In the speach for this year's Nobel Prize, Professor Ulf Danielsson mentions that time goes by a trillionth of a second slower at his feet than at his head. This is an interesting, yet simple calculation that many people get slightly wrong. Obviously, it does not make sense to say that time is slower by an arbitrary trillionth of a second. It must slow by a trillionth of a second in a given time.
My back of the envelope calculation, making a few simplifying assumptions, like the distance between the feet and the head being 1 meter, the radius of the Earth being 6000 km and Schwarzschild radius 1 cm is that time would slow by about one trillionth of a second in one hour. This is probably what Professor Danielson intended to say, but, maybe forgot during the lecture.
Thus, the statement should have been
Time goes by a trillionth of a second an hour slower at my feet than at my head.
I am happy that this year's Nobel Prize is about black holes. I did my PhD on LIGO at Caltech with Kip Thorne and Barry Barish who received the Nobel Prize in 2017 for gravitational wave detection. Andrea Ghez has on a few occasions visited our group and I had the pleasure to meet her. Thus, I can say I have been partial to the densest objects in the universe for decades.
Later, I have used this very calculation Pofessor Danielson mentioned to show propose an avenue to use atomic clocks pick up on minute variations of the speed of time at the surface of the Earth that are linked to underground density perturbations.
Update: After my email, Prof. Danielsson corrected the lecture in both English and Swedish.
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