Perihelion drift in SR and in GR

• lalbatros
In summary: The Universe in a Nutshell, by Stephen Hawking.In summary, the drift of the perihelion in Special Relativity can be calculated by cumulating elementary Lorentz transformations along the Newtonian trajectory, but this calculation produces a much smaller result than the experimental value. On the other hand, General Relativity produces the exact result, within the error bars. The book "Gravitation and Inertia" by Cuifolini and Wheeler may provide more insight into this topic. It is possible to calculate the drift as a perturbation of the Newtonian case using Lorentz transformations, but it is not clear if this applies to the
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?

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lalbatros said:
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.

lalbatros said:
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.

lalbatros said:
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

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.

1. What is perihelion drift in SR and in GR?

Perihelion drift refers to the phenomenon of the point of closest approach between a planet and the sun shifting over time. In special relativity (SR), this drift is caused by the curvature of spacetime due to the planet's motion. In general relativity (GR), it is caused by the curvature of spacetime due to both the planet's motion and the presence of other massive objects.

2. How does perihelion drift differ in SR and GR?

In SR, the perihelion drift is caused solely by the planet's motion, while in GR it is also affected by the curvature of spacetime due to other massive objects. Additionally, the amount of drift predicted by GR is significantly greater than that predicted by SR.

3. What is the significance of perihelion drift in GR?

The observed perihelion drift in the orbit of Mercury was one of the first pieces of evidence for the validity of Einstein's theory of general relativity. It also provides a way to test and refine our understanding of the curvature of spacetime.

4. Can perihelion drift be observed in other planetary orbits?

Yes, perihelion drift has been observed in the orbits of other planets such as Earth and Saturn. However, the effect is much smaller than in Mercury's orbit, making it more difficult to measure accurately.

5. Is perihelion drift the only observable effect of GR on planetary orbits?

No, there are other observable effects of general relativity on planetary orbits, such as the precession of the orbital planes and the Shapiro time delay. These effects are all small, but can be measured with precise observations and have been confirmed by experiments.

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