# Action and reaction in general relativity

• B
Giuly
Hi,

In general relativity, gravitation is not anymore a Force but a deformation of space time. I would like to know what's becomes the 3 law of Newton for gravity that action equal reaction ? When a apple fall on the earth, does "the force" is exactly the same as the one applied on Earth ? Maybe not because the space time deformation is not the same on the gravity center of the apple, and on the gravity center of the Earth ? What becomes 3 law of physics for gravity in general relativity ?

Thanks

Mentor
I would like to know what's becomes the 3 law of Newton for gravity that action equal reaction
The third law is still there and still works, except that because gravity isn’t considered a force the third law isn’t used for gravity.

In Newtonian physics we would say that the Earth exerts a gravitational force on the apple and the apple exerts an equal and opposite force on the earth; the Earth is affected much less than the apple because of its greater mass so we can ignore its negligible acceleration; the net effect is that the apple accelerates towards the earth. The force of the Earth on the apple and the force of the apple on the Earth form a third law pair.

In Einstein’s theory we would say that there are no forces acting on the Earth or the apple. Instead, spacetime is curved in such a way that the unaccelerated freefall paths of the Earth and the apple intersect at the surface of the earth. Because there is no force there is no way to invoke the third law (until the Earth and the apple collide, at which time there are contact forces between the apple and the surface of the Earth to consider).
When a apple fall on the earth, does "the force" is exactly the same as the one applied on earth?
There is no force so that question doesn’t make sense. In general relativity one mass doesn’t pull the other around with a gravitational force so we cannot think in terms of one of them acting on the other. Instead spacetime is curved by the presence of both bodies and there is generally no way to divide the curvature into the part caused by one body and the part caused by the other. (In the case of the Earth and the apple, the Earth is so much more massive that the spacetime curvature is pretty much the same whether the apple is there or not so we might say it’s all caused by the earth, but this approximation will fail if the masses are less dissimilar).

• vanhees71 and Giuly
Mentor
Instead spacetime is curved by the presence of both bodies and there is generally no way to divide the curvature into the part caused by one body and the part caused by the other.
Expanding on this: regardless of how spacetime is curved, locally both energy and momentum are conserved. So, in a local sense Newton’s third law is satisfied everywhere. However, the spacetime curvature makes it difficult to unambiguously write down global definitions for momentum and energy. Since the energy and momentum concepts are fraught globally, it is not even possible to write down generally meaningful global conservation laws.

• dextercioby, vanhees71, PeroK and 1 other person
Giuly
Instead spacetime is curved by the presence of both bodies and there is generally no way to divide the curvature into the part caused by one body and the part caused by the other.
If i understand well, Newton 3 law for gravity will just be a very good approximation, because, if I'm right the deformation of space time at the place of the apple will not be exactly the same as for the place of the earth.

Mentor
Newton 3 law for gravity will just be a very good approximation
As I said above, it is not that simple. Locally it is exact, but globally it isn’t clearly even well defined in general

• vanhees71
Homework Helper
Gold Member
2022 Award
If i understand well, Newton 3 law for gravity will just be a very good approximation, because, if I'm right the deformation of space time at the place of the apple will not be exactly the same as for the place of the earth.
I doubt you could generalise Newton's third law to gravitational interactions very well. We know that Newton's law of gravity is a good approximation to GR in many cases and the third law would apply there - if we allow ourselves to model gravity as a force in those cases.

In general, you have to give up on the idea of gravity being action at a distance with a pair of forces subject to the third law acting remotely on each other. It's not so much whether Newton's third law holds or not. It's that Newton's third law is based on fundamental assumptions about the nature of spacetime that simply do not apply to curved spacetime in general. Especially time-dependent spacetimes.

In general spacetimes, it's not possible to make sense of what the third law is trying to say.

• vanhees71
Giuly
As I said above, it is not that simple. Locally it is exact, but globally it isn’t clearly even well defined in general
In my point of view i disagree with you, i think it's very well defined and that the 3 law of Newton is perfectly respected for general relativity with gravitation, but i can t prove it and as the concept of deformation of space time is not clear for me, i asked this question

• weirdoguy and PeroK
Mentor
If i understand well, Newton 3 law for gravity will just be a very good approximation
It's worse than that - Newton's third law is about forces and gravity isn't a force so cannot be analyzed or understood using the third law.
if I'm right the deformation of space time at the place of the apple will not be exactly the same as for the place of the earth.
That is indeed the case, and it is why the Earth and the apple seem to follow very different trajectories through space - their paths pass through different regions of spacetime with very different curvatures (but be warned that there are many pitfalls, oversimplifications, and hidden assumptions in this seemingly simple statement). However, it is a mistake to think of this difference in curvature as causing subtle differences from the Newtonian third law behavior. If anything, it works the other way around: it is somewhat remarkable that very different local curvatures lead to trajectories that are closely approximated by Newtonian physics.

So bottom line: Newton's third law for local forces works just fine and is exact. However gravity isn't a force so the third law isn't part of how we think about it.

• Dale, vanhees71 and PeroK
weirdoguy
In my point of view i disagree with you, i think it's very well defined and that the 3 law of Newton is perfectly respected for general relativity with gravitation

And what is your basis for this view? You yourself admit you do not understand the very basics of GR (spacetime curvature) so how can you have any opinion on any of that?

• vanhees71
Staff Emeritus
Strategically, trying to shoehorn Newtonian concepts into GR is unlikely to be enlightening, even if you could make it work.

Also, B level means you aren't far enough along to do a GR calculation on your own, so it's not clear to me what purpose this "shoehorning" will serve. It may just place more obstascles in your path when you finally learn GR.

• vanhees71 and PeroK
Mentor
i think it's very well defined and that the 3 law of Newton is perfectly respected for general relativity with gravitation
It's not, and when you think more deeply about it you will start to see the problems.

First, how do you define the gravitational effects to see if they are equal and opposite? The obvious thing to do is to start with ##F=ma##, we know the mass, we measure the acceleration, that let's us calculate a force. We do this for the Earth and the apple and verify that the calculated forces are equal and opposite: Earth has a big ##m## and a small ##a##, apple has a small ##m## and a big ##a## in the opposite direction, ##ma## is the same except for the sign. But there is a catch here: the ##a## that we're using here is a coordinate acceleration not a proper acceleration (search this forum for threads about the difference between the two - it is essential to understanding relativity) so has no unique observer-independent definition. Depending on what we're taking the speeds relative to, we can make the accelerations and hence the quantities ##ma## come out to be whatever we want them to be. If they come out equal and opposite that's just the result of choosing coordinates that made it that way.

Second, we know that the speeds and accelerations will change as the bodies move closer or farther apart, so when we compare ##ma## for the apple and ##ma## for the Earth we have to be careful to take the values at the same time. This is not a problem for classical physics where there is an unambiguous definition of "at the same time" (it's as if there is a big clock somewhere in the universe and we've all agreed to synchronize our own clocks to agree with it) but it is a big problem in relativistic physics. Even in flat spacetime it is not possible to define "at the same time" for spatially separated events (google for "Einstein relativity of simultaneity" - that concept is even more essential to understanding relativity) and the problem is even greater in a curved spacetime.

There's more, but that may be enough to get you started.

• dextercioby, PeterDonis and vanhees71
Gold Member
2022 Award
The third law is still there and still works, except that because gravity isn’t considered a force the third law isn’t used for gravity.
One has to qualify this, however, a bit. In special relativity "the third law" ("actio=reactio") holds in the sense that the total ("global") momentum of a closed system is conserved due to Noether's theorem applied to the translation invariance of space. Newton III, however, cannot hold in the original form, because there cannot be any "action at a distance". The standard way out is the local description of interactions using fields. If you take into account the momentum of both matter and the fields as a closed system, the total momentum is conserved.

In GR it's a notorious problem to formulate energy-momentum conservation, because there's no generally covariant definition of an energy-momentum tensor of the gravitational field. What's for sure left of the global energy-momentum conservation valid in SR is the local generally-covariant law ##\nabla_{\mu} T^{\mu \nu}=0##, where ##T^{\mu \nu}## is the energy-momentum tensor of "matter and radiation" (i.e., everything else than the gravitational field).

Mentor
One has to qualify this, however, a bit.
Yes, although for a B-level thread the simpler qualification might be to generously sprinkle the word "locally" around - within a local inertial frame high school physics just works, and the key conceptual barrier that OP has to overcome here is the unavailability of any global inertial frame.

• vanhees71
Mentor
i think it's very well defined and that the 3 law of Newton is perfectly respected for general relativity with gravitation
Well, everyone is entitled to an opinion. But before dismissing mine you might want to learn enough of the math to back yours up.

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