Perihelion drift in SR and in GR

[lalbatros]

I think that in Special Relativity, the drift of the perihelion can be calculated by cumulating elementary Lorentz transformations along the (Newtonian, unperturbed) trajectory. I read that the result of this calculation is much smaller than the experimental value.

It is also well known that general Relativity produces the exact result, within the error bars.

My questions are:

What happened and what can we learn from that ?

How and why did GR perform better ?

Furthermore, is that not somehow in contradiction with the Equivalence Principle ? Could we not imagine that the EP applied to this system would imply the same result for both SR and GR ?


Thanks a lot for your feedback,

Michel

PS: I would also be interrested by a reference about these calculations, I am a bit lazy !
Specially about calculating the drift as a perturbation of the Newtonian case.
Can that really be done by Lorentz tranformations?

 

[Chris Hillman]

I think that in Special Relativity, the drift of the perihelion can be calculated by cumulating elementary Lorentz transformations along the (Newtonian, unperturbed) trajectory. I read that the result of this calculation is much smaller than the experimental value.

It is also well known that general Relativity produces the exact result, within the error bars.

My questions are:

What happened and what can we learn from that ?

How and why did GR perform better ?

You would probably enjoy reading a book by Cuifolini and Wheeler, Gravitation and Inertia, which will probably further mystify you (it's not as clear as MTW), but the authors do try to answer in passing something like your questions above.

calculating the drift as a perturbation of the Newtonian case.
Can that really be done by Lorentz tranformations?

I am not sure I understand exactly what you have in mind ("cumulating elementary Lorentz transformations along the (Newtonian, unperturbed) trajectory" doesn't quite make sense as stated, as you will probably appreciate after a bit of thought), but the obvious guess is that you are thinking of some computation Einstein made in flat spacetime when he was working toward discovering gtr. E.g, one can imagine a static observer in Minkowski spacetime who holds a string and whizzes a stone around. One can imagine further that he carefully adjusts the force on the string to mimic an inverse-squared central force as closely as possible, and that the stone is following a quasi-elliptical orbit. One can then ask: should the pericenters slowly precess, as happens to a test particle in quasi-Keplerian motion around an isolated massive object in gtr? If I recall correctly, the answer is yes, and the precession turns out to be sometthing like half the rate predicted by gtr, but I'm too lazy to work the computation again right now, in order to check my memory.

is that not somehow in contradiction with the Equivalence Principle ? Could we not imagine that the EP applied to this system would imply the same result for both SR and GR ?

Sorry, I can't figure out what you might have in mind here, but I can say that gtr fully conforms to various versions of "the" equivalence principle, and that this theory is self-consistent. The EP certainly does not imply that the answer in the flat spacetime scenario I outlined above should agree with the gtr prediction for a test particle orbiting an isolated massive object, and I don't see why you might think otherwise.

Chris Hillman

 

[Entropia]

Thank you for your interesting question, Michel.

In Special Relativity (SR), the drift of the perihelion can be calculated by considering the trajectory of a planet in the absence of any external perturbations. This trajectory is calculated using the equations of motion in SR, which are based on the Lorentz transformations. However, this calculation does not take into account the effects of gravity, which is described by General Relativity (GR).

In GR, the equations of motion are based on the Einstein field equations, which take into account the curvature of spacetime caused by the presence of massive objects. This leads to a more accurate prediction of the perihelion drift, which has been confirmed by experiments.

So what happened and what can we learn from this? Essentially, the discrepancy between the results of SR and GR for the perihelion drift shows that SR is not a complete theory of gravity. It does not take into account the effects of gravity on the curvature of spacetime, which is crucial for understanding the behavior of objects in the presence of massive bodies.

GR performs better because it is a more comprehensive theory of gravity, which incorporates the effects of gravity on the curvature of spacetime. This allows for a more accurate prediction of the perihelion drift and other phenomena related to gravity.

Regarding the Equivalence Principle (EP), it is important to note that it applies to objects in free fall in a gravitational field. In the case of the perihelion drift, the planet is not in free fall as it is orbiting around the sun. Therefore, the EP does not necessarily apply in this situation.

As for references, there are many textbooks and articles that discuss the calculation of the perihelion drift in GR. A good starting point would be the book "Gravitation" by Misner, Thorne, and Wheeler, which provides a comprehensive explanation of GR and its predictions for the perihelion drift. Additionally, there are many online resources that discuss the topic in detail.

In summary, the discrepancy between the results of SR and GR for the perihelion drift highlights the limitations of SR in describing gravity. GR, on the other hand, provides a more accurate and comprehensive understanding of gravity and its effects on the behavior of objects in the universe.