When Einstein Was Wrong

To the Editors:

In a recent interesting review of two books on the nature of science [“What Scientists Really Do,” NYR, October 23], Priyamvada Natarajan briefly discusses the role of the theory of relativity in the rate of satellite clocks that circle the earth as part of the GPS system. She notes that the special theory of relativity—which Einstein published in 1905—predicts that moving clocks run slower than identical stationary clocks. This is something that is verified daily when physicists compare the lifetimes of moving unstable particles to the same particles at rest.

But there is a second effect—gravitation—which also predicts slower clock rates. The closer to the surface of the earth, the larger the gravitational potential and the slower the clock. When these two effects are added for the satellite clocks—taking into account the distance of the satellite from the earth and hence the weaker gravitational potential—the satellite clocks run faster.

It is interesting to note that in his 1905 paper Einstein made a prediction about clock rates that is essentially wrong. He writes, “Thence we conclude that a balance clock on the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions.” If the experiment had been carried out in 1905, Einstein would have been surprised to learn that clocks go at sensibly the same rate. There is the effect of gravitation that Einstein did not know about in 1905. The equator clock is further from the gravitation center—the earth bulges—than the pole clock and hence the gravitational effect is smaller.

If the special relativity and gravitational effects are both taken into account, the rates of the two clocks are nearly identical, which is not what Einstein predicted. In fact all the identical clocks on the surface of the earth run at the same rate. One can only wonder what Einstein would have made of this result in 1905.

Jeremy Bernstein
Aspen, Colorado

Priyamvada Natarajan replies:

Jeremy Bernstein is right to note that Einstein himself, using his theory of special relativity, got the prediction of clock speeds wrong in 1905. He incorrectly predicted that due to the earth’s rotation, a clock at the equator would run slower than one at the poles. Einstein did not anticipate his own theory of general relativity, which he would need to get it right—this would come in 1915. In a 2005 article in Physics Today titled “A Small Puzzle from 1905,” the physicists Alex Harvey and Engelbert Schucking pointed out that Einstein made this error by failing to take into account an effect of general relativity positing that clocks more deeply embedded in a gravitational field would run slower. Clocks run slightly faster at the equator compared to the poles because the earth’s rotation produces a slight bulge at the equator. However, the earth is also rotating faster at the equator. These two effects compensate for each other exactly, causing clocks to actually run at the same rate in both locations.

Harvey and Schucking give another explanation for why the rates of polar and equatorial clocks must be the same. They write that in the moving earth frame, both clocks are at rest, and since both clocks are at the same effective gravitational potential, they tick at the same rate. But as I mention in “What Scientists Really Do,” these two effects do not compensate exactly for a clock on earth and one aboard a GPS satellite.

While Einstein did not anticipate his theory of general relativity, a recently discovered unpublished manuscript among his papers at Hebrew University, written in 1931, shows he was curiously prescient in another instance. Despite his initial resistance to Edwin Hubble’s discovery of the expanding universe, and even after his public embrace of it, Einstein struggled privately with the idea. The alternative that he came up with anticipated the steady-state model that would come almost twenty years later (arxiv.org/abs/1402.0132).