per day], on average.
Yes, you're reading that correctly:
477 microseconds [per day], on average.
How much does that add up to, exactly? Let's do some math.
According to online conversion calculators, there are 1,000,000 microseconds in one second.
1,000,000 / 477 = 2,096.436, so it will take about 2,100 days for a clock on Mars to advance one second ahead of a clock on Earth, or 5.75 years.
There are 3,600 seconds in one hour, so — assuming 3,600 seconds x 2,100 days — it would take you 7,560,000 days, or about 20,700 years, to realize an hour's worth of time dilation.

Nearly 21,000 years to lose an hour's worth of time — I think you'll be checked out by then!

As incredibly infinitesimal as that amount of time may seem, it's actually rather vital if we're hoping to one day put human boots on the Martian ground:
[I]f you want to translate the communication infrastructure that we have on Earth to Mars, then we'll have to have the clocks synchronize pretty accurately. If you want to have multiple robots and scale up the operation, then the robots have to talk to each other. And therefore, we'll have to have a different way to communicate. And that's where the calculation comes into play if we are envisioning to becoming an interplanetary species.
Just be sure to adjust your watch accordingly if you happen to find yourself on Mars!

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