Kepler’s Symmetries and Newton’s DE

This is the first of a series of posts about gravity, and the java applet I wrote that simulates Newtonian and relativistic orbits around a (non rotating) black hole. We begin by discussing the relationship between Kepler’s observations and Newton’s universal law of gravitation.

Usually when you write a scientific applet, most of the effort is in the java coding rather than in finding the equations. That is very much true about Newton’s equations of motion around a black hole, which are very easy. His Law of universal gravitation is:
Newton's universal law of gravitation F = G m M / r r
where F is the force, G is the universal constant, M is the mass of the black hole (or other spherically symmetric gravitating body), and m is the mass attracted. We will simplify this. We will use units with GM = 1, let m << M, rewrite Force as mass x acceleration, choose Cartesian coordinates (x,y,z) so that r^2 = x^2 + y^2 + z^2 , only consider motion in 2 dimensions so z=0. The resulting simplified differential equation (DE) is:
Differential equation for Newton's gravity at black hole

Before Newton found his simple equations of motion for gravity, Kepler had observed sufficient planetary motion that he deduced a set of rules known as Kepler’s Laws of Planetary Motion. There are three of these laws (from the Wikipedia article):

1. The orbit of every planet is an ellipse with the sun at one of the foci.

2. A line joining a planet and the sun sweeps out equal areas during equal intervals of time as the planet travels along its orbit. This means that the planet travels faster while close to the sun and slows down when it is farther from the sun.

3. The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axes (the “half-length” of the ellipse) of their orbits. This means not only that larger orbits have longer periods, but also that the speed of a planet in a larger orbit is lower than in a smaller orbit.

The above laws are not written here in mathematical language, but instead in the (more or less) usual English. The above Wikipedia link has a good description of the what these laws look like in mathematics. They’re not simple. Furthermore, each planet has its own set of parameters. Newton’s law, by contrast, applies to all the planets orbiting the sun (by ignoring the relatively small forces between the planets), applies to other systems, (such as the earth and moon) and is quite simple. It was a great success.

This result of Newton is a bit of a miracle. He began with some symmetry relations (in particle physics we would call these things “conserved quantities” or “conservation laws” ) that apply to all orbits of planets around the sun, or the moon around the earth. From them he derived an equation of motion. The miracle is that the equation of motion he derived is much simpler than the conserved quantities themselves. This is the opposite of what goes on in physics today.

Let’s rewrite what happened here in particle physics terms. Kepler took observations. He fitted a formula to those observations. Today we would call the result phenomenology, which is a term from philosophy . The human brain is particularly adept at noticing patterns, or symmetries, and this is what Kepler did. For example, his second law amounts to what we now know as the conservation of angular momentum.

Reducing Kepler’s laws to a universal equation of motion wasn’t easy. Kepler wrote down his laws in 1605. It amazes me that Kepler beat the discovery of the moons of Jupiter by several years. Those moons gave immediate evidence that Kepler’s laws apparently applied to all satellite systems. But the equations of motion did not come easily from these laws. Newton wasn’t even born until 1643 and didn’t explain Kepler’s laws until his 40s. And he was the first. I would think that the primary reason the equations of motion took so long was due to the primitive state of mathematics at the time. Newton had to write his own textbooks.

Einstein’s Symmetries

Newton’s simple equations of motion stood experimental test until Einstein deduced an improvement nearly a century ago, in 1915, that we know today as General Relativity. Einstein began with Special Relativity, and we will began there as well.

The special theory of relativity proposed that the speed of light is the same for all observers. This was wildly incompatible with the assumptions of Newton and it was quite surprising that it was compatible with observations. But there was no equation of motion proposed here. What Einstein did with the special theory of relativity, as with Kepler’s Laws, was propose a symmetry principle. Modern physics explicitly treats the special theory of relativity as a symmetry relation; it goes by Poincare invariance or Lorentz symmetry according as one does or does not (respectively) include translations. The symmetry of special relativity is that the laws of physics be unchanged under boosts and rotations (Lorentz), and also under these plus translations in position and time (Poincare).

To produce general relativity, Einstein began with the special theory of relativity. To the symmetries of special relativity he added the equivalence principle, a principle that holds that the laws of physics be unchanged between experiments made in a gravitational field and experiments made in an accelerated frame of reference.

Einstein’s symmetry laws for the special and general theory of relativity are similar to Kepler’s laws in that they are symmetry relations. And like Kepler’s laws, Einstein’s laws have stood the test of time. It was 82 years after Kepler published his laws that Newton simplified and explained them with an equation of motion. Einstein’s laws have survived 92 years and are still going fairly strong.

Einstein’s Equations of Motion

Einstein’s laws of gravitation and special relativity are symmetry principles similar to those of Kepler in that they are the result of observation. Einstein came up with them in the early 20th century rather than the early 17th century and the more advanced level of mathematics in Einstein’s time allowed his symmetry principles to be put into mathematical language in just a few years. The result is similar to Kepler’s laws in that it is possible to obtain equations of motion for planetary bodies from Einstein’s laws just as Newton was able to find equations of motion corresponding to Kepler’s laws.

In addition to Newton’s laws being written in differential equation form, while Kepler and Einstein’s laws are written as symmetry principles, Newton’s efforts had immediate application for earthly physics while Kepler and Einstein’s gravitation laws did not. Newton’s gravitation revolutionized physics to a much larger degree than Einstein’s gravitation. On the contrary, nearly a century after the discovery of general relativity, few researchers still search for a way of describing quantum mechanics in terms of general relativity. Of those still working on this dry hole, my favorite is Mark Hadley.

In addition, Newton’s gravitation laws was sufficiently accurate that it was immediately clear that it was universal and it was proclaimed as such. Unfortunately for Einstein’s theory, Newton’s was sufficiently accurate that divergences from it are typically small and difficult to observe. As a result of this, even though general relativity has been accepted as the premier theory of gravitation for many years, it is not nearly as well verified by experimental observation.

Several consequences of general relativity have been verified by observation: (1) Gravitational time dilation and frequency shift; (2) Light deflection and time delay by massive bodies; (3) Precession of orbits; (4) Decay of orbits; (5) Geodetic precession; and (6) Frame dragging. A seventh prediction, Gravity Waves, has not yet been observed.

Tests of General Relativity

A scientific theory can never be proved, it can only be shown to be accurate to higher and higher levels. In this the queen of theories is quantum field theory, which predicts the g-factor of the electron to an accuracy measured in parts per trillion. Einstein’s general relativity has had a tougher time. For example, one of the better tested predictions of general relativity is the deflection of light by massive bodies, which has been measured to 0.14 parts per thousand (this from a 103 page review article by Clifford Will).

At the time of the review article referenced above (2001), geodetic precession had been verified to an accuracy of 70 parts per thousand by lunar ranging. More recently, Gravity Probe B has attempted to measure geodetic precession to a much higher accuracy. The results have been much delayed and so far exclude GR at the 1-sigma level.

Frame dragging, also known as gravitomagnetism or the Lense-Thirring effect, had not been seen at the time of the above review article but is being measured by Gravity Probe B. Preliminary numbers also exclude GR at the 1-sigma level. These preliminary results are of course subject to change. Final numbers are now due to be released in May 2008, but previous dates have been delayed before.

GR also predicts gravity waves, but so far these continued to elude the speed-trap radar guns of physicists who expected to already observe them traveling at the speed of light from identifiable sources.

Symmetry vs. Equations of Motion

The present physical theories were established under the principle of “nature can be most easily described by simple symmetries”. Newton’s reduction of Kepler’s laws suggests an alternative asssumption that “nature can be most easily described by simple equations of motion.” For such a theory to have unifying effect to all of physics similar to Newton’s efforts, it needs to be built from the equations of motion that have proved so accurate in quantum mechanics. In short, such a theory needs to be built on the principles of quantum mechanics. In such a rewriting, gravitation must be treated as a force like the other forces of QFT. Of course if this were easy to do it would already have been done.

The first problem in putting Einstein’s symmetry principles into equation of motion into the form of a QFT is that the usual QFT assumes a flat background metric. Fitting QFT to a curved spacetime is difficult, as this encyclopedia article shows, but that is only the first step in combining gravity and quantum mechanics; the problem of how quantum mechanics modifies spacetime is still unsolved. It is this topic that we will discuss in our next post.


Filed under gravity, physics

2 responses to “Kepler’s Symmetries and Newton’s DE

    Vous n’avez jamais prouvé que Newton avait tort.
    Vous n’avez jamais prouvé que le vide n’existait pas.
    Vous n’avez jamais prouvé que les lieux absolus n’existaient pas.
    Vous n’avez jamais prouvé que les déplacements absolus d’un lieu absolu à un autre lieu absolu n’existaient pas.
    Vous n’avez jamais prouvé que d’écoulement du temps n’était pas absolu.
    Et vous n’avez jamais prouvé que les vitesses absolues n’existaient pas.
    Vos preuves sont absurdes :
    You never proved that Newton was wrong.
    You never proved that the vacuum did not exist.
    You never proved that the absolute places did not exist.
    You never proved that absolute displacements of an absolute place to another absolute place did not exist.
    You never proved that of flow of time was not absolute.
    And you never proved that the absolute velocities did not exist.
    Your evidence is absurd:
    Michelson, Morley (et Miller) ont construit leur dispositif en croyant, naïvement que le Soleil était immobile et que la Terre parcourait une ellipse.
    Ce qui est stupide : le Soleil avance !

    Eddington a truqué ses résultats en laissant de côté les photos du Brésil et en écartant celles d’Afrique qui lui déplaisaient.
    Même Hawkings reconnait que les preuves de 1919 n’en sont pas.
    Michelson, Morley (and Miller) built their device by believing, naively that the Sun was motionless and that the Earth traversed an ellipse.
    What is stupid: the Sun advances!

    Eddington faked its results by leaving side the photographs of Brazil and by drawing aside those of Africa which displeased to him.
    Even Hawkings recognizes that the evidence of 1919 is not.
    Mais le plus grave n’est pas là : Einstein raisonnait comme si la lumière passait à côté du Soleil IMMOBILE.
    Ce crétin ne comprenait pas que la lumière des étoiles passait à côté d’un Soleil qui était en train de se déplacer pendant qu’il était froôler par les photons venus de ces étoiles.
    Les partisans de Einstein ne parviennent même plus à comprendre cela : le Soleil avance.
    But most serious is not there: Einstein reasoned as if the light passed beside the MOTIONLESS Sun.
    This cretin did not understand that the light of stars passed beside a Sun which was moving while it was froôler by the photons come from these stars.
    The partisans of Einstein do not manage even any more to include/understand that: the Sun advances.

    I will publish your answer on my blog

    Yanick Toutain

  2. carlbrannen

    Actually, I don’t believe in relativity as a theory. Instead, I think of it as just a useful calculational technique.

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