Does Kepler's 3rd law of planetary motion violate the first postulate of special relativity?

Question

Consider a distant observer traveling at .866 c relative to the solar system along the line that is co-linear with the sun's axis of rotation. According to his/her wristwatch the observer measures the earth's orbital period around the sun to be 730.5 days, correct?

But the observer also measures the major and minor axes of the earth's orbit around the sun to be identical to its major and minor axes in the solar system's rest frame, where the orbital period is only 365.25 days.

So it appears as if Kepler's 3rd law of planetary motion is only valid in the rest frame of the solar system. Does this violate the first postulate of special relativity?

If so then how can Kepler's 3rd law be made frame-invariant?

Answer 1

It appears as if Kepler's 3rd Law of Planetary Motion is only valid in the rest frame of the solar system. Does this violate the First Postulate of Special Relativity?

No. It just means that Kepler’s laws are not actually laws of physics. Instead, they are approximations to the laws of physics in the non-relativistic limit. Specifically, Kepler’s laws are a non-relativistic approximation to general relativity which is the correct law of physics governing gravity.

In fact, experiments have shown that Newtonian gravity, which embodies Kepler’s laws, is incorrect. Numerous observations contradict Newtonian gravity (including Kepler’s laws) and support general relativity as the correct law of gravity.

The first postulate does not imply that everything that is historically called a “law” is valid in all frames. Instead, it asserts that the equations which actually are laws of physics are valid in all frames.

Answer 2

I think the analysis being performed in this Q and A is based on a simple misunderstanding of relativity. The Principle of Relativity (referred to above as the 1st Postulate) states that the laws of physics are valid in any inertial frame of reference locally in that frame. This means that if the observer moving at $0.866c$ (w.r.t the Solar System) does an experiment in a laboratory on his own spacecraft, where everything in the laboratory is stationary with respect to him, all normal physical laws of motion will hold good. Every result in his lab will agree with the results in our labs on Earth. If his entire solar system around him is speeding through the galaxy at $0.866c$ (e.g. like this), Kepler's 3rd Law will hold for that system just as well as for ours.

The Principle of Relativity does not say that far away events and objects will look the same regardless of your state of motion. In fact, the groundbreaking aspect of the theory of relativity was that, in order for the first paragraph to be true, events and objects moving relative to the viewer must look different. For example, in Einstein's original 1905 paper, he shows that a rigid sphere (Sec. 4), seen from a stationary point of view, will be an ellipsoid compressed in the direction of motion when seen from a moving frame. This will apply to the shape of the planetary orbits in the original example (especially if the orbital speeds of the planets are much less than the speed of light, as is the case), as well as the shapes of the planets themselves. Likewise, light which is a certain frequency when viewed in the same frame as the source will be blue-shifted and higher-intensity when the source is moving toward the observer (Sec. 7).

All of these effects stem from an observer viewing an object or event that is far away and/or moving relative to him. But Special Relativity guarantees that for events in your immediate vicinity and stationary with respect to you, the laws of physics do not depend on your rate of motion (with respect to something else). Thus, there is no preferred frame of "absolute rest," because everything is at absolute rest with its immediate environment.

Answer 3

Kepler's third law of planetary motion can be demonstrated to be frame invariant in two ways

1. By dividing the Kepler constant K by $\gamma^2$ which makes Kepler's third law absolutely frame invariant

$$\frac{R^3}{T^2}=\frac{K}{\gamma^2}$$

Let's work the problem to make sure. In the solar system's frame $K$ has the numerical value $7.5x10^{-6}$ when the orbital period $T$ is measured in $days$, the semi-major axis of the orbit $R$ is measured in astronomical units $AU$ and $\gamma = 1$

$$\frac{1AU^3}{365days^2}=\frac{7.5}{1}\times10^{-6}\frac{AU^3}{days^2}$$

While in the distant observer's frame where $\gamma=2$ and hence the Earth's orbital period is 730 days, we have

$$\frac{1AU^3}{730days^2}=\frac{7.5}{4}\times10^{-6}\frac{AU^3}{days^2}$$

Which is also true.

2. By using Kepler's law in its general form as he used it, which automatically compensates for relativistic time dilation. Consider the orbital periods $T$ and semi-major axes $R$ for the Earth and Mars. In this case Kepler's third law can be expressed as

$$\bigg(\frac{T_{Mars}}{T_{Earth}}\bigg)^2=\bigg(\frac{R_{Mars}}{R_{Earth}}\bigg)^3$$

In the frame of the solar system the Earth's orbital period is 365 days, Mars orbital period is 687 days and their ratio is 1.88. In the distant observer's frame where $\gamma=2$, the Earth's orbital period is 730 days, Mars orbital period is 1374 and their ratio is still 1.88. The manner in which the question is set up guarantees that the semi-major axes of the Earth and Mars remain unchanged in both frames. So Kepler's third law of planetary motion is shown again to be strictly frame invariant.

Crucial to these demonstrations is that the solar system's spacetime curvature is ignored. When spacetime curvature is taken into account the above results break down, as can be seen in the Big Ben Paradox? which is a recent and novel challenge to general and special relativity.