Your Head Is Older Than Your Feet
Contents
Einstein’s relativity
The title of this post sounds like clickbait, but it is a fact due to Einstein’s relativity that says time slows down when in motion and when deeper in a gravitational field. Both of these time dilation effects are infinitesimal for human reflexes. For example, two clocks separated by one metre vertically in the earth’s gravitational field runs at different rates by about one part in \(10^{-16}\). The lower clock runs slower by that margin; or conversely, the higher clock runs faster by that margin.
This means that for my height of 1.7 metres, time at the top of my head is gaining about \(1.7 \times 10^{-16}\) seconds over time at the bottom of my feet, each and every second. I’ve been alive for about one billion seconds (i.e., \(10^9\) seconds). Assuming a third of my lifetime is spent sleeping in a horizontal position, where time in my entire body flows at the same rate, my head is older than my feet by about \((1.7 \times 10^{-16}) \times (\frac{2}{3} \times 10^9) \approx 1 \times 10^{-7}\) seconds, or 100 nanoseconds. I find that a rather poetic thought.
Feynman’s lecture
The equation for my rough calculations above came from the legendary Richard Feynman’s lecture titled Curved Space. He explains the curved geometry of space-time in a clever way by imagining a bug walking on the 2D surface of various geometric objects. He derives a simple equation (number 42.6 in the linked site) that I’ll reproduce below in verbose form.
\[\begin{equation} \frac{\text{clock at your feet}}{\text{clock at your head}} = \; \frac{\text{gravity} \times \text{height}}{\text{lightspeed}^2} + 1 \end{equation}\]Gravity and lightspeed are well known constants, while the \(+1\) at the end indicates that a clock at your feet is slower than a clock at your head. So the fraction on the right-hand side is what matters really, and that’s the constant drift of one part in \(10^{-16}\) per metre of height that I mentioned earlier.
Centre of the Earth
Time dilation applies to planets and stars of course. I came across an interesting 2016 paper from the European Journal of Physics, titled The young centre of the Earth. The authors show simple calculations for the Earth and the Sun, whereby their centres are younger than their surfaces by 2.5 years and 39,000 years respectively. Perhaps this is even more poetic than the thought of my feet being younger than my head. Now try to imagine time at the centre of a black hole versus its event horizon!
Accurate clocks
David Wineland shared the 2012 Nobel Prize in Physics after having developed incredibly accurate clocks for more than three decades. His group published some neat results in a 2010 paper in the journal Science, titled Optical Clocks and Relativity. Using clocks that can keep time to a resolution smaller than \(10^{-17}\) seconds, they measured the relativistic time dilation due to uniform motion and gravitational potential. I also strongly recommend the following lecture by Wineland from late 2017.
Is time an illusion?
Since Einstein made time slippery in two important ways, coupled with the time-invariant nature of the laws of physics, it’s worth pondering whether time is an illusion or not. My suspicions are that it indeed is, and I’m not alone in thinking that way. Carlo Rovelli is definitely a poet, as much as a physicist. I have yet to read his latest book The Order of Time. But his lecture at The Royal Institution below is thoroughly enlightening, just as his previous books were. He remarks at one point that “your head is older than your feet”. Go figure!