How can General Relativity explain the Moon drifting apart from Earth

[DLeuPel]

According to various sources, the Moon is separating from Earth 4 cm every year. I’ve searched for the explanation and I’ve found the following:

The friction the seas and oceans from the Earth make with it’s soil causes the Earth’s rotation to slow down. This causes the Moon to speed up because of Newton’s law of action and reaction. Due to this acceleration, the Moon experiences a centrifugal force making it move apart from Earth.

My question is the following: Why does the Moon move apart from Earth according to Einstein’s model of gravity? If mass and energy curve the fabric of space time, does it not mean that the Moon is only rotating Earth because of this deformation and not because of any force acting on it?

 

[PeroK]

According to various sources, the Moon is separating from Earth 4 cm every year. I’ve searched for the explanation and I’ve found the following:

The friction the seas and oceans from the Earth make with it’s soil causes the Earth’s rotation to slow down. This causes the Moon to speed up because of Newton’s law of action and reaction. Due to this acceleration, the Moon experiences a centrifugal force making it move apart from Earth.

My question is the following: Why does the Moon move apart from Earth according to Einstein’s model of gravity? If mass and energy curve the fabric of space time, does it not mean that the Moon is only rotating Earth because of this deformation and not because of any force acting on it?

Imagine you throw a ball horizontally. It falls to Earth in a parabola. If you don't throw it very fast, it doesn't go very far. If you throw is faster, it goes further.

The ball, therefore, is not following a predefined path through spacetime, where there is one and only one path to follow.

The same is true of the Moon and its orbit. There are an infinite number of possible orbits, depending on the speed and distance of the Moon relative to the Earth. And that can and does change over time.

Although, ultimately, gravity can be modeled as the geometry of spacetime, the geometry around the Earth creates a dynamic that is (almost) identical to Netwon's Law of Gravitation. So, in the case of the Earth-Moon system, you can quite happily continue to use Newton's Laws, in the knowldege that you will (and far more easily) get the same results as you would if you applied the equations of General Relativity.

 

[DLeuPel]

Imagine you throw a ball horizontally. It falls to Earth in a parabola. If you don't throw it very fast, it doesn't go very far. If you throw is faster, it goes further.

The ball, therefore, is not following a predefined path through spacetime, where there is one and only one path to follow.

The same is true of the Moon and its orbit. There are an infinite number of possible orbits, depending on the speed and distance of the Moon relative to the Earth. And that can and does change over time.

Although, ultimately, gravity can be modeled as the geometry of spacetime, the geometry around the Earth creates a dynamic that is (almost) identical to Netwon's Law of Gravitation. So, in the case of the Earth-Moon system, you can quite happily continue to use Newton's Laws, in the knowldege that you will (and far more easily) get the same results as you would if you applied the equations of General Relativity.

Thank you for your reply. But I still remain with one question. Why does the Moon accelerate if the Moon is not under a force according to Einstein’s model of gravity ?

 

[PeroK]

Thank you for your reply. But I still remain with one question. Why does the Moon accelerate if the Moon is not under a force according to Einstein’s model of gravity ?

Good question!

First, a short answer. The equations of motion that arise from the spacetime geometry around the Earth are almost identical to Newton's Gravity: objects accelerate towards the Earth, in inverse proportion to the square of distance.

The longer answer is that there is an alternative way to look at Newton's Laws. The original approach was to consider forces: things move according to forces acting on them. However, a French-Italian mathematician, Lagrange, reformulated Newton's Laws according to his "Lagrangian" principle, which is that nature acts in order to minimise (or maximise) certain key quantities. In his version of Newtonian mechanics, objects move in order to minimise the Lagrangian, which is a combination of kinetic and potential energies.

The beauty of Lagrange's insight is that when you consider the curved spacetime of GR, the Lagrangian principle still applies! In this case, an object moves in order to maximise the time it experiences (called its "proper" time).

Motion in GR is explained, therefore, by a combination of the spacetime geometry and the Lagrangian principle.

 

[Nugatory]

Why does the Moon accelerate if the Moon is not under a force according to Einstein’s model of gravity ?

Loosely speaking... The dissipation of energy from the friction effects that you mention means that the stress-energy tensor that appears on the right-hand side of the Einstein Field Equations (google will find these for you, but be warned that the water gets very deep very fast) changes over time, so the curvature of spacetime in the vicinity of the earth/moon system is also changing over time. The moon is following a freefall path (no force, nothing to push the moon off its natural inertial path) through that spacetime, but as the curvature changes so does that freefall path.

That's the "loosely speaking" answer. An exact calculation starting from the Einstein Field Equations would be extraordinarily difficult, and in practice nobody would ever attempt such a thing. Instead we take advantage of the fact that when the gravitational effects are weak (as they are everywhere within our solar system) the EFE's reduce to the "weak field approximation" in which the effects of curvature are indistinguishable from the effects of a $1/r^2$ force, what Newtonian gravity says - and just solve the problem using Newtonian gravity.

 

[DLeuPel]

Loosely speaking... The dissipation of energy from the friction effects that you mention means that the stress-energy tensor that appears on the right-hand side of the Einstein Field Equations (google will find these for you, but be warned that the water gets very deep very fast) changes over time, so the curvature of spacetime in the vicinity of the earth/moon system is also changing over time. The moon is following a freefall path (no force, nothing to push the moon off its natural inertial path) through that spacetime, but as the curvature changes so does that freefall path.

That's the "loosely speaking" answer. An exact calculation starting from the Einstein Field Equations would be extraordinarily difficult, and in practice nobody would ever attempt such a thing. Instead we take advantage of the fact that when the gravitational effects are weak (as they are everywhere within our solar system) the EFE's reduce to the "weak field approximation" in which the effects of curvature are indistinguishable from the effects of a $1/r^2$ force, what Newtonian gravity says - and just solve the problem using Newtonian gravity.

Thank you for your response. It is all clear now

 

[pervect]

As I understand it (not as well as I'd like), we don't pretend to know the stress-energy tensor of the entire interior of the Earth and moon, and then directly use Einstein's field equations Rather, we characterize the external fields of the Earth and moon via a multipole expansion.

Google will find a lot of papers on multipole methods in General relativity which are formulated fully in GR terms.

The advantage of the approach is that we only need to find a fairly small number of multipole moments to characterize a planet , moon, or other source body.

https://ipnpr.jpl.nasa.gov/progress_report/42-196/196C.pdf has a discussion of some practical calculations used in the JPL ephermedies which might be of some interest, especially for the Earth-moon system. But I don't think that the JPL paper necessary use or need the full-GR treatment of multipoles, my reading is that they take a somewhat hybrid approach to the problem, using Newtonian descriptions where they are adequately accurate for the desired calculations.