Mercury Perihelion Precession: Analytic Derivation

[cianfa72]

TL;DR

Analytic derivation of Mercury perihelion precession

HI,

I'm curios about the analytic derivation of Mercury perihelion precession starting from EFE - Einstein Field Equation (or simply just from Schwarzschild solution of the EFE).

Can you advise me about some source or online material to learn it ?

Thanks.

 

[haushofer]

Uhm, how about almost every textbook on GR? Which book(s) on GR did you use to understand the EFE?

 

[PeroK]

Summary:: Analytic derivation of Mercury perihelion precession

HI,

I'm curios about the analytic derivation of Mercury perihelion precession starting from EFE - Einstein Field Equation (or simply just from Schwarzschild solution of the EFE).

Can you advise me about some source or online material to learn it ?

Thanks.

I just searched and found this. Looks like the real deal:

http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Pollock.pdf

 

[haushofer]

I guess I read over the word "analytical".

 

[cianfa72]

Thank you (meanwhile I found it also on Carroll 'Lecture Notes on General Relativity'). As far as I can understand the result is given in the Schwarzschild coordinates ('far-away' coordinate time $t$, reduced radius $r$ and angular displacement $\phi$).

On Carrol there is also a derivation for the apsidal frequency defined as $2\pi$ divided by the time it takes for the ellipse to precess once around.

From a physical point of view to 'interpret' this result I believe we have to employ the 'correct' interpretation for $r$ and $t$ in particular. Now to check the results of the observations about the precession of Mercury perihelion (e.g. to check the calculated apsidal frequency against observation results) which physical 'time' we have to take in account ?

 

[PeroK]

Thank you (meanwhile I found it also on Carroll 'Lecture Notes on General Relativity'). As far as I can understand the result is given in the Schwarzschild coordinates ('far-away' coordinate time $t$, reduced radius $r$ and angular displacement $\phi$).

On Carrol there is also a derivation for the apsidal frequency defined as $2\pi$ divided by the time it takes for the ellipse to precess once around.

From a physical point of view to 'interpret' this result I believe we have to employ the 'correct' interpretation for $r$ and $t$ in particular. Now to check the results of the observations about the precession of Mercury perihelion (e.g. to check the calculated apsidal frequency against observation results) which physical 'time' we have to take in account ?

You are looking for the precession of an orbit per revolution. That's a measurable observable - independent of the theory of gravity you employ.

 

[cianfa72]

In any case, for something on this scale there is effectively a universal time throughout the solar system. There is no need for time measurements of less than a second (at most).

Could you kindly better explain (to me ) this point ?

 

[PeroK]

Could you kindly better explain (to me ) this point ?

We can practically manage with universal time in everyday life and in Newtonian mechanics. The effects of velocity and gravitational-based time dilation are negligible except in a few cases where very precise time synchronisation is required.

If you calculate the period of Mercury in Mercury's proper time, for a clock on Earth and Schwarzschild coordinate time, these will vary negligibly given observations made in days, hours, minutes and seconds.

 

[cianfa72]

If you calculate the period of Mercury in Mercury's proper time, for a clock on Earth and Schwarzschild coordinate time, these will vary negligibly...

Not sure to grasp it: do you mean the relation between Mercury's proper time and (proper time) measured by a clock on the Earth on one hand and the relation between Mercury's proper time and Schwarzschild coordinate time on the other ?

 

[PeroK]

Not sure to grasp it: do you mean the relation between Mercury's proper time and (proper time) measured by a clock on the Earth on one hand and the relation between Mercury's proper time and Schwarzschild coordinate time on the other ?

All of them. Do the calculations.

 

[cianfa72]

All of them. Do the calculations.

Sorry not sure to be able to do the calculation. Anyway I believe we have to consider the Schwarzschild solution for the Sun at the center of solar system and then calculate both the relation between the proper time of a clock on the Earth respect to the Schwarzschild coordinate time and the same for the proper time of Mercury.

 

[PeroK]

You opened an I level thread asking for an analytic derivation:

Summary:: Analytic derivation of Mercury perihelion precession

HI,

I'm curios about the analytic derivation of Mercury perihelion precession starting from EFE - Einstein Field Equation (or simply just from Schwarzschild solution of the EFE).

Can you advise me about some source or online material to learn it ?

Thanks.

Then, you say you can't actually follow any of the mathematics or do any calculations?

sorry not sure to be able to do the calculation.

So, you're just wasting our time?

 

[cianfa72]

So, you're just wasting our time?

No, that was not my intention. I was just interested to the logic to employ to do the calculation.

 

[vanhees71]

You are looking for the precession of an orbit per revolution. That's a measurable observable - independent of the theory of gravity you employ.

True, but measuring this observable can decide between different theories of gravity. So far Einstein's GR passed all high-precision tests (I don't know to which PPN order nowadays, but it's amazing which precision can be achieved nowadays with, e.g., pulsar timing).

 

[PeroK]

True, but measuring this observable can decide between different theories of gravity. So far Einstein's GR passed all high-precision tests (I don't know to which PPN order nowadays, but it's amazing which precision can be achieved nowadays with, e.g., pulsar timing).

The point is that you predict the precession using either Newton or GR, but there is only one experimental measurement. One can, as they did in the 19th century, measure the precession of Mercury without any recourse to GR.

 

[PeterDonis]

I was just interested to the logic to employ to do the calculation.

The logic is that, for the case of the Mercury perihelion precession, the difference between Schwarzschild coordinate time, Mercury's proper time, and the Earth's proper time is negligible, so it doesn't matter which one you want to use for interpretation. Mathematically speaking, the time used in the calculation is Schwarzschild coordinate time; it would be possible to do the calculation explicitly using the proper time of observers on Earth (which is the time ticked by the clocks of the actual observers making the observations), but that would be adding a lot of tedious complication for no useful purpose, given my statement above about the differences between the times being negligible.

Similar remarks apply for this case regarding the difference between the Schwarzschild $r$ coordinate, which is what appears in the math, and proper distance in the radial direction, which is what most people's intuitive interpretation is going to use.

 

[cianfa72]

Similar remarks apply for this case regarding the difference between the Schwarzschild $r$ coordinate, which is what appears in the math, and proper distance in the radial direction, which is what most people's intuitive interpretation is going to use.

ok, consider for instance the proper distance in the radial direction from the center of the Sun to Mercury: it is the spacetime 'length' calculated over the spacelike hypersurface of a given Schwarzschild coordinate time $t$, I guess.
In principle we have to calculate the spacetime length of each spacelike path belonging to that hypersuface joining the center of the Sun and Mercury and then picking the maximum.

 

[FactChecker]

We can practically manage with universal time in everyday life and in Newtonian mechanics. The effects of velocity and gravitational-based time dilation are negligible except in a few cases where very precise time synchronisation is required.

An extreme real-world example being the GPS system, which would have large errors if it did not adjust for gravitational time dilation near the Earth.

 

[PeterDonis]

consider for instance the proper distance in the radial direction from the center of the Sun to Mercury: it is the spacetime 'length' calculated over the spacelike hypersurface of a given Schwarzschild coordinate time , I guess.

Yes. But the difference between this and the simple radial coordinate $r$ of Mercury in Schwarzschild coordinates is negligible. So it works fine to just think of the radial coordinate $r$, which is what appears in the math, as being the same as the proper distance, which is what is easy for your intuition to grasp.

In principle we have to calculate the spacetime length of each spacelike path belonging to that hypersuface joining the center of the Sun and Mercury and then picking the maximum

This is just another way of saying that you want the length of the spacelike geodesic between the center of the Sun and (the center of) Mercury, which is what "spacetime length" means.

 

[cianfa72]

This is just another way of saying that you want the length of the spacelike geodesic between the center of the Sun and (the center of) Mercury, which is what "spacetime length" means.

sorry maybe i was wrong: in case of spacelike geodesics it should the minimum and not maximum

 

[PeterDonis]

in case of spacelike geodesics it should the minimum and not maximum

Yes.