Thursday, December 30, 2010
First a reminder: Precession: Let's remind ourselves the effects coming from general relativity. The first is the precession of the orbit demonstrated for Mercury in the image above. The dotted curve shows what is expected from Newtonian gravity alone: an obit that stays fixed in an elliptical shape forever. The solid line shows how this changes with general relativity: the orbit moves or precesses over time around the sun.
It may help to consider the gyroscope on the right: if you were to put a red dot on the edge of the spinning disk, and watched above, you would not see that it only traces out a circle as it spins, but a precession pattern like the one in the plot above.
Second reminder: A shrunken orbit: The second effect general relativity makes is that of shrinking the orbit, illustrated in the plot above again for Mercury. This is an image of Mercury's potential energy. Planets want to minimize their potential energy and therefore, like a ball on a hill, "roll to the bottom" of their potential. As can bee seen, the bottom of the potential for Newtonian gravity is at a larger radius than for general relativity. Therefore, general relativity forces planets to have a smaller radius.
Note: Both of the effects above have been verified experimentally and are major reasons why general relativity was embraced in the first place.
But is full general relativity really needed? Turning back to the paper, the authors decide to work out these effects in special relativity alone. First, we start with precession. Using special relativity alone you get:
And what about the radius? For special relativity the change in radius is a similar story and becomes:
Always the same factor of 6 huh? I guess so, and I don't off of the top of my head know why it should always been a factor of six. Anyways, despite being off by this factor, it is very interesting that special relativity predicts the same qualitative behavior as general relativity which is absent in Newton. Furthermore, since calculations in special relativity are significantly simpler than for general relativity, this special relativistic calculation is ideal if you are just trying to give a qualitative picture of what effects are relativistic. Perhaps a good one for undergraduates?
Anyone want to take a stab at why it is always a factor of 6???
You can in the comments. :)
Tyler J. Lemmon, & Antonio R. Mondragon (2010). First-Order Special Relativistic Corrections to Kepler's Orbits Submitted to American Journal of Physics arXiv: 1012.5438v1