If clocks run slow in a gravitational field, how can we know the true age of the Solar System and Universe?
I'm a budding astronomer and I wanted to know: The new results from the Cosmic Microwave Background seem to have convinced some astronomers that the Universe is some 13.8 billion years old. However, this sounds a lot like absolute time. If time is measured differently in different frames, how can the Universe have only one age?
Or,similarly, the oldest meteorites seem to imply an absolute age of 4.56 billion years. But how can this be "the" age of the solar system if identical clocks tick at different rates depending on their distance from the Sun? Shouldn't meteorites from Mercury, say, display younger ages because of more slowly ticking radioactive clocks? Thanks much.
This is a really interesting question! You're right that because of general relativity, clocks in a gravitational field will run slower. Therefore, rocks closer to the Sun will have clocks that tick slower than rocks further away, and so, theoretically, Mercury rocks will be younger. However, if you actually work out what the difference in rates is, this is a very small effect, even over the whole history of the Solar System.
For example, if I use a simple calculation (assuming a weak gravitational field, which is OK for the Sun) to compute the time dilation from the surface of the Sun, then I find that clocks on the surface of the Sun will run slow by 6 seconds per year compared with clocks in distant space. In reality the difference between rocks at the distance of Mercury and the distance of, say, the asteroid belt will be smaller than this - the 6 seconds per year is the largest that you would expect over the Solar System. But let's use this large number to see what happens.
We think the age of the Solar System is about 4.5 billion years, so over that time period the difference in time measured at the surface of the Sun and time measured in deep space would be about 850 years. Since the difference is actually less than this from place to place throughout the Solar System, the true number should be even smaller. We don't know the age of the Solar system to a few hundred years accuracy, although that would be impressive! All of our dating techniques have errors, and also the meteorites that we date may have formed at slightly different times. In addition, asteroids may have moved around the Solar System, which would change how slow their radioactiveclocks go. So I guess it's theoretically possible that, in the future, the general relativity effect might limit our ability to date Solar System materials, but I think we probably need some awesome technology before it becomes a problem. (I think planetary scientists would be really happy if they had to worry about this when dating things!)
The same argument applies to the Universe. Although the Universe is much older, we also have bigger errors on what we think the true age is. The fact that we measure things from within the Sun's gravitational environment will give us only a small measurement error. So you're right that the ages of things will be different depending on the reference frame, but the effect is small enough that it can't compete with the inaccuracies in our measurement ability.
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