1. Why Clocks Run Faster in Higher Places
According to the general theory of relativity, time passes more slowly in places with stronger gravity. A clock placed on the ground runs slower compared to a clock placed at a higher elevation [1]. Earth's gravity gets slightly weaker as you move away from the ground. Because gravity is not exactly the same at the bottom and top of a staircase, two identical clocks placed at different heights will tick at slightly different rates [1]. However, this difference is so tiny that we rarely notice it with everyday clocks. The question becomes: how can we detect such a small difference?
2. How Optical Lattice Clocks Detect a 1 cm Height Difference
The optical lattice clock was proposed by Hidetoshi Katori in 2001. It determines one second by measuring the frequency of light absorbed by about 1 million atoms that are cooled to extremely low temperatures and trapped in a lattice made of light [2]. Using many atoms helps reduce variations in measurements, as the average becomes more stable. However, trapping atoms with light can shift their frequency. To solve this, scientists use a special laser wavelength called the "magic wavelength," which minimizes the shift caused by the light trap [2]. The accuracy of an optical lattice clock corresponds to 18 digits, which is a deviation of about 1 second in 10 billion years [6]. Since 10 billion years is about 3 × 10^17 seconds, the error rate is roughly 3 × 10^-18. The change caused by a height difference of 1 cm is about 1.1 × 10^-18, which is nearly the same size. This means the clock's precision is comparable to the size of the change caused by a height difference of 1 cm.
3. Testing Optical Lattice Clocks at Tokyo Skytree
In April 2020, a research team from the University of Tokyo, RIKEN, and other institutions placed one portable optical lattice clock on the ground floor of Tokyo Skytree and another on the observation deck 450 m (about 1,476 feet) up. To show that the clocks could be used outside the laboratory, they were packed in a magnetic shield box with sides of about 60 cm (about 24 inches) and transported [3][7]. The team reported success in verifying the predictions of the general theory of relativity with precision close to previous satellite-based experiments, by comparing the difference in the two clocks' rates with the height difference measured by the Geospatial Information Authority of Japan [3]. If we calculate the size of the difference: since a 1 cm height change causes a rate change of about 1.1 × 10^-18, a height of about 450 m is 45,000 times that, resulting in about 5 × 10^-14. Over one day (86,400 seconds), this equals about 4 × 10^-9 seconds, or about 4 nanoseconds per day. This is a calculated value, not a direct measurement. Four nanoseconds is 4 billionths of a second, but it is large enough for an 18-digit clock to detect.
4. How a 1 mm Height Difference Changes the Time on a Clock
In 2022, a research team at JILA in the United States, led by Jun Ye, confirmed this effect on an even smaller scale. They used about 100,000 ultra-cold strontium atoms arranged in a lattice shaped like stacked pancakes, created with laser light [4]. The entire stack fits within a range of about 1 mm. Within this space, they measured a frequency shift between the atoms in the upper layer and those in the lower layer due to height. The report states that the flow of time changed with height even within the cloud of atoms [4]. Since a 1 cm difference causes a change of about 1.1 × 10^-18, a 1 mm difference should be one-tenth of that, or about 1.1 × 10^-19. The Skytree experiment measured the height of a building, while this experiment measured the thickness of an atomic cloud. The same principle works across these vastly different sizes.
5. Using Clocks as Rulers to Measure Height Differences
If time shifts with height, then if clock precision is sufficient, comparing clocks can help find the difference in gravity between two locations, which means finding the difference in height. This is called "relativistic geodesy," and applications such as monitoring vertical changes in the ground (crust) of a few centimeters are expected [6]. This could provide a new way to check height changes, different from traditional methods using leveling or satellites. However, this is still at the research stage, so it remains an expectation. Another topic is how we define one second. The redefinition of the "second" is planned for 2030 by the General Conference on Weights and Measures. Optical lattice clocks are candidates to replace current cesium atomic clocks [5]. It has not been decided that optical lattice clocks will be adopted; they are only candidates.
6. How to Estimate the Time Difference Caused by Height at Home
If you have a calculator, you can estimate this yourself. The rate of difference can be estimated by multiplying gravitational acceleration (9.8) by height (in meters) and dividing by the speed of light squared (9 × 10^16). For example, if the height difference between the 3rd and 1st floor of an apartment is about 6 m, the calculation is 9.8 × 6 ÷ (9 × 10^16), which is about 6.5 × 10^-16. Over one day (86,400 seconds), this is about 5.6 × 10^-11 seconds, or about 0.06 nanoseconds. What about the difference between the 2nd and 1st floor of your home? Try calculating with different heights to see if the difference doubles when the height doubles. This formula is an approximation for when the height difference is very small compared to the size of the Earth [2]. Reports on these experiments are available in Japanese from the RIKEN press release and the JST feature page.