Shulou(Shulou.com)11/24 Report--

In 1905, Einstein mentioned a thought experiment in his paper on the electrodynamics of moving bodies: two identical clocks, one at the equator and the other at the North Pole. Because of the rotation of the earth, the speed of the equator is faster than the pole, so the clock on the equator is slower than that of the North Pole. Now, we know that this is the time expansion caused by velocity in special relativity.

We know that the speed of the equator is about 464 meters per second. According to the special relativistic time expansion formula, it can be calculated that the equatorial clock is about 100 nanoseconds slower than the Arctic clock every day. This small time difference could not be measured when the special theory of relativity was first proposed, but with the development of the atomic clock, it was found that the time on the earth's surface had nothing to do with the latitude of the earth.

Because there is another factor to consider, which is the gravity of the earth. The earth is not a perfect sphere. Because of its rotation, its poles are slightly flattened and the equator rises. This means that the equatorial surface is farther from the center of the earth than the poles, so objects on the equatorial surface have less gravity than the poles. According to general relativity, the stronger the gravitational field an object experiences, the slower its clock runs. Therefore, the clock in the North Pole travels more slowly than the clock on the surface of the equator.

So who will win these two effects? It turns out that they cancel out completely, and clocks at sea level pass at the same speed regardless of latitude, which has been verified by precise measurements of atomic clocks.

In fact, it is no coincidence that the two effects cancel each other out. According to general relativity, the inherent interval d τ of time in a weak gravitational field can be given by the following formula:

Where Φ _ g is the gravitational potential of the earth. For the clock at rest on the earth's surface, we have dr=d θ = 0 and d Φ = ω dt, which can be substituted into the above formula and arranged to get:

If you look carefully, the formula in parentheses under the root sign is the effective potential.

Φ _ eq= Φ _ g Mel 1 hand 2 ω ²r ²sin ²θ it is the gravitational potential of the earth minus the potential generated by the centrifugal force. The effective potential should be a constant on the surface of the earth, otherwise it will produce a gradient force that we have not experienced. It is precisely because it is a constant that the inherent time interval d τ is the same at the sea level of the earth.

If we want to verify whether the effective potential of the two poles is equal to that of the equator, the following calculation can be referred to:

This article comes from the official account of Wechat: Vientiane experience (ID:UR4351), author: Eugene Wang

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